Loyola Marymount University
Department of Mathematics
Single Subject Matter Program
In Mathematics
Standards of Quality and Effectiveness
Program Document
Approved
Summer, 2004
1
Helen Hawley, Subject Matter Program Consultant
CTC Credentialing Commission
Commission on Teacher Credentialing
Professional Services Division
1812 Ninth Street
Sacramento, CA 95814
Dear Ms. Hawley:
We received your memo of May 14, 2004 informing us that the subject matter review
panel recommended approval of Loyola Marymount University’s document based on the
resubmission of February 20, 2004. Please find below our complete program document
that incorporates all of the changes to Loyola Marymount University’s original
submission.
Thank you for your assistance throughout this process.
Sincerely yours,
Michael D. Grady, Ph.D.
Chairperson
Department of Mathematics
2
Table of Contents
Standards
Page
Preconditions ...................................................
5
Standards Common to All ................................
9
1.
2.
3.
4.
5.
6.
7.
Program Philosophy and Purpose ............
Diversity and Equity ................................
Technology ..............................................
Literacy ....................................................
Varied Teaching Strategies ......................
Early Field Experiences ...........................
Assessment of Subject Matter ..................
Competence
Advisement and Support ..........................
Program Review and Evaluation .............
Coordination ............................................
9
14
18
21
23
27
34
Mathematics Standards....................................
11. Required Subjects of Study......................
11.1 Subject Matter Requirements for ............
Mathematics
Part I: Content Domains
Domain 1: Algebra ...................................
Domain 2: Geometry................................
Domain 3: Number Theory ......................
Domain 4: Probability & Stat. .................
Domain 5: Calculus..................................
Domain 6: History....................................
Part II: Subject Matter Skills & Abilities ..........
12. Problem Solving.......................................
13. Mathematics as Communication ..............
14. Reasoning .................................................
15. Mathematical Connections.......................
16. Delivery of Instruction .............................
55
55
55
8.
9.
10.
38
45
50
56
59
62
64
65
71
74
80
85
89
92
95
Appendices ....................................................... 100
I.
II.
III.
Course Syllabi .......................................... 100
Standards/Course Grids ........................... 168
Bachelor of Arts in Mathematics ............. 171
Single Subject Program
Requirements
3
IV. Mathematics and Science .........................
Teacher Preparation Committee
(MASTeP)
V. Los Angeles Collaborative for .................
Teacher Excellence (LACTE)
VI. LMU Center for Teaching .......................
Excellence program Information
VII. LMU Diversity Requirement ...................
VIII. Description of LMU College ...................
Bound program - Summer 2003
IX. Technology Glossary ...............................
X. Mathematical Association of ...................
America - Preparing
Mathematicians for the
Education of Teachers (PMET)
XI. Mission and Goals of University .............
and the Mathematics Department
XII. Mathematics and Science .........................
(MASTeP) Review Board for
2003-2004
XIII. Catalog Description of Courses ...............
taken from LMU Departmenta;
Websites on August 19, 2003
XIV. LMU Mathematics Faculty CV’s.............
174
175
177
181
182
184
185
186
188
189
195
4
Precondition Narrative
(1) The subject matter preparation program in mathematics at Loyola Marymount
University consists of a minimum of 34 semester hours of core required work plus 21-22
semester hours of breadth designed to broaden the perspective of the student who
completes the program.
(2) Core Requirements
The core requirements and the courses in which they are fulfilled are shown in Table I on
page 7. These requirements are also briefly described below.
Proficiency in Algebra
Students must demonstrate a mastery of high school algebra either by satisfactory
performance on a placement examination or by successfully completing Precalculus
Mathematics (MATH 120) as a prerequisite to enrolling in Calculus I (MATH 131).
Advanced Algebra
The advanced algebra core of the program consists of 6 courses to ensure a rigorous view
of algebra and its underlying structures: the Calculus Sequence (MATH 131, 132, 234),
Methods of Proof (MATH 248), Group Theory (MATH 331), and Senior Seminar for
Future Mathematics Educators (MATH 493). By having students continue to study
algebraic structures throughout their degree program the program ensures that they have
the breadth and depth of knowledge and skills to teach algebra and to provide students
with a wide variety of problem solving situations. Moreover, students in the program
will understand the power of abstraction and symbolism in mathematics using algebra.
Geometry
The geometry core of the program consists of Geometry (MATH 550) and 3 other
courses: Calculus III (MATH 234), History of Mathematics (MATH 490), and Senior
Seminar for Future Mathematics Educators (MATH 493).
Number Theory
The number theory core of the program is integrated into 3 courses: Methods of Proof
(MATH 248), Group Theory (MATH 331), and Senior Seminar for Future
Mathematics Educators (MATH 493).
Calculus
The calculus core of the program consists of the Calculus Sequence (MATH 131, 132, &
234). Calculus and its relationship to the high school curriculum is revisited in Senior
Seminar for Future Mathematics Educators (MATH 493).
5
History of Mathematics
The history core of the program consists of History of Mathematics (MATH 490)
together with three other courses: Workshop in Mathematics I/II (MATH 190/191) and
Senior Seminar for Future Mathematics Educators (MATH 493).
Statistics and Probability
The statistics and probability core of the program consists of Probability and Statistics
(MATH 360).
(3) Breadth and Perspective Requirements
The breadth requirements and the courses in which they are fulfilled are shown in Table
II on page 8. These requirements are also briefly described below.
Mathematical Breadth (18 units)
In order to provide mathematical breadth to the student the program requires 18 more
units of mathematics as indicated in Table II on page 8. Differential Equations (MATH
245) provides breadth in applications and calculus, Linear Algebra (MATH 550)
provides more detailed understanding of advanced algebra and geometry, Numerical
Methods (MATH 282) teaches students to construct and analyze algorithms with
technology, Real Variables (MATH 321) provides students with a deep understanding of
the mathematics underlying calculus and the real number system, Complex Variables
(MATH 357) introduces students to the complex numbers and investigates their
relationship to all fields of mathematics, and the upper division mathematics elective
(MATH XXX) allows students to investigate a topic of interest to them.
Breadth in Related Areas (3 or 4 units)
Each student is required by the department to take at least one science course as a lower
division requirement. They choose from Physics (PHYS 101, 201) or Computer Science
(CMSI 182, 185, 281). This provides the student with breadth and perspective from an
area related to mathematics.
6
Math Placement Exam or
MATH 120 Precalculus
0-3
Number Theory
History of Mathematics
Advanced Algebra
Statistics & Probability
Geometry
Calculus
Semester Units
Algebra Proficiency
Table I. Core Requirements
X
MATH 131
Calculus 1
4
X
X
MATH 132
Calculus II
4
X
X
MATH 190/191
Mathematics I/II
Workshop in
MATH 234
Calculus III
4
Introduction to
3
X
X
3
X
X
MATH 248
Methods of Proof
MATH 331
Theory
Group
2/2
MATH 360
Introduction to Probability & Statistics
3
MATH 490
Mathematics
3
History of
MATH 493
Senior
Seminar for Future Math Educators
3
MATH 550
Concepts of Geometry
3
Fundamental
Total Semester Units:
(depending whether MATH 120 is required)
X
X
X
X
X
X
X
X
X
X
X
34 or
37
7
Table II. Breadth Requirements
COURSES
MATH 245
Differential Equations
MATH 250
Linear Algebra
MATH 282
Elementary Numerical Methods
MATH 321
Real Variables I
MATH 357
Complex Variables
MATH XXX
Upper Division Elective
Science Course chose from:
PHYS 101, 201, CMSI 182, 185, 281
Total Semester Units
SEMESTER
UNITS
3
3
3
3
3
3
3 or 4
21 or 22
8
Standard 1: Program Philosophy and Purpose
The subject matter preparation program is based on an explicit statement of program
philosophy that expresses its purpose, design, and desired outcomes in relation to the
Standards of Quality and Effectiveness for Single Subject Teaching Credential Programs.
The program provides the coursework and field experiences necessary to teach the
specified subject to all of California’s diverse public school population. Subject matter
preparation in the program for prospective teachers is academically rigorous and
intellectually stimulating. The program curriculum reflects and builds on the Stateadopted Academic Content Standards for K-12 Students and Curriculum Frameworks for
California Public Schools. The program is designed to establish a strong foundation in
and understanding of subject matter knowledge for prospective teachers that provides a
basis for continued development during each teacher’s professional career. The
sponsoring institution assigns high priority to and appropriately supports the program as
an essential part of its mission.
The Single Subject Program in Mathematics at Loyola Marymount University is guided
by the Mission and Goals of Loyola Marymount University and the Department of
Mathematics (see Appendix XI), the state-adopted Academic Content Standards for K-12
students and Curriculum Frameworks for California public schools, and the Principles
and Standards of the National Council for Teachers of Mathematics (NCTM).
The Subject Matter Program in Mathematics is based on a program philosophy that
envisions the ideal mathematics teacher as someone who
 communicates mathematical ideas with ease and clarity;
 organizes and analyzes information, solves problems readily, and constructs logical
arguments;
 possesses knowledge and an understanding of mathematics that is considerably
deeper than what s/he teaches;
 enjoys mathematics and appreciates its power and beauty;
 understands how mathematics permeates our world and how the various strands
within mathematics are interwoven;
 is comfortable using technology in the learning, teaching, and "doing" of
mathematics;
 possesses the knowledge, skills, and commitment to teach mathematics to all
students; and
 has the mathematical maturity and attitudes that promote continued growth in
knowledge of mathematics and its teaching.**
**
Adapted from A Call for Change: Recommendations for the Mathematical Preparation of Teachers of
Mathematics; Mathematical Association of America Committee on the Mathematical Education of
Teachers, Washington DC, 1991, and from The Mathematical Education of Teachers vol 11 CBMS
Issues in Mathematics Education Series, 2001.
9
Through required courses (see Preconditions and Appendix II) the program guarantees
that students will obtain the desired depth and breadth of knowledge (see the responses to
Standard 11). Field experience is integral to the program (see the responses to Standard
6). But the program is more than a disjointed set of courses. Certain experiences are
built into the curriculum to assure that students see relationships within the mathematical
topics studied and linkages to the outside world (see responses to Standard 15) and that
the students experience the kinds of teaching and assessment techniques that they are
expected to employ as teachers (see responses to Standards 5, 7 and 16). The program
emphasizes problem solving, communication and reasoning skills (see responses to
Standards 12, 13, and 14) and employs technology routinely (see the responses to
Standard 3). In addition to the field experience component, other elements of the
program (see responses to Standard 2 and Domain 6 of the subject matter requirements
(11.1)) are designed to develop an awareness and appreciation for the diverse
perspectives and groups that contributed to mathematics and are represented in the public
school population the State of California. The university prides itself on its studentoriented emphasis in academics and extra-curriculars. The attention given to student
advisement and support demonstrates this emphasis (see responses to Standard 8).
Continued quality and effectiveness of the subject matter program is provided for through
an ongoing review process (see responses to Standard 9). Finally, as part of a Jesuit
institution the program is held to special standards, since Jesuit educational theory places
the greatest importance on the education of youth. Moreover, in his inaugural address in
1999, the institution's President, Father Robert Lawton, S.J., designated teacher
preparation as one of (only) five institutional priorities (see responses to Standard 10)
chosen as a consequence of Loyola Marymount University's mission and location in the
greater Los Angeles area.
Required Elements
1.1
The program philosophy, design, and intended outcomes are
consistent with the content of the State-adopted Academic Content
Standards for K-12 students and Curriculum Frameworks for
California public schools.
The State-adopted Academic Content Standards for K-12 students and Curriculum
Frameworks for California public schools were foundational material in the development
of the philosophy and goals of the Single Subject Matter Program in Mathematics. The
program was also informed by the Principles and Standards of the National Council of
Teachers of Mathematics. The program described in this document is a revision of a
previously approved (in 1995) subject matter program.
1.2
The statement of program philosophy shows a clear understanding of
the preparation that prospective teachers need in order to be effective
in delivering academic content to all students in California schools.
The Subject Matter Program in Mathematics is based on a program philosophy that
envisions the ideal mathematics teacher as someone who
10








communicates mathematical ideas with ease and clarity;
organizes and analyzes information, solves problems readily, and constructs logical
arguments;
possesses knowledge and an understanding of mathematics that is considerably
deeper than what s/he teaches;
enjoys mathematics and appreciates its power and beauty;
understands how mathematics permeates our world and how the various strands
within mathematics are interwoven;
is comfortable using technology in the learning, teaching, and "doing" of
mathematics;
possesses the knowledge, skills, and commitment to teach mathematics to all
students; and
has the mathematical maturity and attitudes that promote continued growth in
knowledge of mathematics and its teaching.**
1.3
The program provides prospective teachers with the opportunity to
learn and apply significant ideas, structures, methods and core
concepts in the specified subject discipline(s) that underlies the 6-12
curriculum.
As indicated in the grid found in Appendix II, and as described in detail in the
Mathematics Subject Matter Requirements Parts I and II (see Responses to Standard 11),
the program provides coursework in calculus, algebra, geometry, number theory,
probability and statistics, and the history of mathematics. This coursework encompasses
both theory and applications and focuses on problem solving, mathematical reasoning
and communication. It provides an advanced viewpoint on the mathematics content of
grades 6-7 (number sense; algebra and functions; measurement and geometry; statistics,
data analysis, and probability; and mathematical reasoning) and in grades 8-12 (Algebra I
and II, Geometry, Probability and Statistics, Trigonometry, Linear Algebra, Mathematical
Analysis, AP Probability and Statistics, and Calculus) as specified in the Mathematics
Framework.
1.4
The program prepares prospective single-subject teachers to analyze
complex discipline-based issues; synthesize information from multiple
sources and perspectives; communicate skillfully in oral and written
forms; and use appropriate technologies.
The Single Subject Program in Mathematics emphasizes problem solving,
communication, and reasoning skills (see responses to Standards 12, 13, and 14) and
employs technology routinely (see the responses to Standard 3). The program contains
coursework and field experience designed to develop in future teachers an awareness and
**
Adapted from A Call for Change: Recommendations for the Mathematical Preparation of Teachers of
Mathematics; Mathematical Association of America Committee on the Mathematical Education of
Teachers, Washington DC, 1991, and from The Mathematical Education of Teachers vol 11 CBMS
Issues in Mathematics Education Series, 2001.
11
appreciation for the diverse perspectives and groups that contributed to mathematics and
are represented in the public school population the State of California (see responses to
Standards 2 and 6 and to Domain 6 of the Subject Matter Requirements - 11.1). Complex
“discipline-based issues” in mathematics occur naturally in attempts to create and apply
mathematical models to the solution of interdisciplinary problems. The program affords
students many opportunities to develop their skills in analyzing and solving such
problems (see responses to Standard 15).
1.5
Program outcomes are defined clearly and assessments of prospective
teachers and program reviews are appropriately aligned.
The program uses multiple formative and summative measures to assess the subject
matter competence of each student. These include student presentations, research
projects, oral reports, portfolios, field experience journals, observations, and interviews
as well as oral and written examinations. The senior level course Senior Seminar for
Future Mathematics Educators (MATH 493) provides a capstone experience. The
university and the department are committed to offering a student-centered educational
experience and provide a strong support system and personalized advising (see responses
to Standards 7 and 8). The Single Subject Matter Program in Mathematics is reviewed
every four years, with the next review scheduled for 2006, by a special board (see
Appendix XII) containing representatives from LMU faculty in math, science, and
education, community college faculty from LMU's main sources of transfer students,
LMU program graduates, current students and public school personnel. Included in the
review are: program goals and curriculum, other requirements, use of technology,
advising services, student success, assessment procedures and program outcomes (see
responses to Standard 9).
1.6
The institution conducts periodic review of the program philosophy,
goals, design, and outcomes consistent with the following: campus
program assessment timelines, procedures, and policies; ongoing
research and thinking in the discipline; nationally accepted content
standards and recommendations; and the changing needs of public
schools in California.
Loyola Marymount University provides three mechanisms for program review in
mathematics: a university-level committee Academic Planning and Review Committee
(APRC), a departmental committee, and the university-level Math and Science Teacher
Preparation Committee (MASTeP, see Appendix IV). The function and duty of each is
described in detail in the response to Standard 9. The Single Subject Matter Program in
Mathematics is reviewed every four years, with the next review scheduled for 2006, by a
special board (see Appendix XII) containing representatives from LMU math, science
and education faculty, community college faculty from LMU's main sources of transfer
students, LMU program graduates, current students and public school personnel.
Included in the review are: program goals and curriculum, other requirements, use of
technology, advising services, student success, assessment procedures and program
outcomes (see responses to Standard 9).
12
In addition, motivated by assessment requirements for WASC, in 2003 the university
instituted a Learning Outcomes Assessment Plan under the auspices of the Director of
Assessment and Data Analysis. A new Mathematics Departmental Assessment
Committee has begun collecting data and student work samples to investigate specific
questions about the ability of mathematics major graduates to evaluate and communicate
mathematical reasoning.
13
Standard 2: Diversity and Equity
The subject matter program provides equitable opportunities to learn for all prospective
teachers by utilizing instructional, advisement and curricular practices that insure equal
access to program academic content and knowledge of career options. Included in the
program are the essential understandings, knowledge and appreciation of the
perspectives and contributions by and about diverse groups in the discipline.
Loyola Marymount University actively seeks to recruit and retain minority students. The
statistics of the 2002 incoming freshman class attest to the institution's success in this
area: 52% White, 6% African American, 19% Hispanic/Latino, and 10% Asian/Pacific
Islander, with 11% declining to state, and 1% non-resident alien. The freshman class is
61% female and 39% male. Across all four years of the mathematics major for 20032004, the statistics for 46 majors are: 63% White, 2% African American, 15%
Hispanic/Latino, 13% Asian/Pacific Islander, and 7% unknown, with 46% female and
54% male. Amongst the 17 full-time faculty, there are 4 women, 1 Latino, 1 Indian, and 1
Russian "refusenik." A wide variety of pedagogical and instructional approaches are
employed in courses in the program in order to provide equitable access to students of all
backgrounds and to model good teaching practice for future teachers. Several
mathematics courses (Workshop in Mathematics I and II (MATH 190/191) and
History of Mathematics (MATH 490)) specifically address contributions by women and
diverse groups to the discipline.
Required Elements
2.1
In accordance with the Education Code Chapter 587, Statutes of 1999,
(See Appendix A), human differences and similarities to be examined in
the program include, but are not limited to those of sex, race, ethnicity,
socio-economic status, religion, sexual orientation, and exceptionality.
The program may also include study of other human similarities and
differences.
Equity and diversity are experienced daily in the subject matter program due to the
substantial representation of women and ethnicities on the math faculty (Amongst the 17
full-time faculty, there are 4 women, 1 Latino, 1 Indian, and 1 Russian refusenik) and
amongst the mathematics majors (see statistics above for majors).
Numerous
mathematics faculty have clearly demonstrated concern, sensitivity, and commitment to
diverse cultural and ethnic groups, gender and sexual orientation issues, and individuals
with exceptional needs by their membership on various university committees, course
development, and participation in outside activities. Jackie Dewar has been chair of the
univeristy Committee on the Status of Women, and Lily Khadjavi has been a member of
the Intercultural Affairs Committee and is the faculty moderator for the student Gay and
Lesbian Club (Lily Khadjavi). Herbert Medina has developed courses in the American
14
Cultures program, Jackie Dewar developed course in the Women Studies program, and
Lily Khadjavi has developed modules for the elementary statistics course that apply
statistical concepts to the study of social justice issues. Five members of the department
(Jackie Dewar, Lily Khadjavi, Suzanne Larson, Connie Weeks, and Dennis Zill) have
helped organized events such as an annual Expanding Your Horizons Career day to
encourage young women, especially minorities, in grades 6-10 to study math and science.
As a last example, Lily Khadjavi and Ed Mosteig participate in projedcts like the LMU
Science and Engineering Community Outreach Program (SECOP) that recruits
underrepresented high school students to spend two weeks living on the college campus,
attending classes and working on projects to explore the field of engineering and College
Bound's Boeing Academy which is a weekend program that provides college admission
assistance for high school students, with emphasis on the African-American family. In
many of these activities, faculty have encouraged student involvement as well. For
example, students enrolled in the single subject matter program have led one or more
workshops at the Expanding Your Horizons Career day.
The two semester sequence Workshop in Mathematics I/II (MATH 190/191) and the
History of Mathematics (MATH 490) address contributions by women and diverse
groups to the discipline. The Workshop in Mathematics I and II (MATH 190/191)
courses each contains a Mathematical Culture and People Component and a Modern
Mathematics and Careers Component which address cultural, ethnic and gender issues
explicitly through readings, assignments, discussions, interviews and guest speakers. As
the course syllabus for History of Mathematics (MATH 490) indicates, students
encounter the mathematics of Babylonian, Egyptian, Chinese, Indian and Arabian
cultures. Many of the projects/reports assigned are on topics related to contributions of
diverse cultural, ethnic and gender groups; for example, Ramanujan and the meeting
between "untutored" Indian genius and traditional British scholarship; Emmy Noether's
life as a study in the plight of women in early 20th century mathematics; Sofia
Kovalevskaya's life and work.
Cooperation and understanding amongst all students is consciously cultivated beginning
in freshmen level classes and supported throughout by means of the student study lounge
which is located across from faculty offices and adjacent to the department's two
computer labs. In addition, student and faculty participation in a wide variety of Math
Club activities encourages cooperation and understanding.
The university's core curriculum requires every LMU student to take a three unit course
in American Cultures or a course cross-listed with the American Cultures program.
These courses must deal with at least three of Euro-American, Asian-American, AfricanAmerican, Hispanic-American or Native American cultures. (See Appendix VII.)
2.2
The institution recruits and provides information and advice to men and
women prospective teachers from diverse backgrounds on requirements
for admission to and completion of subject matter programs.
15
The department encourages men and women students, and students who are culturally
and ethnically diverse, to enter and complete the subject matter program. Care is taken to
achieve ethnic and gender balance among the off campus speakers invited for career talks
in the Mathematics Workshop courses. The LMU Future Teachers Club and special
programs which have been institutionalized as a result of LMU's participation in the Los
Angeles Collaborative for Teacher Excellence (see Appendix V) such as the Future
Teachers Conference and the Meet the Teachers Roundtable provide information, advice,
and encouragement to male and female prospective teachers from a wide variety of
backgrounds.
2.3
The curriculum in the Subject Matter Program reflects the perspectives
and contributions of diverse groups from a variety of cultures to the
disciplines of study.
The Workshop Courses in Mathematics I and II (MATH 190/191) each contain a
Mathematical Careers and People Component and a Modern Mathematics and
Mathematical Culture Component that address cultural, ethnic and gender issues
explicitly through readings, assignments, discussions, interviews and guest speakers. For
example, in the MATH 190 Mathematical People/Biography Assignment, students
individually research the life of a 20th century mathematician, write a paper, and report to
a small group about their mathematician.
As the course syllabus for History of Mathematics (MATH 490) indicates, students
encounter the mathematics of Babylonian, Egyptian, Chinese, Indian and Arabian
cultures. Many of the projects/reports assigned are on topics related to contributions of
diverse cultural, ethnic and gender groups; for example, Ramanujan and the meeting
between "untutored" Indian genius and traditional British scholarship; Emmy Noether's
life as a study in the plight of women in early 20th century mathematics; Sofia
Kovalevskaya's life and work; developments in ancient Chinese mathematics and
philosophy which parallel - altogether independently - the roots of mathematics as
developed in classical Greece; L. K. Hua and the effects of the Maoist cultural revolution
on Chinese mathematics. A typical supplementary reading list includes titles such as The
Crest of the Peacock: Non-European Roots of Mathematics by G. Joseph, Math Equals
by T. Perl, Women in Mathematics by L. Osen, "Black Women in Mathematics in the
United States," The American Mathematical Monthly. Vol. 88 No. 8 (October 1981),
pp. 592 - 604 by P. Kenschaft, The Man Who Knew Infinity - A Life of the Genius
Ramanujan by R. Kanigel, Women of Mathematics by L. Grinstein and P. Campbell, and
Black Mathematicians by V. Newell (ed.). Thus, there are a variety of ways in which the
History of Mathematics course helps students acquire knowledge, understanding and
appreciation of the perspectives and contributions of diverse cultural, ethnic and gender
groups related to the discipline.
2.4
In the subject matter program, classroom practices and instructional
materials are designed to provide equitable access to the academic
content of the program to prospective teachers from all backgrounds.
16
As described in detail in Standard 5 Varied Teaching Strategies and Standard 16 Delivery
of Instruction, a wide variety of pedagogical and instructional approaches, including
direct instruction, small group work with hands-on materials, technology-based
assignments, cooperative learning, open-ended projects, student-led discussion and oral
presentations, are employed in courses in the program in order to provide equitable
access to students of all backgrounds and to model good teaching practice for future
teachers. Faculty work with the Learning Resource Center on campus to ensure that
disabled students’ learning needs are accommodated.
2.5
The subject matter program incorporates a wide variety of pedagogical
and instructional approaches to academic learning suitable to a diverse
population of prospective teachers. Instructional practices and materials
used in the program support equitable access for all prospective teachers
and take into account current knowledge of cognition and human learning
theory.
A wide variety of pedagogical and instructional approaches, including direct instruction,
small group work with hands-on materials, technology-based assignments, cooperative
learning, open-ended projects, student-led discussion, oral presentations, are employed in
courses in the program in order to provide equitable access to students of all backgrounds
and to model good teaching practice for future teachers.
The Innovations in Teaching Mathematics, Science and Engineering luncheon seminar
series sponsored by the Math and Science Teacher Preparation Committee and held 5
times a year at the university's Center for Teaching Excellence (see Appendix VI for a list
of topics/presentations at the Center) affords faculty teaching in the program the
opportunity to consider and discuss the latest developments in the teaching and learning
mathematics to make sure that all students are well-served.
17
Standard 3: Technology
The study and application of current and emerging technologies, with a focus on those
used in K-12 schools, for gathering, analyzing, managing, processing, and presenting
information is an integral component of each prospective teacher’s program study.
Prospective teachers are introduced to legal, ethical, and social issues related to
technology. The program prepares prospective teachers to meet the current technology
requirements for admission to an approved California professional teacher preparation
program.
Technology is an integral part of any mathematics curriculum. In the Single Subject
Matter program at LMU, technology is interwoven throughout the curriculum. Students
first use technology as part of their subject matter coursework in the freshman Workshop
in Mathematics I/II (MATH 190/191), where students use Excel and graphing
calculators to gather, analyze, and manage information in problem solving exercises. In
the Calculus Sequence (MATH 131, 132, & 234), students are expected to use
Mathematica or other computer algebra systems to solve applied mathematics problems
and are expected to produce mathematically sophisticated reports using these programs.
In addition, in these courses graphing calculators are used extensively by the students for
graphing, modeling, and calculation. The social issues in teaching mathematics with
technology are covered in the Senior Seminar for Future Mathematics Educators
(MATH 493 - see objective 3). Moreover, students receive explicit training in using
mathematics software and in programming in Elementary Numerical Methods (MATH
282). Other courses addressing the technology standard are Ordinary Differential
Equations (MATH 245) and Geometry (MATH 550). This coursework meets the
technology requirements for admission to Loyola Marymount University’s professional
teacher preparation program.
Loyola Marymount students have ready access to the Mathematics Department
Computing Labs which contains 32 computers: 16 G3 iMACs and 16 Pentium III Dells.
In addition to this lab the computer science department has a laboratory with a variety of
IBM machines that math students are able to use and a computer lab with several Sun
Workstations. Through the machines in the labs and with high speed connections in the
dorms students have access to the internet and email. The LMU Mathematics department
also has classroom demonstration capability with G4 Power MAC series computers in 3
of the classrooms dedicated to the Mathematics Department. Each computer in the
computer lab and in the classrooms in the Mathematics Department has access over a
local area network to Microsoft Office, Mathematica, Theorist, FORTRAN, C, Grapher,
and Geometers Sketchpad, and Internet Explorer and Netscape. In addition, all math
majors are expected to own and learn to use the programming ability of a graphing
calculator.
18
Required Elements
3.1
The institution provides prospective teachers in the subject matter
program access to a wide array of current technology resources. The
program faculty selects these technologies on the basis of their effective
and appropriate uses in the disciplines of the subject matter program.
Students are exposed to a wide variety of technology resources in the program, including
explicitly, Excel (Workshop in Mathematics I/II (MATH 190/191), Senior Seminar
for Future Mathematics Educators (MATH 493), Geometer’s Sketchpad (Geometry
(MATH 550)), Mathematica or other computer algebra system (Calculus III (MATH
234) and Elementary Numerical Methods (MATH 282)), graphing calculators
(Calculus Sequence (MATH 131 & 132) and Senior Seminar for Future Mathematics
Educators (MATH 493)), and programming (Elementary Numerical Methods (MATH
282)).
In each case, technology has been chosen that is most appropriate to best address learning
and discovery issues of the coursework. For example, in the National Council of
Teachers of Mathematics (NCTM) on-line electronic examples, Geometer’s Sketchpad is
routinely used to present examples. Likewise, Mathematica has been chosen by a wide
range of mathematics departments to be used as a teaching and learning aid in the core
calculus requirements.
3.2
Prospective teachers demonstrate information processing competency,
including but not limited to the use of appropriate technologies and tools
for research, problem solving, data acquisition and analysis,
communications, and presentation.
Throughout the Single Subject Matter program, students are expected to demonstrate
technological information processing competency. Students are first exposed to the use
of technology as a tool for research and problem solving in the Workshop in
Mathematics I/II (MATH 190/191) and often in Calculus I and II (MATH 131 & 132).
In these courses, students use Excel as a tool for problem solving and research on
numerical patterns and Mathematica and graphing calculators for working with calculus
type problems. In Senior Seminar for Future Mathematics Educators (MATH 493)
as part of their semester research projects, students must show competency in using
technology as a research/problem-solving tool.
Students first learn to use technology for data acquisition in the Workshop in
Mathematics I/II (MATH 190/191), where they use Excel and graphing calculators as
data gathering tools for patterns. In Probability and Statistics (MATH 360) students
learn to use statistical programs for data acquisition and analysis (on computers and
graphing calculators), and in Elementary Numerical Methods (MATH 282), students
must demonstrate competence in using mathematical programming for data analysis and
acquisition.
In Geometry (MATH 550) students are expected to use Geometer’s
19
Sketchpad as a tool for data acquisition in geometry. In particular, they will work on
assignments where they take measurements using Geometer’s Sketchpad and then change
the associated figures in real time to see how measurements change.
The Single Subject Matter program has also integrated the use of technology in
communication and presentation throughout the program. Students first learn to use
Microsoft Word for making mathematical presentations in Workshop in Mathematics
I/II (MATH 190/191). Their final project in Senior Seminar for Future Mathematics
Educators (MATH 493) is to be written in Word, Powerpoint, or with another
appropriate word-processing program. Such presentations occur at other points during
their curriculum as well, often in History of Mathematics (MATH 490) and Geometry
(MATH 550).
3.3:
In the program, prospective teachers use current and emerging
technologies relevant to the disciplines of study to enhance their subject
matter knowledge and understanding.
Again, this element is integrated throughout the Single Subject Matter program
curriculum at Loyola Marymount University. Students use Excel to enhance their
understanding of problem solving in Workshop in Mathematics (MATH 190/191) as
well as in the Senior Seminar for Future Mathematics Educators (MATH 493). In
addition, instructors in the Calculus Sequence (MATH 131,132, & 234) use
Mathematica as a tool for gaining numerical and pictorial representations of calculus
concepts. In Geometry (MATH 550), students use both Geometer’s Sketchpad and other
hands-on and web-based tools to enhance their understanding of both Euclidean and nonEuclidean geometry. Finally, in the Senior Seminar for Future Mathematics
Educators (MATH 493), discussions of the use of technology for enhancing learning and
understanding are interwoven throughout the term as those technologies arise. For
example, one class day is spent showing the students Calculator based Laboratories for TI
graphing calculators, followed by a discussion of the use of these calculators to enhance
students’ subject matter knowledge.
20
Standard 4: Literacy
The program of subject matter preparation for prospective Single Subject teachers
develops skills in literacy and academic discourse in the academic disciplines of study.
Coursework and field experiences in the program include reflective and analytic
instructional activities that specifically address the use of language, content and
discourse to extend meaning and knowledge about ideas and experiences in the fields or
discipline of the subject matter.
All 17 required mathematical content courses in the program involve learning to read,
speak, write, and listen to mathematical reasoning. Three lower division courses have
mathematical discourse and reasoning/proof as a particular focus. The first two are
freshmen year courses Workshop in Mathematics I/II (MATH 190/191) which are
intended to develop students’ abilities to use mathematical language and habits of mind
to analyze, synthesize, and evaluate mathematical problems and experiences. These
courses involve substantial work in problem solving and communicating mathematical
reasons using correct language and notation. Assignments and peer review of others'
writings help students learn to read and evaluate the mathematical writing of others. The
third course, a sophomore course, Introduction to Methods of Proof (MATH 248),
helps students continue to develop their ability to use mathematical language, methods
and notation as they learn additional methods of proof - which is the essential vehicle of
communication of mathematical results. Together these courses prepare students for
more mathematically demanding work in the upper division courses, where they have to
develop abstract proofs on their own, research and write papers, and make presentations
to classes, their instructor, and sometimes other faculty members.
In conjunction with their field experience course Field Experience in Mathematics
(MATH 293), students read professional mathematics teacher education journal articles
and respond in writing in their field experience journals to prompts such as:
 Describe 3 examples of how mathematical language and/or reading skills were
evidenced in the field experience; and
 Summarize in a few sentences each of your readings and describe their relationship to
the fieldwork.
Required Elements
4.1
The program develops prospective teachers’ abilities to use academic
language, content, and disciplinary thinking in purposeful ways to
analyze, synthesize and evaluate experiences and enhance understanding
in the discipline.
As stated above, the freshmen year courses Workshop in Mathematics I/II (MATH
190/191) are designed to develop students’ abilities to use mathematical language and
habits of mind to analyze, synthesize, and evaluate mathematical problems and
experiences. These courses involve substantial work in problem solving and
21
communicating mathematical reasons using correct language and notation. Assignments
and peer review of others' writing help students learn to read and evaluate the
mathematical reasoning and writing of others. In the Problem Solving Component of
these workshop courses, students learn the importance of “Looking Back” after reaching
a solution to a problem. This is a particularly important disciplinary way of thinking in
mathematics that promotes analysis, synthesis and evaluation of the problem solving
experience and leads to deeper understanding.
4.2
The program prepares prospective teachers to understand and use
appropriately academic and technical terminology and the research
conventions of the disciplines of the subject matter.
The Mathematical Communication Component in Workshop in Mathematics I/II
(MATH 190/191) is designed to improve students’ mathematical writing and oral
communication skills. One portion of this component focuses on correct use of technical
terminology with assignments that address various aspects of writing mathematics, such
as, defining variables, giving reasons, honoring the equal sign, watching pronouns.
Building on the work on mathematical reasoning/proof from the freshmen workshop
courses, in the sophomore year, Methods of Proof (MATH 248), students continue to
develop their ability to use mathematical language, methods and notation as they learn
additional methods of proof - which is the gold standard of mathematical discourse and
essential to communicating results of mathematical research.
4.3
The program provides prospective teachers with opportunities to learn
and demonstrate competence in reading, writing, listening, speaking,
communicating and reasoning in their fields or discipline of the subject
matter.
Workshop in Mathematics I/II (MATH 190/191) involve substantial work in
communicating mathematical reasons using correct language and notation. Assignments
and peer review of others' writings help students learn to read and evaluate the
mathematical writing of others. Work in groups and discussion and peer evaluation of
other students’ oral presentations helps students learn to listen to mathematical
reasoning. One portion of the Study Skills Component in Workshop in Mathematics
I/II (MATH 190/191) focuses on learning to read a mathematics text.
Methods of Proof (MATH 248) functions as a “bridge” course, that is a course
designed to prepare students for upper division coursework in which they must exercise
and advance their skills in mathematical discourse. They demonstrate their ability to
communicate their mathematical reasoning both verbally and in writing in upper
division courses such as Geometry (MATH 550), History of Mathematics (MATH
490), and Senior Seminar for Future Mathematics Educators (MATH 493) which
contain assignments requiring them to research and write substantial mathematics papers
and to present their work to the entire class, and in some cases to other faculty members
as well.
22
Standard 5: Varied Teaching Strategies
In the program, prospective Single Subject teachers participate in a variety of learning
experiences that model effective curriculum practices, instructional strategies and
assessments that prospective teachers will be expected to use in their own classrooms.
The faculty of the Mathematics Department of Loyola Marymount University use a wide
variety of instructional strategies and assessments in their teaching. To keep the faculty
abreast of current developments, faculty members are encouraged to attend workshops on
teaching at the LMU Center for Teaching Excellence. In addition, the department sends
individual faculty members to teaching workshops, and the university supports faculty
members attending teaching workshops and conferences. Moreover, faculty are
supported in attending the American Mathematics Society – Mathematical Association of
America Joint Mathematics Meetings where issues in the teaching of mathematics are at
the center of many special sessions and workshops. In the last two years, three faculty
members (Curtis Bennett, Jackie Dewar, and Blake Mellor) have attended Mathematics
Association of America Preparing Mathematicians for the Education of Teachers (PMET)
workshops. In addition, Jackie Dewar facilitated a session at the 2003 Science Education
for New Civic Engagements and Responsibilities (SENCER) conference sponsored by
the Association of American Colleges and Universities, Curtis Bennett was an invited
attendee at two scholarship of teaching and learning meetings (Oxford College at Emory
and University of Illinois), and Jackie Dewar and Curtis Bennett are 2003 Carnegie
Scholars with the Carnegie Academy for the Scholarship of Teaching and Learning
(CASTL) program. All of these activities have been generously supported by the
university. The Mathematics and Science Teacher Preparation (MASTeP) university
committee sponsors presentations at the Center for Teaching Excellence that specifically
address the learning and teaching of prospective teachers (see Appendix VI). In the
2002-2003 academic year, the mathematics department faculty members gave two such
presentations, and the mathematics department sent more than 15 department members to
MASTeP presentations (counting with multiplicity).
Specific learning experiences modeling effective curriculum practices happen in the
Calculus Sequence (MATH 131, 132, & 234) where small group discussion and group
project work is used, in Workshop in Mathematics I/II (MATH 190/191) where
students do individual and group investigations, individual presentations in which they
both present material and lead discussions and practice peer instruction, Methods of
Proof (MATH 248) in which portfolio assessment is practiced, History of Mathematics
(MATH 490) in which students lead discussions, Senior Seminar for Future
Mathematics Educators (MATH 493) in which group work, individual exploration, peer
teaching, and student-centered discussions are used, and Geometry (MATH 550) in
which students conduct explorations, do group work, and receive direct instruction. Of
course, such student-centered instruction techniques occur in most other courses in
addition to direct instruction. Assessment techniques are varied throughout the program
coursework also. Portfolios are used in Workshop in Mathematics I/II (MATH
190/191), Methods of Proof (MATH 248), and Senior Seminar for Future
Mathematics Educators (MATH 493), paper assessments are used in Workshop in
23
Mathematics I/II (MATH 190/191), History of Mathematics (MATH 490), and Senior
Seminar for Future Mathematics Educators (MATH 493). Hands on materials such
as unifix cubes, Rubik’s cubes, dominoes, and calculator based laboratories are used in
Workshop in Mathematics I/II (MATH 190/191), Geometry (MATH 550) and Senior
Seminar for Future Mathematics Educators (MATH 493).
Required Elements
5.1
Program faculty include in their instruction a variety of curriculum
design, classroom organizational strategies, activities, materials and field
experiences incorporating observing, recording, analyzing and
interpreting context as appropriate to the discipline.
Throughout the LMU Single Subject Matter Program in Mathematics students are
exposed to a wide variety of curriculum design and classroom strategies. Students first
encounter the “experimental method” in mathematics problem solving in Workshop in
Mathematics I/II (MATH 190/191) where they learn problem solving strategies that
involve gathering, recording, and analyzing evidence on their own. Using technology
such as graphing calculators, Excel, and Geometers Sketchpad, in Workshop in
Mathematics I/II (MATH 190/191) and Geometry (MATH 550), students embark on
longer-term in-class projects. In addition, students write expository papers in which they
analyze and interpret mathematical results in the Calculus Sequence (MATH 131, 132,
& 234) in Workshop in Mathematics I/II (MATH 190/191), in Senior Seminar for
Future Mathematics Educators (MATH 493), and in other classes.
Finally, in
Probability and Statistics (MATH 360), students interpret and work with data sets.
5.2
Program faculty imply a variety of interactive, engaging teaching styles
that develop and reinforce skills and concepts through open-ended
activities such as direct instruction, discourse, demonstrations, individual
and cooperative learning explorations, peer instructions and studentcentered discussion.
While traditionally one thinks of mathematics teaching as primarily being done by direct
instruction, at the LMU Mathematics Department a wide variety of interactive teaching
methods are used in addition to direct instruction. In the Calculus Sequence (MATH
131, 132, & 234) faculty show the relationships between physical phenomena and
mathematics using demonstrations. In addition cooperative learning explorations occur
during student projects and Mathematica activities. Workshop in Mathematics I/II
(MATH 190/191) use open-ended activities to develop and reinforce skills and concepts
in addition to both individual and cooperative learning explorations, as does Geometry
(MATH 550) and the Senior Seminar for Future Mathematics Educators (MATH
493). Students lead discussions, use peer instruction, and participate in student-centered
discussions in Workshop in Mathematics I/II (MATH 190/191), History of
Mathematics (MATH 490), and Geometry (MATH 550). In addition, Senior Seminar
for Future Mathematics Educators (MATH 493) asks students to interpret grades 6-12
24
student responses, reflect on their fieldwork, and has students interact with each other to
debate the meaning and understanding of mathematical statements and definitions.
5.3
Faculty development programs provide tangible support for subject matter
faculty to explore and use exemplary and innovative curriculum practices.
Loyola Marymount University and its Mathematics Department actively supports faculty
development, particularly as regarding future teacher education. There are three main
avenues of support for faculty development: (1) Faculty attend special meetings
regarding teacher education – Curtis Bennett (2002), Jackie Dewar (2002), and Blake
Mellor (2003) have all attended Mathematical Association of America (MAA) Preparing
Mathematicians for the Education of Teachers (PMET) meetings (see Appendix X); (2)
Faculty attend national meetings with emphasis on the teaching of mathematics – these
include the annual Joint Mathematics Meetings of the AMS-MAA (four faculty attended
last year (2003) and five (2002) the previous year). At these meetings there are many
sessions devoted to teaching and learning in mathematics; and (3) the Center for
Teaching Excellence (CTE) at Loyola Marymount University hosts weekly presentations
on teaching and learning, and faculty in the mathematics department are encouraged to
attend (see Appendix VI). Indeed, the merit documents for the department specifically
mention attending CTE workshops. In addition, the university Mathematics and Science
Teacher Preparation (MASTeP) committee works in conjunction with the CTE to
organize presentations specifically discussing teacher education and learning in
mathematics and the sciences. In 2002-2003, two of these were in mathematics.
The CTE also gives summer faculty grants for faculty finding ways to bring innovative
and exemplary curriculum practices into their classrooms. These grants are the same size
($4000) as the university summer research grants, and they provide faculty with the
resources and an incentive to explore better teaching methods.
Lastly, the Mathematics Department has two Carnegie Fellows with the Carnegie
Academy for the Scholarship of Teaching and Learning (Curtis Bennett and Jackie
Dewar). Loyola Marymount University made a commitment of $15,000 dollars for these
faculty members to be able to undertake their work with the Carnegie Foundation during
2003-2004.
5.4
Program faculty used varied and innovative teaching strategies, which
provide opportunities for prospective teachers to learn how content is
conceived and organized for instruction in a way that fosters conceptual
understanding as well as procedural knowledge.
As discussed in Element 5.2, program faculty use a wide variety of innovative teaching
strategies to help students develop and reinforce skills and concepts in mathematics.
These strategies are used throughout the program. In the department there are two
courses taken primarily by prospective teachers, Senior Seminar for Future
Mathematics Educators (MATH 493) and Geometry (MATH 550). In these courses
innovative strategies are used to examine content taught in the 6-12 curriculum. For
25
example, in sample assignment 3 on the syllabus for Geometry (MATH 550) students
investigate puzzles using geometric figures, use these puzzles to develop several standard
proofs from plane geometry, and then discuss advantages/disadvantages of each proof
argument. Alongside of projects like these, which involve both group and individual
investigative learning, classroom Socratic lectures include time for student-centered
discussions about the teaching and learning of Euclidean geometry. These discussions
typically range over a wide variety of issues, including what are the “learning packets 1”
required of students and teachers for this material. Similarly, in Senior Seminar for
Future Mathematics Educators (MATH 493), various teaching strategies are used
depending on the discussion. For example, the discussion of the difference between a
variable and an unknown is a student-centered discussion in which the professor interacts
only to raise questions. This particular item lends itself to having students discuss what
conceptually lies behind the understanding of a variable.
The writing assignments for Senior Seminar for Future Mathematics Educators
(MATH 493) are chosen to encourage students to focus upon how content is conceived
and organized for instruction. Some assignments encourage the students to look at how
the material they are learning fits into a high school curriculum (see sample assignment
3). Other assignments promote a deeper understanding of the high school curriculum (see
sample assignment 2 and the project) and examine links between conceptual
understanding and procedural knowledge (see sample assignment 1). Teaching in the
class also takes on a kinesthetic role, such as in the presentation of solving a cubic, where
three-dimensional geometric figures are produced. This goes along with a discussion of
how solving the quadratic equation comes from a geometric picture of actually
completing a square. In this case, the class discussion centers around how one learns to
manipulate algebraically with understanding. In addition, Senior Seminar for Future
Mathematics Educators (MATH 493) asks students to interpret grades 6-12 student
responses, reflect on their fieldwork, and has students interact with each other to debate
the meaning and understanding of mathematical statements and definitions.
5.5
Program coursework and fieldwork include the examination and use of
various kinds of technology that are appropriate to the subject matter
discipline.
Technology is integrated throughout the major. Students are introduced to the use of
graphing calculators and Mathematica, a computer algebra system, in the Calculus
Sequence (MATH 131, 132, & 234). They are introduced to the use of Excel as a
mathematics tool in Workshop in Mathematics I/II (MATH 190/191). Students study
programming in Elementary Numerical Methods (MATH 282) and examine and use
Geometer’s Sketchpad in Geometry (MATH 550). In Senior Seminar for Future
Mathematics Educators (MATH 493 – see objective 3), the graphing calculator, Excel,
and Geometer’s Sketchpad are examined as learning/teaching tools in mathematics (see
Appendix IX for descriptions of these software programs).
1
The idea of Learning Packets comes from Knowing and Teaching Elementary Mathematics by
Liping Ma (1999).
26
Standard 6: Early Field Experiences
The program provides prospective Single Subject teachers with planned, structured field
experiences in departmentalized classrooms beginning as early as possible in the subject
matter program. These classroom experiences are linked to program coursework and
give a breadth of experiences across grade levels and with diverse populations. The
early field experience program is planned collaboratively by subject matter faculty,
teacher education faculty and representatives from school districts. The institution
cooperates with school districts in selecting schools and classrooms for introductory
classroom experiences. The program includes a clear process for documenting each
prospective teacher’s observations and experience.
The subject matter program requires that students complete 20 hours of planned
observation, instruction, tutoring or other activities with culturally and linguistically
diverse groups of 12 to 18 year old students as appropriate for future secondary
mathematics teachers. These hours are documented in the required course Field
Experience in Mathematics (MATH 293), which should be completed before the senior
capstone experience Senior Seminar for Future Mathematics Educators (MATH 493).
Students keep a field experience journal which includes: school name, location, grade
level, diversity of classroom/students, teacher's name and position (and certification), and
a dated log entry including time spent on site which summarizes the observation
including mathematical topics discussed, teaching methods used, classroom management
methods observed, use of technology observed, impressions of student response to the
lesson/tutoring/etc. This field experience course also includes associated readings from
professional journals and written reflection on both the readings and the field experience.
As a pre/co-requisite to the senior capstone Senior Seminar for Future Mathematics
Educators (MATH 493), the student's field experience is linked to their capstone
experience. Assignments and class discussion in MATH 493 provide the student with
additional opportunities to reflect on and draw from their field experience.
Field experience placements for mathematics are planned collaboratively by a
subcommittee of the LMU Math and Science Teacher Preparation Committee (MASTeP)
and school district personnel. The MASTeP subcommittee is chaired by the School of
Education’s Clinical Faculty Member in charge of field experiences. Other members of
this committee come from the Department of Mathematics and the Department of Natural
Sciences. All placements are required to involve teachers certified in math.
The assignment of a 200-level course number to the field experience indicates to the
students that their field experience should begin at the lower division. Departmental
advisors of students in the subject matter program discuss the timing of this requirement
with each student.
27
Required Elements
6.1
Introductory experiences shall include one or more of the following
activities: planned observations, instruction or tutoring experiences, and
other school based observations or activities that are appropriate for
undergraduate students in a subject matter preparation program.
Field Experience in Mathematics (MATH 293) requires documentation of 20 hours of
field work experience with students ages 12-18 from diverse populations and in the
presence of a teacher certified in mathematics, along with associated readings and
reflective writing assignments. Completion of MATH 293 is a pre-requisite for Senior
Seminar for Future Mathematics Educators (MATH 493) which provides students
opportunities to further reflect on their field experience.
6.2
Prospective teachers’ early field experiences are substantively linked to
the content of coursework in the program.
The field experience is tightly linked to Field Experience in Mathematics (MATH 293)
for which students are required to keep a field journal and to Senior Seminar for Future
Mathematics Educators (MATH 493) where students further reflect on their field
experience.
Journal prompts ask students to make connections between their
observations and other content courses in the program; to describe and comment on the
use of technology in their field sites; and to observe and describe how mathematical
language and/or reading skills were evidenced in the field experience.
6.3
Fieldwork experiences for all prospective teachers include significant
interactions with K-12 students from diverse populations represented in
California public schools and cooperation with at least one carefully
selected teacher certificated in the discipline of study.
Field Experience in Mathematics (MATH 293) requires documentation of 20 hours of
field work experience with students ages 12-18 from diverse populations and the
presence of a teacher certified in mathematics. Field experience placements for
mathematics are planned collaboratively by a subcommittee of the LMU Math and
Science Teacher Preparation Committee (MASTeP) and school district personnel. The
MASTeP subcommittee is chaired by the School of Education’s Clinical Faculty Member
in charge of field experiences. Other members of this committee come from the
Department of Mathematics and the Department of Natural Sciences. MATH 293
requires the field experience to involve one or more teachers certified in math.
The School of Education at LMU employs Clinical Faculty to maintain relationships with
key personnel in the schools and to find credentialed teachers for LMU students to
28
observe and to practice teach under. The selection criteria for classes and teacher for
observation include:
1) Schools with student and staff populations that reflect the diversity of Southern
California (This ties directly to the mission of LMU.)
2) Teachers that have a deep understanding of the mathematics which they teach
3) Teachers who model sound pedagogical practices
4) Teachers who are able to articulate how theory relates to their practice
5) Teachers who teach to the state-adopted mathematics content standards
We have inserted a memo from Ms. Kimberly Haag below (dated 2/19/04) describing in
detail the selection process from her perspective as a Clinical Faculty member in the
School of Education.
MEMO from LMU Clinical Faculty for Secondary Education, School of Education
TO:
Dr. Jackie Dewar
Professor, Department of Mathematics
FROM:
RE:
Kimberly Haag, B.S., M.A,
Clinical Faculty, Loyola Marymount University
Selection Criteria
DATE: February 20, 2004
Based on my experience as the School of Education Clinical Faculty in charge of
secondary education and the Clinical Faculty representative on the Math and
Science Teacher Preparation Committee, I am able to describe how field
placements are made within the School of Education.
The School of Education (SoE) maintains relationships with administrators and
key personnel at school sites such as Westchester High School in Los Angeles,
North and West High School in Torrance, Culver City High School and
Hawthorne High School. These schools represent the diverse environment of
Southern California both in their student body and on their teaching staff. This
partnership affords us opportunities to place our teachers in classrooms with
exemplary teachers for observation and student teaching. Our master teachers are
fully credentialed and meet the requirements of the No Child Left Behind Act,
they are often tenured in their schools and have mentored teachers before
becoming master teachers. Many of our master teachers are observed by Clinical
Faculty or other SoE faculty before they become master teachers. In these
observations we are looking for teachers who teach to the content standards, have
highly interactive classrooms where higher-level thinking is occurring. We seek
teachers who demonstrate deep conceptual knowledge of their subject matter area,
as well as knowledge of educational theory, and the ability to apply this theory to
the practice of teaching. In addition we want teachers who are reflective
practitioners, collaborative by nature and willing to consider a variety of
29
approaches. Clinical Faculty meet with every master teacher to explain the
expectations the University has for master teachers.
During and at the end of a candidate’s student teaching, each master teacher is
asked to evaluate the candidate in their content knowledge as well as their
pedagogical strategies employed in teaching. Each student teacher, in turn, fills
out a form evaluating the master teacher’s content knowledge and pedagogical
practices. In addition, all master teachers complete a verification of experience
form that is kept on file in the SoE. This form indicates the credentials and
degrees held as well as the pedagogical strategies the teacher employs on a
regular basis. The SoE is fortunate to have many alumni become master teachers,
because candidates beginning in the program are able to observe teachers that
have met these requirements.
Currently, the Clinical Faculty are compiling a notebook that contains lists of
recommended schools and teachers for Subject Matter Program students to
observe. We choose these teachers based on the criteria above and students’
recommendations.
Specific secondary and middle schools with which LMU has regular communication and
partnership are Westchester High School, Culver City Middle School and High School,
Hawthorne High School, West High School in Torrance and North High School in
Torrance as well as Magruder Middle School in Torrance.
Racial/Ethic Demographics for each of these schools is shown below obtained from the
Internet using School Accountability Report Cards (Available at
http://www.cde.ca.gov/ope/sarc/sarclink2.asp?County_Number=19).
The percentage of students is the number of students in a racial/ethnic category divided
by the school's most recent California Basic Educational Data Systems (CBEDS) total
enrollment.
Westchester HS 2001-2
African- American
Amer.-Ind. or Alaska Native
Asian- American
Filipino- American
Hispanic or Latino
Pacific Islander
White (Not Hispanic)
Total Enrollment
Percent
64.7%
0.4%
1.6%
0.5%
25.7%
0.4%
6.6%
1943
30
Culver City HS 2001-2
African- American
Asian- Amer./Pacific Is.
Hispanic or Latino
White/Euro. Amer./Other
Total Enrollment
Percent
23%
13%
36%
27%
1821
Culver City MS 2001-2
African- American
Asian- Amer./Pacific Is.
Hispanic or Latino
White/Euro. Amer./Other
Total Enrollment
Percent
19%
12%
40%
29%
1521
North HS, Torrance 2001-2
African- American
Amer.-Ind. or Alaska Native
Asian- American
Filipino- American
Hispanic or Latino
Pacific Islander
White (Not Hispanic)
Total Enrollment
Percent
7.9%
0.5%
35.4%
2.6%
20.9%
1.1%
31.6%
2071
West HS, Torrance 2001-2
African- American
Amer.-Ind. or Alaska Native
Asian- American
Filipino- American
Hispanic or Latino
Pacific Islander
White (Not Hispanic)
Total Enrollment
Percent
3.4%
0.5%
35.0%
1.7%
11.4%
0.6%
47.4%
1906
31
Magruder MS, Torrance 2001-2
African- American
Amer.-Ind. or Alaska Native
Asian- American
Filipino- American
Hispanic or Latino
Pacific Islander
White (Not Hispanic)
Total Enrollment
Hawthorne HS 2001-2
African- American
Amer-Ind. or Alaska Native
Asian- American
Filipino- American
Hispanic or Latino
Pacific Islander
White (Not Hispanic)
Multiple/No Response
Total Enrollment
6.4
Percent
6.4%
0.4%
24.9%
2.4%
24.4%
1.9%
39.6%
795
Percent
11%
0.1%
1.6%
1.6%
76.3%
2.6%
5.2%
1.5%
2969
Prospective teachers will have opportunities to reflect on and analyze
their early field experiences in relation to course content. These
opportunities may include field experience journals, portfolios, and
discussions in the subject matter courses, among others.
The field experience is organized, amplified, and documented in the required course
Field Experience in Mathematics (MATH 293) for which students keep a field journal.
Journal prompts ask students to make connections between their observations and other
content courses in the program; to describe and comment on the use of technology in
their field sites; to observe and describe how mathematical language and/or reading skills
were evidenced in the field experience. In addition to the field experience journal they
keep, the single subject matter program assures that students further reflect on their field
experience in Senior Seminar for Future Mathematics Educators (MATH 493)
through required assignments and discussion.
32
6.5
Each prospective teacher is primarily responsible for documenting early
field experiences. Documentation is reviewed as part of the program
requirements.
The field experience is documented in the required course Field Experience in
Mathematics (MATH 293) for which students are required to keep a field journal. Field
journal entries include: school name, location, grade level, diversity of
classroom/students, teacher's name and position (and certification), and a daily dated log
entry including time spent on site which summarizes the observation including
mathematical topics discussed, teaching methods used, classroom management methods
observed, use of technology observed, impressions of student response to the
lesson/tutoring/etc. This field journal is first reviewed by the prospective teacher's
departmental advisor and later used as a resource in the capstone experience Senior
Seminar for Future Mathematics Educators (MATH 493).
33
Standard 7: Assessment of Subject Matter Competence
The program uses formative and summative multiple measures to assess the subject
matter competence of each candidate. The scope and content of each candidate’s
assessment is consistent with the content of the subject matter requirements of the
program and with institutional standards for program completion.
Multiple formative and summative assessments occur in the Bachelor of Art in
Mathematics (BAM) degree program that underpins the Single Subject matter program in
mathematics. Since students' coursework in BAM encompasses Standards 11-16 for
Mathematics and the Mathematics Subject Matter Requirements Parts I and II (which
address the Mathematics Content Standards for California Public Schools) and their
coursework is thoroughly evaluated in each course by a variety of means (see Required
Elements below), it follows that students are assessed on each of these Standards and that
assessment is congruent with studies.
While the assessment that occurs in each of our courses is important, there are 3
especially critical points of evaluation for each student within the subject matter program
in mathematics.
(1) The courses Workshop in Mathematics I/II (MATH 190 and 191) function
as skill and confidence builders and provide the students with much assistance in
the freshman year. These courses develop problem solving skills, improve
mathematical writing, and introduce the students to a broad array of modern and
historical topics and career opportunities within mathematics. In these courses,
students write and present a mathematical paper and put together a portfolio.
(2) In the sophomore year of the program Introduction to Methods of Proof
(MATH 248) functions as a bridge course to upper division and hence more
theoretical work. It is a critical course which provides a maturation period for the
students. Faculty teaching this course know that they will put in long hours
reading students' written work and making suggestions for revisions.
(3) The third critical evaluation occurs when the student takes the senior level
capstone course Senior Seminar for Future Mathematics Educators (MATH
493). In addition to providing multiple opportunities to assess integration of
subject matter via class work, students are required to complete original (to them)
research and make oral and written presentations on it. Typically all department
faculty are invited to the presentations.
34
Required Elements
7.1
Assessment within the program includes multiple measures such as
student performances, presentations, research projects, portfolios, field
experience journals, observations, and interviews as well as oral and
written examinations based on criteria established by the institution.
Credit-No credit grading is not allowed in courses in a student's major, minor or core.
Thus all students receive letter grades in their required 17 mathematics degree courses.
At midterm the Registrar's office asks all faculty to submit names of students doing
unsatisfactory work (grades at midterm of C-, D and F) and students are then officially
notified of any "midterm deficiencies."
The assessment process includes a variety of approaches. Each mathematics course in
the program has a final examination or project based on course content. Normally these
examinations/projects are written, but at the instructor's discretion they may have an oral
component or be entirely oral. Tests, quizzes, homework, and class participation are
other commonly used assessment tools. Certain required courses always involve
computer-based assignments (for example, Numerical Methods (MATH 282)). By
choice of the instructor, graphing calculator or computer based assignments frequently
occur in Calculus (MATH 131/132/234), Differential Equations (MATH 245) and
Geometry (MATH 550). Student presentations and projects including both a written and
an oral component are a required part of the Workshop in Mathematics I/II (MATH
190 and 191), of History of Mathematics (MATH 490), of Senior Seminar for Future
Mathematics Educators (MATH 493), and of Geometry (MATH 550). Students
create portfolios in at least three courses in the program (Workshop in Mathematics I/II
(MATH 190 and 191) and Methods of Proof (MATH 248)). For their required Field
Experience in Mathematics (MATH 293) and for the elective Mathematics Internship
(MATH 393) students keep field experience journals.
7.2
The scope and content of each assessment is congruent with the
specifications for the subject matter knowledge and competence as
indicated in the content domains of the Commission-adopted subject
matter requirement.
The course syllabi submitted with this application serve to promote and provide evidence
of congruence of Single Subject Matter program coursework/assessment with the content
domains of the Commission-adopted subject matter requirement (see Appendix I).
7.3
End-of-program summative assessment of subject matter competence
includes a defined process that incorporates multiple measures for
evaluation of performance.
Senior Seminar for Future Mathematics Educators (MATH 493) provides multiple
opportunities to assess integration of subject matter via homework and in-class
35
assignments. Students are required to complete original (to them) research and make oral
and written presentations on it. Typically all department faculty are invited to the
presentations. The student’s final grade in the course determined by the instructor from
this body of work constitutes a summative assessment.
7.4
Assessment scope, process, and criteria are clearly delineated and made
available to students when they begin the program.
The LMU Undergraduate Bulletin describes the three possible degree programs in
mathematics (Bachelor of Arts in Mathematics - BAM, Bachelor of Science in
Mathematics - BSM, and Bachelor of Science in Applied Mathematics - BSAM) and
advises students to choose the BAM degree if they are interested in teaching at the
secondary level as it is designed to meet the single subject matter program requirements
in mathematics.
Incoming freshmen mathematics majors are given academic advising during the
orientation process by a team of math faculty and current math students. Transfer
students or students who change majors have their first advising session with the
Mathematics Department Chairperson.
The three possible degree programs in
mathematics are discussed and students are advised to choose the BAM degree if they
are interested in teaching at the secondary level as BAM together with the Field
Experience component (described below) is designed to meet the Single Subject Matter
Program requirements in mathematics and give the student an undergraduate degree.
Freshmen students are first assigned a faculty specializing in freshman year advising and
then shifted to an advisor who specializes in their degree program in the sophomore year.
On a first meeting with a BAM degree student, the BAM advisor reviews the graduation
requirements in that program (as detailed in the LMU Undergraduate Bulletin) including
the Field Experience component. The advisor recommends that the students begin
meeting the field experience requirements no later than the sophomore year and refers
them to the chair of the Field Experience Subcommittee of Math and Science Teacher
Preparation Committee (MASTeP). The documentation process for a student's field
experience is managed through enrollment in a 0 credit course Field Experience in
Mathematics (MATH 293) which is a pre-requisite to the senior capstone experience
Senior Seminar for Future Mathematics Educators (MATH 493) where the student
has additional opportunities to reflect on the field experience as part of the capstone
coursework.
7.5
Program faculty regularly evaluate the quality, fairness, and effectiveness
of the assessment process, including its consistency with program
requirements.
The Mathematics faculty have a well-deserved reputation for tough but fair grading.
Faculty are generally accommodating to special circumstances, such as, unexpected
travel out of the country or illness and are willing to give a late or make-up exam. The
University also provides for giving an Incomplete grade that allows the student up until
36
about four weeks into the next term to complete the work. In addition, provisions exist
for a student to appeal a final grade and are detailed in the LMU Undergraduate Bulletin.
If the appeal cannot be settled by the student, the faculty member, and the department
chair, then the student may file an appeal with the Dean in writing. The Dean will bring
the concerned parties together and if they fail to reach an agreement, the Dean may
appoint a committee of three impartial parties to investigate the matter and make a
recommendation. The final decision rests with the Dean. It is a testament to the fairness
of grading by the math faculty that no such appeal has been filed in at least the last 20
years.
7.6
The institution that sponsors the program determines, establishes and
implements a standard of minimum scholarship (such as overall GPA,
minimum course grade or other assessments) of program completion for
prospective single subject teachers.
As detailed in the 2003 LMU Undergraduate Bulletin, Mathematics majors (that includes
those in the BAM degree program which underpins the Single Subject Matter Program in
Mathematics) must maintain a minimum cumulative grade point average of C (2.0) in
upper division major requirements; a minimum grade of C (2.0) in each course in the
lower division requirements, in order to graduate. In addition, in order to enroll in any
mathematics course a student must have a minimum grade of C (2.0) in any prerequisite
course.
37
Standard 8: Advisement and Support
The subject matter program includes a system for identifying, advising and retaining
prospective Single Subject teachers. This system will comprehensively address the
distinct needs and interests of a range of prospective teachers, including resident
prospective students, early deciders entering blended programs, groups
underrepresented among current teachers, prospective teachers who transfer to the
institution, and prospective teachers in career transition.
The Subject Matter Program in Mathematics in the College of Science and Engineering is
prominently featured in the 2003 LMU Undergraduate Bulletin under the heading of
Teacher Preparation Programs (p. 288). Students are informed that the Bachelor of Arts
degree in Mathematics is designed to allow completion of the California Preliminary
Single Subject credential in four years, and are advised to inform their departmental
advisor of their interest in teaching and to contact the School of Education for help in
coordinating their programs. In addition, the program allows the student to complete the
mathematics degree, clear credential and a Master of Arts in Teaching Mathematics in
five years including two summers.
The Mathematics department has specialized designated advisors for each of its three
degree programs. Typically four or five faculty are designated Bachelor of Arts advisors,
so each has no more than five prospective teachers to advise.
Incoming freshmen mathematics majors are given academic advising during the
orientation process by a team of math faculty and current math students. Transfer
students or students who change majors have their first advising session with the
Mathematics Department Chairperson.
The three possible degree programs in
mathematics (Bachelor of Arts in Mathematics - BAM, Bachelor of Science in
Mathematics - BSM, and Bachelor of Science in Applied Mathematics - BSAM) are
discussed and students are advised to choose the BAM degree if they are interested in
teaching at the secondary level as it is designed to meet the single subject matter program
requirements in mathematics. Since all lower division mathematics requirements are
identical (except the field experience component), students can still easily opt into the
BAM degree in their junior year. Freshmen students are first assigned a faculty
specializing in freshman year advising and then shifted to an advisor who specializes in
their degree program in the sophomore year. On a first meeting with a BAM degree
student, the BAM advisor reviews the graduation requirements in that program (as
detailed in the LMU Undergraduate Bulletin) including the (new) Field Experience
component Field Experience in Mathematics (MATH 293). The advisor recommends
that the students begin meeting the field experience requirements no later than the
sophomore year and refers them to to the chair of the Field Experience Subcommittee of
Math and Science Teacher Preparation Committee (MASTeP).
Specialized advice about credential programs, requirements and teaching careers is
available from the School of Education. Orientation meetings are held four or five times
a year by the various teacher education credential programs and are advertised across the
38
University campus. Students interested in a teaching career are strongly encouraged to
attend one of these meetings by the end of their sophomore year. The meetings inform
prospective teachers of necessary requirements for entrance into the credential programs,
as well as details of the programs and services available to students through the School of
Education. School of Education faculty are available for individual counseling on
various paths to a teaching credential.
Students expressing an interest in a teaching career are encouraged to attend the Future
Teacher Club meetings and related activities such as the annual Future Teachers
Conference which provide additional information and support for navigating the
credential process.
Loyola Marymount University actively seeks to recruit and retain minority students. The
statistics of the 2002 incoming freshman class attest to the institution's success in this
area: 52% White, 6% African American, 19% Hispanic/Latino, and 10% Asian/Pacific
Islander, with 11% declining to state, and 1% non-resident alien. The freshman class is
61% female and 39% male. Across all four years of the mathematics major for 20032004, the statistics for 46 majors are: 63% White, 2% African American, 15%
Hispanic/Latino, 13% Asian/Pacific Islander, and 7% unknown, with 46% female and
54% male. Amongst the 17 full-time faculty, there are 4 women, 1 Latino, 1 Indian, and 1
Russian "refusenik."
Required Elements
8.1
The institution will develop and implement processes for identifying
prospective Single Subject teachers and advising them about all program
requirements and career options.
As mentioned above, incoming freshmen mathematics majors are giving academic
advising during the orientation process by a team of math faculty and current math
students. Transfer students or students who change majors have their first advising
session with the Mathematics Department Chairperson. The three possible degree
programs in mathematics (Bachelor of Arts in Mathematics - BAM, Bachelor of Science
in Mathematics - BSM, and Bachelor of Science in Applied Mathematics - BSAM) are
discussed and students are advised to choose the BAM degree if they are interested in
teaching at the secondary level as it is designed to meet the single subject matter program
requirements in mathematics. Freshmen students are first assigned a faculty specializing
in freshman year advising and then shifted to an advisor who specializes in their degree
program in the sophomore year. On a first meeting with a BAM degree student, the
BAM advisor reviews the graduation requirements in that program (as detailed in the
LMU Undergraduate Bulletin) including the (new) Field Experience component Field
Experience in Mathematics (MATH 293). The advisor recommends that the students
begin meeting the field experience requirements no later than the sophomore year and
refers them to to the chair of the Field Experience Subcommittee of Math and Science
Teacher Preparation Committee (MASTeP).
39
Transfer students or students who change majors have their first advising session with the
Mathematics Department Chairperson.
The three possible degree programs in
mathematics (Bachelor of Arts in Mathematics - BAM, Bachelor of Science in
Mathematics - BSM, and Bachelor of Science in Applied Mathematics - BSAM) are
discussed and students are advised to choose the BAM degree if they are interested in
teaching at the secondary level as it is designed to meet the single subject matter program
requirements in mathematics. Since all lower division mathematics requirements are
identical, students can still easily opt into the BAM degree in their junior year. After
their initial advising with the department chairperson, transfer students are assigned to an
advisor who specializes in their degree program. On a first meeting with a BAM degree
student, the BAM advisor reviews the graduation requirements in that program (as
detailed in the LMU Undergraduate Bulletin) including the (new) Field Experience
component Field Experience in Mathematics (MATH 293). The advisor recommends
that the transfer students begin meeting the field experience requirements as soon as
possible and refers them to the chair of the Field Experience Subcommittee of the Math
and Science Teacher Preparation Committee (MASTeP).
The university's computerized advisement/registration system connects faculty to the
University Registrar's data-base. Each semester academic advisors in mathematics must
give an electronic go-ahead signal that their advisee has consulted with them before the
student may register. As soon as a student registers in one of the education prerequisites
for the single subject credential program, Socio-Cultural Analysis of Education,
Educational Psychology for the Adolescent Years, or Theories in Second Language
Acquisition, the student is designated as a potential single subject credential student.
This provides an accurate list of single subject credential students to the School of
Education and facilitates communication of important advising information.
Specialized advice about credential programs, requirements and teaching careers is
available from the School of Education. Orientation meetings are held four or five times
a year by the various teacher education credential programs and are advertised across the
University campus and given prominent space on the math department bulletin board
devoted to teaching careers. Math students interested in a teaching career are strongly
encouraged by their departmental advisors to attend one of these meetings by the end of
their sophomore year. The meetings inform prospective teachers of necessary
requirements for entrance into the credential programs, as well as details of the programs
and services available to students through the School of Education. School of Education
faculty are available for individual counseling on various paths to a teaching credential.
8.2
Advisement services will provide prospective teachers with information
about their academic progress, including transfer agreements and
alternative paths to a teaching credential, and describe the specific
qualifications needed for each type of credential, including the teaching
assignments it authorizes.
Students are provided with a wide variety of information from a number of sources given
in parenthesis in the following list. Advisement services include information about the
40
Single Subject Matter Program (LMU Undergraduate Bulletin, departmental advisor,
departmental Bulletin Board for Teaching), course equivalencies (department advisor,
department chairperson and associate dean of the College of Science and Engineering),
financial aid options (Office of Financial Aid, departmental Bulletin Board for Teaching,
Future Teachers Club communications), admission requirements in professional
preparation programs (departmental advisor, School of Education advisor, School of
Education Orientation sessions, departmental Bulletin Board for Teaching), state
certification requirements (School of Education advisor, School of Education Orientation
sessions, Future Teachers Club), field experience placements (departmental advisor,
School of Education advisor, and the chair of the Field Experience Subcommittee of
Math and Science Teacher Preparation Committee (MASTeP), and career opportunities
(Workshop in Mathematics I/II (MATH 190/191) presentations and assignments,
Future Teachers Club, LMU Career Development Services, Meet the Teachers
Roundtable event sponsored by MASTeP, Meet the Districts Career night event
sponsored by School of Education).
8.3
The subject matter program facilitates the transfer of prospective teachers
between post-secondary institutions, including community colleges,
through effective outreach and advising and the articulation of courses
and requirements. The program sponsor works cooperatively with
community colleges to ensure that subject matter coursework at feeder
campuses is aligned with the relevant portions of the State-adopted
Academic Content Standards for K-12 Students in California Public
Schools.
The university has an excellent record of collaboration with community colleges to
articulate academic curricula and to facilitate the transfer of students into the university.
On average, between 10 and 20% of our majors are transfer students.
Moreover, the university and program faculty have an excellent working relationship
with community colleges in the local area. For example, LMU and community college
faculty organize two major events for future teachers throughout the Los Angeles area:
The Future Teachers Conference and the Meet the Teachers Roundtable (see
www.futureteachersconference.org) which originated under the Los Angeles
Collaborative for Teacher Excellence grant NSF-DUE 94-53608 (see Appendix V).
Since 2001, Loyola Marymount University has had an articulation officer, Ms. Alice
Gandara, located in the Registrar’s Office whose responsibility it is to oversee and
facilitate articulation agreements for the university. We have inserted a memo from Ms.
Gandara below (dated 2/5/04) describing in some detail the articulation process from her
41
perspective. In general, the university’s transfer policies parallel those of the UC and
CSU systems. Articulation agreements have been completed with 16 community
colleges including all of the Los Angeles Community College District and our main
community college feeder schools (El Camino College and Santa Monica College).
Agreements with 8 other community colleges in the local area should be in place later
this Spring. These agreements are developed in consultation with the Associate Dean in
the College of Science and Engineering, the chairperson of the Mathematics Department,
and, if needed, the single subject matter program coordinator. Articulation decisions are
based on catalog descriptions but additional information, such as detailed course
descriptions and/or syllabi, is sometimes requested in order to assure equivalence with
coursework in the single subject matter program. This careful scrutiny ensures that
subject matter coursework at feeder schools is equivalent to corresponding work at
Loyola Marymount. Since coursework in the Single Subject Matter program at LMU is
aligned with the relevant portions of the State-adopted Academic Content Standards for
K-12 students in California Public Schools, it follows that transferred courses are as well.
More significantly, LMU program and university faculty forged close working
relationships with community college faculty at the main feeder community colleges as
members of an NSF-funded Collaborative for Excellence in Teacher Preparation. As a
result of their participation in this NSF-CETP (the Los Angeles Collaborative for Teacher
Excellence or LACTE), the main feeder community colleges and LMU are very
cognizant of the critical elements of an excellent teacher preparation program at both
two- and four-year institutions. In addition, we continue to work together on a number of
programs (the annual Future Teachers Conference and Meet the Teachers Roundtable
event) for future K-12 math and science teachers.
42
MEMO from LMU Articulation Officer
TO: Dr. Jackie Dewar
Professor, Department of Mathematics
FROM:
Alice Gandara
Articulation Officer/Transfer Services Manager
RE:
Draft Response for CTC
DATE: February 5, 2004
1. Responsibility of the Articulation Officer:
 To coordinate the process of faculty review and approval of
community college courses for transfer credit and fulfillment of
core and lower division degree requirements.
 To assist in the formulation of university guidelines for the
transferability of courses and fulfillment of core and lower division
degree requirements.
 To update articulation agreements on a yearly basis to incorporate
curriculum changes at the community college and at LMU.
 To maintain the university database that “houses” transfer
information to insure accuracy of Transfer Course Approval forms
and Equivalency Worksheets.
 To review all Transfer Course Approval forms and Equivalency
Worksheets to insure that articulation agreements are honored and
university guidelines are upheld.
2. University Transfer Policies:
 LMU accepts, with few exceptions, all courses approved by the
UC Office of the President (UCOP). Such courses are reviewed by
faculty to determine if they articulate with an LMU course and/or
fulfill a core or lower division major requirement.
 A course that is approved by UCOP and has no LMU equivalent
(such as Oceanography) is automatically approved by the
Articulation Officer.
 Those courses that are transferable only to the CSU system (i.e.,
not reviewed or approved by UCOP) require further scrutiny by
faculty in order to determine their transferability. Once the
transferability of such a course is established, the course is then
reviewed for core and lower division major requirements.
3. Articulated Schools:
 LMU has articulation agreements with the following schools:
1. El Camino College
2. Marymount College
43
3.
4.
5.
6.
7.
8.
8.4
Santa Monica College
Pasadena City College
Glendale Community College
Orange Coast College
Don Bosco Technical Institute
Los Angeles Community College District: Pierce,
Harbor, Southwest, East LA, West LA, City, Valley,
Mission, Trade Tech

LMU is currently working on articulation agreements with:
1.
Foothill-De Anza District: Foothill, De Anza
2.
South Orange County Community College District:
Saddleback, Irvine Valley
3.
Moorpark College
4.
Ventura College
5.
Mt. San Antonio College
6.
Long Beach City College
These should be completed by late Spring, 2004.

LMU’s 5-year goal is to articulate with the community colleges in
Southern California. The long-term goal is to articulate with all
the California community colleges.
The institution establishes clear and reasonable criteria and allocates
sufficient time and personnel resources to enable qualified personnel to
evaluate prospective teachers’ previous coursework and/or fieldwork for
meeting subject matter requirements.
It is the responsibility of the Mathematics department chair and the Associate Dean of
Science to evaluate prospective teachers' previous coursework/transcripts and/or
fieldwork for meeting subject matter requirements.
44
Standard 9: Program Review and Evaluation
The institution implements a comprehensive, ongoing system for periodic review of and
improvement to the subject matter program. The ongoing system of review and
improvement involves university faculty, community college faculty, student candidates
and appropriate public schools personnel involved in beginning teacher preparation and
induction. Periodic reviews shall be conducted at intervals not exceeding 5 years.
Loyola Marymount University provides three mechanisms for program review in
mathematics: the university-level Academic Planning and Review Committee (APRC),
the departmental curriculum committee, and the university-level Math and Science
Teacher Preparation Committee (MASTeP see Appendix IV). The function and duty of
each is described below.
LMU's Academic Planning and Review Committee (APRC) is charged with reviewing all
academic programs. The last review of the mathematics department programs was in
1996-97. The APRC review process is quite extensive and requires a department to
review its mission, analyze the curriculum and compare it to other schools, describe the
curriculum design and review process, faculty governance, recruitment and retention, and
support services and staff. As part of this process the department must prepare a lengthy
review document and be interviewed by an APRC committee member. Information is
gathered about the program's strengths, weaknesses, and needed improvements from
faculty, students, and recent graduates. A final report is prepared by APRC and sent to
the department and to the Academic Vice President.
The Mathematics Department Curriculum Committee (consisting of three math faculty) is
responsible for maintaining up to date course descriptions and syllabi, reviewing and
making recommendations regarding course or curricular changes. Typically, the
department chair refers matters related to curriculum to this committee. A faculty
member can also request of the department chair that a particular matter be considered by
the committee. The committee's recommendations are discussed at department meetings
which typically occur several times a semester. Curriculum matters are decided by a
majority vote. In practice, there is generally unanimous or near unanimous agreement on
most matters. The department chair and the curriculum committee conduct de facto
program reviews responding to concerns about the program that are discovered through
the department chair's exit interviews with students, concerns raised by faculty, and/or
due to changes in national and statewide standards in mathematics. For example, the
Curriculum Committee was the major force in developing the new Bachelor of Arts in
Mathematics degree program in 2002 to improve recruitment of secondary teachers
enrolled in the Single Subject Matter Program in Mathematics.
One responsibility of the university-level committee on Math and Science Teacher
Preparation (MASTeP) is to assist with review and evaluation of the single subject matter
programs in mathematics and in science and for the multiple subject matter
concentrations in mathematics and science. One program is reviewed each year on a
45
rotating four year cycle (2004 Multiple Subject Concentration in Mathematics, 2005
Multiple Subject Concentration in Science, 2006 Single Subject in Mathematics, 2007
Single Subject in Science). The MASTeP Committee which has members from the math
faculty, the science faculty and the education faculty selects a review board which
includes representation from LMU faculty in math, science and education, community
college faculty from LMU's main sources of transfer students (Santa Monica College and
El Camino College), LMU program graduates, current students and public school
personnel. This board reviews and advises on data and reports collected by subject
matter programs or concentrations in review of program goals and curriculum, other
requirements, use of technology, advising services, student success, assessment
procedures and program outcomes, along with quality and effectiveness of partnerships
with K-12 schools and community colleges. (see Appendix XII for a list of the
membership of the 2003-2004 MASTeP Review Board).
In addition, motivated by assessment requirements for WASC, in 2003 the university
instituted a Learning Outcomes Assessment Plan under the auspices of the Director of
Assessment and Data Analysis. A new Mathematics Departmental Assessment
Committee has begun collecting data and student work samples to investigate a specific
question about the ability of mathematics major graduates to evaluate and communicate
mathematical reasoning.
Required Elements
9.1
Each periodic review includes an examination of program goals, design,
curriculum, requirements, student success, technology uses, advising
services, assessment procedures and program outcomes for prospective
teachers.
As described above, the MASTeP Review Board examines the Single Subject Matter
Program in Mathematics every four years, with the next review scheduled for 2006.
Included in the review are: program goals and curriculum, other requirements, use of
technology, advising services, student success, assessment procedures and program
outcomes.
9.2
Each program review examines the quality and effectiveness of
collaborative partnerships with secondary schools and community
colleges.
As described above, the MASTeP Review Board examines the Single Subject Matter
Program in Mathematics every four years, with the next review scheduled for 2006.
Included in the review are: quality and effectiveness of partnerships with K-12 schools
and community colleges. The Review Board contains representatives from K-12 school
districts and community colleges.
46
9.3
The program uses appropriate methods to collect data to assess the
subject matter program’s strengths, weaknesses and areas that need
improvement. Participants in the review include faculty members, current
students, recent graduates, education faculty, employers, and appropriate
community college and public school personnel.
As described above, the MASTeP Committee which has members from the math faculty,
the science faculty and the education faculty selects a review board which includes
representation from LMU math, science and education faculty, community college
faculty from LMU's main sources of transfer students (Santa Monica College and El
Camino College), LMU program graduates, current students and public school personnel.
This board reviews and advises on data and reports collected by subject matter programs
or concentrations in review of program goals and curriculum, other requirements, use of
technology, advising services, student success, assessment procedures and program
outcomes, along with quality and effectiveness of partnerships with K-12 schools and
community colleges. (see Appendix XII for a list of the membership of the 2003-2004
MASTeP Review Board).
The collection of data for the assessment of our program will be done primarily by the
Mathematics Department and the Education College.
The Mathematics Department Assessment Committee has plans to use the following
methods to collect data for assessment (beginning in 2004):
 Annual exit exam – students will complete an exit exam to assess whether they
have mastered important topics and concepts in mathematics.
 Attitudinal survey – surveys assessing students attitudes about and approaches to
mathematics and problem solving.
 Exit Interview – the Mathematics Department Chair conducts exit interviews with
graduating seniors to assess all three majors in the mathematics program covering
student perceptions of the program, advising issues, their first two years, and their
upper division experience (including transitions from lower division to upper
division), and their view of their preparation for their career (including technology
uses), and their general educational experience (including their self-assessment of
the change in their ability to function as a member of a team).
The College of Education uses the following methods to collect information:
 Candidate Progress Reports on each student teacher – This report covers student
teacher’s content and pedagogical knowledge as assessed by the master teacher,
the mentor teacher, and the university supervisor.
 Candidate Final Evaluations on each student teacher – This is similar to the
progress report, except for when it is administered.
This information will be passed along to the MASTeP Review Board. In addition the
review board will look at student demographics such as gender, ethnicity, years to
graduation, percent of Bachelor of Arts in Mathematics (BAM) majors, and persistence in
47
BAM major (as collected by the university). In addition, MASTeP will attempt to
discover how many of our graduates continue teaching 2-years, 5-years, and 10-years
after graduation. The community college faculty on the MASTeP Review Board will
give input on the quality and effectiveness of the program’s partnership with community
colleges.
The MASTeP Review Board will analyze this information to make a formative
assessment of the program, specifically identifying strengths and weaknesses. Based on
this analysis, the board will provide recommendations to the Mathematics Department for
changes to improve the program and the quality of our students’ preparation for future
teaching.
9.4
Program improvements are based on the results of periodic reviews, the
inclusion and implications of new knowledge about the subject(s) of study,
the identified needs of program students and school districts in the region,
and curriculum policies of the State of California.
Periodic reviews of the program and curriculum policies of the State of California do
foster program improvements. For example, issues raised in preparing the single subject
program document in 1994 resulted in the department institutionalizing in departmental
course descriptions and syllabi: (1) Portfolio assignments are given in both semesters of
Workshop in Mathematics I /II (MATH 190/191), and Introduction to Methods of
Proof (MATH 248); and (2) Oral presentations are to be made by every student both in
History of Mathematics (MATH 490) and in the capstone course Senior Seminar for
Future Mathematics Educators (MATH 493). Previously, these were simply practices
that individual faculty members employed in these courses. Another example occurred
during the writing of this year's document. In examining Standard 6 Field Experience, it
became clear that a more formal method of incorporating the field experience into the
program was required. As a result, a new course Field Experience in Mathematics
(MATH 293) was proposed by faculty writing the document, and approved by the
department.
As mentioned above, the Mathematics department's Curriculum Committee was the
major force in developing the new Bachelor of Arts in Mathematics degree program to
serve the needs of prospective secondary teachers enrolled in the Single Subject Matter
Program in Mathematics. This was as a result of examining the overall mathematics
program and curriculum and attempting to better serve the three different clientele
groups: future teachers, students heading off to graduate school, and students going to
work as industrial mathematicians (NOTE: Separate degree programs were developed for
the other two groups as well).
The Dean of the College of Science and Engineering conducts yearly exit interviews with
all graduating students. These interviews are an important source of information about
various programs within the College which the Dean shares with the appropriate
departments. Because of the demonstrated usefulness of the data gathered by the Dean's
48
interviews, the mathematics department chair also conducts exit interviews. The
Mathematics Department maintains informal contact with many of its graduates, by
inviting them back to give career talks, to attend the national math honor society (Pi Mu
Epsilon) initiation ceremony each year, and to attend special retirement dinners or
building dedications. These informal contacts are another important source of
information about how well our program prepares its students. For example, recent
graduates of the single subject matter program who were classroom teachers commented
how grateful they were to have been introduced to Excel through their teaching
assistantship assignment in the mathematical literacy course which is part of the
math/science core for students not required to take a mathematics course as part of their
major. As a result, using Excel as a tool to collect data and look for patterns has been
incorporated into activities in Workshop in Mathematics I/II (MATH 190/191).
49
Standard 10: Coordination
One or more faculty responsible for program planning, implementation and review
coordinate the Single Subject Matter Preparation Program. The program sponsor
allocates resources to support effective coordination and implementation of all aspects of
the program. The coordinator(s) fosters and facilitates ongoing collaboration among
academic program faculty, local school personnel, local community colleges and the
professional education faculty.
Two university-level committees are concerned with teacher preparation.
(1) The university-wide Teacher Education Committee (TEC) serves as the umbrella
group that ensures collaboration between the School of Education and the rest of the
University. The Academic Vice President serves as the Chair of the Committee. A math
faculty member representing the Mathematics Subject Matter Program is a member of the
Teacher Education Committee.
(2) The Mathematics and Science Teacher Preparation Committee (MASTeP) was
established in April 2000 to institutionalize and carry forward improvements and
activities originated by the Los Angeles Collaborative for Teacher Excellence (LACTE
see Appendix V). LACTE was a $5,500,000 project funded by the National Science
Foundation for years 1995-2000 which joined ten colleges and universities with the goal
of improving K-12 math and science teacher preparation in the greater Los Angeles area.
The responsibilities of MASTeP include:
 Coordinate math and science teaching oriented internships;
 Run the Innovations in Math/Science/Engineering Luncheon Seminar Series;
 Organize the annual Meet the Teachers Roundtable event and assist with the annual
Future Teachers Conference;
 Promote and moderate the Future Teachers Club (an important vehicle for recruiting
prospective teachers);
 Serve as advisory boards for the single subject matter programs in mathematics and in
science;
 Assist with coordination of the field experiences component of the single subject
matter programs in math and science;
 Develop additional programs to enhance K-12 teacher preparation in Mathematics
and Science; and
 Facilitate communication and coordination between subject matter faculty and
professional education faculty.
Membership consists of nine faculty with at least two members each from the areas of
Mathematics, Science, and Education. In addition representatives of the multiple subject
program, community colleges, and local schools and school districts are invited to
meetings of this committee.
A third departmental-level committee is also concerned with mathematics teacher
preparation.
50
(3) The Mathematics Department “School of Education Liaison Committee” (SEL) is
charged with facilitating communication between the Mathematics Department and the
School of Education on matters related to the single subject matter program and
mathematics concentration in the multiple subject matter program. SEL also monitors
curriculum needs of the program and the concentration. This Committee has 3 full time
math faculty as members.
Required Elements
10.1
A program coordinator will be designated from among the academic
program faculty.
The program coordinator for the Single Subject Matter Program in Mathematics is one of
the three mathematics faculty on the Mathematics department's School of Education
Liaison (SEL) committee and is on the MASTeP Committee as a member or ex-officio
member. Currently, Jackie Dewar is program coordinator.
10.2
The program coordinator provides opportunities for collaboration by
faculty, students, and appropriate public school personnel in the design
and development of and revisions to the program, and communicates
program goals to the campus community, other academic partners, school
districts and the public.
The program coordinator makes use of the two university level committees (TEC and
MASTeP) and one departmental level committee (SEL) described above to facilitate
collaboration and communication with other entities within and outside of the university.
10.3
The institution allocates sufficient time and resources for faculty
coordination and staff support for development, implementation and
revision of all aspects of the program.
Teacher preparation is one of 5 areas that the President of Loyola Marymount University
has declared at his inauguration in 1999 the institution will focus on. To that end, the
institution has committed time and resources to two university-level committees: TEC
and MASTeP. University deans and the academic vice president are participating
members of TEC. The MASTeP committee described above has an annual budget of
$12,500 to support paid internships, a faculty luncheon seminar series focusing on
innovations in teaching math, science and engineering, the Future Teachers Club and the
Meet the Teachers Roundtable event. In 2002, the mathematics department was allowed
to hire a faculty member at an advanced rank with experience in teacher preparation
(Curtis Bennett).
Since the number of students in our program is relatively small, secretarial support within
the department is adequate, and advising duties are shared, there is currently no need for
release time for the program coordinator. Support equivalent to 75% of a summer
51
research grant is being split between the two faculty members who are writing this
document. Moreover, the university readily supports faculty attendance at national, state,
and local conferences and workshops related to teacher education.
10.4
The program provides opportunities for collaboration on curriculum
development among program faculty.
At the department level the School of Education Liaison Committee (SEL) is charged
with facilitating communication between the Mathematics Department and the School of
Education on matters related to both the single subject matter program and the multiple
subject matter progra. SEL also monitors curriculum needs of the program. This
Committee has 3 full time math faculty as members. Within in the department the
Curriculum Committee provides another important avenue for collaboration on
curriculum development.
10.5
University and program faculty cooperate with community colleges to
coordinate courses and articulate course requirements for prospective
teachers to facilitate transfer to a baccalaureate degree-granting
institution.
Through the legacy of the Los Angeles Collaborative for Teacher Excellence - LACTE grant (a $5,500,000 project funded by the National Science Foundation for years 19952000 which joined ten colleges and universities in the greater Los Angeles area with the
goal of improving K-12 math and science teacher preparation), the university and
program faculty have an excellent working relationship with community colleges in the
local area (see Appendix V). Currently the subject matter program faculty (and the
MASTeP Committee) co-plan two major events for future teachers throughout the Los
Angeles area: The Future Teachers Conference and the Meet the Teachers Roundtable
(see www.futureteachersconference.org).
In addition, the university has an excellent record of collaboration with community
colleges to articulate academic curricula and to facilitate the transfer of students into the
university. On average, ten to twenty percent of our majors are transfer students.
As evidence that university and program faculty cooperate with community colleges to
coordinate courses and articulate course requirements for prospective teachers to
facilitate their transfer to LMU, we include the following memos.
52
EMAIL MEMO from Santa Monica College Math Department Chair
Subject: Coordination and Articulation on Teacher Preparation: SMC and LMU
Date: Wed, 4 Feb 2004 23:59:34 -0800
Thread-Topic: Coordination and Articulation on Teacher Preparation: SMC and
LMU
Thread-Index: AcPrvf4obC9OzA4SREm8txTSq2l59A==
From: "MANION_FRAN" <MANION_FRAN@smc.edu>
To: <jdewar@lmu.edu>
X-OriginalArrivalTime: 05 Feb 2004 22:24:19.0946 (UTC)
FILETIME=[CC7A44A0:01C3EC36]
Dear Dr. Dewar,
This purpose of this memo is to confirm that Loyola Marymount University
mathematics faculty do cooperate with the Santa Monica College Mathematics
Department on matters related to the preparation of K-12 mathematics teachers. I
believe that our two departments have developed a good working relationship
through our participation in an NSF-funded Collaborative for Excellence in
Teacher Preparation during 1995-2001. Our ongoing collaboration on a number of
projects continues to benefit the future K-12 math and science teachers at both of
our institutions.
When questions or problems arise regarding coordination or articulation of
courses, I can easily contact you or Dr. Michael Grady, the Mathematics
Department Chair, and I have confidence that we can reach a mutually acceptable
resolution. I have no doubt that together we will continue to facilitate transfer of
prospective teachers between our institutions.
If the Commission on Teacher Credentialing has any questions regarding these
matters, they may contact me by phone 310-434-4722 or by email:
manion_fran@smc.edu
Fran Manion
Chair, Mathematics
Santa Monica College
53
MEMO from El Camino College Division of Mathematical Sciences Dean
February 4, 2004
Dr. Jackie Dewar
Department of Mathematics
Loyola Marymount University
1 LMU Drive
Los Angeles, CA 90045
Dear Dr. Dewar,
I am writing to confirm that Loyola Marymount University mathematics faculty
continues to cooperate with the El Camino College Mathematics Department on
matters related to K-12 mathematics teacher preparation. The two departments
have developed a good working relationship through our participation in the NSFfunded Los Angeles Collaborative for Teacher Excellence (LACTE) during 19952001 (see www.lacteonline.org).
Our ongoing collaboration on a number of projects continues to benefit the future
K-12 math and science teachers at both of our institutions. When questions or
problems arise regarding coordination or articulation of courses, I can easily
contact you or Dr. Michael Grady, the Mathematics Department Chair, and I have
confidence that we can reach a mutually acceptable resolution. I have no doubt
that together we will continue to facilitate transfer of prospective teachers
between our institutions.
If the Commission on Teacher Credentialing has any questions regarding these
matters, they may contact me by phone 310-660-3200 or by email:
dgoldberg@elcamino.edu.
Sincerely,
Donald Y. Goldberg
Dean
Division of Mathematical Sciences
El Camino College
Torrance, CA
54
Standard 11: Required subjects of study
In the program, each prospective teacher studies and learns advanced mathematics that
incorporates the Mathematics Content Standards for California Public Schools:
Kindergarten through Grade Twelve (1997) and the Mathematics Framework for
California Public Schools: Kindergarten Through Grade Twelve (1999). The curriculum
of the program addresses the Subject Matter Requirements and standards of program
quality as set forth in this document.
The course of study of the Single Subject Matter program in mathematics is listed in
Appendix III. The coursework in the department that is required reflects the Subject
Matter Requirements as can be seen by looking at Grid 1 in Appendix II. Indeed, most of
the material in the Subject Matter Requirements is covered in more than one course in the
department to provide students with the breadth and depth of knowledge in each area that
come from seeing a topic multiple times in multiple contexts. Even those topics only
covered once in the grid are often covered multiple times. For example the grid lists that
(2.2a) Plane Euclidean Geometry on similarity and congruence is only explicitly covered
in Senior Seminar for Future Mathematics Educators (MATH 493) and Geometry
(MATH 550), however, problems are always assigned in the Calculus Sequence (MATH
131, 132, 234) requiring students to solve problems involving similarity and congruence.
A similar statement holds true for most other topics.
In addition, the curriculum meets the standards of quality. Loyola Marymount
mathematics students earning the Bachelor of Science (BSM) degree often matriculate
into top mathematics Ph.D. programs in the country. Single Subject Matter program
students take identical lower division coursework and many of the same upper division
courses as the BSM students. For the most part, the courses use standard undergraduate
mathematics texts. Program courses cover advanced mathematics incorporating the
Mathematics Content Standards for Public Schools. One can check this explicitly as each
standard in Grid I corresponds to a Content Standard for Public School. In addition,
Senior Seminar for Future Mathematics Educators (MATH 493) and Geometry
(MATH 550) both incorporate discussions directly related to the standards.
Required Elements
11.1
Required coursework includes the following major subject areas of
study: algebra, geometry, number theory, calculus, history of
mathematics, and statistics and probability. This coursework also
incorporates the content of the student academic content standards
from an advanced viewpoint (see Attachment to Standard 11: Required
subjects of study page 18). Furthermore, infused in required
coursework are connections to the middle school and high school
curriculum.
55
The required coursework in the Single Subject Matter program in mathematics at LMU
requires all of the areas of study as can be seen in Grid I in Appendix II. Below each
subject area of study is dealt with individually.
Domain 1.
Algebra
Candidates demonstrate an understanding of the foundations of the algebra contained in
the Mathematics Content Standards for California Public Schools: Kindergarten
Through Grade Twelve (1997) as outlined in the Mathematics Framework for California
Public Schools: Kindergarten Through Grade Twelve (1999) from an advanced
standpoint. To ensure a rigorous view of algebra and its underlying structures,
candidates have a deep conceptual knowledge. They are skilled at symbolic reasoning
and use algebraic skills and concepts to model a variety of problem-solving situations.
They understand the power of mathematical abstraction and symbolism.
The algebra standard consists of four subtopics: structures, polynomials, functions, and
linear algebra. The LMU Single Subject Matter program in mathematics covers all four
of these in detail as is seen below. Overall, the program aims to give the students a firm
background in mathematics to ensure a rigorous view of algebra and its underlying
structures. To this effect, algebra is covered explicitly in nine courses (covering 28 credit
hours): the Calculus Sequence (MATH 131, 132, 234), Methods of Proof (MATH 248),
Linear Algebra (MATH 250), Real Analysis (MATH 321), Group Theory (MATH
331), Complex Analysis (MATH 357), and Senior Seminar for Future Mathematics
Educators (MATH 493). By having students continue to study algebraic structures
throughout their degree program the program ensures that they have the breadth and
depth of knowledge and skills to teach algebra and to provide students with a wide
variety of problem solving situations. Moreover, students in the program will understand
the power of abstraction and symbolism in mathematics using algebra.
1.1
Algebraic Structures
a. Know why the real and complex numbers are each a field, and that particular
rings are not fields (e.g., integers, polynomial rings, matrix rings)
b. Apply basic properties of real and complex numbers in constructing
mathematical arguments (e.g., if a < b and c < 0, then ac > bc)
c. Know that the rational numbers and real numbers can be ordered and that the
complex numbers cannot be ordered, but that any polynomial equation with
real coefficients can be solved in the complex field
Algebraic structures are explicitly dealt with in five required classes in the curriculum:
Methods of Proof (MATH 248), Real Analysis (MATH 321), Group Theory (MATH
331), Complex Analysis (MATH 357), and Senior Seminar for Future Mathematics
Educators (MATH 493). The field properties of the real numbers are covered most
explicitly in Senior Seminar for Future Mathematics Educators (MATH 493) as is
shown in sample assignment 2, which expects students to generate all of the ordered-field
properties of the real numbers and apply basic properties of real and complex numbers in
56
constructing mathematical arguments. The properties of the complex numbers are
studied extensively in Complex Analysis (MATH 357). Finally, the order properties of
these fields are covered in Senior Seminar for Future Mathematics Educators
(MATH 493), Real Analysis (MATH 321), and Complex Analysis (MATH 357). In
addition, the other courses touch on these topics as can be seen from the course syllabi in
Appendix I. As discussed below, connections between the collegiate mathematics and
grades 6-12 mathematics are infused in the course of study. This is particularly true of
Senior Seminar for Future Mathematics Educators (MATH 493). Examples of
functions and problems from high school are infused in the Calculus Sequence (MATH
131, 132, & 234) also.
1.2
Polynomial equations and Inequalities
a. Know why graphs of linear inequalities are half planes and be able to apply
this fact (e.g., linear programming)
b. Prove and use the following: The Rational Root Theorem for polynomials with
integer coefficients
The Factor Theorem
The Conjugate Roots Theorem for polynomial equations with real
coefficients
The Quadratic Formula for real and complex quadratic polynomials
The Binomial Theorem
c. Analyze and solve polynomial equations with real coefficients using the
Fundamental Theorem of Algebra
Polynomial equations and inequalities are covered explicitly in the Calculus Sequence
(MATH 131, 132, & 234), Complex Analysis (MATH 357), and Senior Seminar for
Future Mathematics Educators (MATH 493). In addition, other courses such as
Differential Equations (MATH 245), History of Mathematics (MATH 490), and Real
Analysis (MATH 321) touch on these topics in homework sets and lecture. Graphs of
linear inequalities are covered explicitly in the Calculus Sequence (MATH 131, 132, &
234), while the main theorems on polynomials (Rational Root Theorem, Factor Theorem,
Conjugate Roots Theorem, Quadratic Formula, and Binomial Theorem) are covered
explicitly in Senior Seminar for Future Mathematics Educators (MATH 493) and
Complex Analysis (MATH 357). Finally, the Fundamental Theorem of Algebra is
covered in detail in Complex Analysis (MATH 357) and reviewed in Senior Seminar
for Future Mathematics Educators (MATH 493) when discussing algebraic and
transcendental numbers.
57
1.3 Functions
a. Analyze and prove general properties of functions (i.e., domain and range,
one-to-one, onto, inverses, composition, and differences between relations and
functions)
b. Analyze properties of polynomial, rational, radical, and absolute value
functions in a variety of ways (e.g., graphing, solving problems)
c. Analyze properties of exponential and logarithmic functions in a variety of
ways (e.g., graphing, solving problems)
Functions are covered in depth throughout the curriculum. Courses explicitly including
functions and their properties in their syllabi are the Calculus Sequence (MATH 131,
132, & 234), Methods of Proof (MATH 248), Group Theory (MATH 331), Complex
Analysis (MATH 357), and Senior Seminar for Future Mathematics Educators
(MATH 493). General properties of functions are covered in detail in Methods of Proof
(MATH 248) and the Calculus Sequence (MATH 131, 132, & 234). In addition, in
Group Theory (MATH 331) and Complex Analysis (MATH 357) students are expected
to work with the domain and range of functions as part of their study of these subjects.
Indeed, a common problem in Group Theory (MATH 331) is to prove that the
composition of one-to-one and onto functions is one-to-one and onto. The properties of
polynomial rational, radical, and absolute value functions are then studied in numerous
ways and from a variety of viewpoints including graphing, problem solving, and by
studying growth in the Calculus Sequence (MATH 131, 132, & 234). Polynomial
equations are dealt with more generally in Complex Analysis (MATH 357) and Senior
Seminar for Future Mathematics Educators (MATH 493). In the latter course,
students study polynomials intensively when proving the existence of transcendental
numbers by basic arguments on polynomials and their growth rates. This is then brought
back to the 6-12 curriculum when students are asked to examine what elements of the
proof can be discussed in which 6-12 classes, and why they might be important to cover
there. (For example, the idea that at a given point, the “slope” of a polynomial is
bounded can be talked about at various levels. In algebra 2 a teacher might simply point
out that the graph of a polynomial cannot be vertical at any point, while in calculus, a
teacher can point out the importance of the mean value theorem in bounding the growth
of a polynomial nearby a given point.) Finally, the properties of exponential and
logarithmic functions are covered in detail in the Calculus Sequence (MATH 131, 132,
& 234), Complex Analysis (MATH 357) and Senior Seminar for Future Mathematics
Educators (MATH 493). The first and last of these courses concentrates on real valued
exponential and logarithmic functions, where the growth and graph are studied, students
solve problems using them, and discuss the uses of logarithms in understanding the
Richter scale.
Complex Analysis (MATH 357) then broadens and deepens the
students’ understanding by generalizing these functions to the complex numbers.
58
1.4
Linear Algebra
a. Understand and apply the geometric interpretation and basic operations of
vectors in two and three dimensions, including their scalar multiples and
scalar (dot) and cross products
b. Prove the basic properties of vectors (e.g., perpendicular vectors have zero
dot product)
c. Understand and apply the basic properties and operations of matrices and
determinants (e.g., to determine the solvability of linear systems of equations)
Linear algebra is covered in detail in Linear Algebra (MATH 250), a class specifically
designed to teach topics involving vectors, their basic properties, and matrices and
determinants. In addition, the Calculus III (MATH 234) spends a great deal of time
working with vectors, their operations, and their geometric properties. Finally, Senior
Seminar for Future Mathematics Educators (MATH 493) also covers matrices,
vectors, and determinants, using these to prove facts about which numbers can be
constructed with ruler and straightedge as well as to discuss algebraic numbers and
finding polynomials having specified roots. Indeed, one problem in MATH 493 involves
having students link inverting a 2  2 matrix with rationalizing the denominator of a
fraction involving the sum of an integer and a square root. This prepares them for linking
a 3  3 matrix with how one might rationalize a denominator involving a cube root.
Domain 2.
Geometry
Candidates demonstrate an understanding of the foundations of the geometry contained
in the Mathematics Content Standards for California Public Schools: Kindergarten
Through Grade Twelve (1997) as outlined in the Mathematics Framework for California
Public Schools: Kindergarten Through Grade Twelve (1999) from an advanced
standpoint. To ensure a rigorous view of geometry and its underlying structures,
candidates have a deep conceptual knowledge. They demonstrate an understanding of
axiomatic systems and different forms of logical arguments. Candidates understand,
apply, and prove theorems relating to a variety of topics in two- and three-dimensional
geometry, including coordinate, synthetic, non-Euclidean, and transformational
geometry.
Not surprisingly, geometry is most centrally covered by the course Geometry (MATH
550) in the LMU curriculum. Every aspect of the subject matter requirements is
explicitly covered in this class, however, specific items from the geometry strand are also
covered in Calculus III (MATH 234), Linear Algebra (MATH 250), Complex
Analysis (MATH 357), History of Mathematics (MATH 490), and Senior Seminar for
Future Mathematics Educators (MATH 493) as discussed below.
The connections of geometry to the 6-12 curriculum are discussed throughout these main
courses covering the geometry standard. In particular, starting with Calculus III (MATH
234), students are introduced to the question of where do the formulas for volumes that
59
they learned in 6-12 classes come from. Geometry (MATH 550) explicitly links ideas to
the geometry students see in the 6-12 curriculum, as the course provides for both formal
and informal proofs of many of the ideas that arise in a high school geometry class. For
example, sample assignment 3 on the angle sum of a triangle begins this linking, while
problems such as finding the relationship between the sum of the angles of a spherical
triangle and the area of that triangle, call on understandings of surface area and ratio and
proportion in that curriculum.
A major goal of Senior Seminar for Future
Mathematics Educators (MATH 493) is the infusion of connections between the high
school curriculum and collegiate mathematics. This course links geometry topics to both
geometry and algebra in the high school curriculum. In particular, in this class the
following discussions take place: the history of the finding of values for pi, good
fractional approximations for this number and where they originate, links between the
Euclidean algorithm for finding greatest common divisors and geometric constructions,
etc.
2.1
Parallelism
a. Know the Parallel Postulate and its implications, and justify its equivalents
(e.g., the Alternate Interior Angle Theorem, the angle sum of every triangle is
180 degrees)
b. Know that variants of the Parallel Postulate produce non-Euclidean
geometries (e.g., spherical, hyperbolic)
Parallelism is covered specifically in Geometry (MATH 550) where students investigate
the implications of the parallel postulate and the variants producing hyperbolic and
spherical geometry. Indeed, one class day is spent having students generate all the
different things that parallel might mean and how one might generalize the idea of
parallel to curves2. In general, the program has the goal of building a rigorous view of
geometry and its underlying structures in its students. Geometry (MATH 550) is
specifically designed to promote student understanding of different forms of logical
arguments and to cover the major topics of two- and three-dimensional geometry.
2.2
Plane Euclidean Geometry
a. Prove theorems and solve problems involving similarity and congruence
b. Understand, apply, and justify properties of triangles (e.g., the Exterior Angle
Theorem, concurrence theorems, trigonometric ratios, Triangle Inequality,
Law of Sines, Law of Cosines, the Pythagorean Theorem and its converse)
c. Understand, apply, and justify properties of polygons and circles from an
advanced standpoint (e.g., derive the area formulas for regular polygons and
circles from the area of a triangle)
d. Justify and perform the classical constructions (e.g., angle bisector,
perpendicular bisector, replicating shapes, regular n-gons for n equal to 3, 4,
5, 6, and 8)
e. Use techniques in coordinate geometry to prove geometric theorems
2
This is suggested in the first chapter of the text Mathematics for High School Teachers: an Advanced
Perspective by Usiskin, Peressini, Marchisotto, and Stanley.
60
Plane Euclidean geometry is covered in Geometry (MATH 550), Senior Seminar for
Future Mathematics Educators (MATH 493), and History of Mathematics (MATH
490). Students encounter similarity and congruence in all of these courses as these topics
underlie many proofs in Euclidean geometry and in the classical constructions. The
general properties of triangles (exterior angle theorem, concurrence theorems,
trigonometric ratios, law of sines, law of cosines, Pythagorean theorem, and its converse)
are covered in Geometry (MATH 550) and Senior Seminar for Future Mathematics
Educators (MATH 493). Geometry has specific investigative projects for proving many
of these theorems (see sample homework 1 for example). The senior seminar focuses
more specifically on those items needed to understand constructing numbers (like
quotients and products), although some time is spent on the Pythagorean triangle and
classifying Pythagorean triples using both the difference of squares method and the
method of intersecting a line through (-1,0) and the unit circle. Similarly, properties of
polygons and circles are covered in these two classes, both of which study the definition
of pi in some detail (see the perimeter, area, and volume topic from the MATH 550
syllabus). The classical constructions are then covered in Geometry (MATH 550),
Senior Seminar for Future Mathematics Educators (MATH 493), and History of
Mathematics (MATH 490). In Geometry (MATH 550) almost all the constructions are
covered, while in Senior Seminar for Future Mathematics Educators (MATH 493)
the main constructions are reviewed to prepare students to construct a regular pentagon,
the hardest of the classical polygon constructions, which the students work through as
part of a project (which also involves the golden ratio). History of Mathematics
(MATH 490) covers these topics from a historical perspective when studying ancient
Greek mathematics and then the impossibility of constructing segments of certain
lengths. Finally, in Geometry (MATH 550) coordinate systems are used to show
students that some theorems can be proved much more easily with the appropriate choice
of coordinate system.
2.3
Three-Dimensional Geometry
a. Demonstrate an understanding of parallelism and perpendicularity of lines
and planes in three dimensions
b. Understand, apply, and justify properties of three-dimensional objects from
an advanced standpoint (e.g., derive the volume and surface area formulas for
prisms, pyramids, cones, cylinders, and spheres)
Three-dimensional geometry is mainly covered in the two classes Calculus III (MATH
234) and Geometry (MATH 550). Parallelism is first encountered in the curriculum in
Calculus III (MATH 234) in the Vectors in 3-space section of the course. In Geometry
(MATH 550) three-dimensional parallelism is covered in the Euclidean 3-space section.
Calculus III (MATH 234) derives the volume and surface area of standard 3dimensional objects using the tools of calculus. Hence this is the point at which students
discover such important ideas as that the 1/3 in the volume area for a pyramid, cone, and
other “point” objects arising from the integration of an x 2 term (problems leading
students to this idea are also often given in Calculus II (MATH 132) also). Meanwhile,
Geometry (MATH 550) uses Cavalieri’s principle to derive volumes of objects after
61
starting with the notion that 6 congruent right tetrahedrons can be put together to form a
cube. Between the two classes the volume and surface areas are found of the standard
three-dimensional objects including prisms, cones, cylinders, and spheres.
2.4
Transformational Geometry
a. Demonstrate an understanding of the basic properties of isometries in twoand three-dimensional space (e.g., rotation, translation, reflection)
b. Understand and prove the basic properties of dilations (e.g., similarity
transformations or change of scale)
Transformational geometry is explicitly covered in two classes: Linear Algebra (MATH
250) and Geometry (MATH 550). In addition, the topics arise as pieces of other topics
in Calculus III (MATH 234), where vector projections are studied, Group Theory
(MATH 321), where collections of transformations are used as examples of groups,
Complex Analysis (MATH 357), where complex transformations are studied, and
Senior Seminar for Future Mathematics Educators (MATH 493), where one uses
transformational geometry as a way of analyzing how to help students think about
complex numbers in the complex plane (e.g., thinking of the complex number i   1 as
representing a quarter turn in the complex plane, so that i 2  1 represents a half-turn, or
a reversal of direction). In general, understanding the basic properties of isometries in
two- and three-dimensions is covered most fully in the Geometric Transformations and 3Dimensional Euclidean geometry sections of Geometry (MATH 550). They are also
covered in detail in the Linear Transformation section of Linear Algebra (MATH 250).
Domain 3.
Number Theory
Candidates demonstrate an understanding of the number theory and a command of the
number sense contained in the Mathematics Content Standards for California Public
Schools: Kindergarten Through Grade Twelve (1997) as outlined in the Mathematics
Framework for California Public Schools: Kindergarten Through Grade Twelve (1999)
from an advanced standpoint. To ensure a rigorous view of number theory and its
underlying structures, candidates have a deep conceptual knowledge. They prove and
use properties of natural numbers. They formulate conjectures about the natural
numbers using inductive reasoning, and verify conjectures with proofs.
Number theory is explicitly covered in three classes in the Single Subject matter
program: Methods of Proof (MATH 248), Group Theory (MATH 331), and Senior
Seminar for Future Mathematics Educators (MATH 493). Topics in number theory
arise in other courses, most explicitly in Workshop in Mathematics (MATH 190/191),
where elementary ideas from number theory are often used in problems that students
investigate, and in particular induction problems, where patterns are conjectured and then
those conjectures are verified with proofs or shown false with counterexamples. One of
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the main purposes of Senior Seminar for Future Mathematics Educators (MATH
493) is for students to develop a rigorous view of number theory with deep conceptual
knowledge as well as an understanding of how numbers developed and their underlying
structures. In Methods of Proof (MATH 248), students use the context of number theory
to learn inductive reasoning and proofs.
3.1
Natural Numbers
a. Prove and use basic properties of natural numbers (e.g., properties of
divisibility)
b. Use the Principle of Mathematical Induction to prove results in number
theory
c. Know and apply the Euclidean Algorithm
d. Apply the Fundamental Theorem of Arithmetic (e.g., find the greatest common
factor and the least common multiple, show that every fraction is equivalent to
a unique fraction where the numerator and denominator are relatively prime,
prove that the square root of any number, not a perfect square number, is
irrational)
Students first encounter the properties of natural numbers in problem solving exercises in
Mathematics Workshop I (MATH 190) and again when studying mathematical
induction in Mathematics Workshop II (MATH 191). They continue this study in the
Elementary Number Theory topic in Methods of Proof (MATH 248). They encounter
divisibility, modular arithmetic, divisibility tricks, the Euclidean Algorithm, and
mathematical induction, which they used to prove properties of the natural numbers. In
Group Theory (MATH 331) students review topics from number theory that are needed
to deepen their understanding of modular arithmetic. Thus, students cover the Euclidean
Algorithm, modular arithmetic, use induction to prove results like the existence of a least
positive integer n such that an=mb+1 if a and b are relatively prime. In Senior Seminar
for Future Mathematics Educators (MATH 493), the Euclidean algorithm and
divisibility tricks take center stage at several points, in particular during the first section
of the course, Rational Numbers and Irrationality Proofs, where students learn multiple
methods to show that the square root of a non-square natural number is irrational. These
lessons are taught to encourage students to formulate conjectures about the natural
numbers using inductive reasoning (see sample assignment 1 for example). In all of
these classes, connections to the high school curriculum are made apparent. In particular,
in both Group Theory (MATH 331) and Senior Seminar for Future Mathematics
Educators (MATH 493), the need for the Euclidean Algorithm for finding greatest
common divisors is promoted by having students think about how they found greatest
common divisors in their 6-12 curriculum classes. Moreover, in Senior Seminar for
Future Mathematics Educators (MATH 493) much time is spent asking students to
think about arguments on the natural numbers and how one can convey their essence to
grade 6-12 students. The purpose of parts g and h of Sample Assignment 1 are exactly
this.
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Domain 4.
Probability and Statistics
Candidates demonstrate an understanding of the statistics and probability distributions
for advanced placement statistics contained in the Mathematics Content Standards for
California Public Schools: Kindergarten Through Grade Twelve (1997) as outlined in
the Mathematics Framework for California Public Schools: Kindergarten Through
Grade Twelve (1999) from an advanced standpoint. To ensure a rigorous view of
probability and statistics and their underlying structures, candidates have a deep
conceptual knowledge. They solve problems and make inferences using statistics and
probability distributions.
Probability and Statistics are covered primarily in the course Probability and Statistics
(MATH 360). In addition, often the student projects in Senior Seminar for Future
Mathematics Educators (MATH 493) involve probability and statistics and Pascal’s
triangle and its properties are often discussed in Workshop in Mathematics I/II (MATH
190/191). The main goal of Probability and Statistics (MATH 360) is for students to
gain a rigorous view of probability and statistics and their underlying structures. The
course aims to give students a deep conceptual knowledge so that all students can work
through and understand the statistics that they will use in life as scientists, teachers, and
citizens. For future mathematics educators, there is a further goal is that they will gain an
understanding of statistics sufficient to allow them to teach an advanced placement
statistics course well. Throughout the course, examples are used that can be transferred to
grade 6-12 probability and statistics classes.
4.1
Probability
a. Prove and apply basic principles of permutations and combinations
b. Illustrate finite probability using a variety of examples and models (e.g., the
fundamental counting principles)
c. Use and explain the concept of conditional probability
d. Interpret the probability of an outcome
e. Use normal, binomial, and exponential distributions to solve and interpret
probability problems
Students study probability in the Probability section and the section titled “Probability
Mass Functions, Probability Density Functions, and Important Distributions” of
Probability and Statistics (MATH 360). In the Probability section, students prove and
apply the basic principles of permutations and combinations. This leads to a discussion
of interesting examples and a variety of models via the fundamental counting principles
(the multiplication rule, the addition rule, and basic inclusion-exclusion). Conditional
probability is covered as an explicit section in this course (in addition, see the selected
final exam questions), and students are expected to both use and explain conditional
probability. The Probability section also has students interpreting probabilities of
outcomes. The Probability Mass Functions, Probability Density Functions, and
Important Distribution section has students studying and using important distributions
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(such as the normal, binomial, and exponential distributions) to solve and interpret
answers to probability problems.
4.2
Statistics
a. Compute and interpret the mean, median, and mode of both discrete and
continuous distributions
b. Compute and interpret quartiles, range, variance, and standard deviation of
both discrete and continuous distributions
c. Select and evaluate sampling methods appropriate to a task (e.g., random,
systematic, cluster, convenience sampling) and display the results
d. Know the method of least squares and apply it to linear regression and
correlation.
e. Know and apply the chi-square test
Statistics is the centerpiece of the Basic Concepts section, the Sampling Method and
Graphical Presentation Section, the Random Variable section, the Hypothesis section,
and the Least Squares section of Probability and Statistics (MATH 360). The Basic
Concepts section of the course is where students first encounter the idea of mean, median,
and mode of both discrete and continuous distributions. Students are asked to work with
these concepts with both real world data together with data chosen for its statistical
interest. Following this, students study the various methods describing the breadth of
distributions, namely: quartiles, range, variance, and standard deviation. In the second
section of the course, Sampling Methods, Graphical Presentations and Comparison of
Data Sets, students illustrate some of these ideas using box plots and other graphical
methods, while at the same time they discuss sampling methods. The method of least
squares and linear regression is studied in the last section of the class, Least Squares,
Linear Regression and Correlation. The chi-square test is studied in the hypothesis
testing section of the class.
Domain 5.
Calculus
Candidates demonstrate an understanding of the trigonometry and calculus contained in
the Mathematics Content Standards for California Public Schools: Kindergarten
Through Grade Twelve (1997) as outlined in the Mathematics Framework for California
Public Schools: Kindergarten Through Grade Twelve (1999) from an advanced
standpoint. To ensure a rigorous view of trigonometry and calculus and their underlying
structures, candidates have a deep conceptual knowledge. They apply the concepts of
trigonometry and calculus to solving problems in real-world situations.
Calculus, being one of the most important subjects in mathematics and its applications
occupies a great deal of the Single Subject Matter program in mathematics at LMU.
Courses explicitly applying to this domain are the Calculus Sequence (MATH 131, 132,
& 234), Differential Equations (MATH 245), Numerical Methods (MATH 282), Real
Analysis (MATH 321), Complex Analysis (MATH 357), Senior Seminar for Future
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Mathematics Educators (MATH 493), and even Geometry (MATH 550). An effect of
covering calculus in so many courses is that candidates develop deeper and deeper
understandings of the underpinnings of the subjects and in each delving gain deeper
conceptual knowledge. This also allows them multiple chances to use calculus in solving
both real-world problems (as they do in the Calculus Sequence (MATH 131, 132, &
234) and Differential Equations (MATH 245)) and see the applications of calculus to
mathematics in general. Throughout the lower division courses, problems and examples
in the coursework are often identical to problems from the 6-12 curriculum. Senior
Seminar for Future Mathematics Educators (MATH 493) then takes the infusion of
discussions of the 6-12 curriculum to new depths as, for example, students are asked to
evaluate different methods of describing integration by parts for high school seniors, and
which methods might best be used when. These discussions are clothed in the context of
further mathematical investigations, such as a proof that the numbers e and pi are
irrational.
5.1
Trigonometry
a. Prove that the Pythagorean Theorem is equivalent to the trigonometric
identity sin2x + cos2x = 1 and that this identity leads to 1 + tan2x = sec2x and
1 + cot2x = csc2x
b. Prove the sine, cosine, and tangent sum formulas for all real values, and
derive special applications of the sum formulas (e.g., double angle, half
angle)
c. Analyze properties of trigonometric functions in a variety of ways (e.g.,
graphing and solving problems)
d. Know and apply the definitions and properties of inverse trigonometric
functions (i.e., arcsin, arccos, and arctan)
e. Understand and apply polar representations of complex numbers (e.g.,
DeMoivre's Theorem)
Trigonometry is explicitly covered in the Calculus Sequence (MATH 131, 132, 234),
Complex Analysis (MATH 357), and Geometry (MATH 550). In the Calculus
Sequence (MATH 131, 132, 234) the basic trigonometric identities and their relationships
to the Pythagorean Theorem are discussed on several occasions. Trigonometry is first
covered in Calculus I (MATH 131) when discussing the derivatives of the arcsin, arcos,
and arctan functions. Then in Calculus II (MATH 132) it arises in the discussions of
solving trigonometric integrals. This is even touched on briefly in Geometry (MATH
550) when the Pythagorean theorem arises. Geometry (MATH 550) then continues the
discussion, proving the angle sum formulas for sine, cosine, and tangent for all real
numbers. In the Calculus Sequence (MATH 131, 132, and 234) students discuss how
the double angle and half angle formulas are special cases of the more general formula.
The trigonometric functions are then studied in detail in the Calculus Sequence (MATH
131, 132, 234) where students graph the functions, use them to solve problems ranging
from maximum-minimum problems to harmonic motion problems and three-dimensional
geometry problems. The Calculus Sequence (MATH 131, 132, 234) also has students
knowing and applying the definitions of the inverse trigonometric functions. Students
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calculate derivatives of the functions by implicit differentiation, solve trigonometric
integrals with substitution, and examine the graphs of these functions. Students study the
polar representation of complex numbers twice in the Single Subject Matter program.
They are first introduced to the polar representation and DeMoivre’s Theorem in
Complex Analysis (MATH 357) and then they see it again in Senior Seminar for
Future Mathematics Teachers (MATH 493) where the polar representation is used in
solving the general cubic equation, in taking roots of complex numbers, and in looking at
two by two matrices over the real numbers in a new way.
5.2
Limits and Continuity
a. Derive basic properties of limits and continuity, including the Sum,
Difference, Product, Constant Multiple, and Quotient Rules, using the formal
definition of a limit
b. Show that a polynomial function is continuous at a point
c. Know and apply the Intermediate Value Theorem, using the geometric
implications of continuity
Limits and continuity are most explicitly covered in the Calculus Sequence (MATH 131,
132, 234), Real Analysis (MATH 321), and Complex Analysis (MATH 357). Students
also touch on these topics in Senior Seminar for Future Mathematics Educators
(MATH 493) when discussing Dedekind cuts (the formal definition of the real line) as
well as in various proofs in the course. The main discussion of the properties of limits
and continuity are first carried out in the Calculus Sequence (MATH 131, 132, 234)
where students must address these issues very carefully. The formal definition is then
used to refine students’ knowledge of limits and continuity in Real Analysis (MATH
321), where students are asked to create elementary proofs of these properties based on
the definition. In Complex Analysis (MATH 357) in the Complex Functions and
Mappings section, students then revisit the notion of limit and continuity in a new domain
to further their understanding. Polynomials are shown to be continuous in both the
Calculus Sequence (MATH 131, 132, 234) and Real Analysis (MATH 321). Again, in
calculus continuity of polynomials is done in a less rigorous fashion, followed by the
students filling in the details in Real Analysis (MATH 321). Finally, the Intermediate
Value Theorem is used in the Calculus Sequence (MATH 131, 132, 234) and Real
Analysis (MATH 321) as well as in Senior Seminar for Future Mathematics
Educators (MATH 493) where this is linked to the necessity for using Dedekind cuts to
define the real numbers. Of course, it is most carefully covered in Calculus I/II (MATH
131/132) in the section on Limits and Continuity.
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5.3
Derivatives and Applications
a. Derive the rules of differentiation for polynomial, trigonometric, and
logarithmic functions using the formal definition of derivative
b. Interpret the concept of derivative geometrically, numerically, and analytically
(i.e., slope of the tangent, limit of difference quotients, extrema, Newton’s
method, and instantaneous rate of change)
c. Interpret both continuous and differentiable functions geometrically and
analytically and apply Rolle’s Theorem, the Mean Value Theorem, and
L’Hopital’s rule
d. Use the derivative to solve rectilinear motion, related rate, and optimization
problems
e. Use the derivative to analyze functions and planar curves (e.g., maxima,
minima, inflection points, concavity)
f. Solve separable first-order differential equations and apply them to growth
and decay problems
Derivatives and their applications are covered extensively in the Calculus Sequence
(MATH 131, 132, & 234), Differential Equations (MATH 245), Numerical Methods
(MATH 282), and Real Analysis (MATH 321). In addition, these topics are touched
upon in Complex Analysis (MATH 357) and Senior Seminar for Future Mathematics
Educators (MATH 493). Students first derive the rules of differentiation for polynomial,
trigonometric, and logarithmic functions in the Calculus Sequence (MATH 131, 132, &
234) in the derivatives and applications portion of the class and then repeat these
derivations more formally in Real Analysis (MATH 321). They learn to interpret
derivatives geometrically, numerically, and analytically in the derivatives and
applications section of Calculus I (MATH 131), where they look at the derivative as a
slope of a tangent, a limit, and as an instantaneous rate of change. Maxima and minima
problems are also covered in this section (also see sample problem 1). Newton’s method
is addressed both in Calculus I (MATH 131) and in Numerical Methods (MATH 282).
Rolle’s Theorem, the Mean Value Theorem and L’Hopital’s rule are covered in the
Calculus Sequence (MATH 131, 132, & 234) and then revisited in Real Analysis
(MATH 321). Moreover, a central idea in Calculus I (MATH 131) is for students to be
able to interpret continuous functions geometrically and analytically. In the class
students learn to understand and explain graphically what makes a function discontinuous
as well as to understand the types of functions that require careful analytical inspection to
determine continuity. Students learn to use derivatives to solve related rates problems,
rectilinear motion problems, and optimization problems in calculus in the section on
derivatives and their applications. Moreover, these real world problems are covered in
detail in Differential Equations (MATH 245) where applications take a much more
central stage. One of the main goals of Calculus I (MATH 131) is for students to learn
how to analyze planar curves using the derivative. This means that students are asked to
find minima, maxima, inflection points, concavity, and other features of planar curves by
using the derivative. One typical final exam question involves giving students a graph of
the derivative of a function and asking students to classify the critical points of the
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function, find the intervals of concavity, find the intervals where the function is
increasing and decreasing, and finally sketch the graph of the original function. Students
first encounter separable differential equations and solve them in Calculus III (MATH
234) in the section on partial derivatives where students learn to use Lagrange multipliers
to integrate. First-order separable differential equations are much more central, however,
in Differential Equations (MATH 245), where they are among the first differential
equations that are studied. They are first studied in the lecture on separation of variables.
Moreover, as can be seen from the first of the sample final exam problems, students are
specifically expected to be able to apply these to growth and decay models. Indeed, a
major goal of Differential Equations (MATH 245) is for students to learn to model
physical situations using differential equations.
5.4
Integrals and Applications
a. Derive definite integrals of standard algebraic functions using the formal
definition of integral
b. Interpret the concept of a definite integral geometrically, numerically, and
analytically (e.g., limit of Riemann sums)
c. Prove the Fundamental Theorem of Calculus, and use it to interpret definite
integrals as antiderivatives
d. Apply the concept of integrals to compute the length of curves and the areas
and volumes of geometric figures
Integrals and their applications are covered most thoroughly in the Calculus Sequence
(MATH 131, 132, & 234), and then selected topics from integration, generalizations to
complex space, and more formal proofs of their properties are covered in Differential
Equations (MATH 245), Real Analysis (MATH 321), Complex Analysis (MATH 357),
and Senior Seminar for Future Mathematics Educators (MATH 493). In both
Calculus I (MATH 131) and Calculus II (MATH 132), students derive definite integrals
of the standard algebraic functions using the formal definition of the integral. This is
repeated with a more formal treatment of the evaluation of the limit in Real Analysis
(MATH 321). The integral is interpreted geometrically as the area under the graph of the
function in Calculus I (MATH 131) and Calculus II (MATH 132), and then this is
explicitly revisited in Senior Seminar for Future Mathematics Educators (MATH
493). In this last class, the interpretation is done in the special case of understanding the
natural logarithm of a number a as the integral of the function f ( x)  1 from 1 to a .
x
This is followed by a discussion of how to bring this to the high school curriculum as a
geometric way to understand the number e . Calculus II (MATH 132) spends a great
deal of time in the curriculum (see the Integrals and Applications Section) on recognizing
the integral as a limit of a Riemann sum since much of the course is spent modeling
problems, and then discovering that the model leads to a Riemann sum and its limit. This
notion is revisited more formally in Real Analysis (MATH 321), where the limit is
treated more formally. The Fundamental Theorem of Calculus is a central idea of
calculus, and thus it is part of the entire Calculus Sequence (MATH 131, 132, & 234),
where it is proved in the first two courses. Moreover, it is then revisited as a major idea
in Real Analysis (MATH 321) and proved even more formally there. The fundamental
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theorem is then used throughout Calculus II (MATH 132) to evaluate integrals and
interpret them as anti-derivatives. Finally, in Senior Seminar for Future Mathematics
Educators (MATH 493) the Fundamental Theorem and its role in understanding the
relationship between integration by parts and the product rule is discussed in detail when
discussing irrationality proofs for e and  . Arc length, surface area, and volumes using
integrals are first covered in Calculus II (MATH 132). The topic is explicitly revisited
in two courses in the curriculum. The first time is when the concept is generalized to
three dimensions in Calculus III (MATH 234 in the section on Multiple Integrals). Here
students improve their understanding of using integrals to find volumes and areas by
working with multiple integrals rather than single integrals. Near the end of the
mathematics curriculum, the students see the topic one more time in Geometry (MATH
550 see perimeter, area, and volume section), where they use polynomial limits and
Cavalieri’s principle to calculate length of curves, surface areas, and volumes and then
compare these techniques to the more formally defined methods from calculus.
Following this is a discussion of how each fits into the 6-12 curriculum.
5.5
Sequences and Series
a. Derive and apply the formulas for the sums of finite arithmetic series and
finite and infinite geometric series (e.g., express repeating decimals as a
rational number)
b. Determine convergence of a given sequence or series using standard
techniques (e.g., Ratio, Comparison, Integral Tests)
c. Calculate Taylor series and Taylor polynomials of basic functions
Sequences and series are studied at several points in the Single Subject Matter program in
Mathematics at LMU. They are explicitly part of the syllabus in Calculus II (MATH
132), Differential Equations (MATH 245), Real Analysis (MATH 321), and Complex
Analysis (MATH 357). In addition, these topics are discussed as background material in
both Senior Seminar for Future Mathematics Educators (MATH 493) and Geometry
(MATH 550). Students first derive and apply the formulas for finite arithmetic series and
finite and infinite geometric series in Calculus II (MATH 132 – see sequence and series
section). Finite and infinite geometric series are revisited in Real Analysis (MATH 321),
where they are used as the basic examples for understanding formal proofs of
convergence. Both arithmetic and geometric series are then discussed in Senior Seminar
for Future Mathematics Educators (MATH 493) as issues arising in the irrationality
proofs for e. In Euler’s first irrationality proof for e, the argument follows from looking
at the sum of the inverse factorials and comparing it to a geometric series. As a result,
this provides an excellent point for review of the basic finite and infinite series and
calculating their sums. Methods for determining the convergence of sequences and
series are covered in detail over several weeks in Calculus II (MATH 132 see the
sequence and series topic). This topic is then reviewed in Real Analysis (MATH 321)
where several of the tests are given more formal proofs. The comparison test is also
discussed in Senior Seminar for Future Mathematics Educators (MATH 493) and
used in the proof that e is irrational. Taylor series and Taylor polynomials of basic
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functions are explicitly covered in Calculus II (MATH 132) and are then reviewed and
used in Differential Equations (MATH 245), Complex Analysis (MATH 357) and
Senior Seminar for Future Mathematics Teachers (MATH 493). Calculus II (MATH
132) spends roughly two to three lectures on Taylor series (and Taylor polynomials) and
introduces students to the Taylor series for the standard functions, such as f ( x)  e x ,
g ( x)  cos x , and h( x)  sin x . In Differential Equations (MATH 245), Taylor series
techniques are used to solve complicated differential equations that do not fall to other
techniques. Moreover, Taylor polynomials are covered in this course to motivate a
discussion on the level of accuracty of approximate solutions arrived at by using series.
Domain 6.
History of Mathematics
Candidates understand the chronological and topical development of mathematics and
the contributions of historical figures of various times and cultures. Candidates know
important mathematical discoveries and their impact on human society and thought.
These discoveries form a historical context for the content contained in the Mathematics
Content Standards for California Public Schools: Kindergarten Through Grade Twelve
(1997) as outlined in the Mathematics Framework for California Public Schools:
Kindergarten Through Grade Twelve (1999; e.g., numeration systems, algebra,
geometry, calculus).
History of Mathematics is integrated throughout classes in the Single Subject Matter
program in mathematics at LMU with historical notes and discussions of famous
mathematicians. Moreover, the walls of the mathematics department at LMU are covered
with posters discussing a wide variety of historical topics in mathematics. In the Single
Subject Matter program, however, there are four classes where history is explicitly part of
the syllabus: Workshop in Mathematics I/II (MATH 190/191), History of
Mathematics (MATH 490), and Senior Seminar for Future Mathematics Educators
(MATH 493). In these four courses, history is dealt with in different ways. In
Workshop in Mathematics I/II (MATH 190/191), historical developments in
mathematics are taught with an effort to bring students to understand what are the
important steps in mathematics and how they have affected society, science, and the
development of mathematics. The purpose here is so that students starting the major can
understand the importance of mathematical development in modern society and thus gain
a richer view of the importance of mathematics. In addition, in these two courses,
students write biographies of 20th century mathematicians giving them a view of recent
mathematical developments. A third aspect of the history covered in Workshop in
Mathematics I/II (MATH 190/191) is the emphasis on the contributions of diverse
cultural, ethnic, and gender groups to mathematics. History of Mathematics (MATH
490) is the course in the curriculum in which historical topics are most centrally covered.
This course is primarily for future mathematics educators, and in the course, students
learn about the historical context of mathematics and, in particular, the mathematics
covered in the Mathematics Framework for the California Public Schools. The last
course specifically dealing with history of mathematics is Senior Seminar for Future
Mathematics Educators (MATH 493). History is interwoven throughout this course, as
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a course goal is to discuss the understanding of number through mathematics history.
Thus when studying number as ratio, the Pythagoreans are discussed briefly, when
addressing number as length, a great deal of time is spent on Euclidean mathematics
together with a mention of the Chinese mathematics understanding of the Pythagorean
Theorem (called the GioGu theorem in China), while the discussion of number as
solution leads to a discussion of the Renaissance rebirth of mathematics in Italy and the
work of Cardano, Bombelli, and others in solving equations, together with a discussion of
the complex numbers (due to Bombelli) and the controversy surrounding them. Moving
forward to the work on Dedekind cuts, the historical imperative of putting mathematics
on a firm foundation is discussed. Thus, this course aims to put much of the mathematics
covered in the K-12 curriculum into a historical context.
6.1
Chronological and Topical Development of Mathematics
a. Demonstrate understanding of the development of mathematics, its cultural
connections, and its contributions to society
b. Demonstrate understanding of the historical development of mathematics,
including the contributions of diverse populations as determined by race,
ethnicity, culture, geography, and gender
As stated above, there are four main courses in the LMU Single Subject Program in
mathematics, which address the issue of mathematics history: Workshop in
Mathematics I/II (MATH 190/191), History of Mathematics (MATH 490), and Senior
Seminar for Future Mathematics Educators (MATH 493). Students are first exposed
to an understanding of the development of mathematics, its cultural connections, and its
contributions to society in Workshop in Mathematics I/II (MATH 190/191, see the
Modern Mathematics and Mathematics Culture component) where mathematics’
contributions to society, the cultural component of mathematics, and the development of
mathematics are addressed. This is then fully fleshed out in History of Mathematics
(MATH 490), the centerpiece to the Single Subject Matter programs coverage of this
thread. Students are expected to demonstrate an understanding of the historical
development of mathematics including the contributions of diverse populations based on
race, ethnicity, culture, geography, and gender in both Workshop in Mathematics I/II
(MATH 190/191) and History of Mathematics (MATH 490). Workshop in
Mathematics I/II (MATH 190/191) is particularly concerned with diversity issues as can
be seen in the Modern Mathematics and Mathematics Culture component. In this class,
historical vignettes discussing mathematics done by diverse cultures, etc, and speakers
and videos are chosen to show how every segment of the world has contributed to
mathematics. History of Mathematics (MATH 490) then deepens this perspective and
asks students to demonstrate an understanding of the historical development. Moreover,
as Sample Final Exam problems 5 and 9 (on part I) shows, students in this class are
expected to be able to present compelling arguments for a non-Eurocentric view of
mathematics, emphasizing the importance of diverse cultures. One can note that the
historical development and contributions by diverse populations is central to this class by
noticing that objectives 2-8 of the History of Mathematics (MATH 490) syllabus deal
with these issues. This idea is visited one last time in Senior Seminar for Future
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Mathematics Educators (MATH 493), where students look specifically at the
development of number, and ideally see how this development should affect teaching at
the 6-12 level.
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Part II: Subject Matter Skills and Abilities Applicable to the Content Domains in
Mathematics
Candidates for Single Subject Teaching Credentials in mathematics use inductive and
deductive reasoning to develop, analyze, draw conclusions, and validate conjectures and
arguments. As they reason, they use counterexamples, construct proofs using
contradictions, and create multiple representations of the same concept. They know the
interconnections among mathematical ideas, and use techniques and concepts from
different domains and sub-domains to model the same problem. They explain
mathematical interconnections with other disciplines. They are able to communicate
their mathematical thinking clearly and coherently to others, orally, graphically, and in
writing, through the use of precise language and symbols.
Candidates solve routine and complex problems by drawing from a variety of strategies
while demonstrating an attitude of persistence and reflection in their approaches. They
analyze problems through pattern recognition and the use of analogies. They formulate
and prove conjectures, and test conclusions for reasonableness and accuracy. They use
counterexamples to disprove conjectures.
Candidates select and use different representational systems (e.g., coordinates, graphs).
They understand the usefulness of transformations and symmetry to help analyze and
simplify problems. They make mathematical models to analyze mathematical structures
in real contexts. They use spatial reasoning to model and solve problems that cross
disciplines.
Reasoning, proving, explaining, communicating and problem solving are an integral part
of every course in the LMU Single Subject Matter program in mathematics. In every
class, students are expected to solve both routine and complex problems (as can be seen
from each class’s objectives).
Moreover, analyzing problems through pattern
recognition and analogies, formulating and proving (and disproving) conjectures and
testing conclusions for reasonableness and accuracy are threaded throughout all of the
upper division mathematics courses. The upper division courses, as well as many of the
lower division courses, emphasize the use of different representational systems, the use of
transformation and symmetry to help solve problems, and the use of spatial reasoning to
model and solve problems that cross disciplines.
Problem solving and reasoning is threaded throughout the curriculum, where it plays a
vital role in each course as problem solving in context. Students first encounter the
abstract notion of mathematics problem solving in Workshop in Mathematics (MATH
190/191), where one of the four components of the year-long course sequence is problem
solving. In these courses, students solve both routine and complex problems by drawing
from a variety of strategies. As can be seen by a quick perusal of the Problem Solving
Components of both MATH 190 and MATH 191, students learn a variety of heuristic
strategies. In addition, this sequence is the first place that students encounter the notion
of making mathematical statements and conjectures and proving them as is shown in the
Mathematical Writing, Verbal Communication, and Study Skills component of the two
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courses. This course sequence further helps students see the interconnectedness of
mathematics with other disciplines. These ideas are mirrored in the Calculus Sequence
(MATH 131, 132, & 234) where students learn to see mathematics as a model for
problems arising in the disciplines of Physics (rectilinear motion problems), Biology
(growth and decay), Chemistry (mixing problems/related rates), and other disciplines. In
addition, they learn to use concepts from other domains (for example complex analysis,
probability and statistics, and linear algebra) to model similar problems. As the sample
assignments from the Calculus Sequence (MATH 131, 132 & 234), Differential
Equations (MATH 245), and Complex Analysis (MATH 357) show, students are
expected to be able to communicate and explain these connections to other disciplines, as
well as communicate their mathematical thinking orally, graphically, and in writing.
Throughout all other courses, students must be able to communicate their thinking clearly
and coherently, and the final project in Senior Seminar for Future Mathematics
Educators (MATH 493) provides a final test of this. Student use of inductive and
deductive reasoning to develop, analyze, draw conclusions, and validate conjectures is
also a common thread in the program, starting with Workshop in Mathematics (MATH
190/191) as has already been discussed and then being further developed in Methods of
Proof (MATH 248), where students are asked to develop their ability to formulate and
evaluate conjectures and develop proofs and counterexamples. This idea is then seen in
other courses, but perhaps most clearly in Senior Seminar for Future Mathematics
Educators (MATH 493) and Geometry (MATH 550). In Geometry (MATH 550), the
course has many individual and group exploration projects where the students are asked
to develop geometric conjectures (often with guidance) and in the end prove them (or
reformulate them and then prove them). The final project in Senior Seminar for Future
Mathematics Educators (MATH 493) is an open-ended project that requires students to
formulate their own theorems and prove them, and thus for students to successfully
complete the program, they must demonstrate the ability to formulate and analyze
conjectures in a wide variety of ways.
The solution of complex and routine problems is required in all mathematics courses.
Workshop in Mathematics (MATH 190/191) exposes students to a wide variety of
strategies (see the Problem Solving Component), with persistence and reflection specific
approaches taught throughout. Methods of Proof (MATH 248) then teaches students
proper techniques of proof and developing counterexamples with more routine problems
used for learning, while the advanced mathematics courses often require quite
complicated arguments to be formulated and presented by students. Finally, Senior
Seminar for Future Mathematics Educators (MATH 493) promotes the importance of
persistence and reflection by assigning students a semester long research/problem solving
project, in which they are asked to produce their own conjectures and theorems.
Candidates are taught to use different representational systems (coordinates, graphs,
figures, manipulatives, etc.) in a variety of courses in the curriculum, starting with
Workshop in Mathematics (MATH 190/191) and the Calculus Sequence (MATH 131,
132, & 234). In Workshop in Mathematics (MATH 190/191) in the Problem Solving
component, students are taught to draw a picture and choose a good coordinate system.
In addition, the use of multiple representations via pursuing parity, trying other
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approaches, and considering special cases is emphasized in this component. The
Calculus Sequence (MATH 131, 132, & 234) approaches the issues of multiple
representations focusing on the idea of function. In these classes, students must
demonstrate an understanding of how to go back and forth between the graph of a
function, a table for the function, and the equation for the function. Moreover, they are
taught to use symmetry to translate functions so that they are “centered” at better
locations for analysis. A central tenet of the Calculus Sequence (MATH 131, 132, &
234) and Differential Equations (MATH 245) is the importance of using mathematics to
model real situations as one can see this by looking at the sample homework problems
and projects for these courses. Later courses in mathematics also look at using a variety
of models to represent physical (and computer based) problems. Linear Algebra
(MATH 250) (see sample problem (a2)) has students using linear algebra to model
Markov processes via transition matrices. In Complex Analysis (MATH 357)
applications to real-world phenomena are discussed at the end of every topic. In Group
Theory (MATH 331) using groups and modular arithmetic to model error
detection/correction problems is a recurring topic. All of these speak to the students’
mastery of using modeling to solve problems from other disciplines. Finally, using
spatial reasoning to solve problems that cross disciplines is absolutely crucial in the
Calculus Sequence (MATH 131, 132, & 234), Differential Equations (MATH 245),
and Complex Analysis (MATH 357).
11.2
Required coursework exposes underlying mathematical reasoning,
explores connections among the branches of mathematics, and provides
opportunities for problem solving and mathematical communication.
The 17 required mathematics content courses in the program provide students with ample
opportunities for problem solving, mathematical communication, examining
mathematical reasoning, and making connections across different branches of
mathematics.
Mathematical reasoning: Each of the 17 mathematics content courses in the program
exposes the fundamental role of reasoning in mathematics. However, three courses
especially come to mind in addressing this Standard.
The first two courses are
Workshop Course in Mathematics I/II (MATH 190/191) because of the emphasis on
problem solving and communicating reasoning in these courses. Students are expected to
give justifications for their mathematical statements. Beginning freshmen often comment
that this was the first course that required them to do so. The third course, Methods of
Proof (MATH 248), is positioned in the sophomore year as the transition from the
Calculus sequence to more theoretical upper division courses. Proof techniques
(induction, proof by cases, contradiction, contrapositive method, proof by
counterexample, the pigeonhole principle, combinatorial methods) are the major thrust of
this course. Course topics which serve as a vehicle for learning and practicing proof
techniques are elementary number theory, elementary set theory, relations, functions, and
cardinality. Skill in making valid mathematical arguments and knowledge of these topics
is considered essential for the required upper division coursework in advanced calculus
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Real Variables (MATH 321), abstract algebra Group Theory (MATH 331), Geometry
(MATH 550) and Probability and Statistics (MATH 360).
Connections across different branches of mathematics: Because the department has a
carefully thought out sequence of courses and enforces prerequisites, later courses build
on earlier courses and students see the interrelationship among different branches of
mathematics. For example, in Complex Variables (MATH 357) students see that
complex numbers are a generalization and extension of real numbers and that complex
variable techniques can be used to evaluate real integrals that arise in calculus. In
History of Mathematics (MATH 490) and Senior Seminar for Future Mathematics
Educators (MATH 493) students see the algebraic concepts of rings and fields used to
answer questions about geometric constructions. Senior Seminar for Future
Mathematics Educators (MATH 493) explores other connections, including the links
between linear algebra and complex numbers, between the geometric notion of the
Euclidean algorithm and the algebraic version of the Euclidean algorithm, and between
translation of equations and the quadratic formula.
Problem solving: Beginning in the freshman year, two courses are designed to focus on
problem solving (especially in non-routine settings), communication, reasoning, and
making connections. Those courses are the first-year Workshop in Mathematics I/II,
(MATH 190/191). The Problem Solving Component of the two semester sequence for
freshmen Workshop in Mathematics I/II, (MATH 190/191) discussions draw heavily
on Polya's four step process and his heuristic approach and on Alan Schoenfeld's research
on metacognitive aspects of problem solving. Heuristics such as understand the problem,
break mind set, simplify the problem, draw a figure or make a model, collect and
organize data, look for a pattern, work backwards, be persistent, try another approach,
look back after a solution is found, etc., are each illustrated by a set of non-routine
problems. For the first semester the problems generally involve only the most elementary
mathematical concepts. This is intentional so that the students can concentrate on the
problem solving PROCESS and not be frustrated by needing mathematical concepts that
they have not mastered. By illustrating the process with fun, nonstandard problems at an
elementary level, students can more easily become self-aware and analyze their own
strengths and weaknesses. In addition, the initial use of such problems seem to build
confidence. These problem sets are worked on in class either individually or in small
collaborative groups. The success of the collaborative approach convinces students that a
solution can be found and fosters perseverance on their part when they work individually.
In the second semester, the topics and methods are more advanced, (for example, use the
pigeonhole principle, pursue parity, use special cases, try fewer variables, argue by
contradiction, use the principle of mathematical induction) but still do not require
calculus. Students are always expected to justify conclusions. Special attention is given
to Polya's fourth step - Looking Back - which involves not only generalizing solutions,
but also looking for other proofs or approaches for the same problem and stating and
solving related problems. Classroom discussions of control issues help students analyze
and become aware of their mathematical knowledge and behaviors. Lessons, including
strategies, techniques, mindsets, persistence and confidence, learned in this two-semester
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sequence in the freshman year form a strong foundation for the rest of the program.
Other courses in the major continue to provide problem solving opportunities. For
example, in the Calculus Sequence (MATH 131/132/234) and Differential Equations
(MATH 245) students work on solving applied problems from engineering and physics
(see sample assignments from these courses). The final project in the Senior Seminar
for Future Mathematics Educators (MATH 493) requires the students to inquire into a
problem new to them, investigate it, and then apply it to the 6-12 curriculum. As this
project is a full-term long assignment, it strongly encourages perseverance in the students
and provides a vehicle for summative assessment.
Mathematical communication: At the lower division level each student takes a two
semester sequence Workshop in Mathematics I/II (MATH 190/191) which has a
Mathematical Communication Skills Component. Students must write up the solution to
one problem discussed in class each day. These solutions are reviewed by peers and/or
the instructor and, if needed, a re-write is required. In doing group work and in listening
to presentations made by other groups students listen to and evaluate the reasoning of
others. In each semester of the workshop courses they are also required to write an
expository paper and make an oral presentation on a mathematical topic of their choice
each semester. Another lower division course with an emphasis on communication is
Methods of Proof (MATH 248). This course emphasizes methods of proof and affords
students many opportunities to write and re-write proofs until their reasoning and writing
are correct and clear. At the upper division level, in History of Mathematics (MATH
490) every student gives an oral and written project report. Every student is required to
take as a capstone course Senior Seminar for Future Mathematics Educators (MATH
493) which requires a project consisting of both a written and an oral report which draws
on their prior coursework. Geometry (MATH 550) has students write up both formal
and informal arguments in geometry for a variety of grade levels.
11.3
Required courses are applicable to the requirements for a major in
mathematics. Remedial classes and other studies normally completed in
K-12 schools are not counted in satisfaction of the required subjects of
study.
The Single Subject Matter Program in Mathematics rests on the Bachelor of Art in
Mathematics (BAM see Appendix III) degree program at Loyola Marymount University.
All coursework is applicable to the requirements for the Bachelor of Art degree in
Mathematics. No remedial coursework or other studies normally completed in K-12
schools counts toward the degree.
11.4
The institution that sponsors the program determines, establishes and
implements a standard of minimum scholarship for coursework in the
program.
As detailed in the LMU Undergraduate Bulletin, Mathematics majors (that includes those
in the BAM degree program which underpins the Single Subject Matter Program in
Mathematics) must maintain a minimum cumulative grade point average of C (2.0) in
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upper division major requirements; a minimum grade of C (2.0) in each course in the
lower division requirements, in order to graduate. In addition, in order to enroll in any
mathematics course a student must have a minimum grade of C (2.0) in any prerequisite
course.
11.5
Required coursework includes work in computer science and/or related
mathematics such as: 1) discrete structures (sets, logic, relations and
functions) and their application in the design of data structures and
programming; 2) design and analysis of algorithms including the use of
recursion and combinations, and 3) use of the computer applications and
other technologies to solve problems.
The Single Subject Matter Credential in Mathematics program at Loyola Marymount
University requires students to take Numerical Methods (MATH 282). This course
requires students to write programs in FORTRAN and Mathematica to solve problems.
To do so, they are required to use logic and a deep understanding of function to help them
design the programs. In addition, when studying Newton’s methods and other iterative
algorithms, the students must design and analyze algorithms that require the use of
recursion and other combinatorial methods. In Linear Algebra (MATH 250) students
solve recursive problems involving Markov chains and recursion. Numerical Methods
(MATH 282) also requires the students to use technology to solve problems. Other
coursework in the program also requires students to use technology to solve problems. In
particular, the Calculus Sequence (MATH 131, 132, 234) contains assignments that
require students to figure out how to solve problems using technologies. Similarly, in
Senior Seminar for Future Mathematics Educators (MATH 493) the semester-long
projects require students to use technology to investigate problems and solve them.
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Standard 12: Problem Solving
In the program, prospective teachers of mathematics develop effective strategies for
solving problems both within the discipline of mathematics and in applied settings that
include non-routine situations. Problem-solving challenges occur throughout the
program of subject matter preparation in mathematics. Through coursework in the
program, prospective teachers develop a sense of inquiry and perseverance in solving
problems.
The 17 required mathematics content courses in the program provide students with ample
opportunities to use a variety of strategies to formulate and solve problems. However,
two courses warrant special mention as focusing on problem solving (especially in nonroutine settings), developing a sense of inquiry, and fostering perseverance in solving
problems. Those courses are the first-year Workshop in Mathematics I/II, (MATH
190/191).
The Problem Solving Component of the two-semester sequence for freshmen Workshop
in Mathematics I/II, (MATH 190/191) is designed to improve the students' problem
solving skills and accordingly their confidence levels. This component of these courses
addresses problem solving strategies including both heuristics and control issues.
Discussions draw heavily on Polya's four step process andheuristic approach and on Alan
Schoenfeld's research on metacognitive aspects of problem solving. Heuristics such as
understand the problem, break mind set, simplify the problem, draw a figure or make a
model, collect and organize data, look for a pattern, work backwards, be persistent, try
another approach, look back after a solution is found are each illustrated by a set of non
routine problems. For the first semester the problems generally involve only the most
elementary mathematical concepts. This is intentional so that the students can
concentrate on the problem solving process and not be frustrated by needing
mathematical concepts that they have not mastered. By illustrating the process with fun,
nonstandard problems at an elementary level, students can more easily become self-aware
and analyze their own strengths and weaknesses. In addition, the initial use of such
problems seem to build confidence. These problem sets are worked on in class either
individually or in small collaborative groups. The success of the collaborative approach
convinces students that a solution can be found and fosters perseverance on their part
when they work individually. In the second semester, the topics and methods are more
advanced, (for example, use the pigeonhole principle, pursue parity, use special cases, try
fewer variables, argue by contradiction, use the principle of mathematical induction) but
still do not require calculus. Students are always expected to justify conclusions. Special
attention is given to Polya's fourth step - Looking Back - which involves not only
generalizing solutions, but also looking for other proofs or approaches for the same
problem and stating and solving related problems. Classroom discussions of control
issues help students analyze and become aware of their mathematical knowledge and
behaviors. Awareness of metacognitive issues is key to a problem solver making the
transition from novice to expert. Creating a classroom in which students feel free to
express themselves is absolutely essential to achieving this goal.
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Other courses in the major also provide problem solving opportunities. For example, in
the Calculus Sequence (MATH 131/132/234) and Differential Equations (MATH 245)
students work on solving applied problems from engineering and physics (see sample
assignments from these courses). The final project in the Senior Seminar for Future
Mathematics Educators (MATH 493) requires the students to inquire into a problem
new to them, investigate it, and then apply it to the 6-12 curriculum. As this project is a
full-term long assignment, it affords students a last opportunity to develop and practice
perseverence.
Required Elements
12.1 In the program, each prospective teacher learns and demonstrates the
ability to:
Place mathematical problems in context and explore their relationship with
other problems.
Throughout the program, students are expected to place mathematical problems in
context and explore their relationships with other problems. In Workshop in
Mathematics (MATH 190/191), a major problem solving objective is for students to
learn to ask more general questions after solving problems, encouraging them to look for
relationships of problems to other problems. In the Calculus Sequence (MATH 131,
132, & 234) assignments require students to view problems in the context of applications
and explore the relationships between modeling issues and calculus problems. For
example, in the pipeline project in Calculus I (MATH 131), students must find a good
model for the problem and then solve a minimization problem for that model. Later
courses look at connections and relationships between problems quite naturally. In
Complex Variables (MATH 357), students relate the problems of complex analysis to
questions about functions and calculus. Moreover, they place problems on fractional
linear transformations in the context of linear algebra and can relate problems between
these two fields. In Geometry (MATH 550) students encounter geometry problems and
place them in a variety of contexts. For example, one classroom investigation relates the
problem of finding the area of a spherical triangle in terms of the angle sum to the Euler
number of a convex polyhedron in Euclidean 3-space. An objective of Geometry
(MATH 550) is for students to solve geometric problems using coordinate systems, the
key to which is placing geometry problems in context and exploring their relation to
algebra problems. History of Mathematics (MATH 490) is another place where
students constantly explore placing problems in context and learn to understand the
importance of relating different types of problems. In studying the solution of the cubic
equation for example, students learn the importance of algebraic reasoning and applying
it to solve problems of geometry. In studying the impossibility theorems, they learn to
place the question of constructing numbers into the context of ring theory (something
also done in Senior Seminar for Future Mathematics Educators (MATH 493)).
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The course in which students are most clearly expected to explore relationships between
problems is Senior Seminar for Future Mathematics Educators (MATH 493). In this
course, assignments ask them to relate problems between geometry, algebra, fractions,
and solving equations. For example in one assignment, students relate the solution to the
high school geometry problem in measuring the height of a flagpole using similar
triangles to the question about whether constructible numbers form a field. This type of
problem has them place the question of whether the constructible numbers form a field in
an unexpected context. In their course projects, students are then expected to
demonstrate this ability, which forms part of their summative assessment.
12.2 In the program, each prospective teacher learns and demonstrates the
ability to:
Solve mathematical problems in more than one way when possible.
Students encounter the expectation of solving problems in more than one way at the
beginning of the program in the Problem Solving Component of the Workshop in
Mathematics I/II, (MATH 190/191) in the freshman year. Much attention is given to
Polya's 4th Step - Looking Back - which discusses attempting to solve problems in more
than one way. Other exercises ask students to find both algebraic and geometric
solutions to problems like:
Problem: Let r be any real number. How many solutions does the system of
equations x = r and x2 + y2 = 1 have? First try solving this problem without
drawing any pictures. Then try solving it without doing any algebra.
This problem is then followed up with a reflective writing assignment:
Comment on the two approaches (algebra only vs. diagram only) to solving this
problem. What were the advantages/disadvantages of each? Which way was
easier for you? Why?
Seeking additional solutions continues to be stressed throughout the curriculum, but
especially in Methods of Proof (MATH 248), where more than one proof is
offered/discussed for a given result, in Geometry (MATH 550), where a variety of
approaches to solving a problem are discussed (see sample assignment 3 for example),
and in the capstone course Senior Seminar for Future Mathematics Educators
(MATH 493), where three different proofs, each emphasizing a different property of the
rational numbers, are given for the irrationality of n for n a non-square integer, and
two proofs are given for the irrationality of the number e . In other situations, students
see and discuss the value of 3 different derivations of the quadratic formula, and they
offer up different solutions to homework problems that are discussed in class.
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12.3 In the program, each prospective teacher learns and demonstrates the
ability to:
Generalize mathematical problems in more than one way when possible.
Generalizing a solution to a problem is the heart of Polya's 4th Step - Looking Back - and
is emphasized in the Problem Solving Component of the Workshop in Mathematics
I/II, (MATH 190/191) in the freshman year. As an example, here are two exercises from
MATH 190:
 What is the sum of the angles in any convex hexagon? Can you generalize? Can you
justify your answer in more than one way?
 When a, b, c, d are consecutive integers in that order, bc – ad = 2 always holds.
(a) Try to prove this.
(b) Write at least two “looking back” questions.
And here are two reflection prompts from MATH 190:
 List as many “looking back” questions as you can, and for each one, explain what the
purpose is for asking that question.
 Has your view of looking back changed as a result of this class? If so, how?
Prospective teachers are expected to generalize mathematical problems in more advanced
courses as well. For example one assignment in Geometry (MATH 550) has them look
at multiple generalizations of the notion of parallel and how various results might
generalize under each of these. In the final project in Senior Seminar for Future
Mathematics Educators (MATH 493) students are expected to generalize their initial
problem and its solution.
12.4 Use appropriate technologies to conduct investigations and solve
problems.
Students are introduced to a wide variety of technologies useful for conducting
investigations and solving problems as they progress through the Single subject program.
These include: graphing calculators and Excel in Workshop in Mathematics I/II
(MATH 190/191) and in the Calculus sequence (MATH 131/132/234), the computer
algebra system Mathematica in the Calculus sequence (MATH 131/132/234) and in
Numerical Methods (MATH 282), Geometers Sketchpad in Geometry (MATH 550).
For example, in Geometry (MATH 550), students use Geometers Sketchpad to uncover
the relationship between the ratio of the length of a side of a triangle to the sine of the
opposite angle and the circumradius of the triangle (see sample assignment 1 in MATH
550); the syllabus for Calculus II (MATH 132) contains a sample problem that
"appropriately" employs Mathematica as a tool for problem solving in the calculus
sequence. Finally, the project in Senior Seminar for Future Mathematics Educators
(MATH 493) requires students to do an initial investigation that requires the sensible use
of technology. For example, one project question asks students to find rational
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approximations of certain real numbers. The project asks students to start by finding
which fractions with denominator less than 10, 20, 30, etc. best approximate 2 . In this
case, students have learned to use Excel to find approximations and look for patterns. For
example, last year one student discovered the “convert to a fraction” command on Excel
on her own and began an exploration potentially heading for continued fractions because
of it.
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Standard 13: Mathematics as Communication
In the program, prospective teachers learn to communicate their thinking clearly and
coherently to others using appropriate language, symbols and technologies. Prospective
teachers develop communication skills in conjunction with mathematical literacy in each
major component of a subject matter program.
In every mathematics course in the program students are required to write and/or speak
about mathematics using appropriate mathematical terminology. However, several
courses which focus on developing or demonstrating these skills warrant special mention.
At the lower division level each student takes a two semester sequence Workshop in
Mathematics I/II (MATH 190/191) which has a Mathematical Communication Skills
Component. Students must write up the solution to one problem discussed in class each
day. These solutions are reviewed by peers and/or the instructor and, if needed, a rewrite is required. Students are often given reflective writing prompts related to their
problem solving work (for example, "List several things you can do to make sure you
understand a problem," or "Describe a situation in which you found yourself breaking
mind set in order to solve a mathematical or an every-day problem"). They are also
required to write an expository paper and make an oral presentation on a mathematical
topic of their choice each semester.
Another lower division course with an emphasis on communication is Methods of Proof
(MATH 248). This course emphasizes methods of proof and affords students many
opportunities to write and re-write proofs until their reasoning and writing are correct and
clear.
At the upper division level, in History of Mathematics (MATH 490) every student gives
an oral and written project report. Every student is required to take as a capstone course
Senior Seminar for Future Mathematics Educators (MATH 493) which requires a
project consisting of both a written and an oral report which draws on their prior
coursework. Geometry (MATH 550) has students write up both formal and informal
arguments in geometry for a variety of grade levels.
Required Elements
13.1
Articulate mathematical ideas verbally and in writing, using appropriate
terminology.
Students are required to do this in every mathematics course. As described in detail
above, several courses have a special focus on mathematical writing: Workshop in
Mathematics I/II (MATH 190/191), Introduction to Methods of Proof (MATH 248),
History of Mathematics (MATH 490), and Senior Seminar for Future Mathematics
Educators (MATH 493).
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13.2
Where appropriate present mathematical explanations suitable to a
variety of grade levels.
Several courses in the program afford the opportunity to present or examine mathematical
explanations suitable to a variety of grade levels. In Geometry (MATH 550) students see
a variety of solutions to problems and both present and discuss grade levels they are
appropriate for. For example, in the angle sum of a triangle project, students are expected
to write a formal proof of the problem and then discuss how to bring its level down to
other grade levels. In another example from this course, in finding the area of a spherical
triangle in relation to its angle sum, students will work with arguments along the way that
are suitable to high school students studying 3-dimensional geometry (arguing that the
area of the triangle with three 90 degree angles on a sphere of radius 1 is  / 2 for
example). In Senior Seminar for Future Mathematics Educators (MATH 493),
students are explicitly asked in assignments to work with a variety of grade levels. In the
assignment on constructing square roots, they are expected to produce a proof appropriate
to a high school geometry class. When discussing solving the cubic equation, students
present multiple derivations of the quadratic formula suited to three different 6-12
classes. Two final exam problems for the class are
1) Give three derivations of the quadratic formula. For each summarize where
you might use it in a classroom. You may assume that certain derivations
should only be done in special cases, or that they should only be done after the
quadratic equation has been presented some other way. Be as specific as you
can, and support your answers with some evidence. You may want to look at
high school texts or talk with high school teachers about this problem.
2) After discussing the period of fractions with a class, a student asks you the
following question: ``I noticed that on my calculator, when I multiply any
two numbers that have different prime factors, then the period of 1 over the
product is the larger of the two periods of 1 over the numbers. Why is this
true?'' (Professorial comment: this is not in general true.) Find the smallest
product mn for which the student’s statement is false. Give an explanation
appropriate to an algebra 1 student for why this is not in general true.
In History of Mathematics (MATH 490) in the process of using "historical" methods to
solve problems and contrasting these approaches to modern day methods, students will
encounter mathematical concepts and explanations suitable to a variety of grade levels.
For example, when learning that, until the advent of algebraic notation, a geometric
representation of algebraic identities was the only possible representation, students
present and analyze justifications for A(B+C) = AB + AC or (A+B)2 = A2 + 2AB + B2
that are suitable for middle school and high school.
The examples using non-routine but elementary problems and approaches to problem
solving based on Polya's work that are considered in Workshop in Mathematics I
(MATH 190) involve mathematical explanations that are suitable to a variety of grade
levels also.
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The department offers many other opportunities outside of coursework to examine or
present mathematical explanations suitable to a variety of grade levels. Through the
elective mathematics internship opportunities and the required Field Experience in
Mathematics (MATH 293) students have the opportunity for placement as tutors or
teaching assistants in local middle schools, or as tutors for after school sessions at a local
high school. The department has funding for paid teaching assistants on campus in
evening help sessions for Precaclulus or Calculus, and for computer lab
assistants/discussion leaders in the mathematical literacy course required of all LMU
students whose program does not otherwise require a mathematics course. These
internships and teaching assistantships afford the students the chance to communicate
mathematics at a variety of levels from grades 6 through 14.
Because of LMU mathematics department faculty involvement in activities such as
Family Math, Math for Girls, and Expanding Your Horizons Career Day, students are
offered volunteer opportunities to present workshops or assist in mathematical activities
with students in grades K through 12.
13.3
Present mathematical information in various forms, including but not
limited to models, charts, graphs, tables, figures, and equations.
Because of the breadth of the coursework (calculus, numerical methods, differential
equations, elementary number theory and logic and proofs, linear algebra, abstract
algebra, real variables, complex variables, probability and statistics, geometry, history of
mathematics) required by the program (a minimum of 17 mathematics content courses in
all), students encounter a broad range of assignments that require them to communicate
mathematical information in various forms, including but not limited to models, charts,
graphs, tables, figures, and equations. In particular, in Probability and Statistics
(MATH 360) students use charts and figures. In the Calculus Sequence (MATH 131,
132, & 234) projects require the students to make presentations using equations, graphs,
and tables. Students’ final projects in Senior Seminar for Future Mathematics
Educators (MATH 493) are explicitly graded on how they present information in charts,
tables, graphs, figures, and equations. Students in this course also get a chance to present
arguments using algebra tiles and other physical models of algebraic ideas.
13.4
Analyze and evaluate the mathematical thinking and strategies of others.
Frequent opportunities to analyze and evaluate the thinking of other students occur in
Workshop in Mathematics I/II (MATH 190/191) because of the focus on working in
groups and having students present their solutions to the whole class. In Methods of
Proof (MATH 248) students are often asked to evaluate a "sample proof" generated by
the instructor or taken from the textbook exercises. Additional opportunities occur
throughout the curriculum.
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Since 1985 the mathematics department has provided a centrally located study area for
students. As hoped, this area has become a gathering and study place for mathematics
students and promotes student discussion and interaction.
13.5
Use clarifying and extending questions to learn and to communicate
mathematical ideas.
In applying Step 1 Understand the Problem and Step 4 Look Back of Polya's four step
process, which is introduced in the Problem Solving Component of Workshop in
Mathematics I/II (MATH 190/191), students frequently ask clarifying and extending
questions. Instructors in every course model the use of clarifying and extending
questions to learn and communicate mathematics; in addition, this "method" is further
analyzed and discussed and practiced in Senior Seminar for Future Mathematics
Educators (MATH 493).
13.6
Use appropriate technologies to present mathematical ideas and concepts.
In Numerical Methods (MATH 282) students are introduced to procedural programming
through small FORTRAN programming assignments and they also use the Computer
Algebra System Mathematica. Throughout the freshman and sophomore years many of
our calculus and differential equations instructors use Mathematica as a device for
teaching concepts both in classroom demonstrations and as a laboratory tool to enhance
the students' understanding of the concepts developed in the classroom. In Workshop in
Mathematics I/II (MATH 190/191), students are introduced to Excel and to the use of
the Equation Editor in Microsoft Word to produce mathematical documents. In
Geometry (MATH 550) Geometers Sketchpad is used.
Students have ready access to the Mathematics Department Computing Labs which
contains 32 computers: 16 G3 iMACs and 16 Pentium III Dells. In addition to this lab
the computer science department has a laboratory with a variety of IBM machines which
our students are able to use and a computer lab with several Sun Workstations. Students
have access to the internet and email through the machines in the labs and with high
speed connections in the dorms. Classroom demonstration capability is provided by G4
Power MAC series computers in 3 of the classrooms dedicated to the Mathematics
Department.
Each computer in the computer lab and in the classrooms in the Mathematics Department
has access over a local area network to Microsoft Office, Mathematica, Theorist,
FORTRAN, C, Grapher, and Geometers Sketchpad, and Internet Explorer and Netscape.
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Standard 14: Reasoning
In the program, prospective teachers of mathematics learn to understand that reasoning
is fundamental to knowing and doing mathematics. Reasoning and proof accompany all
mathematical activities in the program.
Each of the 17 mathematics content courses in the program emphasizes the fundamental
role of reasoning in mathematics. However, three courses especially come to mind in
addressing this Standard. The first two courses are Workshop Course in Mathematics
I/II (MATH 190/191) because of the emphasis on problem solving and communicating
reasoning in this sequence. Students are expected to give justifications for their
mathematical statements, and beginning freshmen often comment that this was the first
course that required them to do so. The third course, Methods of Proof (MATH 248), is
positioned in the sophomore year as the transition from the Calculus sequence to more
theoretical upper division courses. In the words of one faculty member, its goal is to
"teach aspiring mathematicians how to do proofs." Proof techniques (induction, proof by
cases, contradiction, contrapositive method, proof by counterexample, the pigeonhole
principle, combinatorial methods, etc.) are the major thrust of this course. Course topics,
which serve as a vehicle for learning and practicing proof techniques, are elementary
number theory, elementary set theory, relations, functions, and cardinality. Skill in
making valid mathematical arguments and knowledge of these topics is considered
essential for the required upper division coursework in advanced calculus Real Variables
(MATH 321), abstract algebra Group Theory (MATH 331), History of Mathematics
(MATH 490), and Geometry (MATH 550). Therefore, each of these courses has
MATH 248 as a prerequisite. In the Calculus Sequence (MATH 131, 132, 234) and in
Probability and Statistics (MATH 360) students employ both formal and informal
reasoning verbally in class and on written assignments. In the capstone Senior Seminar
for Future Mathematics Educators (MATH 493), student must demonstrate their
ability to make and assess formal and informal mathematical arguments.
Required Elements
14.1
Formulate and test conjunctures using inductive reasoning, construct
counter-examples, make valid deductive arguments, and judge the validity
of mathematical arguments in each content domain of the subject matter
requirements.
Specific evidence that students are expected to formulate and test conjectures, construct
counterexamples, make valid arguments and judge the validity of mathematical
arguments in each content domain of the subject matter requirements appears in the
corresponding course syllabi as indicated in the chart below.
89
Content Domain
Course Syllabi
Location
Algebraic Structures
MATH 331
Objective 3
Prerequisite
MATH 493
Course description
Grading - Project
Objective 10
Sample homework
Geometry
MATH 550
Grading
Objective 6
Prerequisite
Number Theory
MATH 248
Course description
Topics-Logic/Proof language
Topics-Number Theory
Objectives 1, 2, 3, 4
Final Exam Review Problems
Probability and Statistics
MATH 360
Objectives 1, 3, 8
Calculus
MATH 131
MATH 132
MATH 234
MATH 321
Objectives 1, 3, 8
Topics- Integrals/Applications
Objectives 2, 6
Topics-Sequences
Topics-Functions/Continuity
Objective 3
History of Mathematics
MATH 490
Sample exam questions Part II
Objectives 7, 9, 10
Prerequisite
14.2
Present informal and formal proofs in oral and written formats in each
content domain of the subject matter requirements..
Specific evidence that student are expected to present informal and formal proofs in oral
and written formats in each content domain of the subject matter requirements appears
in the corresponding course syllabi as indicated in the chart below
90
Content Domain
Algebraic Structures
Course Syllabi
MATH 331
MATH 493
Location
Objective 3
Prerequisite
Course description
Grading - Project
Objective 10
Sample homework
Geometry
MATH 550
Grading
Objective 6
Prerequisite
Number Theory
MATH 248
Course description
Topics-Logic/Proof language
Topics-Number Theory
Objectives 1, 2, 3, 4
Final Exam Review Problems
Probability and Statistics
MATH 360
Objectives 1, 3, 8
Calculus
MATH 131
MATH 132
MATH 234
MATH 321
Objectives 1, 3, 8
Topics- Integrals/Applications
Objectives 2, 6
Topics-Sequences
Topics-Functions/Continuity
Objective 3
History of Mathematics
MATH 490
Sample exam questions Part II
Objectives 7, 9, 10
Prerequisite
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Standard 15: Mathematical Connections
In the program, prospective teachers of mathematics develop a view of mathematics as
an integrated whole, seeing connections across different mathematical content areas.
Relationships among mathematical subjects and applications are a consistent theme of
the subject matter program’s curriculum.
The minimum required 17 mathematics content courses and 1 science/computer science
in the program give the student broad exposure to the connections between various
mathematical topics and their applications in a wide variety of disciplines. Additionally,
the department provides numerous opportunities to hear guest lecturers from industry
through the Mathematics Workshop talks and the College of Science and Engineering's
Sigma Xi Friday afternoon lecture series.
Required Elements
15.1 Illustrate,
applications.
when
possible,
abstract
mathematical
concepts
using
Students are expected to use applications when explaining mathematical concepts in
many courses. For example, in the Calculus Sequence (MATH 131, 132, 234 (see
objective 7 of 234)) the derivative is explained by reference to the tangent line, to the rate
of change of the function, to velocity when the function represents position, to
acceleration when the function represents velocity and the definite integral is explained as
representing the area under a curve. Linear Algebra (MATH 250), Complex Analysis
(MATH 357), and Geometry (MATH 550) are other courses that have this as an explicit
objective.
15.2
Investigate ways mathematical topics are inter-related.
In Workshop in Mathematics I/II (MATH 190/191) a number of activities encourage
consideration of the inter-relationship of mathematical topics. Here is a problem that
examines an algebraic approach and a geometric approach to the same question.
Problem: Let r be any real number. How many solutions does the system of
equations x = r and x2 + y2 = 1 have? First try solving this problem without
drawing any pictures. Then try solving it without doing any algebra.
This problem is then followed up with a reflective writing assignment:
Comment on the two approaches (algebra only vs. diagram only) to solving this
problem. What were the advantages/disadvantages of each? Which way was
easier for you? Why?
Because the department has a carefully thought out sequence of courses and enforces
prerequisites, later courses build on earlier courses and students see the interrelationship
92
of mathematical topics. For example, in Complex Variables (MATH 357) students see
that complex numbers are a generalization and extension of real numbers and that
complex variable techniques can be used to evaluate real integrals. In History of
Mathematics (MATH 490) and Senior Seminar for Future Mathematics Educators
(MATH 493) students see the algebraic concepts of rings and fields used to answer
questions about geometric constructions.
In the Senior Seminar for Future
Mathematics Educators (MATH 493) other connections are explored, including the link
between linear algebra and complex numbers, the links between the geometric notion of
the Euclidean algorithm and the algebraic version of the Euclidean algorithm, and the
link between translation of equations and the quadratic formula for example.
15.3
Apply mathematical thinking and modeling to solve problems that arise in
other disciplines.
Students apply mathematical thinking and modeling to solve problems that arise in other
disciplines in many courses. In the Calculus Sequence (MATH 131, 132, 234) many
physical applications such as rectlinear motion, free-falling bodies, related rates, area,
volumes, liquid pressure, work, growth and decay, and mass properties are considered.
In Differential Equations (MATH 245) students encounter problems involving circuits,
population growth and mechanical systems.
In Complex Variables (MATH 357)
students see applications of mathematics to fluid flow dynamics, conformal mapping and
temperature distribution problems. In Probability and Statistics (MATH 360) the
Poisson distribution is applied to problems involving traffic and telephone waiting times,
Bayes formula used to consider medical diagnosis, hypothesis testing and confidence
intervals are applied to political and opinion polls.
15.4 Recognize how a given mathematical model can represent a variety of
situations.
The program shows students how a given mathematical model can represent a variety of
situations. An example was given above in Element 15.1 involving the derivative.
Another example is the exponential function used in calculus (or the differential equation
y' = ky at the sophomore level) to describe exponential growth or decay, Newton's law of
cooling, continuous compounding of interest, and the concentration of a solute in a cell.
The sine and cosine functions are seen to represent a variety of situations in the Calculus
sequence (MATH 131, 132, 234) and Differential Equations (MATH 245) including
harmonic motion, temperature fluctuations, pendulum motion.
15.5
Create a variety of models to represent a single situation.
The program begins to develop students' ability to create a variety of models to represent
single situations in the Problem Solving Component of Workshop in Mathematics I/II
(MATH 190/191) when it encourages them to find more that one representation of a
problem and to seek out more than one solution method. In Methods of Proof (MATH
248) students find that they can view a function dynamically as a rule of correspondence
93
associating elements in one set with elements in another set or statically as a relation given
by a subset of a Cartesian product of two sets.
15.6
Understand the interconnectedness of topics in mathematics from an
historical perspective.
The program requires students to show evidence of the knowledge of interconnectedness
of topics of mathematics from a historical perspective in several courses. In Geometry
(MATH 550) non-Euclidean geometries are developed, and their discovery and the effect
in had on the rest of mathematics and the axiomatic method is discussed. Many
connections are discussed in History of Mathematics (MATH 490); for example,
students encounter the interplay of abstract algebra and geometry in considering the
impossibility proofs for the 3 classical Greek construction problems. In Senior Seminar
for Future Mathematics Educators (MATH 493) topics are frequently presented from a
historical perspective. For example, the relationship between Dedekind’s definition of
the real numbers and the need for a firm logical foundation for mathematics.
94
Standard 16: Delivery of Instruction
In the program, faculty use multiple instructional strategies, activities and materials that
are appropriate for effective mathematics instruction.
A wide variety of pedagogical and instructional approaches, including direct instruction,
small group work with hands-on materials, technology-based assignments, cooperative
learning, open-ended projects, student-led discussion, oral presentations, are employed in
courses in the program in order to provide equitable access to students of all backgrounds
and to model good teaching practice for future teachers. In addition, faculty work with
and are assisted by the Learning Resource Center on campus to ensure that disabled
students’ learning needs are accommodated.
Specific examples of effective mathematics instructional practices throughout the
curriculum follow: In the Calculus Sequence (MATH 131, 132, & 234) small group
discussion and group project work is used, in Workshop in Mathematics I/II (MATH
190/191) students do individual and group investigations, individual presentations in
which they both present material and lead discussions and practice peer instruction, in
Methods of Proof (MATH 248) portfolio assessment is practiced, in History of
Mathematics (MATH 490) students lead discussions, in Senior Seminar for Future
Mathematics Educators (MATH 493) group work, individual exploration, peer
teaching, hands-on manipulatives, and student-centered discussions are used, and in
Geometry (MATH 550) and Probability and Statistics (MATH 360) students conduct
explorations, employ technology, do group work, and receive direct instruction.
Required Elements
16.1
Is taught in a way that fosters conceptual understanding as well as
procedural knowledge.
When polled for the 1994 Subject Matter in Mathematics Program document, faculty
unanimously indicated that they believe that they were asking more conceptual questions
and fewer rote question on exams that they did a decade ago. In the ensuing 9 years, the
department has experimented with a variety of instructional methods, reform calculus
texts, technological advances, and attended numerous faculty development workshops
both on and off campus. From 1995-2001 as participants in the Los Angeles
Collaborative for Excellence in Teaching - LACTE (an NSF funded project - NSF-DUE
94-53608 see Appendix V), seven department members developed or revised six
mathematics courses and disseminated their work, methods, and materials through the
LACTE website (www.lacteonline.org), conferences, and in textbook form.
The instructional and assessment methods used by the Loyola Marymount University
faculty across the mathematics curriculum (and described in detail in Standards 3 through
7 and 12 through 15 as well as in this standard) are designed to foster deep conceptual
understanding as well as procedural knowledge.
95
A wide variety of pedagogical and instructional approaches, including direct instruction,
small group work with hands-on materials, technology-based assignments, cooperative
learning, open-ended projects, student-led discussion, oral presentations, are employed in
courses in the program. Many of these pedagogical and instructional practices are known
to foster conceptual understanding.
Specific examples of effective mathematics instructional practices throughout the
curriculum follow: In the Calculus Sequence (MATH 131, 132, & 234) small group
discussion and group project work are used (e.g., MATH 131 project 2, and MATH 132
project 1), in Workshop in Mathematics I/II (MATH 190/191) students do individual
and group investigations, individual presentations in which they both present material and
lead discussions and practice peer instruction, in Methods of Proof (MATH 248)
portfolio assessment is practiced, in History of Mathematics (MATH 490) students lead
discussions, in Senior Seminar for Future Mathematics Educators (MATH 493)
group work, individual exploration, peer teaching, hands-on manipulatives, and studentcentered discussions are used, and in Geometry (MATH 550) and Probability and
Statistics (MATH 360) students conduct explorations, employ technology, do group
work, and receive direct instruction.
In the Calculus Sequence (MATH 131, 132, & 234), students do group project work (see
MATH 131 project 2, and MATH 132 project 1). To successfully complete these
projects, students must explore the deeper concepts underlying the calculus tools they are
learning. Moreover, the assignment requires students to explain these concepts in their
own words. The work required to successfully explain concepts in this way fosters deep
understanding.
In Workshop in Mathematics I/II (MATH 190/191) the students lead discussions, work
in groups, do investigations, and explore concepts such as mathematical induction, the
pigeonhole principle, and arguing by cases. Moreover, students complete short writing
assignments about these topics. These classes, in particular, are filled with group work
assignments requiring students to interpret their content knowledge and apply it in
unusual places. Again, the work of bringing one’s knowledge to bear on a problem
unlike that which the concept was learned to address is known to improve students’
conceptual understandings.
The Senior Seminar for Future Mathematics Educators (MATH 493) is centered
around building a deeper understanding of the mathematics needed for teaching. In
addition to completing a semester-long project, students are asked to analyze difficulties
in learning basic mathematical concepts like the addition of fractions. This analysis
allows the students to gain a deeper understanding of notions like equivalence classes,
and how these deeper understandings affect the learning of their future students. Another
assignment asks the students to program a calculator (or spreadsheet) to perform long
division. This assignment fosters deeper understanding of that algorithm, and by
requiring the students to unpack their own understanding of long division, teaches
students a methodology for coming to deeper understanding about seemingly elementary
topics. Finally, the teaching of the course encompasses group work, student directed
96
instruction, the use of manipulatives for understanding the solutions of quadratic and
cubic equations, all of which build up different levels of deep understanding.
The final example we give here is from Geometry (MATH 550). Students in this class
work on projects involving Geometer’s Sketchpad to allow them to explore relations in
open-ended problems. Other projects (see the spherical triangles project and the angle
sum project) from the class include projects encouraging students to use models to
explore and gain deeper understanding). All of these projects are started in class for one
full class period, so that the instructor can promote and guide students to make
conjectures, formulate hypotheses, test them, and prove them.
Note that all assignments specifically mentioned above are included in the attachment
below of course syllabi and sample problems.
16.2
Incorporates a variety of instructional formats including but not limited to
direct instruction, collaborative groups, individual exploration, peer
instruction, and whole class discussion led by students.
At the LMU Mathematics Department a wide variety of interactive teaching methods are
used in addition to the traditional method of direct instruction. In the Calculus Sequence
(MATH 131, 132, & 234) demonstrations are used to introduce the relationships between
physical phenomena and mathematics; and in addition, student projects and Mathematica
activities are approached through cooperative learning. Workshop in Mathematics I/II
(MATH 190/191), Geometry (MATH 550) and Senior Seminar for Future
Mathematics Educators (MATH 493) frequently use open-ended activities in addition
to both individual and cooperative learning explorations. Students lead discussions, use
peer instruction, and participate in student-centered discussions in Workshop in
Mathematics I/II (MATH 190/191), History of Mathematics (MATH 490), and the
Senior Seminar for Future Mathematics Educators (MATH 493). In addition, Senior
Seminar for Future Mathematics Educators (MATH 493) asks students to interpret
grade 6-12 student responses, reflect on their fieldwork, and has students interact with
each other to debate the meaning and understanding of mathematical statements and
definitions.
16.3
Provides for learning mathematics in different modalities, e.g., visual,
auditory, and kinesthetic.
Throughout the courses in the Single Subject Matter Program, direct instruction, work in
groups, and discussion and peer evaluation of other students’ oral presentations provides
opportunities for learning mathematics in an auditory mode. Classroom computer
demonstrations and traditional blackboard use address visual learners. Teaching in
Senior Seminar for Future Mathematics Educators (MATH 493) and Geometry
(MATH 550) often employs both visual and kinesthetic modalities, such as in the use of
puzzles in Geometry (MATH 550) to discover and demonstrate properties of triangles
and in the presentation of solving a cubic in Senior Seminar for Future Mathematics
97
Educators (MATH 493), where three-dimensional geometric figures are produced. The
latter leads to a discussion of how solving the quadratic equation comes from a geometric
picture of actually completing a square. In this case, the class discussion centers around
how one learns to manipulate algebraically with understanding. Other examples of
visual or kinesthetic modalities being used are spatial puzzle exercises in Workshop in
Mathematics I/II (MATH 190/191), demonstrations with decks of cards and dice in
Probability and Statistics (MATH 360), dominoes to model the Principle of
Mathematical Induction in Methods of Proof (MATH 248), centroid demonstrators in
calculus, and so on.
16.4
Develops and reinforces mathematical skills and concepts through openended activities.
Open-ended activities are frequently used to develop and reinforce skills and concepts;
for example, in Workshop in Mathematics I/II (MATH 190/191) when students explore
problem solving, in Geometry (MATH 550) when students investigate properties of
triangles and develop proofs that are suggested by their investigations, and in Senior
Seminar for Future Mathematics Educators (MATH 493) when students complete an
original (to them) research project.
16.5
Uses a variety of appropriate technologies.
Technology is integrated throughout the program. Students are introduced to the use of
graphing calculators and the computer algebra system, Mathematica, in the Calculus
Sequence (MATH 131, 132, &, 234). They use graphing calculators and Excel as tools
in Workshop in Mathematics I/II (MATH 190/191) to collect generate data when
looking for patterns, examining conjectures and seeking counterexamples. Students
study programming in Numerical Methods (MATH 282) and examine and use
Geometer’s Sketchpad in Geometry (MATH 550). In the Senior Seminar class (MATH
493 – see objective 3), the graphing calculator, Excel, and Geometer’s Sketchpad are
examined as learning/teaching tools in mathematics.
16.6
Includes approaches that are appropriate for use at a variety of grade
levels.
In the department there are three courses taken primarily by prospective teachers:
History of Mathematics (MATH 490), Senior Seminar for Future Mathematics
Educators (MATH 493), and Geometry (MATH 550). In these courses innovative
strategies are used to examine content taught in the 6-12 curriculum. For example, as
shown in sample assignment 3 on the syllabus for Geometry (MATH 550), students
investigate puzzles using geometric figures, use these puzzles to develop several standard
proofs from plane geometry, and then discuss advantages/disadvantages of each proof
argument. Part of the purpose of the student investigation is for the students to analyze
the various conceptual and organizational methods for understanding the angle sum of a
triangle in Euclidean space. Alongside of projects like these, which involve both group
and individual investigative learning, classroom Socratic lectures include time for
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student-centered discussions about the teaching and learning of Euclidean geometry.
These discussions will range over a wide variety of issues, what are the “learning
packets3” required of students and teachers for this material.
In History of Mathematics (MATH 490) in the process of using "historical" methods to
solve problems and contrasting these approaches to modern day methods, students will
encounter mathematical concepts and explanations suitable to a variety of grade levels.
For example, when learning that, until the advent of algebraic notation, a geometric
representation of algebraic identities was the only possible representation, students
present and analyze justifications for A(B+C) = AB + AC or (A+B)2 = A2 + 2AB + B2
that are suitable for middle school and high school.
Similarly, in Senior Seminar for Future Mathematics Educators (MATH 493),
various teaching strategies are used depending on the discussion. For example, the
discussion of the difference between a variable and an unknown is a student-centered
discussion in which the professor interacts only to raise questions. This particular item
lends itself to having students discuss what conceptually lies behind the understanding of
a variable. The writing assignments for this class are chosen to require students to focus
upon how content is conceived and organized for instruction, making the students look at
both 6-12 student ideas (see sample assignment 3 from MATH 493), looking at how
material they are learning can fit into a high school curriculum or requires them to
conceptualize the high school curriculum better (see sample assignment 2 and the
project), and think about what conceptual understanding links to the procedural
understanding (see sample assignment 1).
3
The idea of Learning Packets comes from the work of Li-Ping Ma on Teaching and Knowing of
Mathematics.
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APPENDIX I:
Course Syllabi
100
MATH 131 Calculus 1 (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: James Stewart, Single Variable Calculus Early Transcendentals (4th ed) 1999.
Credit Hours: 4 hours
Catalog Description: Limits, continuity, derivatives of algebraic and transcendental
functions, applications of the derivative, antiderivatives, introduction to the definite
integral, Fundamental Theorem of Calculus.
Prerequisites: MATH 120 or placement.
Topics:
Polynomial equations and inequalities:
 Graphs of linear inequalities. (SMR 1.2 a)
 Rational Root Theorem (SMR 1.2 b)
 The Factor Theorem. (SMR 1.2 b)
Functions:
 Properties of functions (domain, range, one-to-one, onto, inverses,
composition) (SMR 1.3 a)
 Properties of rational, radical, and absolute value functions. (SMR 1.3 b)
 Properties of exponential and logarithmic functions. (SMR 1.3 c)
Trigonometry:
 Review trigonometric identities, relate them to the Pythagorean Theorem,
and use them for simplifying answers. (SMR 5.5 a)
 Analyze properties of trigonometric functions using graphing, derivatives,
and problem solving. (SMR 5.1 c)
 Use the definitions and properties of the inverse trigonometric functions to
solve problems in calculus. (SMR 5.1 d)
Limits and Continuity
 Basic properties of limits and continuity including Sum, Difference,
Product, Constant multiple theorems. (SMR 5.2a)
 Derivatives of polynomials from the definintion. (SMR 5.3b)
 Intermediate Value Theorem.
Derivatives and Applications:
 Differentiation of polynomials, trigonometric definitions, and exponential
functions (including derivations of results). (SMR 5.3 a).
 Geometric, numerical and analytic understandings of the derivative.
(SMR 5.3 b)
 Newton’s Method. (SMR 5.3 b)
 Interpret continuous functions geometrically and use and apply Role’s
Theorem, the Mean Value Theorem, and L’Hospitals room. (SMR 5.3 c)
101

Apply notions of the derivative to solve real world problems (related rates,
rectilinear motion, max-min problems, growth and decay problems).
(SMR 5.3 d)
 Use derivatives to analyze planar curves. (SMR 5.4e)
Integrals and Applications
 Integrals of standard functions from formal definition. (SMR 5.4 a).
 Interpret the concept of an integral geometrically and analytic
properties. (SMR 5.4 b)
 Prove the Fundamental Theorem of Calculus and use it to interpret
definite integrals as antiderivatives. (SMR 5.4 c)
Instruction and Technology: In this course we may occasionally break into small
groups to work on exploratory assignments and problems. Students will be expected to
actively participate in these activities. Students should have a graphing calculator (TI-83
recommended) for this course as we will make use of one throughout the term. We will
also use Mathematica for investigation, problem solving, and computation.
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation.
Objectives:
(1) Students will be able to clearly communicate differential calculus arguments in
everyday and mathematical languages. In particular, they will be able to solve these
problems in context and explore their relation with other problems. (4.1, 4.2, 4.3,
12.1, 14.1, 14.2)
(2) Students will be able to use technologies appropriately to investigate and solve
problems in differential calculus and enhance their understanding. (3.2, 3.3, 12.3)
(3) Students will be able to articulate mathematical ideas from differential calculus in
writing using appropriate terminology. (13.1, 14.2)
(4) Students will be able to present mathematical information in various forms and use
appropriate technologies to present these ideas and concepts. (13.3, 13.6)
(5) Students will be able to use the ideas of differential calculus to model and solve
problems arising in other disciplines. (15.3)
(6) Students will be able to solve problems in differential calculus. (11.4)
(7) Students will place differential calculus problems in context. (12.1)
(8) Students will study and present informal proofs in calculus. (14.1, 14.2)
(9) Students will recognize how the derivative can be used to model a variety of
situations (growth, rectilinear motion, etc.) (15.6)
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Sample Assignments and Exam Questions:
1. Project 2: Pipeline Construction
This project must be done in a group of three or four students. Every member of
the group will receive the same grade. The final project must be typed. Plots and
graphs should be generated using Mathematica (or another program, such as
Excel) and pasted into the document. Equations should be typeset (using, for
example, the Microsoft Equation Editor or Mathematica). All answers should be
in complete sentences, written in clear and grammatically correct English. You
must explain all your work.
The coastal city of Mellor has been growing rapidly over the last decade and is
experiencing severe water shortages. To fill this need, city leaders have been
arguing for the construction of a water pipeline from the city of Bennett, which
has much greater rainfall. After years of lobbying, the state legislature has finally
allocated funds to build the pipeline. Several different construction companies are
putting in bids – your team has been hired as consultants by one of these
companies to help design their pipeline.
Mellor lies in the foothills of a mountain range and Bennett sits on a plateau
higher in the range. The terrain between the two cities consists of a series of hills
and valleys, as shown below.
Height hundreds of feet
10
8
6
4
2
2.5
5
7.5
10
12.5
15
17.5
Distance miles
The pipeline needs to follow a due west-east path, so there will be no detours
around the mountains. The company that has hired you has provided the
following information and requirements:

Due to the difficulties of building supports on inclines, the supports for the
pipeline will be built at the peaks of the hills and the base of the valleys. This will
103
involve either tunneling through the peaks, or building support columns from the
peak/valley. There will, of course, be a support at either end, as the road enters
each city.

The hills begin at Mellor, where the altitude is 0 feet above sea level. The
peaks occur 2 miles, 6 miles, 12 miles and 16 miles from Mellor (along a direct
line to Bennett). The altitudes are 400 feet, 300 feet, 600 feet and 1000 feet,
respectively. The valleys occur 4 miles, 8 miles and 14 miles from Mellor, at
altitudes of 100 feet, 0 feet and 400 feet, respectively. Bennett is 18 miles from
Mellor, at an altitude of 700 feet above sea level.

The pipeline must be built in straight segments, each with a constant slope.
Due to the added construction difficulties, the design should involve no more than
three separate segments. The slope can only be changed at a support.

The cost (in dollars) of building a support column is 100 times the square
of the height of the column (in feet) (since the support must become thicker as
well as taller). Similarly, the cost (in dollars) of tunneling through a hill is 100
times the square of the height of the peak above the tunnel (since the hill becomes
wider as you move down from the peak). This height is measured from the point
in the tunnel that lies directly below the peak.

The first support (in Mellor) is built where the pipeline connects to
Mellor’s water system. You may choose to connect your pipeline either to the
city reservoir, requiring a support 180 feet high, or directly to the water treatment
plant, requiring a support 30 feet high.

The state has mandated that no bid may exceed $38 million. There are
also rumors that the governor will veto the project if the cost exceeds $33 million,
so the company would like a bid lower than this if at all possible.
Your fee will be determined by several criteria. Most important, of course, is the
cost of your design, which will be compared to the other designs submitted.
However, the construction company is also planning to use your proposal to help
decide whether it will hire you for future contracts, so they are interested in your
methodology as well as your results. Be sure to explain why you made the
choices you did, and show the details of all your calculations.
Your report should be appropriate for the company’s CEO, who doesn’t know
much calculus. Presentation counts! For this project, the reports must be typeset
in MS Word or some other word-processing program – not Mathematica. The
report should use calculus, but it should also explain your reasoning for someone
that has very little background in calculus. The report should also include
appropriate graphs as visual aids. A grading rubric with a more detailed
explanation of the expectations for the project will be handed out soon.
104
Selected final exam questions:
a. A right circular cylinder is inscribed in a cone of height h and base radius
r. Find the largest possible volume of such a cylinder. (objectives 1, 6,
and 8)
b. Use Newton’s method (and a calculator) to solve the equation
cos( x)  x  2 in radians accurate to 5 decimal places. State the formula
you are using, your initial guess, and the first 2 approximations.
(objectives 1, 2, and 6)
105
MATH 132 Calculus 2 (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: James Stewart, Single Variable Calculus Early Transcendentals (4th ed) 1999.
Credit Hours: 4 hours
Catalog Description: Techniques of integration, numerical methods of integrations with
error analysis, applications of the integral, improper integrals, infinite series, an
introduction to parametric equations and polar coordinates.
Prerequisites: MATH 131.
Topics:
Trigonometry:
 Review trigonometric identities, relate them to the Pythagorean Theorem,
and use them for solving integrals. (SMR 5.1 a)
 Analyze properties of trigonometric functions using graphing, integrals,
and problem solving. (SMR 5.1 c)
 Use the definitions and properties of the inverse trigonometric functions to
solve problems in calculus. (SMR 5.1 d)
Integrals and Applications:
 Derive definite integrals of standard algebraic functions using the formal
definition of the integral. (SMR 5.4a)
 Interpret the concept of a definite integral geometrically, numerically, and
analytically (SMR 5.4b)
 Prove the Fundamental Theorem of Calculus. (14.2, SMR 5.4c)
 Apply the integral to calculate the length of curves and the areas and
volumes of geometric figures. (SMR 5.4 d)
Sequences and Series:
 Derive and apply the formulas for the sums of finite arithmetic series and
finite and infinite geometric series. (SMR 5.5 a)
 Determine the convergence of a given sequence or series using
Comparison Test, Ratio Test, Integral Test, Limit Ratio Test (and other
tests as necessary). (SMR 5.5b)
 Calculate Taylor series and Taylor polynomials of basic functions. (SMR
5.5c)
Instruction and Technology:
In this course we may occasionally break into small groups to work on exploratory
assignments and problems. Students will be expected to actively participate in these
activities. Students should have a graphing calculator (TI-83 recommended) for this
course as we will make use of one throughout the term. We will also use Mathematica
for investigation, problem solving, and computation.
106
Grading:
Grades will be determined by a combination of homework, exams, out of class projects,
and class participation.
Objectives:
(1) Students will be able to clearly communicate calculus arguments in everyday and
mathematical languages. In particular, they will be able to solve these problems in
context and explore their relation with other problems. (4.1, 4.2, 4.3, 12.1, 14.1, 14.2)
(2) Students will be able to use technologies appropriately to investigate and solve
problems in calculus and enhance their understanding. (3.2, 3.3, 12.3)
(3) Students will be able to articulate mathematical ideas in calculus in writing using
appropriate terminology. (13.1, 14.2)
(4) Students will be able to present mathematical information in various forms and use
appropriate technologies to present these ideas and concepts. (13.3, 13.6)
(5) Students will be able to use the ideas of integral calculus to model and solve problems
arising in other disciplines. (15.3)
(6) Students will be able to recognize how integration and Riemann sums can model a
variety of physical situations (15.6)
107
Sample Assignments and Exam Questions:
(a) Project 1:
You are starting a company that applies a special coating on surfboards to speed
up the ride. It costs \$200 to cover $10$ square feet of board (only the bottom of
the surfboard needs to be coated). You need to figure out a pricing system for the
surfboards that come into your shop. Upon measuring, you find that the standard
long board is an ellipse defined by the equation
20.25 y 2  x 2  20.25
where x and y are measured in feet. You also have that the standard short board is
given (approximately) by the equations:
1.4(e (x1)2  1 ) if 0  x  3

e
4
f (x)   mx  b
3
if
 x5
4
 1 (x  5)2
if 5  x  6

where m 
3
5
, and b  1 
, and g(x)=-f(x), between x=0 and x=6.
4
4e
Remembering the Flat Fish Noserider, a new surfboard design that was featured
in a Los Angeles Times article last September 22nd, you want to have a price for
it. Unfortunately, all you have to go on is the picture in the paper. (See picture
next page.)
You need to establish prices for coating each of these boards, which means you
need to know the cost of the coating process. Using integration techniques and
approximation techniques from class, find an estimate for coating each board that
you is within \$10 of the actual cost. How would your estimates change if you
need to be within \$1 of the actual cost? Explain how you arrived at these
estimates, and how well you can guarantee that they are within the specified level.
You should do the long board without using Mathematica (although you can
check your answers). The short board, will require estimating an integral that
cannot be integrated with simple functions. In this case, explain which
approximation technique you are using, do {\bf not} simply ask Mathematica to
numerically integrate.
For this project, you may work in groups of 3 or 4, but I want each individual to
hand in their own final write up. Your write-up should address all the questions
indicated, and a rubric will be given out after spring break. You will receive
some points for the difficulty (and/or ingenuity) of your own design. In general,
you may use Mathematica to compute complicated sums, but do not simply have
it calculate for you at this point.
(This project speaks to all 6 course objectives.)
108
MATH 190 Workshop Course in Mathematics I (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Developing Successful Math Majors by J. Dewar, S. Larson, and T. Zachariah
Credit Hours: 2 hours
Catalog Description: Study skills, analytical and problem solving skills, technical
writing, recent files of study and advances in mathematics, mathematical career
opportunities.
Prerequisites: None.
Topics:
Problem Solving: (SMR Part II)
 Polya’s Four Step Approach to Problem Solving
 Heuristic Strategies:
o Understand the Problem (Deductive Reasoning)
o Break Mind Set (Multiple Representations)
o Simplify the Problem (Analogies, Pattern Recognition)
o Make a Model/Draw a figure (Spatial Reasoning, Multiple
Representations)
o Collect and Organize Data/Look for a Pattern (Using Appropriate
Technology, Pattern Recognition, Inductive Reasoning) (3.1, 3.2,
3.3)
o Working Backwards
o Try Another Approach (Multiple Representations, Persistence)
o Look Back (Reflection)
Mathematical Writing, Verbal Communication and Study Skills: (4.1, 4.2,
4.3, 13.1, 13.3, 13.4, 13.5, 13.6, 14.1, 14.2; SMR Part II)
Good Mathematical Communication (Verbal and Written)
o Organizing an Argument
o Giving Reasons for Each Step
o Defining Variables
o Proper Mathematical Notation (using different letters for different
things, honoring the equal sign)
o Writing in Complete Sentences
o Displaying Equations
o Clear Writing (balance between words and symbols, pronouns with
clear antecedents)
 Identifying, Correcting and Avoiding Commonly Made Mathematical
Writing Errors.
 Peer Review of Writing
109



Using a chalkboard, overhead projector, and PowerPoint
Using visual aids
Using Equation Editor in Microsoft Word and/or TeX to produce
mathematical documents
 Study Skills
o Reading Mathematics
o Working in Groups
o Asking Questions
o Learning from homework
o Studying for a Test
o Approaching Long-term Projects
Modern Mathematics and Mathematical Culture:
 Recent Advances and Modern Applications of Mathematics
o Guest speakers (2.3)
o Videos (2.3)
o Internet Resources
 Historical Development of Mathematics.
 Contributions by diverse cultural, ethnic, and gender groups to
Mathematics (2.3)
 Professional Societies in Mathematics (AMS, AWM, MAA, NCTM,
SIAM)
Mathematical Careers and People:
 Biographies of 20th Century Mathematicians (2.3)
 Mathematical Careers
o Guest speakers (2.3)
o Videos (2.3)
o Internet Resources
Instruction and Technology:
We will employ many relatively new pedagogical innovations. These include working
cooperatively in small groups (see text §2.5), emphasizing writing mathematics (see
Chapter 3), attending to the cognitive aspects of problem solving (see text §2.4), and
utilizing a variety of non-standard assessment techniques. Student self-assessment (see
text §3.3.N and 3.6.C), peer review (see text §3.4), reflective writing prompts (see text
§3.3.R) and preparation of a portfolio (see text §3.3.N) are just some of the alternative
assessment methods used in the courses. Evaluation criteria (see text §3.3.N, 3.3.O and
3.3.P) are shared with students in advance of collecting major assignments.
When appropriate, technology is seamlessly incorporated into the problem solving
sessions (for example, Excel, Mathematica, and graphing calculators). Guest speakers
address the application of technology in industry or research. Technology’s effect on the
development of modern mathematics is also discussed (for example, text §4.4.A and
4.4.F). The efforts to acquaint beginning majors with modern mathematics (see Chapter
4), the attempt to provide career-related information (see Chapter 5), and the emphasis on
mathematicians as people (see Chapter 5) are other novel aspects of these courses.
110
Two important features of the course are “working in groups,” and maintaining “a
relaxed, friendly atmosphere.” Past students comment that working in groups was one of
the strong points of the course, and cite the chance to learn from other students and
learning to work with other students as important benefits. Also, students credit the
relaxed classroom atmosphere for encouraging self-expression and interaction with other
math majors. In addition, graduating students say they got to know one another and
began the networking and study support groups which helped them in the major. (2.5,
3.1, 3.2, 3.3, 5.1, 5.2, 5.4, 5.5, 16.1, 16.2, 16.3, 16.4, 16.5, 16.6)
Grading:
This course has four components: 1) Problem Solving, 2) Mathematical
Communication and Study Skills, 3) Modern Mathematics and Mathematical Culture,
and 4) Mathematical Careers and People. Each component will count equally towards
your final grade. Active participation in class discussions and problem solutions is
essential to your benefiting from this course. Each week you will have at least one
problem solution to write up, and possibly one problem solution to revise. Short written
responses to various reading assignments or in-class presentations will be assigned. A
paper (minimum 4 pages) on a mathematical topic of interest to you will be assigned.
Maintain a section in your 3-ring binder to store all of your written assignments. At the
end of the semester, you will be asked to create a portfolio of your mathematical work.
So you will want to have all your written assignments available to choose selections for
the portfolio. The portfolio will include three or four pieces of your work that you think
show the scope of your improvement in problem solving and writing mathematics and an
explanation why you chose those pieces. (S 7.1, 7.2)
Objectives:
The course objectives fall into several categories:
Problem Solving Objectives (11.2, 12.1, 12.2, 12.3, 12.4, 13.5, 14.1)
 To improve problem solving skills
 To develop students’ ability to formulate and test conjectures by using inductive
reasoning, constructing counterexamples, and making deductive arguments
 To utilize technology appropriately
 To help students learn to monitor their progress toward a solution
 To develop the habit of “looking back” at the end of a problem and seek multiple
ways of solving and generalizing the problem
 To develop confidence in problem solving abilities
111
Study Skills Objectives
 To develop good study skills in mathematics
 To encourage the formation of math study groups
Communication Objectives (4.1, 4.2, 4.3, 13.1, 13.3, 13.4, 13.5, 13.6, 14.2)
 To give students the opportunity to discuss and evaluate mathematical reasoning
of their own and of others
 To improve students’ use of charts, graphs, figures, and equations to present
mathematical arguments
 To improve mathematical writing skills of both formal and informal arguments
 To provide the student with the experience of writing a mathematical paper
 To develop the ability to make an oral presentation in mathematics
Mathematical Career Objectives (6.4, 8.1, 8.2)
 To impart information about mathematical careers
 To encourage students to pursue a math-related career, including teaching
Modern Mathematics Objectives (15.2)
 To inform students about modern applications of mathematics
 To let students know that the discipline of mathematics is alive and developing
and consists of inter-related fields
Mathematical Culture Objective (2.1, 2.3, 15.6)
 To increase students’ awareness of the history and people of mathematics
 To inform students of the contributions of a diverse cultural, ethnic and gender
groups to mathematics
Community-building Objective
 To help new students “feel at home” in the mathematics department
Days and Topics List:
First Semester Sample Syllabus for Developing Successful Math Majors
Week
1.
Source(text)
Appendix 3
§2.2.L
§2.2.C
§5.5, [4]
Topics Covered
Introduction
Course surveys4
Problem Solving Pretest
Problem Solving: Understanding the Problem
Mathematical Careers: Video Math, Who Needs It?
2.
§3.6.A
§2.2.B
§3.4.A-3.4.C
Study Skills: Overview
Problem Solving: Breaking Mind Set
Writing Skills: How to Peer Review
3.
§3.2.A
§2.2.D
§5.2.G
Writing Skills: Examining Writing Samples
Problem Solving: Simplify the Problem
Mathematical Careers Discussion: Based on Careers that Count
4
Course survey instruments to measure initial and/or final attitudes, confidence levels, and
knowledge about problem solving approaches, study skills, mathematical careers and people are
located in Appendix 3.
112
§3.6.B
Study Skills: Select and Practice a Skill
4.
§3.3.C
§2.2.E
§5.2.I
Writing Skills: English - Writing in Sentences
Problem Solving: Make a Model/Draw a Figure
Mathematical Career Speaker: Actuarial careers
5.
§4.4.B
§3.6.C
§2.2.F
§3.3.D
Mathematical Culture: Video segment on Linear Perspective from Life by
the Numbers
Study Skills: Self-Test
Problem Solving: Collect and Organize Data; Look for a Pattern
Writing Skills: Strike a Balance Between Words and Symbols
6.
§5.2.I
§3.3.B
§2.2.G
Mathematical People: Guest speaker gives a math autobiography
Writing Skills: Incorrect Math
Problem Solving: Work Backwards
7.
§3.3.A
§2.2.H
§5.2.A
Writing Skills: Common Errors
Problem Solving: Persistence
Mathematical People: Discussion of biographies of mathematicians
8.
§5.2.I
§3.7, [17]
§3.3.F
Mathematical Career Speaker: Secondary teaching careers
Problem Solving: The Candy Problem from Math Horizons
Writing Skills: Use Different Letters for Different Things
9.
§3.3.E
§2.2.I
§4.4.D
Writing Skills: Honor the Equal Sign
Problem Solving: Another Approach
Modern Mathematics: Presentation on Apportionment
10.
§5.2.I
§2.4
§4.4.E
Mathematical Culture Speaker: History of Calculus and Study Abroad
opportunity
Problem Solving: Cognitive Aspects
Modern Mathematics: Presentation on Graph Theory
11.
§3.3.G
§2.2.J
§5.2.I
Writing Skills: Defining Terms
Problem Solving: Look Back
Study Skills Speaker: A senior math major gives advice
12.
§3.3.H
§2.2.K
§5.2.C
Writing Skills: Give Reasons
Problem Solving: Putting It All Together
Mathematical People: Video segment on Richard Tapia from
Breakthrough: Profiles of Scientists of Color
13.
§3.3.F
§2.2.K
Writing Skills: Using Different Letters for Different Things
Problem Solving: Additional practice and discussion
Study Skills: “Preparing for Finals” presentation by Learning Resource
Center Math Specialist
14.
§3.3.O, 3.4.F
Student Presentations of Math Papers
15.
Appendix 3
§2.2.L
Course evaluations and course surveys
Problem Solving Post-test
Discussion of Problem Solving Post-test
113
MATH 191 Workshop Course in Mathematics II (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Developing Successful Math Majors by J. Dewar, S. Larson, and T. Zachariah
Credit Hours: 2 hours
Catalog Description: (Continuation of MATH 190 - Workshop Course in Mathematics I) Study
skills, analytical and problem solving skills, technical writing, recent fields of study and advances
in mathematics, mathematical career opportunities.
Prerequisites: MATH 190 or consent of instructor.
Topics:
Problem Solving: (4.1, 4.2, 4.3, 14.1, 14.2; SMR Part II)
 Polya’s Four Step Approach to Problem Solving
 Heuristic Strategies for solving problems and constructing proofs
o Draw a Diagram (Spatial Reasoning, Multiple Representations)
o Use the Pigeonhole Principle
o Pursue Parity (Multiple Representations)
o Break the Problem Down by Cases
o Special Cases (Min/Max, Inductive Reasoning, Multiple Representations)
o Fewer Variables (Pattern Recognition, Inductive Reasoning) (3.1, 3.2, 3.3)
o Mathematical Induction (Using Appropriate Technology, Pattern
Recognition, Inductive Reasoning) (3.1, 3.2, 3.3)
o Contradiction (Multiple Representations, Persistence)
o Look Back (Reflection, Generalization)
Mathematical Writing, Verbal Communication and Study Skills: (4.1, 4.2, 4.3, 13.1,
13.3, 13.4, 13.5, 13.6, 14.1, 14.2; SMR Part II)
 Good Mathematical Communication (Verbal and Written)
o Organizing an Argument
o Giving Reasons for Each Step
o Defining Variables
o Proper Mathematical Notation (using different letters for different things,
honoring the equal sign)
o Writing in Complete Sentences
o Displaying Equations
o Clear Writing (balance between words and symbols, pronouns with clear
antecedents)
 Identifying, Correcting and Avoiding Commonly Made Mathematical Writing
Errors.
 Peer Review of Writing
 Using a chalkboard, overhead projector, and PowerPoint
 Preparing a Poster Paper Presentation
 Using visual aids
114

Using Equation Editor in Microsoft Word and/or TeX to produce mathematical
documents
 Study Skills
o Reading Mathematics
o Working in Groups
o Asking Questions
o Learning from homework
o Studying for a Test
o Approaching Long-term Projects
Modern Mathematics and Mathematical Culture:
 Recent Advances and Modern Applications of Mathematics
o Guest speakers (2.3)
o Videos (2.3)
o Internet Resources
 Historical Development of Mathematics (15.6)
 Contributions by diverse cultural, ethnic, and gender groups to Mathematics (2.3)
 Professional Societies in Mathematics (AMS, AWM, MAA, NCTM, SIAM)
Mathematical Careers and People:
 Interviews of Math Faculty Members (2.3)
 Mathematical Careers
o Guest speakers (2.3)
o Videos (2.3)
o Internet Resources
Instruction and Technology: We will employ many relatively new pedagogical innovations.
These include working cooperatively in small groups (see text §2.5), emphasizing writing
mathematics (see Chapter 3), attending to the cognitive aspects of problem solving (see text
§2.4), and utilizing a variety of non-standard assessment techniques. Student self-assessment (see
text §3.3.N and 3.6.C), peer review (see text §3.4), reflective writing prompts (see text §3.3.R)
and preparation of a portfolio (see text §3.3.N) are just some of the alternative assessment
methods used in the courses. Evaluation criteria (see text §3.3.N, 3.3.O and 3.3.P) are shared
with students in advance of collecting major assignments.
When appropriate, technology is seamlessly incorporated into the problem solving sessions (for
example, Excel, Mathematica, and graphing calculators). Guest speakers address the application
of technology in industry or research. Technology’s effect on the development of modern
mathematics is also discussed (for example, text §4.4.A and 4.4.F). The efforts to acquaint
beginning majors with modern mathematics (see Chapter 4), the attempt to provide career-related
information (see Chapter 5), and the emphasis on mathematicians as people (see Chapter 5) are
other novel aspects of these courses.
Two important features of the course are “working in groups,” and maintaining “a relaxed,
friendly atmosphere.” Past students comment that working in groups was one of the strong points
of the course, and cite the chance to learn from other students and learning to work with other
students as important benefits. Also, students credit the relaxed classroom atmosphere for
encouraging self-expression and interaction with other math majors. In addition, graduating
students say they got to know one another and began the networking and study support groups
which helped them in the major. (2.5, 3.1, 3.2, 3.3, 5.1, 5.2, 5.4, 5.5, 16.1, 16.2, 16.3, 16.4, 16.5,
16.6)
115
Grading: This course has four components: 1) Problem Solving, 2) Mathematical Communication
and Study Skills, 3) Modern Mathematics and Mathematical Culture, and 4) Mathematical Careers
and People. Each component will count equally towards your final grade. Active participation in class
discussions and problem solutions is essential to your benefiting from this course. Each week you will
have at least one problem solution to write up, and possibly one problem solution to revise. Short
written responses to various reading assignments or in-class presentations will be assigned. At the end
of the semester you will present a poster paper to math students and faculty on a mathematical topic of
interest to you. Maintain a section in your 3-ring binder to store all of your written assignments. At
the end of the semester, you will be asked to create a portfolio of your mathematical work. So you will
want to have all your written assignments available to choose selections for the portfolio. The portfolio
will include three or four pieces of your work that you think show the scope of your improvement in
problem solving and writing mathematics and an explanation why you chose those pieces. (7.1, 7.2)
Objectives:
The course objectives fall into several categories:
Problem Solving Objectives (11.2, 12.1, 12.2, 12.3, 12.4, 13.5, 14.1)
 To improve problem solving skills
 To develop students’ ability to formulate and test conjectures by using inductive
reasoning, constructing counterexamples, and making deductive arguments
 To utilize technology appropriately
 To help students learn to monitor their progress toward a solution
 To develop the habit of “looking back” at the end of a problem and seek multiple ways of
solving and generalizing the problem
 To develop confidence in problem solving abilities
116
Study Skills Objectives
 To develop good study skills in mathematics
 To encourage the formation of math study groups
Communication Objectives (4.1, 4.2, 4.3, 13.1, 13.3, 13.4, 13.5, 13.6, 14.1, 14.2)
 To give students the opportunity to discuss and evaluate mathematical reasoning of their
own and of others
 To improve students’ use of charts, graphs, figures, and equations to present mathematical
arguments
 To improve mathematical writing skills of both formal and informal arguments
 To provide the student with the experience of writing a mathematical paper
 To develop the ability to make an oral presentation in mathematics
Mathematical Career Objectives (6.4, 8.1, 8.2)
 To impart information about mathematical careers
 To encourage students to pursue a math-related career, including teaching
Modern Mathematics Objectives (15.2)
 To inform students about modern applications of mathematics
 To let students know that the discipline of mathematics is alive and developing and
consists of inter-related fields
 To provide an opportunity to research a mathematical topic of interest to the student
Mathematical Culture Objective (2.1, 2.3, 15.6)
 To increase students’ awareness of the history and people of mathematics
 To inform students of the contributions of a diverse cultural, ethnic and gender groups to
mathematics
Community-building Objective
 To help new students “feel at home” in the mathematics department
Days and Topics List:
Second Semester Sample Syllabus for Developing Successful Math Majors
Week
Source (text)
Topics Covered
1.
Introduction
Appendix 3
Pre-course survey for new students
§2.3.K
Problem Solving Pre-test
§2.2.A, 2.3.A
Problem Solving: Review of Polya’s Four Steps
§3.2 Writing Skills: Review of writing skills discussed last semester
2.
§3.6.F
§2.3.B
§5.2.E
§3.4A-3.4C
3.
4.
§3.6.D
§3.3.S
Study Skills: Making New Semester Resolutions
Problem Solving: Draw a Diagram
Mathematical Careers: Discussion based on information found on the Internet or in
Jobs Related Almanac
Writing Skills: Examining Writing Samples and Peer Review
§5.2.I
Study Skills: Reading a Math Textbook (1)
Problem Solving: Comparing Two Approaches to the Same Problem – Using Algebra
vs. Drawing a Diagram
Mathematical Career Speaker: NASA Astronaut
§3.6.F
Study Skills: Review New Semester Resolutions
117
§5.2.H
§2.3.H
§4.4.F
Mathematical People: Choose student interview teams and corresponding faculty to
interview
Problem Solving: Inductive Approach
Modern Mathematics: Presentation on the Mandelbrot Set and video Nothing But
Zooms
5.
§3.6.E
§2.3.H
§3.3.P
Study Skills: Reading a Math Textbook (2)
Problem Solving: Mathematical Induction
Writing Skills: Introduce Poster Paper Assignment
6.
§3.3.M
§5.2.H
§2.3.F
Writing Skills: Writing in Other Courses
Mathematical People: Discussion based on faculty interviews
Problem Solving: Special Cases
7.
§3.3.I
§2.3.G
§5.5, [5]
Writing Skills: Watch Those Pronouns
Problem Solving: Fewer Variables
Mathematical Careers: Video and discussion - Operations Research + You = Exciting
Career
8.
§2.3.E
§3.3.J
§5.2.I
Problem Solving: Break Down By Cases
Writing Skills: Putting It All Together (1)
Mathematical Culture and People Speaker: On women in the history of mathematics
9.
§2.3.D
§4.4.C
§4.4.A
§5.2.I
§2.3.I
§3.3.P
Problem Solving: Pursue Parity
Math Careers and People: Video about Florence Nightingale The Passionate
Statistician
Modern Mathematics: Discussion of GIMPS
Modern Mathematics Speaker: Biostatistics
Problem Solving: Contradiction
Writing Skills: Meet students to discuss poster paper status
11.
§3.3.K
§2.3.C
§5.5, [2]
Writing Skills: Putting It all Together (2)
Problem Solving: Pigeonhole Principle
Mathematical People: Video about Polya Let Us Teach Guessing
12.
§3.3.L
§2.3.J
Writing Skills: Putting It all Together (3)
Problem Solving: Putting It All Together
13.
§2.3.J
§4.6, [7]
Problem Solving: Additional practice and discussion
Modern Mathematics Speaker: Knot Theory and video Not Knots
14.
§3.3P, 3.4.G
Student Poster Paper Presentations
15.
Appendix 3
§2.3.K
Course evaluations and course surveys
Problem Solving Post-test
Discussion of Problem Solving Post-test
10.
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MATH 234 Calculus 3 (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: James Stewart, Multivariable Calculus (2nd ed) 2001.
Credit Hours: 4 hours
Catalog Description: Partial derivatives, multiple integrals, three-dimensional spaces, vectors in
two- and three- dimensional space, line integrals, Green’s theorem.
Prerequisites: MATH 132.
Topics:
Vectors and Three-Space
 Cartesian coordinates and vectors in space (SMR 1.4a)
 Dot products (SMR 1.4b)
 Cross products (SMR 1.4b)
 Vector projections (SMR 2.4a)
 Equations of lines, planes (SMR 2.3a)
 Angles between lines, planes, and parallel, perpendicular conditions (SMR 2.3a)
 Functions and surfaces (SMR 2.3ab, 1.3a)
 Cylindrical Coordinates (SMR 2.3b)
 Spherical Coordinates (SMR 2.3b)
Vector Functions
 Curves in space (SMR 1.3b)
 Derivatives and integrals of vector functions (SMR 5.3ab)
 Arc Length (SMR 5.4d)
 Curvature (SMR 5.4d)
 Motion in space (SMR 5.3d)
 Parametrically-defined surfaces (SMR 2.3b)
Partial Differentiation
 Functions of Several Variables (SMR 1.3a)
 Limits and continuity (SMR 2.3a)
 Partial derivatives (SMR 5.3a)
 Tangent planes and linear approximations (SMR 2.3a)
 The chain rule for several variables (SMR 5.3ab)
 Directional derivatives and the gradient vector
 The second derivative test for functions of two variables (SMR 5.3d)
 Lagrange multipliers and constrained max-min problems (SMR 5.3f)
Multiple Integrals
 Double integrals over rectangular regions (SMR 5.4ad)
 Double integrals over general regions (SMR 5.4ad)
 Area and volume by double integrations (SMR 2.3b)
 Double integrals in polar coordinates
 Applications of double integrals (SMR 2.3b)
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 Triple integrals (SMR 2.3b)
 Surface area (SMR 2.3b)
 Integration in cylindrical and spherical coordinates (SMR 2.3b)
Vector Calculus
 Vector fields (SMR 1.4ab)
 Line integrals
 The fundamental theorem for line integrals (SMR 5.4c)
 Green’s Theorem
Instruction and Technology: In this course we may occasionally break into small groups to
work on exploratory assignments and problems. Students will be expected to actively participate
in these activities. Students should have a graphing calculator (TI-83 recommended) for this
course, as we will make use of one throughout the term. We will also use Mathematica for
investigation, problem solving, figure making, and computation.
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation.
Objectives:
(1) Students will develop their skills in writing solutions to multivariate calculus problems in
appropriate mathematical language. In addition, they will learn to use these skills to assist
them in analyzing, synthesizing and enhancing understanding of multivariate calculus.
(4.1, 4.2, 4.3, 13.1)
(2) Students will examine the underlying mathematical reasoning behind multivariate
calculus, and they will learn to solve problems in multivariate calculus. (11.2, 14.1)
(3) Students will place multivariate calculus problems in context and explore their
relationships with other problems. (12.1)
(4) Students will present information concerning functions of several variables in a variety of
forms, including equations and figures. (13.3)
(5) Students will use appropriate technologies to present ideas and concepts from multivariate
calculus. (13.6)
(6) Students will examine and present formal and informal arguments and solutions in
multivariate calculus. (14.1, 14.2)
(7) Students will analyze and illustrate concepts from multivariate calculus using rectilinear
motion. (15.1)
(8) Students will apply mathematical thinking and modeling using multivariate calculus
techniques to solve problems arising in physics and engineering. (15.3)
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Sample Homework Project
The group projects in this course will consist of multi-part questions that depend on answers that
require theoretical explanations, computations that necessitate the use of mathematical software,
and creative approaches. You should try to have every person at every meeting. If you wish to
meet with me to discuss your project, you must have more than half of your group present at the
time (e.g., if your group has three people, you must have at least two people to discuss the project
with me).
You should type the final copy of the paper. If you use Microsoft Word, I recommend using
their equation editor.
Your grade will depend on the following items:
 Mathematical Correctness
 Completeness of Derivations and Explanations
 Inclusion of Tables, Graphs, Diagrams to Complement WrittenExposition
 Quality of Writing (Sentence Structure, Grammar, Diction)
 Creativity in Presentation
Geodesic Project: Your goal of this project is to write Mathematica code that
accomplishes the following: given the location of two cities in the world, (i) graphically
construct the shortest flight path connecting the cities, and (ii) compute the length of this
path. The path of shortest distance between two points is called a geodesic. Below is a
picture of the shortest flight path between New York and Los Angeles. Note that, due to
the curvature of the earth, the path is not a straight line.
(Picture not included with this document)
1. Look up latitude and longitude (using any references at your disposal, including
the internet), and describe this system in a few paragraphs of your own. Feel free
to add as much detail as you wish.
2. A great circle of a sphere is defined to be a circle on the sphere of maximal radius.
For example, if we approximate the earth as a sphere, then the equator and all of
the longitudinal lines form an incomplete collection of great circles.
Mathematically, the great circles of a given sphere can be described as the curves
of intersection of the given sphere with various planes through the center of the
sphere. Suppose we take a sphere centered at the origin with radius r. Create a
vector parameterization of the intersection of this sphere with the plane through the
origin given by the equation Ax+By+Cz=0.
3. Although the earth is not a perfect sphere, we can use a sphere as an approximate
model. Look up the radius of the earth. Denote this radius by  . Then transpose
earth into xyz-space such that
a. (0,0,0) is the center of the earth,
b. the north pole lies on the positive z-axis, and
c. the longitudinal line at 0 degrees lies in the xz-plane.
4. Given a point on earth with longitude a  and latitude b  , find spherical
coordinates (  ,  , ) for this point. What is the distance along the sphere between
points that have spherical coordinates (  ,1 ,1 ) and (  , 2 , 2 ) ? Be very careful!
I'm not asking you to use the standard distance, but rather the distance that you
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would actually fly to travel between these points. Your answer should be the
length of an arc of a great circle.
5. Write a Mathematica function that given two locations in terms of latitude and
longitude, the code outputs the distance between them. Write Mathematica code
that draws the earth as a sphere, and then draws a boldface arc connecting two
locations given by longitudinal and latitude locations. Look up the latitude and
longitude of a pair of your favorite cities, apply the Mathematica code you just
wrote to compute the distance between them and draw a picture.
122
MATH 245 – Ordinary Differential Equations
Instructor:
Office Hours:
email:
WebPage:
Text: Zill and Cullen, Differential Equations with Boundary-Value Problems, Brooks/Cole
Publishing Co., 2001 (5th edition)
Credit Hours: 3 hours
Catalog Description: Differential equations as mathematical models, analytical, qualitative and
numerical approaches to differential equations and systems of differential equations, and Laplace
transform techniques.
Prerequisites: MATH 132
Topics:
Introduction:
 Properties of functions (domain, range, one-to-one, onto, inverses, composition)
(SMR 1.3 a)
 Properties of exponential and logarithmic functions. (SMR 1.3 c)
 Properties of trigonometric functions (SMR 5.1c)
 Understanding functions as solutions to differential equations
Simple Differential Equations:
 Direction Fields (analyzing curves) (SMR 5.2e)
 Separation of Variables
 Linear Systems of Differential Equations
 Using Substitution to Solve Differential Equations (including trigonometric)
 Understanding equations involving derivatives
 Numerical Methods to solve differential equations
 Applying separable equations to growth and decay problems. (SMR 5.3f)
Modeling with Differential Equations:
 Modeling rectilinear motion problems. (SMR 5.2d)
 Modeling Growth and Decay (SMR 5.3f)
 Modeling predator-prey problems.
More Complicated Differential Equations:
 Higher order equations
 Annihilators
 Variation of Parameters
 Cauchy-Euler Equations
 Transforms
o Laplace
o Inverse
 Advanced topics.
123
Instruction and Technology: The use of computer technology is essential for the modern study
of differential equations. Students should have a graphing calculator (TI-83 recommended), and
will use it for class work and homework assignments throughout the semester. In addition,
students will collaborate on projects using Mathematica.
Grading:
Grades will be determined by a combination of homework, exams and out of class
projects.
Objectives:
(1) Students will understand the meaning and concept of a differential equation.
(2) Students will be able to use technology to investigate the solutions to differential
equations numerically.
(3) Students will be able to solve exactly linear differential equations with constant
coefficients.
(4) Students will be able to solve exactly various types of first order differential equations.
(5) Students will use differential equations to model situations in the physical and social
sciences.
(6) Students will be able to clearly communicate, in writing, arguments involving differential
equations, using both mathematical formalism and everyday language.
Sample Assignments and Exam Questions:
(a) Flowerpot Project
Math 245 – Group Project 1: Flowerpots
This project will be done in a group of 3-4 people. You will be studying the rate at which water
drains out of flowerpots. The project is due on Friday, February 28.
A flowerpot can be viewed as a truncated cone, as shown below. The radius of the top is R, the
radius of the bottom is r, and the height of the pot is H. The initial height of the water is h0, and
the radius of the hole in the bottom is Rh.
Question: How long will it take the water in the pot to drain out?
We will begin by collecting data. You will be provided with flowerpots and rulers.
1. Measure your pot. Record: the height of the pot, the diameter of the base, the diameter of
the top and the diameter of the hole(s).
2. Fill the pot with water. Using a ruler, measure the height of the water at several times as it
drains. You should take at least 5 measurements. Do this at least three times. You will
probably want to have one person holding the pot, one reading the ruler, one reading the
stopwatch, and one recording the data.
3. Change the number of holes in the bottom of the pot, and repeat (2).
4. Get a new pot, of a different size, and repeat (2) and (3).
Now, we want to model the situation. Make some reasonable conjectures about the rate at which
water drains (constant? Proportional to the height of the water? Something else?). For each
model, set up and solve a differential equation. Compare your data with the solutions, and discuss
how well it fits. If the fit is poor, you need to find a better model!
124
Your final project should be a detailed and well-written explanation of how you arrived at your
solution (including descriptions of any rejected models, and why they were rejected). It must be
typed. Graphs should be created in Excel or Mathematica (or some other program) and pasted in.
You may use Mathematica to help with your computations, but a Mathematica printout is not an
acceptable format. Your data should be included as an appendix.
(b) Selected Final Exam questions
1. At the beginning of the new millennium (January 1, 2001), the city of Rollem, CA had
50,000 inhabitants. The birth rate in Rollem is 15 births per 1000 people per year, and the
death rate is 10 deaths per 1000 people per year. Moreover, each year 500 people move
out of Rollem and 200 people move in. Assume these rates stay constant.
(a) Find a differential equation for the total population P(t) of Rollem, where t is the
number of years into the new millennium.
(b) Solve the differential equation, and find the population in Rollem on January 1, 2021.
(c) What will happen to the population over the long term?
2
2. Find the general solution to the differential equation y  2 y  5y  10x .
3. A force of 2 Newtons stretches a spring 4 meters. A mass of 0.5 kg is suspended from the
spring, and the system is placed in a fluid which provides a damping force numerically
equal to the instantaneous velocity of the mass. The mass is set in motion, and you
observe that the displacement from the equilibrium is 3 meters after 1 minute and 1 meter
after 2 minutes. What is the displacement after 4 minutes? Is the system underdamped,
overdamped or critically damped?
Schedule:
Week
January 13
Monday
No class
Wednesday
Introduction (1.1, 1.2)
Friday
Mathematica tutorial
January 20
Martin Luther King
Day
Direction Fields (2.1)
January 27
Separation of
Variables (2.2)
Substitution (2.5)
Modeling with DE’s
(1.3)
Last day to Add/Drop
classes
Linear Equations (2.3)
Catch-up
Hand out Project 1
Modeling (3.3)
February 3
February
10
February
17
February
24
Modeling (3.1)
Numerical Solutions
(2.6)
Modeling (3.2)
Review
Midterm 1
March 3
March 10
Spring Break
Superposition (4.4)
March 17
Cauchy-Euler (4.7)
4.1, continued
Exact Equations (2.4)
Higher-order linear
equ’ns (4.1)
Reduction of order (4.2) Constant Coefficients
(4.3)
Project 1 due
Spring Break
Spring Break
Annihilators (4.5)
Variation of Parameters
(4.6)
Elimination (4.8)
Nonlinear Eq’ns (4.9)
125
March 24
Modeling (5.1)
Modeling (5.1)
March 31
April 7
April 14
Cesar Chavez Day
Review
Inverse Transform
(7.2)
Additonal operations
(7.4)
Modeling (5.3)
Midterm 2
Translation Th’ms (7.3)
Review
Final Exams
Review
Final Exams
April 21
April 28
May 5
Dirac Delta Function
(7.5)
Last day to Withdraw
Modeling (5.2)
Hand out Project 2
Catch-up
Laplace Transform (7.1)
Good Friday
Systems of Linear
Equ’ns (7.6)
Project 2 due
Review
Final Exams
126
MATH 248 Methods of Proof (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Fletcher and Patty, Foundations of Higher Mathematics, 2nd ed.
Credit Hours: 3 hours
Catalog Description: Number theory, sets, functions, equivalence relations, cardinality, methods
of proof, induction, contradiction, contraposition.
Prerequisites: MATH 132 Calculus II, or consent of instructor.
Topics
Logic and the Language of Proofs (1.3, 1.4, 4.1, 4.2, 4.3, 13.1, 13.4, 13.5, SMR 3.1a, SMR Part
II)
 Propositions, conditional statement, hypothesis, conclusion, truth tables, logical equivalence,
expressions and tautologies, quantifiers, forming negations, contrapositives, and converses
 Methods of Proof: Direct, Bi-Conditional, Contrapositive, Indirect proof (Contradiction),
Proof by cases, Existence proofs, Counterexamples
Elementary Number Theory (1.3, 11.1, 11.5, SMR 1.1b, SMR 3.1abcd, SMR Part II)
 Natural numbers and integers and their properties: even, odd, prime, composite, multiples,
factors, divisors, Fundamental Theorem of Arithmetic, sequences, recursion, Division
Algorithm, greatest common divisor, least common multiple, relatively prime, Euclidean
algorithm, Principles of Mathematical Induction and the Least Natural Number Principle
Sets, Operations, and Properties (1.3, SMR Part II)
 Set, element, member, subset, proper subset, element chase proofs, indexing families, equal
sets, cardinality of a finite set, empty set, power set, intersection and union of sets,
DeMorgan’s Laws, disjoint sets, complement, universal set, set difference, Cartesian product
of sets, representing set relations with Venn diagrams
Relations (1.3, 11.5, SMR Part II)
 Binary relations, relation from a set A to a set B, relation on a set A, graphing relations by
Cartesian graphs and by directed graphs, reflexive, symmetric, transitive relations,
equivalence relations, equivalence classes, partitions, properties of partitions and equivalence
classes, congruence modulo n, congruence classes, congruence class arithmetic
Functions (1.3, 11.5, SMR 1.3.a, SMR1.3b, SMR Part II)
 Function viewed as a rule, function viewed as a binary relation, how to prove a relation is a
function, domain, range, codomain, equal functions, one-to-one (injective) function, onto
(surjective) function, bijective function, function composition, composition of one to one &
127
onto functions, inverse relation, inverse function, composition property of a function and its
inverse, how to find an inverse function, images and pre-images of sets under functions
Cardinality (1.3, SMR Part II)
 Cardinality of a set, numerically equivalent sets, finite sets, infinite sets, countable set,
uncountable set, products of countable sets, cardinality of N, Z, Q, R and (0,1), Cantor
diagonalization argument, Cantor's Theorem
Objectives:
1) To develop the student’s ability to understand and evaluate the validity of mathematical
arguments. ( 1.3, 1.4, 4.1, 4.2, 4.3, 11.5, 13.1, 13.4, 13.5, 14.1, 14.2, 15.1, 15.2, 15.4,
15.5)
2) To develop the student’s ability to construct mathematical proofs (including proofs by
induction, contradiction and contraposition) and counterexamples. ( 1.3, 1.4, 3.2, 3.3, 4.1,
4.2, 11.5, 12.3, 13.1, 13.4, 14.1, 14.2)
3) To improve his/her ability to communicate mathematics verbally and in writing; ( 1.3, 1.4,
4.1, 4.2, 11.5, 13.1, 13.4, 14.1, 14.2, 15.1, 15.2, 15.4, 15.5)
4) To gain an understanding of certain topics from basic set theory, elementary number theory,
relations, and functions (see Topics list above) that are basic to the study of advanced
mathematics. (11.5, SMR 1.1b, SMR 3.1abcd, SMR Part II)
.
Instruction and Technology: A variety of instructional formats will be used. These will be
chosen from direct instruction (lecture), collaborative groups, individual exploration, peer
instruction and whole class discussion led by students, as deemed appropriate by the instructor (
5.1, 5.2, 5.4, 16.1, 16.2, 16.3, 16.4)
Assessment: Grades will be determined by a combination of homework (including re-writes*),
quizzes, exams, class preparation/participation, and a portfolio (see Portfolio Assignment below).
( 7.1, 7.2)
*Certain "proof" problems that are initially flawed as to content or reasoning or lack clarity of
writing will be returned with comments/suggestions to the student for a second (and sometimes
third) attempt. These are known as "re-writes."
128
Portfolio Assignment Given at the End of the Semester
View this assignment as an opportunity for self-reflection about the progress you have made in the
course, for clarifying what you know and don't know, and as an aid in preparing to study for the final.
The Assignment: Select four pieces of your work from the semester on at least two different topics
that show how your mathematical understanding and proof writing have improved. At least one pair
(two pieces) should be rewrites of the same proof.
Structure your portfolio as follows; put it in a slim folder:
Table of Contents
Introduction - In the portfolio you are to demonstrate how your problem solving and
mathematical proof writing have improved. In this section comment in general terms what
improvement or change you believe you have made in these two areas.
Main Portion - In separate paragraphs, discuss each of the “pieces” of work you include.
Describe, specifically, why they were included and how they demonstrate your improvement.
Conclusion - Reflect on this portfolio assignment as a culminating exercise for the semester.
What did you learn from looking over and presenting a self-analysis of your work?
Appendix - Copies of the “pieces” of work. Number or letter them for easy reference.
What I expect: Evidence that you took the assignment seriously in terms of time, thought, and effort
you expended; very specific analyses of mistakes you made or misconceptions that you had which
were eventually corrected, as exhibited by certain problems included in the portfolio. Your essay is
word-processed and should exhibit good grammar, spelling, punctuation. See below for descriptors
of outstanding responses and of poor responses.
What an outstanding response would exhibit: It would be clear that a significant amount of effort and
time was invested by the student in the portfolio; essay would contain a thoughtful self-reflection and
self-analysis; student would refer to mathematical topics using correct notation and terminology;
student would analyze errors made; may comment on instructor's corrections; student would present
specific evidence that misunderstandings are eventually cleared up; English grammar, spelling, and
punctuation would be essentially perfect and portfolio neatly presented.
Some characteristics of a poor response: It would be clear that little reflection, effort or time were
invested by the student; self-analysis is brief and shallow; comments are very general in nature; little
or no use of mathematical terminology; little specific evidence offered for improvement; little or no
reference to particular errors made; English grammar, spelling, or punctuation may be poor; portfolio
is carelessly put together; student failed to follow directions: did not submit requested number of
work samples; essay did not address submitted samples.
Final Exam Review Problems
The following five problems were chosen to address major topics in the course (logic, sets,
elementary number theory, induction, relations, function properties) and they require you to write
proofs or find counterexamples. Working on them should help you review for the final (but do
not limit your review to these problems only) and can earn you extra credit homework points.
129
You will be graded on correctness and clarity of your solution. A solution key will be available
on the last day of class to aid you in studying for the final.
1. Consider the relation S defined on the set of natural numbers N by
xSy iff 3 divides xy, for natural numbers x and y.
Determine with a proof or a counterexample whether or not S is
(a) reflexive, (b) symmetric, or (c) transitive.
2. Let A, B, C, D be any sets. Prove or disprove that ( A  B)  (C  D)  ( A  C )  ( B  D) .
3. Let f be a function from the Reals to the Reals and x 0 be any real number. The following is
the definition of f being continuous at x 0 :
The function f is said to be continuous at x 0 if for each positive 

such that | f ( x)  f ( x0 ) |  whenever | x  x0 |  .
cal

structure (i.e. to reveal any hidden quantifiers and the conditional statements it contains).
(b) Write in "useful" English what it means for f not to be continuous at x 0 .
4. Prove that for the equivalence relation congruence mod 11,
[10n] = [1] if n is an even natural number,
and
[10n] = [-[1] if n is an odd natural number.
Explain how this result is the basis for a method of finding the remainder when a number is
divided by 11.
Illustrate your explanation with the number 10,472.
5. Let f : A  B and g : B C . We discussed the following two results in class:
If gof is one to one, then f is one to one.
If gof is onto, then g is onto.
Find examples to show that the converse of each of these does not hold. In your examples
identify clearly A, B, C and f and g.
Days and Topics List:
Class Hours
8
Topics
Logic, Propositions, Quantifiers
Direct proof
Bi-Conditional proofs
130
Contrapositive proofs
Indirect proof
Proof by cases
Existence proofs
Counterexample
8
Elementary number theory, even odd, prime, composite, Fundamental
Theorem of Arithmetic, Mathematical induction, recursion, Least
Natural Number Principle,
Division Algorithm, gcd, lcm, relatively
prime, Euclidean algorithm
4
Sets, Operations, and Properties
Counting
Indexed Sets
Algebra of Sets
8
Binary Relations
Equivalence Relations
Partitions
Congruence modulo n
Order Relations
7
Functions (binary relation point of view)
Injections, Surjections, Projections
Composition of Functions
Inverse of a Function
Images and Inverse Images
Finite and Infinite Sets
Countable and Uncountable Sets
4
Review and Testing
_____
39
131
MATH 250 Linear Algebra (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Howard Anton, Elementary Linear Algebra (8th ed) 2000. or Stephen Leon, …
Credit Hours: 3 hours
Catalog Description: Systems of linear equations, Gauss and Gauss-Jordan elimination,
matrices, and matrix algebra, determinants. Linear transformations of Euclidean space. General
vector spaces, linear independence, inner product spaces, orthogonality. Eigenvalues and
eignevectos, diagonalization. General linear transformations.
Prerequisites: MATH 248 or consent of instructor.
Topics:
Systems of Linear Equations:
 Introduction and geometric interpretation to systems of linear equations (SMR
1.4a)
 Gaussian elimination (1.4b,c)
 Gauss-Jordan elimination
Matrices:
 Matrix operations (SMR 1.4 c)
 Elementary Matrices and their relationship to Gaussian elimination and inverses of
matrices (SMR 1.4 c)
 Finding inverses of matrices and their relationship to solving systems of linear
equations (SMR 1.4 c)
Determinants:
 The determinant function (SMR 1.4 c)
 Different methods for evaluating the determinant of a matrix (SMR 1.4 c)
 Properties of the determinant and relationship to matrix invertibility and solving
systems of linear equations (SMR 1.4 c)
Euclidean Vector Spaces
 Vectors in 2-space and 3-space (SMR 1.4 a)
 Scalar multiplication, dot product, cross product and their properties (SMR 1.4 a,
b)
 Vectors in n-space
General Vector Spaces:
 Definitions and basic properties
 Subspaces
 Linear Independence, basis, and dimension
 Row space, column space, and nullspace of a matrix (SMR 1.4 c)
Inner Product Spaces
 Inner products, angle and orthogonality in inner product spaces
 Orthonormal bases
Eigenvalues, Eigenvectors and Diagonalization
132
 Eigenvalues of matrices
 Eigenvectors of matrices
 Diagonalization
Linear Transformations
 Orthoganol Transformations (Rotations and Reflections) (SMR 2.4a)
 Dilations (SMR 2.4b)
 General transformations.
Instruction and Technology: In this course we may occasionally break into small groups to
work on exploratory assignments and problems. Students will be expected to actively participate
in these activities. Students make use of Mathematica for investigation, problem solving, and
computation.
Grading: Grades will be determined by a combination of homework, quizzes, exams, out of
class projects, and class participation.
Objectives:
(1) Students will be able to clearly communicate linear algebra arguments in everyday and
mathematical language. In particular, they will be able to solve these problems in context and
explore their relation with other problems. (4.1, 4.2, 4.3, 12.1)
(2) Students will be able to use technologies appropriately to investigate and solve problems in
linear algebra and enhance their understanding. (3.2, 3.3, 12.3)
(3) Students will be able to articulate mathematical ideas from linear algebra in writing using
appropriate terminology. (13.1)
(4) Students will be able to present mathematical information in various forms and use
appropriate technologies to present these ideas and concepts. (13.3, 13.6)
(5) Students will be able to use the ideas of linear algebra to model and solve problems arising in
other disciplines. (15.3)
(6) Students will be able to solve problems in linear algebra. (11.4)
(7) Students will place linear algebra problems in context. (12.1)
(8) Students will present informal and formal proofs in linear algebra. (14.2)
(9) Students will recognize how matrices and linear algebra can be used to model a variety of
situations (15.6)
(10) Students will explore the underlying mathematical reasoning behind linear algebra. (11.2)
(11) Students will see the design and analysis of recursion problems in linear algebra. (11.5)
(12) Students will use appropriate technologies to conduct investigations and solve problems in
linear algebra. (12.4)
(13) Students will illustrate concepts involving eigenvalues and eigenvectors using
applications. (15.1)
133
Sample Assignments and Exam Questions:
(a) Sample Assignment
Use Mathematica to do the following problems.


1) Find the matrix A which has eigenvectors b1  (4,1,4) , b2  (1,1,2) ,

b3  (5,1,3) and associated eigenvalues 2, 2, -3 respectively.
2) A population of rabbits raised in a research laboratory has the following
characteristics.
i. Half of the rabbits survive their first year. Of those, half survive their
second year. The maximum life span is three years.
ii. During the first year the rabbits produce no offspring. The average
number of offspring is 6 during the second year and 8 during the third
year.
The laboratory population now consists of 24 rabbits between 0 and 1 years
old, 24 between 1 and 2 years old, and 20 between 2 and 3 years old. The

current “age distribution vector” is x1  (24,24,20) T and the “age transition
0 6 8
matrix” is: .5 0 0.

0 .5 0

Note that the first entry of the age distribution vector gives the number of
rabbits between 0 and 1 year old, the second entry gives the number of
rabbits between 1 and 2 years old, and the third entry gives the number of
rabbits between 2 and 3 years old. The age transition matrix has the
property that the result of multiplying it by an age distribution vector is the
age distribution vector of the following year. For example, after one year,
0 6 824 304
the age distribution vector will be: x 2  Ax1  .5 0 024 12 , and

0 .5 0

20
 
12 

there will be 304 rabbits between 0 and 1 years old, 12 rabbits between 1
and 2 years old, and 12 rabbits between 2 and 3 years old. From the age
distribution vectors, we see that the percentage of rabbits in the three age
groups changes each year. Suppose that the laboratory prefers a stable
growth pattern, one in which the percentage in each age group remains the
same each year. For this stable growth pattern to be achieved, the (n+1)st
age distribution vector must be a scalar multiple of the nth age distribution
vector. That is xn1  Axn  xn for some  . Find an initial age
distribution vector y1 that will lead to a stable age distribution for the
rabbit population.
(b) Selected final exam questions:
Consider a system of the form
x1 
m1 x3  b1
 m1 x1  x 2  m1 x3  b1
 m2 x1  x 2  m1 x3  b2
134
where m1 , m2 , b1 , b2 are constants.
a) Under what conditions will this system of linear equations have a unique
solution? Explain.
b) Under what conditions will this system of linear equations have a no
solution? Explain.
c) Under what conditions will this system of linear equations have infinitely
many solution? Explain.
Prove or give a counterexample to each of the following statements.
o AB = BA for all n n matrices A, B.
o If A is a singular n n matrix, then at least one entry of A must be 0.
o If V is an inner product space and v  V then v, v  0 .
o Suppose A, B, and C are n n matrices. If AB = AC, then B = C.
o The sum of two 3 3 nonsingular matrices is nonsingular.
o The product of three nonsingular matrices is nonsingular.
Which of the following vectors in  3 (with the Euclidean inner product)





v1  (1,1,2) , v2  (3,1,2) , v3  (2,4,1) , v4  (1 / 2,0,1 / 4) , v5  (1 / 2,1 / 2,1)
are:
a)
Orthogonal?
b)
In the same direction?
c)
In opposite directions?
135
MATH 282 Elementary Numerical Methods (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text:
Credit Hours: 3 hours
Catalog Description: Computer solutions of applied mathematical problems using FORTRAN
and Mathematica. Nonlinear equations, differentiation, integration.
Prerequisites: MATH 131.
Topics:
Programming/Computer Arithmetic
 Introduction to FORTRAN
 Introduction to Mathematica
 Design of Algorithms using discrete structures.
Solution Methods
 Measuring Errors
 Bisection Method
 Secant Method
 Newton’s Method
 General Root Finding
 Analysis of algorithms
o Iteration
o Recursion
Interpolation and Numerical Differentiation
 Lagrange Form
 Difference Forms
 Interpolating Polynomials with Mathematica
 Numerical Differentiation
 Cancellation Error
Derivatives and Integrals
 Using Mathematica to derive derivative formulas
 Accurate computation of derivatives in FORTRAN.
 Elements of numeric integration
 Deriving integration formulas with Mathematica
Linear Algebra
 Linear systems of equations
 Gaussian elimination
 Solving systems of equations in Mathematica
 FORTRAN solution of linear equations
136
Instruction and Technology: This is a class in numerical methods. Technology is used
throughout. Students will learn to program using Mathematica and Fortan.
Grading: Grades will be determined by a combination of homework (often involving
programming), exams, out of class projects, and class participation.
Objectives:
(1) Students will be able to discrete structures in programming and creating algorithms (11.5)
(2) Students will be able to use technologies appropriately to investigate and solve problems
involving numerical analysis. (3.2, 3.3, 12.3)
(3) Students will be able to use technology to find derivatives numerically and with computer
algebra systems. (13.1)
(4) Students will be able to use iteration and recursion to design and analyze algorithms. (11.5)
(5) Students will be able to use computer applications to solve problems in calculus, linear
algebra, and approximation of roots. (11.5)
137
MATH 293 Field Experience (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text:
Credit Hours: 0 hours
Catalog Description: Planned observation, instruction or tutoring experiences appropriate for
future secondary mathematics teachers; related professional reading and reflection (6.1)
NOTE: Completion of this field experience is a pre- or co-requisite for MATH 493 and
opportunities to further reflect on your field experience will be given as part of your MATH 493
coursework. (6.2, 6.4, 6.5)
Prerequisites: None.
Topics:
Documentation of 20 hours of field work experience with students ages 12-18 from
diverse populations and at least one teacher certified in mathematics, including
reflection and associated reading (6.3)
 Document your 20 hours of field experience in a field experience journal using the
following format.
Format: For each fieldwork experience give school name, location, grade level,
diversity of classroom/students, teacher's name and position (and certification), and a
daily dated log entry including time spent on site which summarizes the observation
including mathematical topics discussed, teaching methods used (5, 5.1, 5.5),
classroom management methods observed, use of technology observed, impressions of
student response to the lesson/tutoring/etc.
 Associated Reading:
Choose three readings from the focus issues of The Mathematics Teacher on Algebraic
Thinking (February 1997) , History (November 2000), Mathematics Teaching in
Middle School on Algebraic Thinking (February 1997), Data and Chance (March
1999), or from other readings suggested by the teacher you are observing.
 Reflection (6.4)
Reflect on the fieldwork experience/associated readings using the reflection prompts
below as labels.
Reflection prompts:
 List 3 to 5 “major” lessons learned from your field experience;
 Describe whether and how the experience changed the likelihood of becoming
a teacher;
 Describe at least 3 connections between LMU coursework and the field
experience (6.2);
138




Summarize in a few sentences each of your readings and describe their
relationship to the fieldwork;
Describe how technology was employed in the instructional process and
comment on if/how in your judgement it enhanced the students' learning (5.5)
Describe 3 examples of how mathematical language and/or reading skills were
evidenced in the field experience (S4, 4.1)
Give at least one suggestion to improve the field experience for future LMU
students.
139
MATH 321 Real Variables I (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Stephen Abbot, Understanding Analysis, Springer-Verlag, New York, 2001.
Credit Hours: 4 hours
Catalog Description: The real number system; least upper bound; sequences, Cauchy sequences;
functions; limits of functions; continuity; derivatives; Riemann integration.
Prerequisites: MATH 248.
Topics:
Number Systems
 Natural Numbers
o Well-Ordering Principle (SMR 3.1b)
o Induction.
 Rational Numbers
o Definitions
o Field Axioms (SMR 1.1a)
 Real Numbers
o Definitions – Dedekind cuts
o The Real numbers are an ordered field. (SMR 1.1ac)
o Basic properties of the real numbers. (e.g., a<b, c<0, then ac>bc) (SMR
1.1b)
o Completeness Axiom and its consequences (e.g., lest upper bound
property).
Sequences
 Derive the formulas for sums of finite and infinite geometric series. (SMR 5.5a)
 Limits of Sequences.
 Limit Theorems/Tests. (14.1, 14.2, SMR 5.5b)
 Proofs with sequences (epsilon-delta proofs) (14.1, 14.2).
 Subsequences
 Cauchy Sequences.
 Monotonicity.
Functions and Continuity
 Continuous Functions (including proof that polynomial functions are continuous).
(14.1, 14.2, SMR 5.2b)
 Properties of Continuous Functions.
o Rule derivations for continuous functions (sum, difference, product,
constant multiple, and quotient rules for limits and application to
continuous functions) (SMR 5.2a)
o Intermediate Value Theorem (14.1, 14.2, SMR 5.2c)
 Uniform Continuity
 Limits of Functions
Derivatives
140




Basic properties of the derivative proved from the definition.
o Derivatives of basic functions. (SMR 5.3a)
o Derivative rules (sum, product, chain, etc.)
L’Hopital’s rule.
Rolle’s Theorem (SMR 5.3c)
Mean Value Theorem (SMR 5.3c)



Definition of the Riemann Integral, Riemann sums. (SMR 5.4b)
Integrals of standard algebraic functions using the definition. (SMR 5.4a)
Fundamental Theorem of Calculus. (14.1, 14.2, SMR 5.4c)
Integration
Instruction and Technology: This course is a standard first year analysis course. The most
appropriate instruction methods will be used to enhance student learning. Mathematica or other
programs may be used to help students acquire an intuitive understanding of ideas in analysis.
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation.
Objectives:
(1) Students will develop their skills in writing solutions to real analysis problems in
appropriate mathematical language. In addition, they will learn to use these skills to assist
them in analyzing, synthesizing and enhancing understanding of multivariate calculus.
(4.1, 4.2, 4.3, 13.1)
(2) Students will examine the underlying mathematical reasoning behind calculus using the
tools of real analysis, and they will learn to solve problems in real analysis. (11.2)
(3) Students will examine and present formal and informal arguments and solutions of real
analysis problems. (14.1, 14.2)
(4) Students will learn to use real analysis to formally understand intuitive ideas from
calculus. (15.3)
Sample Final Exam Questions:
1. Define what it means for the sequence (a n ) to diverge.

2. Prove that the series
r
k
converges if | r | 1 and diverges if | r | 1.
k 0
3. Let f ( x)  x 3 .

a. Using the -  definition of continuity, prove that f is continuous at any
c  0.
b. By computing f ' (c ) directly prove that f is differentiable at any c   .
4. A fixed point of a function is a value x where f ( x)  x . Assume f :[a,b] [a,b] is
continuous and differentiable on (a, b) .
a. Prove that f has a fixed point. (Hint: Consider g ( x)  f ( x)  x .)
b. Prove that if f '(x) 1 for all x  (a,b) then the fixed point is unique.
141
MATH 331 Elements of Group Theory (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Joseph Gallian, Contemporary Abstract Algebra 4th edition.
Credit Hours: 3 hours
Catalog Description: Group Theory. Binary operations, subgroups, cyclic groups, factor groups,
isomorphism, homomorphism, and Cayley’s theorem.
Prerequisites: MATH 248.
Topics:
Basics
 One-to-One functions. (SMR 1.3a)
 Onto functions. (SMR 1.3a)
 Compositions and closure. (SMR 1.3a)
 Mathematical Induction (SMR 3.1b)
 Euclid’s Lemma and Divisibility (SMR 3.1a)
 Fundamental Theorem of Arithmetic (SMR 3.1d)
 Euclidean Algorithm (SMR 3.1c)
Group Axioms and Examples
 Group Axioms
 Elementary Examples
o Cyclic groups.
o Dihedral groups. (SMR 2.4a)
o Modular Arithmetic groups.
o GL(2,R), GL(3,R).
o Two and Three dimensional Isometry groups. (SMR 2.4a)
 Abelian Groups
Permutation Groups
 Permutations, definition of groups. (SMR 4.1a)
 Orbit-Stabilizer Theorem
 Cayley’s Theorem
Cosets and Factor Groups
 Definitions of Cosets
 Lagrange’s Theorem
 Fermat’s Little Theorem
 Euler’s Theorem
 Normal Subgroups
 Factor Groups
 Isomorphism Theorems
Structure of Groups
 Cauchy’s Theorem
 Class Equation and p-groups
142


Sylow’s Theorems
Fundamental Theorem of Abelian Groups
Instruction and Technology: The course will consist primarily of lectures, however we may
occasionally break into small groups to work on exploratory assignments and problems. Students
will be expected to actively participate in these activities. There may be occasional use of
computer algebra systems.
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation.
Objectives:
(1) Students will develop their skills in writing solutions to problems in group theory using
appropriate mathematical language. In addition, they will learn to use these skills to assist
them in analyzing, synthesizing and enhancing understanding of group theory. (4.1, 4.2,
4.3, 13.1)
(2) Students will examine the underlying mathematical reasoning behind group theory, and
they will learn to solve problems in group theory. (11.2)
(3) Students will examine and present formal and informal arguments and solutions in group
theory. (14.1, 14.2)
(4) Students will use group theory and permutation group arguments to solve and model
problems involving counting. (15.3)
(5) Students will see applications of group theory in modern mathematics and science. In
particular, students will see the application of modular arithmetic and group theory in
error-correction and cryptography.
143
MATH 357 Complex Variables (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text:
Credit Hours: 3 hours
Catalog Description: Complex variables; analytic functions, Laurent expansions and residues;
evaluation of real integrals by residues; integral transforms.
Prerequisites: MATH 234.
Topics:
Complex Numbers and the Complex Plane:
 Define complex numbers and elementary operations. Derive algebraic properties
(e.g., show complex numbers form a field). (SMR 1.1 a, 1.1 b)
 Define the complex plane and derive geometric properties of complex numbers
including the (SMR 1.1 b)
 Discuss polar form of complex numbers and associated formulas for powers and
roots. (SMR 1.1 b, 5.1 e)
 Introduce the basic topology of the complex plane including notions of open,
closed, connected, and a domain.
 Discuss applications to second order differential equations arising in electrical
engineering. (15.3)
Complex Functions and Mappings:
 Analyze complex functions both algebraically and as mappings of the complex
plane.
 Introduce and analyze the complex linear, power, and reciprocal functions.
 Define and study the notions of multiple valued and branch cuts.
 Introduce the theoretical notions of limit and continuity of complex functions.
Discuss relation to limits of real multivariable functions. Prove various algebraic
properties of limits. (SMR 5.2a,b,c)
 Discuss applications of complex functions to vector fields associated to planar
flows of a fluid. (15.1, 15.3)
Analytic Functions:
 Define and study the concepts of differentiability and analyticity of complex
functions. (SMR 5.3a)
 Prove the Cauchy-Riemann equations and demonstrate their applications. (SMR
5.3a)
 Introduce and study harmonic functions.
 Discuss applications to gradient fields associated to electrostatics, fluid flow, and
heat flow. (15.1, 15.3, 15.4)
Elementary Functions
 Define and analyze the complex exponential, logarithmic, power, trigonometric,
and hyperbolic functions. (1.3a,b,c)
144

Introduce and study branches of inverse complex trigonometric and hyperbolic
functions. (5.1c,d)
 Discuss applications of elementary complex mappings to Dirichlet problems
arising in the study of electrostatics, fluid flow, and heat flow. (15.1, 15.3, 15.4,
15.5)
Integration in the Complex Plane
 Define the concept of a complex integral and compare and contrast it to the
concept of real integrals. (5.4a)
 State and prove the Cauchy-Goursat Theorem. Discuss theoretical applications of
the theorem.
 State and prove Cauchy’s integral formulas, and discuss theoretical applications
including Louiville’s Theorem, the Fundamental Theorem of Algebra, Morera’s
Theorem, and the Maximum Modulus Theorem.
 Prove the Fundamental Theorem of Algebra and the conjugate roots theorem.
(1.1c, 1.2 bc, 5.4d)
 Discuss applications to circulation and net flux in a planar fluid flow. (15.1, 15.3)
Series and Residues
 Define complex sequences and series. Introduce Taylor and Laurent series. (5.5c)
 Analyze zeros and poles of complex functions using series.
 Define residues and prove the Residue Theorem.
 Investigate applications of the Residue Theorem to the evaluation of real integrals.
 Prove the Argument Principle and Rouche’s Theorem. Discuss applications of
summing infinite series.
 Discuss applications to Laplace and Fourier transforms.
Conformal Mappings
 Define the concept of a conformal mapping. (2.4ab)
 Introduce and study linear fractional and Schwarz-Christoffel transformations.
 Present and discuss the Poisson integral formulas.
 Discuss applications to boundary-value problems and streamlining.
Instruction and Technology: This course will consist mostly of classroom presentation with
occasional in class activities. Students are expected to keep up with readings and ask questions.
Students will need access to a computer algebra system (available in the mathematics computer
lab).
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation.
Objectives:
(1) Students will be able to clearly communicate arguments concerning the complex numbers in
everyday and mathematical languages. In particular, they will be able to solve these problems
in context and explore their relation with other problems. (4.1, 4.2, 4.3, 12.1)
(2) Students will be able to use technologies appropriately to investigate and solve problems
involving complex numbers and enhance their understanding of complex numbers. (3.2, 3.3,
12.3)
(3) Students will be able to articulate mathematical ideas in writing using appropriate
terminology. (13.1)
145
(4) Students will be able to use the ideas of complex numbers and complex analysis to model and
solve problems arising in other disciplines. (15.3)
(5) Students will be able to recognize how complex analysis can be used to model a variety of
physical situations (15.6)
(6) Students will show an understanding of the foundations and proofs of complex analysis.
(11.2)
(7) Students will solve problems in complex analysis and apply complex variable methods to
solving problems arising in other disciplines. (11.2, 15.3)
(8) Students will be able to discuss how the study of complex variables relates to other topics in
mathematics like differential equations, calculus, linear algebra, and geometry. (15.2)
(9) Students will be able to illustrate concepts of complex variables using applications from other
disciplines. (15.1)
146
Sample Assignments:
1. Section 1.3 homework: A. Read Section 1.3. B. Answer the following questions. Write
complete sentences. (i) Explain the difference between arg(z) and Arg(z). (ii) In Section
1.1 we saw that z1  z 2 if and only if Re( z1 )  Re( z 2 ) and Im( z1 )  Im( z 2 ) . Suppose we
write z1 and z 2 in polar form, z1  r1 (cos1  i sin 1 ) and z 2  r2 (cos 2  i sin  2 ) . If
z1  z 2 , then what must be true about  1 and  2 ? How about r1 and r2 ? (iii) If z is any
1
3 
z is in the
complex number, then use Equation (6) to describe where the point   i
2
2 
complex plane. C. Do problems 1, 3, 6, 7, 9, 11, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28,
34, 35, 40, 42, 43 in Exercises 1.3.
2. Computer Lab Assignment: Reread part (ii) of the Remarks at the end of Section 2.6. In
Problems 57-60, use a CAS to show that the given function is not continuous inside the
unit circle by plotting the image of the given continuous parametric curve. (Be careful,
Mathematica and Maple plots can sometimes be misleading.) 57. f ( z )  z  Arg ( z ) ,
1 1
z (t )   
3it ,  1  t  1 .
2 2

147
MATH 360 Introduction to Probability and Statistics (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Robert Hogg and Elliot Tanis, Probability and Statistical Inference (6th ed) 2001.
Credit Hours: 3 hours
Catalog Description: Descriptive statistics, probability, discrete and continuous random
variables, limit theorems, sampling distributions, estimations of parameters, nonparametric
methods, hypothesis testing, linear regression.
Prerequisites: MATH 123 or 132.
Topics:
Basic Concepts, Numerical Characteristics:
 Frequency, histograms, discrete and continuous type data
 Mean, median, mode (SMR 4.2 a)
 Range, percentiles, quartiles, variance, standard deviation (SMR 4.2 b)
Sampling Methods, Graphical Presentations and Comparison of Data Sets:
 Random sampling, systematic sampling, and convenience sampling (SMR 4.2 c)
 More about histograms, stem-and-leaf display, two-sided stem-and-leaf displays
(SMR 4.2 c)
 Boxplots (SMR 4.2 b, c)
Probability:
 Definition and basic properties.
 Methods of enumeration – multiplication principle, permutations, combinations
(SMR 4.1 a)
 Finding and interpreting probabilities (SMR 4.1 b, d)
 Conditional probability (SMR 4.1 c)
 Independent events and Bayes theorem
Random Variables:
 Discrete and continuous random variables
 Mathematical expectation, mean, variance, standard deviation (SMR 4.2ab)
Probability Mass Functions, Probability Density Functions and Important
Distributions:
 Probability mass, probability density and cumulative distribution functions and
their relationship to each other
 Discrete distributions: hypergeometric, binomial, Poisson. (SMR 4.1 e)
 Continuous distributions: uniform, normal, exponential, gamma, Cauchy, beta,
chi-square. (SMR 4.1 e)
 Using the distributions
148
Distributions of Two Random Variables, Independent Random Variables
Central Limit Theorem and Law of Large Numbers
Confidence Intervals:
 For means, difference of two means, variances, proportions (4.2a)
 Sample size
Hypothesis Testing
 Tests for means, proportions
 Chi-square test (SMR 4.2 e)
Least Squares, Linear Regression and Correlation (SMR 4.2 d)
Instruction and Technology: In this course we occasionally break into small groups to work on
exploratory assignments and problems. Students will be expected to actively participate in these
activities. Students should have a calculator for this course as we will make use of one throughout
the term. We will also use Mathematica and/or Excel for investigation, problem solving, and
computation. At times, students are asked to view and participate in statistical related internet
sites.
Grading: Grades will be determined by a combination of homework, quizzes, exams, out of
class projects.
Objectives:
(1) Students will be able to clearly communicate probability and statistical arguments in everyday
and mathematical languages. In particular, they will be able to solve these problems in
context and explore their relation with other problems. (4.1, 4.2, 4.3, 12.1, 14.1, 14.2)
(2) Students will be able to use technologies appropriately to investigate and solve problems in
probability and statistics and enhance their understanding. (3.2, 3.3, 12.3)
(3) Students will be able to articulate mathematical ideas from probability and statistics in writing
using appropriate terminology. (13.1, 14.1, 14.2)
(4) Students will be able to present mathematical information in various forms and use
appropriate technologies to present these ideas and concepts. (13.3, 13.6)
(5) Students will be able to use the ideas of probability and statistics to model and solve problems
arising in other disciplines. (15.3)
(6) Students will be able to solve problems in probability and statistics. (11.4)
(7) Students will place probability and statistics problems in context. (12.1)
(8) Students will examine and present formal and informal proofs in probability and statistics.
(14.1, 14.2)
(9) Students will be able to collect, classify and represent data in appropriate ways both using
technology and without. (3.3)
(10) Students will be able to explain what a random variable is, and classify random variables
and discrete, continuous, or neither. (11.5)
(11) Students will be able to recognize connections between real data, probability functions,
probability density functions and distribution functions. (15.2, 15.4)
(12) Students will be able to compute and interpret probabilities using methods of enumeration,
distributions, and probability functions. (11.5)
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(13) Students will have a conceptual understanding of the Central Limit Theorem and its
consequences and will be able to apply the theorem in practical situations. (11.2)
(14) Students will be able to explain the conceptual context, to calculate, and to interpret
methods of inferential statistics. (11.2)
150
Sample Exam Questions:
Selected final exam questions:
Let $A_1$ and $A_2$ be the events that a person is left-eye dominant or right-eye
dominant, respectively. When a person folds their hands, let $B_1$ and $B_2$ be the
events that their left thumb and right thumb, respectively, are on top. A survey in one
statistics class yielded the following table.
A1
A2
Totals
B1
5
14
19
B2
7
9
16
Totals
12
23
35
If a student is selected randomly, find
a)
$P(A_1|B_1)$
b)
$P(A_2|B_2)$
c)
If the students had their hands folded and you hoped to select a right-eye
dominant student, would you select a "right thumb on top" or a "left thumb on top"
student? Why?
A manufacturer of automobile batteries claims that the distribution of the lengths of life
of its best battery has a mean of 54 months, and a standard deviation of 6 months.
Suppose a consumer group decided to check the claim by purchasing a sample of 50 of
these batteries and subjecting them to a test that determines their lives. Assuming the
manufacturer's claim is true, what is the probability the consumer group's sample has a
mean life of 52 or fewer months? How believable is the manufacturer's claim? Explain.
A random sample of 50 cups from a certain coffee dispensing machine yields a mean of
6.9 ounces per cup and a sample standard deviation of .12 ounces. Test at the .05 level of
significance the null hypothesis that, on the average, the machine dispenses $\mu = 7.0$
ounces against the alternative hypothesis that, on the average, the machine dispenses
$\mu< 7.0$ ounces. Explain what your result means.
In a survey of 1,000 American households, 75% claimed to have made a financial
contribution to charity in the past year.
1. Assuming that these households were a simple random sample from the population
of all American households, calculate a 95% confidence interval for the proportion
of the population who made a financial contribution to charity in the past year.
2. Would the interval have been wider, narrower, or the same width, if 520
households had been sampled and 75% claimed to have made a financial
contribution to charity in the past year? Explain.
A screening test for a certain disease has been found to detect the presence of the disease
98% of the time when administered to an afflicted person. In 6% of the cases, a well
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person will be incorrectly diagnosed as having the disease by this test. Studies have
shown that 4% of the population have the disease. What is the probability that a patient
actually has the disease, given that they tested positive?
A new type of band has been developed by a dental laboratory for children who have to
wear braces. The new bands are designed to be more comfortable, and hopefully provide
more rapid progress in realigning teeth. An experiment was conducted to compare the
mean wearing time necessary to correct a specific type of misalignment between the old
braces and the new bands. One hundred children were randomly assigned, fifty to each
group. The children wearing the old type of braces had a mean wearing time needed of
410 days and a standard deviation of 45 days. The children wearing the new type of bands
had a mean wearing time needed of 380 days and a standard deviation of 60 days.
1. Find a 95% confidence interval for the difference in mean wearing time for the two
types of braces.
2. Is there sufficient evidence to conclude that the new bands do not have to be worn
as long as the old braces? Explain.
152
MATH 490 History of Mathematics (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: An Introduction to the History of Mathematics, (6th edition) by Howard Eves.
Supplementary readings and/or assignments address both European and non
European roots of mathematics as well as contributions by women.
Credit Hours: 3 hours
Catalog Description: The development of mathematics from historical and cultural viewpoints
including both European and non-European roots of mathematics as well as contributions by
women.
Prerequisites: Prerequisite MATH 248 Introduction to Methods of Proof
Topics:
Ancient Mathematics: Mathematics from the early ages. We will discuss the fertile
crescent, Chinese, Indian, Egyptian, Greek, Mayan, Roman and Arabian mathematics.
(SMR 6.1ab)
Renaissance Mathematics: The rebirth of mathematics in Europe: Cardano, Bombelli,
Pascal, Descartes, Newton, Leibniz, et al. (Roughly 1400-1700 CE). (SMR 6.1ab)
The Golden Age of Mathematics: Mathematics in the seventeen and eighteen hundreds:
Euler, Galois, Abel, Gauss, Lie, Kovalevsky, et al. (SMR 6.1ab)
Modern Mathematics: Mathematics in the 20th century: Hilbert, Ramanujan, Noether, et
al. (SMR 6.1ab)
Five important strands are addressed: (1) the development and properties of systems
of numeration; (2) the historical development of algebraic notation from rhetorical, to
syncopated, to symbolic; (3) the development of the calculus beginning with the ideas of
Eudoxus and Archimedes, moving to the work of Cavelieri, and then Newton and Leibniz,
and culminating in the rigorous foundation provided by Cauchy; (4) the role of axiomatic
systems and proof in the development of the discipline of mathematics; (5) the
contributions of diverse cultures, ethnicities and genders to mathematics. (2.3, 15.6, SMR
6.1ab)
Instruction and Technology: A variety of instructional formats will be used. These will be
chosen from direct instruction (lecture), collaborative groups, individual exploration, peer
instruction and whole class discussion/presentation led by students, as deemed appropriate by the
instructor (2.4, 2.5, 5.1, 5.2, 5.4, 16.1, 16.2, 16.3, 16.4, 16.5, 16.6). Students should have access
to and be able to use a graphing calculator, Geometers Sketchpad, Mathematica and the Internet.
Students should also have access to NCTM journals such as The Mathematics Teacher and
Teaching Mathematics in Middle School.(3.1, 3.2, 3.3)
Assessment: Grades will be determined by a combination of homework, exams, class
participation. Assignments include at least one oral presentation and one written paper. (4.1, 4.2,
4.3, 7.1, 7.2)
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Objectives:
(1) Students will be able to communicate mathematical ideas in writing and orally using
mathematical language including symbols and using everyday language. (13.1, 14.1, 14.2,
SMR Part II)
(2) Students will understand the historical use of geometric concepts and relationships to model
mathematical ideas and real-world constructs. (15.1, 15.6, SMR 6.1ab)
(3) Students will understand the history of axiomatic systems, including the systems for
Euclidean and non-Euclidean geometry. (SMR 6.1ab)
(4) Students will understand the historical development of the concepts of number and
numeration systems. (SMR 6.1ab)
(5) Students will understand the historical development of algebraic notation. (SMR 6.1ab)
(6) Students will understand the evolution of the ideas of calculus. (SMR 6.1ab)
(7) Students will understand the role that "proof" has played in the history of mathematics.
(SMR 6.1ab)
(8) Students will have a knowledge of the historical development of mathematics, including the
contributions of underrepresented groups and diverse cultures. (2.3, SMR 6.1ab)
(9) Students will examine mathematical concepts and explanations suitable to a variety of grade
levels. (13.2, 14.1)
(10) Students will solve problems using historical methods/reasoning and contrast these
approaches with modern methods/reasoning. (12.1, 12.2, 12.3, 12.4, 13.4, 14.1, 14.2)
Days and Topics List:
Class Chapter in Eves
Hours
2
1
2
2
2
3
3
4
2
2
5
6
3
7
2
8
3
3
9
10
6
11,12
Content
Numerical Systems. Primitive Counting. Number Bases. Written Numbers.
Grouping Systems. Numerical Systems. Hindu-Arabic Numerical Systems.
Arbitrary Bases
Babylonian and Egyptian Mathematics. Commercial Mathematics.
Geometry. Algebra. Rhind Papyrus.
Pythagorean Mathematics. Axiomatic Systems. Pythagorean Theorem and
Pythagorean Triples. Irrational Numbers. Geometric Forms of Algebraic
Identities. Geometric Solution of Quadratic Equations.
Duplication, Trisection and Quadrature. Euclidean Tools. The Three Famous
Problems. π.
Euclid and His Elements.
Greek Mathematics After Euclid. Archimedes. Diophontus. Eratothenes.
Hypatia. Greek Trigonometry. Greek Algebra.
Chinese, Hindu, and Arabian Mathematics. Arithmetic. Algebra. Geometry.
Trigonometry.
European Mathematics, 500 to 1600. The Dark Ages. Fibonacci. Beginnings
of Algebraic Symbolism. Cubic and Quartic Equations.
The Dawn of Modern Mathematics. Logarithms. Galileo. Kepler. Pascal.
Analytic Geometry and Other Precalculus Developments. Descartes. Fermat.
Agnesi..
The Calculus and Related Concepts. Zeno's Paradoxes. Eudoxus' Method of
Exhaustion. Archimedes' Method of Equilibrium. The Beginnings of
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4
13-15
5
--------39
Integration. Cavalieri's Method of Indivisibles. The Beginnings of
Differentiation. Newton. Leibniz, Du Chatelet.
Selected topics from 18th, 19th and 20th Century Mathematics: Liberation
of Geometry and Algebra, Arithmetization of Analysis.
Testing and Student Presentations
Each student is required to make an oral presentation and write a paper.
Sample Exam Questions:
Part I. Historical/Cultural Development Questions:
1. (a) Give the prevailing view of what prompted the initial and then later the further development of
mathematics. (b) What are two alternative theses for the development of mathematics? (c) In what
way did the availability of writing materials influence what we do and don’t know about the history of
mathematics (cite examples in your response to this part)?
2. Define irrational number. How would the Pythagoreans have likely encountered the irrational
number 2 geometrically ? Why was the discovery of the existence of irrational numbers such a
crisis for the Pythagorean philosophy? What effect did this discovery have on the development of
Greek mathematics?
3. List the 3 major topics treated in Euclid's Elements. What is the significance of the title of the
book? Cite (at least) two indicators that this book has been extremely influential in mathematics. Why
was it such an influential book?
4. Describe the three impossible Euclidean constructions in words and in pictures. Two of them can
be shown to be equivalent to constructing particular numbers. Explain these numerical equivalences.
What influence did these construction problems have on the development of Greek Mathematics?
5. Many calculus textbooks leave the impression that calculus was invented by Newton in
England and Leibniz on the continent (independently) all on their own. Give arguments for and
against the view that they each were solely responsible for the development of the calculus.
6. Give arguments for and against the Eurocentric view of the history of the development of
mathematics.
7. Answer the “who, how and why” questions about calculus being placed on a firm foundation.
8. Describe Hilbert's dream of proving the consistency and completeness of mathematics and
Godel's destruction of that dream. (Be sure to define the terms consistent and complete as they
apply to mathematical systems in your answer.)
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9. List major contributions to the development of mathematics by Greek, Hindu, Chinese and
Arabian cultures. Assess the relative merits of the mathematics done by these four cultures.
Part II.
Mathematical Content Questions

In a positional numeration system to base 4, let 0, 1, 2, 3 be represented by A, B, C, D. Express
the numbers 50, 501, 78, and 625 in this system.

Let x be a perfect number. Prove that any multiple of x is abundant.

Prove that a real number x is rational if and only if x and 1 are commensurable.

Draw and label a figure showing Euclid's method of solving x2 + 7x = 44. Indicate where in this
construction process it would be necessary to (a) square a length and (b) take the square root of a
length. Show both roots in the figure (or draw a second figure for the second root).

Give a geometric (not an algebraic) explanation of Bhaskara's "Behold" picture proof of the
Pythagorean Theorem. (See Fig 60 from Eves 6th ed.)
x2 y2
Find the slope of the tangent line to the ellipse 9 + 16 = 1 at (x0, y0) (a) by Newton's
method, (b) by Fermat’s method.
Find the division of the stakes in a game of chance between two equally skilled players A and
B where A needs one more point to win and B needs three more points to win (a) using
Fermat's method; (b) using Pascal's method.
Two hypothetical planets are moving about the sun in elliptical orbits having equal semimajor axes. The semiminor axis of one is half that of the other. How do the periods of the
planets compare? Justify your answer.



156
MATH 493 Senior Seminar for Future Mathematics Educators (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Advanced Mathematics for High School Teachers, Curtis D. Bennett and David Meel,
2001. http://myweb.lmu.edu/cbennett/Portfolio/Assignments/BKPD.pdf.
Credit Hours: 3 hours
Course Description: This is a capstone course in mathematics. It is specifically designed for students
with an interest in education. As a capstone course, it will integrate the topics of many undergraduate
mathematics courses (including some you probably haven’t taken ). In this course, we will discuss
various understandings of numbers and their strengths and weaknesses. We shall work towards a firm
understanding of the intermediate value theorem and the definition of the real numbers. The topics
discussed are first touched upon in the high school curriculum, but rarely discussed in any detail there. In
this course, we will build up from our previous knowledge and the high school curriculum to more
advanced topics. The main idea of the course is to relate advanced mathematics to the high school
curriculum. One other important idea in this class is the idea of using a variety of representations of
“numbers” (including matrices as a generalization of number) and what each representation helps you best
understand. In each topic we will also often reflect on how students at all levels understand and learn
material, and how this relates to both the mathematics and its teaching. The course project will also
provide significant evidence for a final summative assessment of your work in the Bachelor of Arts in
Mathematics program. (SMR part II, 5.4, 6.2, 6.4, 7.3, 11.2, 13.2, 15.2, 15.4, 15.5, 15.6)
Prerequisites: MATH 248 Methods of Proof, MATH 293 Field Experience (Corequisite is
acceptable)
Topics:
Rational Numbers and Irrationality Proofs:
 Division Algorithm and Euclidean Algorithm. (SMR 3.1a,c)
 Foundations of rational number system – basic properties of integers, why addition
and multiplication are defined the way they are. (SMR 1.1ac, 3.1ad)
 Irrationality proofs for square roots of non-square integers – least terms, minimal
denominator, limit proof, rational root theorem. (SMR 1.2b, 3.1d)
 Irrationality of e and π - exponential functions, infinite series for e, arithmetic and
geometric series, integration by parts, trigonometric functions. (SMR 1.3c, 3.1d,
5.4c)
Number as Length – constructible numbers:
 Euclidean algorithm on the real line. (SMR 3.1 c)
 Construction of rational lengths and lengths of square roots – basic constructions,
AAA similarity, similar triangles. (SMR 2.2 a, b,d)
 Constructible numbers form a field. (SMR 1.1a)
 Construction of a regular pentagon. (SMR 2.2d)
 Polynomial rings, extension/quotient fields – polynomial rings are not fields, factor
theorem. (SMR 1.1a, 1.2b)
 Categorization of constructible numbers.
Number as Root of a Polynomial (Algebraic Numbers):
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
Using matrices to understand irrational numbers (complex numbers as matrices,
extension fields, inverting complicated numbers) (SMR 1.4c)
 Solving quadratic equations (over real and complex numbers) – geometric,
algebraic, and function/graph proofs (SMR 1.2b, 1.3b)
 Solving a depressed cubic equation.
o Using geometric deconstruction of a cube (Cardano’s technique). (SMR
1.3b)
o Roots of complex numbers, quadratic equation for complex numbers, polar
representation of complex numbers. (SMR 1.1a)
o History of complex numbers. (SMR 6.1b)
 Fundamental Theorem of Algebra. (SMR 1.2c)
 Transcendental Numbers – Liouville’s proof
o Mean Value Theorem. (SMR 5.3c)
o A use of the derivative of a polynomial. (SMR 5.3a)
o The Liouville numbers.
Real numbers and Dedekind cuts:
 Axiom system for the real numbers (using their basic properties in constructing
mathematical arguments, real numbers can be orders, but complex numbers can’t).
(SMR 1.1bc)
 Intermediate Value Theorem – history and importance. (SMR 5.3c, 6.1a)
 Formal definition of the real numbers. (SMR 1.1b)
The numbers π and e:
 Area of a circle – Archimedes’ way. (SMR 2.2 c)
 Exponential functions and the logarithm – three definitions of e and their
equivalence, analyzing properties in a variety of ways. (SMR 1.3 c)
Instruction and Technology: This class will use a wide variety of instructional methods.
Roughly one-fifth of the days will be spent doing investigative work individually and/or in teams
of students. Moreover, we will use a variety of learning modalities including visual, auditory, and
kinesthetic. Students will be expected to use Excel, programmable graphing calculators, and
other appropriate mathematical technology as assignments dictate (this will be particularly true in
the research projects – see below). (2.4, 2.5, 3.1, 3.2, 3.3, 5.1, 5.2, 5.4, 5.5, 12.4, 16.1, 16.2, 16.3,
16.4, 16.5, 16.6)
Grading:
Tests: Two tests, (midterm and a final) will be given during the semester. The final will be a
take-home exam. It is likely the final exam will include a student portfolio for the course. (7.1)
Homework: For each section covered in lecture there will be a homework assignment. The
homework problems are designed to promote deeper understandings of important mathematical
concepts (that you should learn). As a result, conferencing with other students in the class is
permissible; however, copying of work from any source is not. Individuals may work as a group,
but the responses to homework problems should be written up individually. (11.2, 4.3)
Project: A project accounting for 30% of your final grade will be assigned. The paper describing
your project will be due on Monday, Dec. 4 and an oral presentation the following week. Each
week of the term, however, I will expect a progress report in writing, or via email (see attached
list of questions to be answered in each progress report). While it is acceptable (although not
158
recommended) if once in a while you report having done little work on the project, you should be
working on the project every week. It is expected (and highly recommended) that you will meet
with me occasionally during the term to discuss the project.
The problem you choose is a starting point. As is the case in mathematics research, there is no
set stopping point for your project, although there are minimum expectations. For each problem,
I have a vision of what you will accomplish during the term, but this is only a vague outline. I
fully expect the best projects to stray (sometimes wildly) from this outline, and I will work very
hard to help you decide where you should proceed after completing the first part of the project.
There is no set guideline for the number of pages for your project. In mathematics, some of the
best papers have been fewer than 10 pages long. Of course, one of the most famous papers in
algebra took up an entire journal. In your paper, you should discuss some of the background of
the project and the mathematical information in the project in addition to writing up complete
proofs of the theorems with which you end up. At some stage, I may give you more background
information on your project, although this will only happen after you have completed the first
portion of it. You should also either think about how the project could be used in a classroom or
about the connections of your problem with high school mathematics.
Projects will be completed either individually (by graduate students) or in teams of two to three
students. In the case of a group, a single write-up is necessary addressing the indicated
question(s) asked and any additional questions that you raise, or that are raised by the professor.
Projects will be graded based upon (a) communication, (b) visual representation, (c)
computations, (d) proofs, (e) decision making, (f) interpretation of results, (g) conclusions drawn
and supported (including relationship to the high school curriculum), (h) overall presentation
(including oral presentation), and (i) overall impression. (4.1, 4.2, 4.3, 7.1, 12.1, 12.2, 12.3, 12.4,
13.1, 13.2, 13.3, 13.5, 13.6, 14.1, 14.2, 15.4)
Writing Assignments: From time to time in the semester, writing assignments will be given
requiring reflection upon material presented in class or assigned for reading and its relation to the
high school curriculum and your field experiences. These will be graded and scored in with the
homework. The point values of individual writing assignments will vary depending on the
amount of work required. (7.1)
Class Participation and Discussion: Occasionally, the class will work on problems in groups, and
frequently there will be in class discussions. A participation and discussion grade will be
assigned based on students contributing in a meaningful way to group work and discussions.
(Such contribution need not be confined to time during class. Comments delivered to the
instructor outside of class often contribute valuably to what happens in class.)
Summative Assessment: Included in this class will be a summative assessment of students’
subject-matter competence. Such assessment will be carried out using tools like content
knowledge surveys, analysis of the oral and written project presentations, and a portfolio
assessment of the students work. (7.3)
Objectives:
(1) Students will learn about the mathematical development of number systems including the
contributions of a diverse variety of cultures. (2.3)
(2) Students will learn to use appropriate technology for solving routine and complicated
mathematics problems, as a research tool in mathematics, and for presentation and
communication of mathematics. (3.1, 3.2)
(3) Students will learn to use appropriate technology to enhance their mathematics subject matter
knowledge. Moreover, they will learn to use technologies available in the K-12 school
159
systems as tools for investigation for problems found in those grade levels, and discuss issues
related to using these technologies. (3.3, 5.5, 13.2, 13.6)
(4) Students will improve their use of appropriate academic language, content, and disciplinary
thinking in purposeful ways to analyze, synthesize and evaluate experiences, and enhance
understanding in mathematics. They will further improve their understanding and use of
mathematics terminologies and research conventions. In particular, they will investigate the
role of definition, proof, and theorem in leading to deeper understandings of number systems.
(4.1, 4.2, 4.3, 13.1)
(5) Students will reflect on field experiences and other K-12 material in reference to how it
relates to the development of number system in mathematics and its history as well as in
higher level mathematics classes. (6.4, 13.2)
(6) Students will investigate underlying mathematical reasoning for each of the topics in this
class, and they will explore the connections between the branches of mathematics and how
they relate to the K-12 curriculum. (11.2)
(7) Students will explore mathematical problems in context and explore their relationship with
other mathematical problems. (12.1)
(8) Students will explore class problems and project problems in multiple ways, including
solving them in multiple ways, generalizing them in multiple ways, and using varied and
appropriate technology to investigate them. (12.2, 12.3, 12.4)
(9) Students will investigate how to present mathematical explanations appropriate to a wide
variety of audiences (including a variety of grade levels). Such explanations will incorporate
various forms, such as the use of graphs, charts tables, etc. (in project presentations), using
clarifying questions to learn, communicate, and extend mathematical ideas, and using
appropriate technologies to present mathematical ideas and concepts. (13.2, 13.3, 13.5, 13.6)
(10) Students will formulate and test conjectures using inductive reasoning, construct counterexamples, make valid deductive arguments, and judge the validity of mathematical arguments
in relationship to their project and course material. In addition they will present formal and
informal proofs throughout. (14.1,14.2)
(11) Students will gain a deep appreciation for how mathematics topics from the major are
inter-related, from both a modern and historical perspective. (15.2, 15.6)
(12) Students will learn to create a wide variety of models to represent a given number system,
and recognize how different models can represent a variety of situations. (15.4, 15.5)
(13) Students will gain experience in both written and oral presentation of mathematics. (9.4)
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Sample Homework Questions:
1. You will need to use a programmable calculator or Excel for this assignment
1
a. Calculate the period of the decimal representations of
for each integer n less
n
than 60. (You should use Excel or a calculator program to do this in general.)
b. If n is prime, what can you say about the period of the decimal representation of
1
. Be as specific as possible. Why might this be so?
n
a
c. Prove that if a and b are positive integers, then the decimal representation of
b
has period strictly less than b . In doing this, please refer to the description of
division given in class discussion on Wednesday. That is, note that in class
discussion, we pointed out that long division is an iterative process (you do the
same thing over and over again). What is it about that process that forces the
fraction to repeat?
d. Translate this proof into an explanation appropriate to a skeptical high school
student. I.e., think about how you can answer the question: How do you know
1
repeats?
97
a
e. If a and b are positive integers, find a condition that forces
to have a
b
terminating decimal representation. Prove your answer.
f. Look at your expression of difficulties in lieu of the discussion from Wednesday.
What is the relationship of difficulties and the eventual algorithm? What changes
1
in the algorithm if you work in another base? Express as a decimal in base 3.
5
g. Solve the following: In a certain town, two-thirds of the men are married and
two-fifths of the women are married. Assuming all marriages are between one
man and one woman (and everyone is monogamous), what fraction of the people
in the town are married? What are the key ideas in the solution of this problem?
Can you make your ``proof'' simple?
h. Now, find a student that is not in the sciences (ideally someone who has not taken
calculus), and try and help them solve the problem. Reflect upon their difficulties
(if any) in solving the problem. Also explain how you helped them work through
the problem. (This portion of the response should be well-written, spell-checked,
and make sense.)
2. Real number homework: So far we have talked about rational numbers (and their
decimal equivalents), constructible numbers, algebraic numbers, and transcendental
numbers. We have now completed the set of real numbers
i. Sketch a diagram relating the natural numbers, the integers, the rational numbers,
the constructible numbers, the algebraic numbers, and the transcendental numbers.
Explain the linkages and include, if necessary, other sets of numbers needed to fill
out the picture.
j. The guiding principle in defining the real numbers is the number line. What are
all the properties that the set of real numbers should have? You should come up
161
with at least twelve properties that are expected of the set (Hint: Think about the
entire field).
k. After trying to come up with answers to this question, pick up the pink sheet and
see how many of the needed properties you have identified. On that pink sheet,
identify (in words) the ramification or meaning of each axiom (think about how
you would identify these meanings to high school students).
l. Prove
i. If x<y, then -y<-x. Hint: Assume x<y and add the same thing to both sides
of the inequality in order to establish that -y<-x.
ii. 0<1. Although this statement seems obvious, you need to establish this
result using the pink sheets axioms. What are the three possible ways of
relating 0 and 1? Use the axioms to establish that what you know is
impossible is truly impossible.
1 1
m. Prove: If 0<x<y, then 0   .
y x
i. Assume 0<x<y. What are the possibilities for the relationship between 0
1
and ?
x
ii. Argue why, from the set of axioms, two of these possibilities cannot be
true.
1
iii. What does this mean about ?
y
iv. Using 0<x<y, multiply each element of the compound inequality by the
1 1
same factor in order to establish 0   . Explain why this is legal.
y x
v. If x<y and z<0, then yz<xz. Hint: Assume x<y and z<0. Use axiom 11
and a previous result to establish yz<xz.
3. Left to their own devices, some children will develop an addition algorithm that starts
by adding the left-most term first and then moving successively to the right. Discuss
the relative merits of this algorithm as compared to the standard algorithm for
terminating decimals. What about for non-terminating decimals? How do the above
problems relate to this question?
162
MATH 550 Geometry (template syllabus)
Instructor:
Office Hours:
email:
WebPage:
Text: Euclidean and Non-Euclidean Geometries, M. Helena Noronha, Prentice-Hall, 2002.
Credit Hours: 3 hours
Catalog Description:
Prerequisites: MATH 248 Methods of Proof
Topics:
Neutral Geometry:
 Basic axioms of neutral geometry – Euclid’s first four axioms, axioms of incidence
and betweenness and the continuity principles. (2.1a)
 Neutral triangles – SSS congruence, alternating interior angle theorem, and the
triangle inequality. (2.2 a,b)
 Basic geometric constructions – angle bisector, perpendicular bisector, replication
of shapes. (2.2 d)
Euclidean Plane Geometry:
 The parallel postulate – Equivalents to the parallel postulate including the alternate
interior angle theorem, the angle sum theorem, the mutual perpendicular line
theorem. (2.1 a)
 Euclidean triangles and circles – concurrence theorems, exterior angle theorem,
AAA similarity theorem, Pythagorean theorem (and its converse), geometric mean
theorem. (2.2 a,b, 5.1)
 Trigonometric functions – sine, cosine, law of sines, and law of cosines. (2.2 b,
5.2b)
 Euclidean geometric constructions – rational number construction, basic geometric
shapes. (2.2d)
Geometric Transformations:
 Rigid motions – isometries in 2-dimensional space (rotations, reflections,
translations). (2.4 a)
 Similarities and inversions – dilations, similarity transformations. (2.4 b)
 Coordinate systems – definitions, proving theorems using coordinate systems. (2.2
e)
Euclidean 3-space:
 Axiom system for 3-dimensional geometry.
 Perpendicular and parallel lines and planes. (2.3 a)
 Rigid motions in 3-space – isometries in 3-dimensional space. (2.4a)
Perimeter, Area, and Volume:
 Perimeter and circumference – derive perimeter formulas for circles from polygons
compare to the method using integration. (2.2 c, 5.4d)
 Area – compute areas of polygonal regions (signed and unsigned) and use to find
area of circles. (2,2 c)
 Volumes and surface areas – find volumes of three-dimensional objects,
Cavalieri’s principle. (2.3 b, compare to using calculus)
163
Non-Euclidean Geometries
 Spherical geometry – change of axioms, area of triangles, AAA congruence
theorem. (2.1 b)
 Hyperbolic geometry – change of axioms, area of triangles, meaning of parallel.
(2.1 b)
Instruction and Technology: In this course, we will be using classroom Socratic lectures
roughly 2 days a week and in-class activity/group work 1day/week. We will use the Lénárt
sphere for investigations in spherical geometry and students will construct a model of hyperbolic
space for investigations into hyperbolic geometry. Students will also need to familiarize
themselves with Geometer’s Sketchpad, which we will use for some activities. (5.1, 5.2, 5.4, 5.5)
Grading: Grades will be determined by a combination of homework, exams, out of class
projects, and class participation. In addition, students in this class will write a short paper and
give an oral presentation expanding on a topic that may be taught in a high school geometry
class. Give careful attention to mathematical reasoning in this paper and presentation. Topics for
this paper may come from student field work, issues of Mathematics Teacher, the on-line NCTM
standards, or other reasonable sources. (6.2, 6.4, 7.1, 14.1, 14.2)
Objectives:
(1) Students will be able to solve routine and complex problems in geometry drawing form a
variety of strategies. (SMR part II)
(2) Students will be able to clearly communicate arguments in geometry in everyday and
mathematical languages both orally and in writing. In particular, they will be able to solve
these problems in context and explore their relation with other problems. (4.1, 4.2, 4.3, 12.1)
(3) Students will be able to use technologies (including Geometer’s Sketchpad and the Lénárt
Sphere) appropriately to investigate and solve problems involving Euclidean and nonEuclidean geometry and enhance their understanding of these geometries. (3.1, 3.2, 3.3, 5.5,
12.4)
(4) Students will be able to articulate mathematical ideas orally and in writing using appropriate
terminology and technologies. (13.1, 13.6)
(5) Students will show an understanding of the foundations and proofs of Euclidean geometry.
(11.2)
(6) Students will solve problems in Euclidean and non-Euclidean geometry and evaluate and
present formal and informal proofs of theorems in Euclidean and non-Euclidean geometry
both orally and in writing. (11.2, 14.1, 14.2, 15.3)
(7) Students will be able to discuss how geometry relates to other topics in mathematics like
linear algebra, complex analysis, and calculus. (15.2)
(8) Students will be able to discuss how the discovery of non-Euclidean geometries led to
foundational questions in other fields of mathematics. (15.6)
(9) Students will see varied teaching strategies in geometry and see how they help content be
conceived and organized for instruction, fostering conceptual understanding and procedural
knowledge. (5.2, 5.4)
(10) Mathematics education students will reflect on and analyze their early field experiences
related to geometry. (6.4)
(11) Students will see multiply ways to solve problems in Euclidean geometry, including
axiomatically and using coordinate systems. (12.2)
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(12) Students will be able to present mathematical explanations in geometry appropriate to a
variety of grade levels. (13.2)
(13) Students will formulate and test conjectures in Euclidean and non-Euclidean geometry and
judge the validity of mathematical arguments. (14.1)
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Sample Assignments:
1. Geometry Project: The Sine Law - In this project, our goal is to find a relationship of the
Law of Sines to another geometric invariant of a triangle. A second goal is to familiarize
you with Geometer's Sketchpad, a program that allows you to investigate geometrical
figures. For the first few steps, there are some instructions, but for later steps you will
need to use the help menu (contents and search might be a good thing to think about).
a. Start Geometer's Sketchpad on the computer. Create a triangle using sketchpad.
(To do this: Step~1: Plot three points with the mouse. Step~2: select all three
points at the same time (using the select (arrow) button, click on each point) while
holding down the shift key. Step~3: Use the construct menu to construct segment.
This will give you a triangle).
b. Use the label button (finger with letter) and label each of the points.
c. Measure each of the angles of the triangle (Hint: what will need to be selected so
that Sketchpad knows what angle you want to measure?)
d. Measure each segment.
e. Use Sketchpad to calculate the quotient of the segment with the sine of the angle
opposite.
f. Click on a single vertex of the triangle and drag it around. What happens? What
happens with the measurements?
g. Construct the circumscribed circle and the inscribed circle for the triangle. (Hint,
what bisectors might you need to construct to get the centers of these circles?)
h. Measure the radii of these two circles.
i. Copy down the measurements for 6 different triangles. What relationships (if any)
do you notice between any of your numbers? What might you want to calculate as
part of your sketch to double check this?
j. Prove your conjecture above. The picture below will help. You may use the
theorem that the measure of an inscribed angle is equal to half the measure of the
central angle marking off the same chord.
2. Geometry Project: Area of triangles on sphere - Remember that a straight line on a
sphere is a great circle (or a diameter). Thus one of the curiosities of elliptical (spherical)
geometry is that triangles have angle sums different from  radians (or 180 degrees). The
purpose of this project is to investigate the relationship between the angle sum and the
area of the triangle. For this purpose, all angles need to be measured in radians. We will
start with some basic angle sum formulas.
a. What is the surface area of a sphere of radius 1?
b. What is the angle sum of the triangle formed by the intersection of the northern
hemisphere, the Eastern Hemisphere, and the longitudinal line of 90 degrees east?
What is the area of that region?
c. Find several other triangles whose area you can calculate easily and make a chart
of their angle sums and areas.
d. What is a reasonable conjecture for the relationship between a spherical triangles
area and its angle sum?
e. Given that a lune on a sphere (the region bounded by two great circles) has front
and back area equal to 4 multiplied by the angle between the two circles, calculate
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the area of each of the three lunes formed when taking the great circles associated
to a triangle (based on angles in the triangle).
f. Use this to prove your conjecture.
3. Geometry Project: Triangle Angles Sum - Cut out four congruent triangles (try not to
make them special) and label the angles of each. Arrange them in such a way as to form
one big triangle. Draw the arrangement below.
a. Will the picture be essentially the same whatever triangle you cut out? Explain
b. Looking at this figure, state two theorems about triangles that this motivates.
c. One of the theorems that can be motivated from this picture is the theorem that the
sum of the angles of a Euclidean triangle is 180 degrees or pi radians, where we
are given that these are the measure of an angle which forms a straight line. Use
the picture to give a formal proof of this theorem. You may use the theorems
relating the angles cut by a transversal intersecting two parallel lines.
d. There are two common ways of addressing these issues in the high school
curriculum: the first is to suggest that students should tear off the angles of a
triangle and put them together to see a straight line, the second is to perform the
formal proof by drawing an auxiliary line at a vertex parallel to the side opposite
the vertex. The former can be problematic as it does not necessarily lead to a
proof, and the latter is unmotivated. Explain how you might use the puzzle to
bridge the gap of these two ideas, and how all three might be used in a school
curriculum.
e. Prove that for all triangles, if you bisect the three sides and connect the bisection
points then you will get 4 congruent triangles, each similar to the original triangle.
For this you may use that there exists a similar triangle of half the size of the given
triangle and the standard congruency theorems for triangles.
f. Using Geometer's Sketchpad, take any convex quadrilateral and bisect each of its
sides and connect them. What sort of figure do you get? (No proof required)
4. Geometry Project: Pythagorean Theorem to Angle Sum Formulas –
a. Cut out four congruent right triangles. Label the sides and arrange them so that
they form a large square with an inner square. What is the area of the big square?
What is the area of the triangles? Give two descriptions of the area of the inner
square. Explain how this proves the Pythagorean Theorem.
b. Now cut out two pairs of congruent right triangles, each pair having hypotenuse 1
unit long. Again, label one of the angles. Label the other angles and the sides of
the triangles in terms of this one angle. Arrange the triangles so they form a
rectangle. What is the shape of the interior angle now? Draw the picture below
and label all angles (including those on the interior region). What is the area of the
whole rectangle? What are the lengths of the sides in terms of the angles? Give
the area of the interior region in two different ways. Remember that
sin(  / 2   )  cos( ) and sin(    )  sin(  ) .
c. Write a formal proof of the Pythagorean Theorem using the first figure.
d. Write a formal proof for the angle sum formula for sine and/or cosine using the
figure.
e. Use this and the angle theorems for the value of sine and cosine for angles greater
than pi radians to give a general proof of the angle sum formulas for sine and
cosine using any angles  and  .
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5. Lists of basic geometry proofs from the text.
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MATH 550
x
x
MATH 493
x
MATH 490
MATH 357
x
x
x
MATH 360
MATH 331
MATH 293
MATH 282
MATH 250
MATH 248
MATH 245
MATH 234
MATH 321
structures
a
b
c
1.2 Polynomials
a
b
c
1.3 Functions
a
b
c
1.4 Linear Algebra
a
b
c
MATH 191
Course
Algebra
MATH 190
MATH 131
MATH 132
Appendix II:
Standards/Course grids:
1.1
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Geometry
2.1 Parallelism
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2.2 Euclidean
a
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2.3 3D geom
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2.4 Transform
a
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Number Thry
3.1 Naturals
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Calculus
5.1
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Trig
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Limits & Cont
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Derivatives
a
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f
Integrals
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Sequences
a
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MATH 490
MATH 360
MATH 357
MATH 331
MATH 321
MATH 293
MATH 282
MATH 250
MATH 248
MATH 245
MATH 191
MATH 234
MATH 190
MATH 550
4.2
Prob
a
b
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d
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Statistics
a
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d
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MATH 493
4.1
MATH 132
MATH 131
Course
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History
6.1 Chron & topic dev
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SMR II
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Sta
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MATH 131
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6 7
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MATH 132
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MATH 190
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MATH 191
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MATH 234
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MATH 245
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MATH 248
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MATH 250
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MATH 282
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MATH 293
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MATH 321
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MATH 331
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MATH 357
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MATH 360
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MATH 490
MATH 493
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MATH 550
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Appendix III
Appendix IV
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Appendix I
Appendix II
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Appemdix V
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Appendix VI
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Appendix VII
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Appendix VIII
Appendix IX
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Appendix X
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Appendix XII
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Appendix XIII
Appendix XIV
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Appendix III:
LMU Bachelor of Arts in Mathematics Single Subject Program
Requirements
These are the BAM degree requirements for the Mathematics major in the Single Subject
program.
This major is designed for students who are interested in pursuing a career in teaching
mathematics at the secondary (i.e., high school) level. With the help of her/his advisor
the student may design a schedule carefully so that s/he can complete the California
Preliminary Single Subject (Secondary) CLAD Teaching Credential during her/his four
years at LMU. Furthermore, a program may be designed that allows a student to
complete the mathematics degree, credential, and a Master of Arts in Teaching
Mathematics at LMU in five years including two summer sessions.
General Major Requirements
Students must complete the corresponding Bachelor of Arts University Core
requirements as defined by the College of Science and Engineering; students will
choose the proper sequence of University Core courses in consultation with their
advisor.
Mathematics majors and minors are not permitted to enroll in any mathematics course
without a minimum grade of C (2.0) in that course's prerequisite. A minimum grade of
C (2.0) is required in each course in the lower division major requirements. A minimum
cumulative grade point average of C (2.0) is required in the upper division major
requirements for graduation.
Lower Division Requirements:
MATH 131, 132, 190, 191, 234, 245, 248, 250, 282, 293; one science course chosen
from PHYS 101, 201 and CMSI 182, 185, 281. A second CMSI or PHYS course also
may be counted towards the Bachelor of Arts degree in Mathematics (please see upper
division requirements below).
Upper Division Requirements (11 courses):
MATH 321, 331, 357, 360, 490, 493, 550, one additional 3-unit upper division MATH
elective chosen in consultation with his/her advisor; EDUC 488; and two 3-unit EDUC
courses chosen from the list of requirements for the preliminary single subject
secondary credential or one such EDUC course and one course from PHYS 101,201 or
172
CMSI 182, 185, 281,282 or CHEM 110, 112, 220 or BIOL 101, 102, 201, 202.
Core Requirements
CATEGORY
UNITS
CHOOSE FROM:
American
Cultures
3 units Any course cross-listed as AMCS.
College Writing
3 units ENGL 110. A minimum grade of C is required.
Communication
or Critical
Thinking
3 units
Critical/Creative
Arts
Choose one from Critical Arts - ANIM 100, ARHS 200, 201, 202,321,
340, 345; DANC 281,381; FILM 210, 314, 315; INDA 100MUSC 102,
104, 303, 365; THEA 240, 245, 331, 336, 341, 346, 348, 349, or
6 units 430.
Choose one from Creative Arts - ANIM 12; ART 150, 151, 153, 278,
280, 350; DANC 163; ENGL 205, 311, 312; FILM 260; MUSC 105,
106; SCWR 220 or THEA 110
History
This requirement is satisfied by choosing one course from the
6 units Western Civilization sequence HIST 100 or 101, and one from
Contemporary Societies HIST 152, 162, 172, 182, or 192.
Literature
The prerequisite for these courses is the successful completion of
ENGL 110.
3 units
Choose from CLAS 200, 210, 220; ENGL 130, 140, 150; FNLT 180;
THEA 240, 245, 331, 336, 341, 346, 348, 430.
Philosophy
Lower division: PHIL 160. Those who transfer to LMU with a
6 units minimum of 60 units are exempt from taking PHIL 160.
Upper division: Choose either PHIL 320 or 330.
Choose from COMM 100,110,130,140
or COMM 206 or PHIL 220
Students must select form one of the following options:
1. Select two courses from ECON 100, 110, 120; GEOG 100; POLS
135, 155; PSYC 100; SOCL 100 or 105. Courses must be from
different departments.
Social Sciences
2. Select one course from AFAM 115; APAM 117; CHST 116 or
WNST 100, and one course from ECON 100, 110, 120; GEOG 100;
6 units
POLS 135, 155; PSYCH 100; SOCL 100 or 105.
3. Select two courses from the same department. The first is
selected from ECON 100, 110, 120; GEOG 100; POLS 135, 155;
PSYC 100; SOCL 100 or 105. The second course in the same
department is selected from upper division courses that the student
is qualified to take.
Theological
Studies
Lower-division: Choose the 100 level series of THST courses.
Students who transfer to LMU with a minimum of 60 units are
6 units exempted from the lower division requirement.
Typical Plan of Study
173
Below is a template of an 8-semester plan for a math major in the Bachelor of Arts Single Subject
Matter Program requiring a total of 124-126 semester units. It outlines the typical sequence of
courses, assuming a hypothetical student enters LMU as a first-year student and has no AP or
prior University credit for courses.
Freshman Year
Fall Semester
Spring Semester
MATH 131: Calculus I (4)
MATH 190: Workshop I (2)
ENGL 110: College Writing (3)
Core Requirement (3)
Core Requirement (3)
MATH 132: Calculus II (4)
MATH 191: Workshop II (2)
Science Requirement (3/4)
Core Requirement (3)
Core Requirement (3)
TOTAL:15/16 units
TOTAL: 15 units
Sophomore Year
Fall Semester
Spring Semester
MATH 234: Calculus III (4)
MATH 248: Intro to Methods of
Proof(3)
Core Requirement (3)
Core Requirement (3)
Core Requirement (3)
TOTAL: 16 units
MATH 245: Differential Equations (3)
MATH 250: Linear Algebra (3)
MATH 282: Elem. Numerical Methods (3)
MATH 293: Field Experience (0)
Core Requirement (3)
Core Requirement (3)
TOTAL:15 units
Junior Year
Fall Semester
Spring Semester
MATH 321: Real Variables I (3)
EDUC 4xx or second science (3/4)
MATH 357: Complex Variables (3)
Core Requirement (3)
Elective (3)
TOTAL: 15/16 units
MATH 490 History of Mathematics (3)
MATH 331: Group Theory (3)
Math Requirement - elective (3)
Core Requirement (3)
Elective (3)
TOTAL:15 units
Senior Year
Fall Semester
MATH 550 Geometry (3)
MATH 360 Probability and Statistics (3)
EDUC 488: Math Methods for Secondary
Teaching (3)
Core Requirement (3)
Elective (3)
Elective (3)
TOTAL: 18 units
Spring Semester
MATH 493 Senior Seminar for Future
Mathematics Educators (3)
Math Requirement - elective (3)
EDUC 4yy (3)
Core Requirement (3)
Elective (3)
TOTAL:15 units
Students are encouraged to consult frequently with their advisor about the sequence of classes.
174
Appendix IV:
Mathematics and Science Teacher Preparation Committee (MASTeP)
Special Committee on Mathematics and Science Teacher Preparation (MASTeP)
Objectives
 To maintain the improved educational programs, opportunities and support for pre-service K12 math and science teachers at LMU;
 To support and further enhance curricular and pedagogical improvements;
 To maintain collaborative connections and to continue on-going collaborative activities: on
campus, with other LACTE institutions (NSF-DUE 94-53608), and with Los Angeles
educational institutions and schools.
Responsibilities
 Coordinate internships
 Run the Innovations in Math/Science/Engineering luncheon seminar series
 Organize the annual Meet the Teachers Roundtable event and assist with the annual Future
Teachers Conference
 Moderate the Future Teachers Club (FTC)
 Serve as advisory boards for the Secondary Math and Science single subject programs
 Assist with coordination of the field experience component of the Secondary Math and
Science single subject programs
 Develop additional programs to enhance K-12 teacher preparation in Mathematics and
Science
Membership
Term of membership to the committee is three years with a rotating membership schedule.
Membership will consist of nine faculty with at least two members each from the areas of
Mathematics, Science, and Education.
175
Appendix V:
Los Angeles Collaborative for Teacher Excellence (LACTE)
Participating Institutional Partners List

California State University, Dominguez Hills and El Camino College

California State University, Los Angeles and East Los Angeles College

California State University, Fullerton and Fullerton College

Loyola Marymount University and Santa Monica College

Occidental College and Glendale Community College
The Los Angeles Collaborative for Teacher Excellence (LACTE) was a five-year, $5.5 million
project, funded by the National Science Foundation in 1995-2000, to enhance the science and
mathematics education of pre-service teachers. LACTE brought together ten Los Angeles area
colleges, universities, and community colleges to join the national effort in promoting excellence
and innovation in science and mathematics education. The project's goals were to improve K-12
science and mathematics teaching through better training of new teachers, and to increase the
number of students, especially minorities, who choose a career in teaching.
Internships
LMU initiated science and mathematics teaching internship opportunities at the California
Science Center, UCLA Ocean Discovery Center, and several Catholic and public schools. Fortyfive LMU students have benefited from these internship experiences and earned over $60,000.
The University has committed $12,500 annually to support internships post-LACTE.
LACTE students report that internships are the most influential factor in their ultimate decision to
teach and one of the most valuable aspects of the project for them.
Scholarships
Between 1996 and 2001, 42 LMU students received LACTE NSF-funded scholarships totaling
$89,500 to help them prepare for a career in K-12 math or science teaching.
LACTE Student Group Activities
LMU had one of the most active student populations in LACTE with over 100 LMU students
participating in a variety of activities such as, field trips, Internet workshops on Math/Science
Teaching, and regular student group meetings with guest speakers or videos. More than 20 LMU
students made presentations at conferences or led workshops at the annual Expanding Your
Horizons in Math and Science Career Day for girls in grades 6-10. In 1999-2000, the LACTE
students received official LMU club status as the MAST (Math and Science Teachers) Club. In
2002, the MAST Club reorganized as the Future Teachers Club to broaden its appeal. The
University has committed $500 annually to support the club.
176
Meet the Teachers Roundtable
This event, initiated at LMU, has connected nearly 100 students annually from college campuses
throughout Los Angeles with talented local schoolteachers who are good role models for the
teaching of mathematics and science. At the Roundtable future teachers experience hands-on
math and science lessons at a variety of grade levels and can make appointments to visit the
classrooms of the role model teachers. The University has committed $750 annually in support of
this event post-LACTE.
Impact on Students
Through its various activities and events, LACTE provides students the opportunity to connect
with current teachers and to learn from them. The internships give them essential early teaching
experience under the guidance of a talented mentor teacher. In addition, the experience of
planning or presenting at a conference provides the LACTE students with new skills, contacts and
confidence that will help them to become teacher-leaders in their own schools.
Faculty Development
With LACTE support, over 20 LMU math and science faculty have attended local, regional and
national conferences related to math and science teaching. Many of these faculty have
implemented new educational approaches and adopted a more student-centered teaching style.
LACTE has sponsored monthly luncheon discussions and presentations on campus focused on
educational issues in mathematics and science. In 1999 a more formal seminar series, titled
Innovations in Mathematics, Science and Engineering Education, was initiated. This luncheon
seminar meets 6 times per year in the new Center for Teaching Excellence and is open to the
entire university community. The University has committed $1200 annually to fund this seminar
series which is now being organized by the Math and Science Teacher Preparation Committee
(MASTeP).
Program and Curriculum Development
Through LACTE 14 LMU faculty developed or revised 10 math and science courses ranging
from precalculus to environmental science. Revisions in the requirements for Liberal Studies
majors to concentrate in math or science, fostered by the LACTE program, have resulted in a
significant increase in the number of liberal studies majors electing math or science as their area
of concentration. LACTE also provided impetus and initial funding to offer separate methods
courses to meet the disparate needs of preservice secondary math and secondary science teachers.
The University has committed to continuing these improved and expanded offerings.
177
Appendix VI:
LMU Center for Teaching Excellence program information
CTE EVENT SCHEDULE
CTE Schedule
Spring, 2003
January 21 (Tuesday) (convocation hour)
Co-sponsored with Mission and Identity Committee - Gathering of alumni of Western
Conversations (co-sponsored by Committee on Mission and Identity, chaired by Mike Horan,
Theology)
February 5 (Wednesday) (3 - 5 PM) Integrating Ethics into Curricula
How can instructors incorporate Ethics’ Education into a wide range of courses to train students in
a variety of professions, and to motivate students to become active citizens in their community?
Elaine E. Englehardt is Vice President for Scholarship and Outreach, and Professor of
Philosophy, at Utah Valley State College. She has recently completed several books in Ethics
with Harcourt College Publishers. For the past fifteen years, she has written and directed several
large grants in ethics education funded by FIPSE, and NEH, and is a professor for EDNET’s
television series “Ethics and Values.” Recently, she received the Governor's Award for
excellence. In 2001 she received the national Theodore M. Hesburgh Award. She has also
received the Distinguished Service Award by the Utah Academy for Sciences, Arts and Letters,
and has been selected as the prestigious Utah Professor of the Year
February 6 (Thursday) 3-5 PM Art and Scholarship of Teaching for Second Year Faculty
February 13 (Thursday) (convocation hour) Jane Crawford, Presentation of CTE Grant: “The One-Room
School House: Latinists and Historians”
Dr. Crawford will discuss the use of diverse class groups that help students with different learning
perspectives (Latin language study and History) integrate course material. Latin students read
sources in the original language while historians used translations. These divergent perspectives
led to broader learning experiences for all students in the course on the Julio-Claudians.
February 14 (Friday) Grant applications are due to CTE
February 18 (Tuesday) (convocation hour) Authentic Problems and Learning - Curt Bennett (Mathematics)
Curt Bennett, Ph.D., a 2000 Carnegie Fellow with the Carnegie Academy for the Scholarship of
Teaching and Learning, will report on how the use of authentic questions in a course can change
courses and deepen student understanding. Afterward, he will lead a discussion on: what are
authentic questions, what is the role of authentic questions in classes, and when is authentic too
authentic?
Co-sponsored by the Special Committee on Math-Science Teacher Preparation
4:40 - 6 PM Philosophy Graduate Students meet
February 19 (Wednesday) 3 - 5 PM “Collaborative Learning Strategies” by Barbara Burke, Cal Poly
Pomona, co-sponsored by the LRC (Greg Kozwolski contact person)
3 - 5 PM in the Executive Conference Center - Uhall 1857
Would you like to develop some strategies that would: engage students, increase student
participation, promote cooperation among students, and
enhance learning of the course material?
Barbara Burke is a nationally known expert on collaborative learning strategies and has
conducted workshops at many universities in our area. She will discuss several simple strategies
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that you could use to increase active learning in your class, to encourage students to learn
together, and to help one another in a noncompetitive environment. She will lead participants in
exercises designed to promote this style of learning. Most importantly, she will give tips to ensure
that student teams promote learning and mastery for all the students on a team.
February 20 (Thursday) (convocation hour) "Involving Students in Research - the Data are already there.
Part 1 - Hidden Treasures from the Center for the Study of Los Angeles”.
Presenters: Center for the Study of Los Angeles Director, Fernando Guerra, and Assistant Director
Mara Marks, Library Curator Clay Stalls, and Michael Engh, S.J., History department,
Did you know that your students can study the original documents of city
and state political leaders, community organizers, women reformers, L.A.'s
earliest families, RebuildLA, and educational reform efforts of LEARN ( Los Angeles Educational
Alliance for Restructuring Now) and LAAMP (Los Angeles Annenberg Metropolitan Project)?
Learn about these rich resources for your classes found in LMU's Center for the Study of Los
Angeles. Presenters will discuss teaching strategies using primary source materials.
February 25 (Tuesday) (convocation hour) "Involving students in research - the data are already there:
Part 2 - Using the Inter-university Consortium for Political and Social Science databases"
Presenters: Jim Faught, Sociology, Glen Johnson-Grau, Library, Matt Streb, Political Science
This database gives students and faculty access to an amazing variety of data that can be
analyzed by faculty and/or students - Census 2000, Election 2000, health and medical care,
criminal justice, aging, historical records, and a wide variety of educational data from the
International Archives of Education. There are even modules for using data sets to teach research
methods courses.
February 26 (Wednesday) (3 - 5 PM) Ron Barrett, Psychology, Shane Martin, Education, and Abbie
Robinson Armstrong, Intercultural Affairs: “Risks and tips for teaching diversity information - the
Discussion Continues.”
Come and join in a discussion about methods of incorporating diversity information into courses,
and suggestions for avoiding pitfalls in the process! Those who attended a similar discussion
section during the fall semester recommended that we continue the discussion on this very
important topic.
March 3 - 6 (Spring break)
March 11 (Tuesday) 4:40 - 6 PM Philosophy Graduate Students meet
March 13 (Thursday) (convo) Art and Scholarship in Teaching
Committee on Excellence in Teaching meets to review grant applications in Grant’s conference room.
March 18 (Tuesday) (convo) Marcia Albert (LRC) Classroom Assessment Workshop: How to Help
Students Succeed in the Classroom. Gain an understanding of the student-centered model and
how it is related to classroom assessment techniques.
March 20 (Thursday) Math-Science Teacher Preparation Committee Presentation - Caroline Viviano of
Natural Sciences. Teaching with your mouth shut!
March 25 (Tuesday) (convo) Orientation for president’s institute members
4:40 - 6 PM Philosophy
Graduate Students meet
March 27 (Thursday) 3 - 5 PM Art and Scholarship of Teaching for Second Year Faculty
April 7th, Monday, 12 – 1 PM “Human Being or Human Doing” by special guest speaker, Deborah
Grubbe, PE, Corporate Director of Safety and Health of the DuPont Corporation.
Ms. Grubbe, will share with us some details about how leading companies are working with
universities to promote intercultural sensitivity. In her remarks about diversity, she will address how
differences in culture actually hold some answers about how to think about our lives in a new way.
Please RSVP by April 3rd.
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April 8th, Tuesday, 12:15 – 1:30 PM “Improving Teaching and Learning Effectiveness with Body
Linguistics” by Dr. Whitey Brewer (Engineering and Production Management).
Body Linguistics provides an innovative new approach to clarify and enhance nonverbal
communication to maximize effectiveness in all teaching and learning endeavors. Effective team
building principles integrate the tutorial and experiential aspects of education. This humanistic and
systematic methodology transforms the art of team building into a science that can be used to
create synergy in the student/teacher relationship. This event is co-sponsored by the Special
Committee on Math and Science Teacher Preparation.
4:40 - 6 PM Philosophy Graduate Students meet
April 9th, Wednesday, 12 – 1 PM “Mapping the Road Ahead: Writing Learning Objectives that Work”
A faculty panel of Mel Mendelson (Chair, Mechanical Engineering), Judy Scalin (Co-Chair, Dance),
Kelly Wahl (Director of Assessment , LMU), and Trisha Walsh (Psychology), will discuss how they
have successfully used learning objectives to outline their goals and assess how they have met
them. Those who attend the workshop will learn how to write clear learning objectives that
facilitate documenting the success of a course or program.
April 10th, Thursday, 12:15 – 1:30 PM “Art and Scholarship of Teaching for Second Year Faculty”
April 16th, Wednesday 2 - 4 PM “Success Stories: Connecting Course Content and Service Activities”
This presentation includes faculty panelists KarenMary Davalos, (Chicana/o Studies), Vicki Graf
(Education), Michael Horan (Theological Studies), Yvette Lapayese (Education), and Pam Rector
(Center for Service and Action). Have you always wanted to incorporate community-based
learning into your course work? Do you need encouragement to take the first step? Find out more
about service learning from colleagues who have connected their course content to service.
Receive sample syllabi and reflection activities. Come and join the conversation!
April 22nd, Tuesday, 12:15 – 1:30 PM “How to incorporate the ‘Color of God’ into your classes”
Jennifer Abe-Kim (Psychology) and Douglas Burton-Christie (Theological Studies), Co-Chairs of
the Fall 2003, Bellarmine Forum, will discuss how the ‘Color of God’ topic for Fall, 2003, might be
integrated into the design of Fall classes. Faculty from all Colleges and Schools are welcome!
April 22nd, Tuesday, 2 - 5 PM “Faculty Showcase for Technology Integration: Part One.”
Faculty from around the university will demonstrate how they have integrated technology into their
classes.
Co-sponsored by ITS.
April 23rd, Wednesday, 2 - 5 PM “Faculty Showcase for Technology Integration: Part Two.”
Faculty from around the university will demonstrate how they have integrated technology into their
classes.
Co-sponsored by ITS.
May 19 (Monday) 9 AM - 5 PM President’s Institute
May 20 (Tuesday) 9 AM - 5 PM President’s Institute
May 21 (Wednesday) 9 AM - 5 PM President’s Institute
May 22 (Thursday) 9 AM - 5 PM President’s Institute
May 23 (Friday) 9 AM - 5 PM President’s Institute
CTE Grant Information
The LMU Center for Teaching Excellence announces this year's round of faculty development grants.
These grants focus entirely on the development of teaching. Faculty members can use these grants for
projects that design new strategies or techniques for their classes, new ways of integrating the curriculum
in their classes, or new ways of measuring student achievement in their classes. Because this year funds
are available from the Academic Computing Committee, projects involving web page design will not be a
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high priority for the committee this time. Examples of proposals the committee finds appealing are one that
designed a portfolio in support of Dance classes, a Woman's Study class that constructed assessment
tools to gauge student attitudes and how they change (or don't change) when the materials of a course
conflict with a student's own belief system, and a Chemistry class that developed strategies and
assessment tools for measuring how students retain information from a Math class and transfer that
information to the Chemistry class. Proposals that grow out of the Center's events are especially welcome.
These awards are limited to tenured and tenure-track faculty who are teaching full time during the grant
period. The grant sum is $3500, and the project may be completed during the summer or the following
academic year. These applications will be reviewed by the members of the Committee on Excellence in
Teaching. Proposals are due by February 14, 2003.
These awards are central to the mission of the Center and we hope that you will give them your serious
consideration because they encourage teaching development and innovation. And since you will be
expected to share the project's outcomes with the rest of the faculty, these awards contribute to the
dialogue on teaching.
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Appendix VII:
LMU Diversity requirement.
American Cultures Core Curriculum Requirement
One course (3 semester hours) is required of all majors in the College of Science and Engineering
Mission of American Cultures
(http://bellarmine.lmu.edu/Departments/AMCS/program.html)
Courses in American Cultures studies will enhance students' familiarity with and
appreciation of several of the diverse cultural groups that comprise the multi-ethnic society of the
United States. The in-depth, comparative and interdisciplinary study of the cultures, behaviors,
experiences and inter-group relations of the following groups-African American, Asian/Pacific
Islander American, Chicano/Latino American, European American and Native American- will
provide students with some of the strategies and help them gain competencies and sensibilities
that will enable them to contribute to and thrive in a culturally diverse world. (It is understood
that the above categories do not include the entirety of peoples comprising the United States of
America. Moreover, it is recognized that a rich variety of cultures is also represented within these
broad groups.)
Students will also strengthen their knowledge and awareness of their own ethnic or
cultural group. They will also develop their own creative and critical faculties, their own
analytical and affective responses to various forms of cultural expression. This approach would,
by definition, coax students to challenge the boundaries of ethnicity, culture, and academic
discipline. In so doing, students will not only improve their intergroup communication skills, they
will also become better able to see, appreciate, and respect the perspectives of others- factors that
are essential to the creation of a more understanding and just society.
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Appendix VIII:
Description of LMU College Bound program - Summer 2003
The College of Science and Engineering Hosts the Boeing Engineering Academy
For the second year in a row, LMU’s College of Science and Engineering has hosted the Boeing
Engineering Academy (BAE). The BAE is sponsored by College Bound and financially
supported by the Boeing Corporation. The participants are 15 selected, high-achieving AfricanAmerican high school juniors from Southern California.
College Bound is a non-profit organization of concerned parents working together to supplement
the day to day educational programs offered to their children in both public and private schools.
Offering a unique approach to college counseling, College Bound provides development of study
skills, parental participation and guidance in the admission and financial aid application process.
The purpose of the College Bound program is to provide college admission assistance for
students, with emphasis on the African-American family, through workshops, mentoring and
identification of applicable resources.
The BAE is a monthly study/mentoring program designed to develop study skills in the
participants and to introduce them to engineering as a college major. These goals are
accomplished through two components, mathematics and engineering. This year, the
mathematics component was taught by Ed Mosteig, Assistant Professor of Mathematics at LMU,
and the engineering component was taught by Joe Callinan, Professor Emeritus of Mechanical
Engineering. Two LMU undergraduate engineering students, Kadeen Vaughn and Basil Etefia,
served as teaching assistants and conducted PowerPoint and Excel workshops for the students.
The mathematics component of the Boeing Engineering Academy consisted of a series of lectures
interlaced with intensive group work. One of the main purposes of the math component was to
encourage the students to express their ideas to one another. To this end, they collaboratively
worked through many problems in groups. After one group solved a problem, they would
compare and explain their answers to another group. In addition, students individually presented
their solutions in front of the class.
Mathematically, the students were exposed to a variety of themes and concepts. Many of the
exercises were exploratory in nature, requiring the students to form conjectures about general
mathematical patterns. For example, they were shown the iterative process (originally used by
the ancient Egyptians) of replacing a number by the average of itself and double its reciprocal.
Their goal was to determine how this process converges, and then to explain why this behavior
occurs. Often times, their studies focused on concepts from calculus. Without the notion of a
derivative or integral, they were required to qualitatively and quantitatively describe instantaneous
velocity and compute areas under a graph. They applied these ideas to problems of motion in
space as well as fully investigating standard applications such as finding the minimal factory costs
of constructing soda cans.
Overall, exposure to a variety of mathematical ideas was to introduce them to various facets of
mathematics. Most importantly, they were put in the position of exploring unfamiliar
mathematics, thus allowing them to approach problems creatively without heavily relying upon a
standard set of problem-solving techniques. One of the most common complaints of college
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freshman is that homework no longer precisely mimics the examples in the book and those done
in class. By exposing the students to problems that require creative thinking at an early stage,
they will be better prepared to tackle such problems in college.
The goal of the engineering component was to expose the students to the practice of engineering
through hands-on activities. The topics covered included basic concepts (physical properties and
dimensional units), analysis (Newton’s second law, the physics of constant acceleration rectilinear
motion, graphical calculus and the use of Excel spreadsheets), engineering experimentation,
teamwork, design, and written & oral communications. “Flight” was selected as the theme of the
engineering component in recognition of the 100th anniversary of the Wright Brothers first
powered, controlled flight of a heavier than air vehicle. Since the class met only once a month,
the instructor and students communicated with one another via email and snail-mail.
Fundamental physical properties (e.g., velocity, acceleration, mass, force, work, etc.) and the
primary SI dimensional units (mass, length and time) were defined. The dimensional units of all
the properties were then derived using the primary units and the definition of these properties.
The students applied these concepts to the verification of the dimensional integrity of various
equations describing aircraft and rocket flight. Using the tools of analysis described above, the
students were asked to determine such engineering results as the height attained by a particular a
research rocket (analysis) and the runway length required for the takeoff of a Boeing 757
commercial airliner (design). In the laboratory the students conducted wind tunnel tests to
determine the lift and drag coefficients of an airfoil and they conducted propulsion tests on a
small turbojet engine to determine its performance. They were required to write engineering
reports discussing the results of their experiments. Also, each student was required to make a
PowerPoint presentation on the topic of “themselves.”
The engineering component concluded with a view of “flight” from the pilot’s perspective. Mr.
Oscar York, President of the Los Angeles Chapter of the Tuskegee Airmen and a mechanical
engineer, made an inspirational and cogent presentation on the Tuskegee Airmen, his personal
experiences and his advice to the young students of the Boeing Engineering Academy.
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Appendix IX:
Technology Glossary




Mathematica – A Computer Algebra System, used by many different universities and
corporations. Single Subject Program students in mathematics at LMU will see
Mathematica integrated into one or more of the courses they take.
Graphing Calculator – Graphing calculators are used at middle schools, high schools, and
beyond to help students learn to put graphical information together with algebraic
information. Students at LMU will see graphing calculators used as instructional tools in
the Calculus Sequence (MATH 131, 132, & 234) and in Senior Seminar for Future
Mathematics Educators (MATH 493) in addition to other classes.
Excel – A spreadsheet program that is available on nearly all computers and is used at
many levels of education. Students at LMU will see Excel used as an instructional tool in
Workshop in Mathematics I/II (MATH 190/191).
Geometer’s Sketchpad – A geometry program that can be used at the middle school, high
school, and collegiate level. Use of Sketchpad is endorsed in the NCTM principles and
standards. LMU students will have some instruction involving Geometer’s Sketchpad in
Geometry (MATH 550).
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Appendix X:
Mathematics Association of America Preparing Mathematicians for the
Education of Teachers (PMET)
A growing set of national reports calls for better preparation of the nation's
mathematics teachers by mathematics faculty. To help meet this need, the
Mathematical Association of America (MAA) has a multi-dimensional
program: Preparing Mathematicians to Educate Teachers (PMET).
The PMET program has four major components:
1.
Faculty Development - Workshops and minicourses will help mathematicians to be better
prepared to provide high-quality mathematical education to teachers.
2.
Information and Resources - PMET will provide the mathematics community with information
about the mathematical education of teachers by multiple means, including talks, articles, and
websites with course resources.
3.
Regional Networks - PMET will build an infrastructure of regional networks to help initiate,
support and coordiante efforts at individual institutions to improve the mathematical education of
teachers. Initially, PMET will concentrate activities in five states-- California, Nebraska, New
York, North Carolina, and Ohio --in order to build model networks.
4.
Mini-grants - PMET will support efforts by mathematicians at individual institutions to imporve
their teacher education programs and to develp new instructional materials.
Read about PMET in the March, 2003 issue of MAA's FOCUS magazine:
Preparing Mathematicians to Educate Teachers (PMET)
an article by Victor J. Katz and Alan Tucker
PMET is funded by a grant from the National Science Foundation (DUE-0230847).
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Appendix XI:
Mission and Goals of
the University and the Mathematics Department
Mathematics Department Program Goals and Objectives
In general terms, the mission of the Mathematics Department is to provide the students of Loyola
Marymount University the best mathematics education possible. More specifically, the mission of
the Mathematics department has two components:
 Prepare the mathematics majors at LMU for graduate study in mathematics, for teaching
mathematics, and/or for a career in a field which uses mathematics.
 Provide the students from other academic disciplines the mathematics education necessary for
success in their chosen discipline.
Common Goals and Objectives for the Majors in all Three Degree Programs Mathematics
Each goal is stated and followed by a list of objectives that will lead to the accomplishment of
that goal.
I. Develop the Content Knowledge of Each Student
Students should
 understand the fundamental concepts and applications of single variable, multivariable, and
vector calculus and differential equations
 understand elementary numerical methods and have an awareness of the mathematical uses of
a computer algebra system (CAS)
 know basic properties of logic and elementary methods of proof
 understand basic linear algebra and its applications
 understand the theory of single variable calculus
 understand basic probability and basic statistical methods
II. Develop the Problem Solving Skills of Each Student
Students should
 be able to understand and create rigorous mathematical arguments
 be able to solve mathematical problems using a variety of tools including the library, the
Internet, and appropriate technologies
III. Develop the Communication Skills of Each Student
Students should
 be able to read and communicate mathematics both in written form and orally
 be able to work both independently and as part of a team
IV. Develop Each Student as a Life-long Learner
Students should
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

possess a personal motivation and enthusiasm for further study
possess an awareness of the ethical issues in mathematics
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Appendix XII
Math and Science Teacher Preparation
Review Board Members 2003-2004
Dr. Curt Bennett
Associate Professor of Mathematics
Loyola Marymount University
Los Angeles, CA 90045
Mrs. Kathy Clemmer (LMU School of Ed grad)
Chair of Mathematics Department El Segundo High School
Adjunct Professor of Education
Loyola Marymount University
Los Angeles, CA 90045
Rosemary Connolly
Principal
St. Anastasia Elementary School
Los Angeles, CA 90045
Dr. Judy Kasabian (LMU math and School of Ed grad; former high school math teacher)
Professor of Mathematics
El Camino College
Torrance, CA
Dr. James Landry
Professor of Chemistry and Biochemistry
Chair of Natural Sciences
Loyola Marymount University
Los Angeles, CA 90045
Professor Fran Manion
Math Department Chair
Santa Monica College
Santa Monica, CA
Tammy Swanson (LMU grad)
Mathematics Department Chair
Venice High School (LAUSD)
Venice CA
A current student to be chosen during fall term 2003.
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Appendix XIII:
Catalog Description of Courses taken from LMU Departmental Websites
on August 19, 2003
MATH 120: PRECALCULUS MATHEMATICS (3 semester hours)
Functions; polynomial, rational, trigonometry, exponential, and logarithmic functions.
Prerequisite: MATH 101 or Mathematics Placement Examination
MATH 131: CALCULUS I (4 Semester Hours)
Limits, functions, continuity, derivatives of algebraic and transcendental functions, applications
of the derivative, antiderivatives, introduction to the definite integral, Fundamental Theorem of
Calculus.
Prerequisite: MATH 120 or Mathematics Placement Examination
MATH 132: CALCULUS II (4 Semester Hours)
Techniques of integration, numerical methods of integration with error analysis, applications of the
integral, improper integrals, infinite series, an introduction to parametric equations and polar
coordinates.
Prerequisite: MATH 131 or equivalent
MATH 190: WORKSHOP IN MATHEMATICS I (2 Semester Hours)
Study skills, analytical and problem solving skills, technical writing, recent fields of study and
advances in mathematics, mathematical career opportunities.
MATH 191: WORKSHOP IN MATHEMATICS II (2 Semester Hours)
A continuation of MATH 190.
MATH 198: SPECIAL STUDIES (0-4 Semester Hours)
Courses having a special syllabus and description not listed in the Bulletin.
MATH 199: INDEPENDENT STUDIES (1-4 Semester Hours)
Individualized study arranged by a student with a faculty member and approved by the Chairman
of the Department and the Dean.
MATH 234: CALCULUS III (4 Semester Hours)
Partial derivatives, multiple integrals, three-dimensional space, vectors in two- and threedimensional space, line integrals, Green's theorem.
Prerequisite: MATH 132 or equivalent
MATH 245: ORDINARY DIFFERENTIAL EQUATIONS (3 Semester Hours)
Differential equations as mathematical models, analytical, qualitative, and numerical approaches
to differential equations and systems of differential equations, and Laplace transform techniques.
190
Prerequisite: MATH 132 or equivalent
MATH 248: INTRODUCTION TO METHODS OF PROOF (3 Semester Hours)
Number theory, sets, functions, equivalence relations, cardinality, methods of proof, induction,
contradiction, contraposition.
Prerequisite: MATH 132
MATH 250: LINEAR ALGEBRA (3 Semester Hours)
Systems of linear equations, Gauss and Gauss-Jordan elimination, matrices and matrix algebra,
determinants. Linear transformations of Euclidean space. General vector spaces, linear
independence, inner product spaces, orthogonality. Eigenvalues and eigenvectors,
diagonalization. General linear transformations.
Prerequisite: MATH 248 or consent of instructor
MATH 282: ELEMENTARY NUMERICAL METHODS (3 Semester Hours)
Computer solutions of applied mathematical problems using FORTRAN and Mathematica. Nonlinear
equations, differentiation, integration.
Prerequisite: MATH 131
MATH 293 FIELD EXPERIENCE (0 Semester Hours)
Planned observation, instruction or tutoring experiences appropriate for future secondary
mathematics teachers; related professional reading and reflection.
NOTE: MATH 293 is a new course which will appear in the Undergraduate Bulletin in 20042005.
MATH 321: REAL VARIABLES I (3 Semester Hours)
The real number system, least upper bound, sequences, Cauchy sequences, functions, limits of
functions, continuity, derivatives, and Riemann integration.
Prerequisite: MATH 248
MATH 322: REAL VARIABLES II (3 Semester Hours)
Infinite series, uniform convergence, power series, and improper integrals.
Prerequisite: MATH 321
MATH 331: ELEMENTS OF GROUP THEORY (3 Semester Hours)
Group theory. Binary operations, subgroups, cyclic groups, factor groups, isomorphism,
homomorphism, and Cayley's theorem.
Prerequisite: MATH 248
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MATH 332: ELEMENTS OF THE THEORY OF RINGS AND FIELDS (3 Semester
Hours)
Rings, integral domains, fields, ideals, factor rings, polynomial rings, and unique factorization
domains.
Prerequisite: MATH 331
MATH 350: ADVANCED LINEAR ALGEBRA (3 Semester Hours)
Vector spaces over an arbitrary field, dual spaces, Cayley-Hamilton theorem, invariant subspaces,
canonical forms for matrices, inner product spaces over C, the spectral theorem.
Prerequisite: MATH 250
MATH 355: METHODS OF APPLIED MATHEMATICS (3 Semester Hours)
Series solutions and special functions. Orthogonal functions and Fourier series, partial differential
equations and boundary value problems.
Prerequisites: MATH 234 and 245
MATH 357: COMPLEX VARIABLES (3 Semester Hours)
Complex variables; analytic functions, Laurent expansions and residues; evaluation of real
integrals by residues; integral transforms.
Prerequisite: MATH 234
MATH 360: INTRODUCTION TO PROBABILITY AND STATISTICS (3 Semester
Hours)
Descriptive statistics, probability, discrete and continuous random variables, sampling
distributions, estimations of parameters, nonparametric methods, hypothesis testing, linear
regression.
Prerequisite: MATH 123 or 132
MATH 366: DISCRETE METHODS (3 Semester Hours)
An introduction to graph theory; trees; coloring; Eulerian circuits. Combinatorics; permutations,
and combinations;
recurrence relations.
Prerequisite: MATH 248 and junior standing.
MATH 393: MATHEMATICS INTERNSHIP (1-3 Semester Hours)
Research or applied mathematical work conducted in an industrial, business, or government
setting on a project designed jointly by an on-site supervisor and a departmental faculty member.
Enrollment is subject to available opportunities and approval of the department chair. Suitable
opportunities in an educational setting are also acceptable.
MATH 398: SPECIAL STUDIES (1-4 Semester Hours)
Courses having a special syllabus and description not listed in the Bulletin.
192
MATH 399: INDEPENDENT STUDIES (1-4 Semester Hours)
Individualized study arranged by a student with a faculty member and approved by the Chairman
of the Departmentand the Dean.
MATH 471: TOPOLOGY (3 Semester Hours)
An introduction to metric and topological spaces; continuity and homeomorphism; separation
properties; connectivity and compactness; examples and applications.
Prerequisite: MATH 321
MATH 490: HISTORY OF MATHEMATICS (3 Semester Hours)
The development of mathematics from historical and cultural viewpoints including both European
and non European roots of mathematics as well as contributions by women.
Prerequisite: MATH 248
MATH 491: SENIOR MATHEMATICS SEMINAR (3 Semester Hours)
Subject matter is chosen by the instructor. Coursework will involve student presentations to the
class.
MATH 493: SENIOR SEMINAR FOR FUTURE MATHEMATICS EDUCATORS (3
Semester Hours)
Topics in high school mathematics are examined from an advanced standpoint by developing and
exploring extensions and generalizations of typical high school problems, by making explicit
connections between these problems and upper division mathematics courses, and by providing
historical context. Current issues in secondary mathematics education will be investigated.
Written and oral presentations are required.
Prerequisite: Senior standing or consent of instructor.
MATH 495: MATHEMATICAL MODELING (3 Semester Hours)
Introduction to various modeling techniques, design and implementation of algorithms,
organization and presentation of results, introduction to problem solving using computer algebra
systems.
Prerequisite: Senior standing or consent of the instructor
MATH 498: SPECIAL STUDIES (1-3 Semester Hours)
Courses having a special syllabus and description not listed in the Bulletin.
MATH 499: INDEPENDENT STUDIES (1-3 Semester Hours)
Individualized study arranged by a student with a faculty member and approved by the Chairman
of the Department and the Dean.
MATH 550: FUNDAMENTAL CONCEPTS OF GEOMETRY (3 Semester Hours)
Euclidean and non-Euclidean planar geometries, axiomatic systems, synthetic and analytic
representations, relationships
with algebra, and selected topics and applications.
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Prerequisite: MATH 248 and 250
MATH 560: ADVANCED TOPICS IN PROBABILITY AND STATISTICS (3 Semester
Hours)
Material to be covered will be determined by the instructor. Consult with the instructor for the
specific topics in probability and statistics that will be covered in any given semester.
Prerequisite: MATH 360
MATH 561: COMPUTATIONAL METHODS IN LINEAR ALGEBRA (3 Semester Hours)
Numerical solutions of linear systems of equations, Gauss elimination and iterative methods
eigenvalues and eigenvectors.
Prerequisites: CMSI 182 or 185 or MATH 282, and MATH 250 or consent of the instructor.
MATH 562: NUMERICAL ANALYSIS (3 Semester Hours)
Numerical solutions of non-linear equations, interpolation, numerical differentiation, integration,
solution of differential equations.
Prerequisites: CMSI 182 or 185 or MATH 282, and MATH 245 or consent of the instructor.
MATH 568: MATHEMATICAL METHODS OF OPERATIONS RESEARCH (3 Semester
Hours)
Linear and dynamic programming, network analysis, inventory control.
Prerequisite: MATH 360
MATH 575: INTRODUCTION TO ORBIT DETERMINATION (3 Semester Hours)
A brief introduction to Banach and Hilbert spaces, the Projection Theorem, linear minimum
variance estimates, the Kalman filter, variational equations and orbit determination examples.
The final exam is a computer problem that involves tracking a spacecraft orbiting an asteroid by
means of Doppler measurements.
Prerequisites: MATH 245 and 250 and consent of the instructor.
MATH 582: ANALYSIS OF ALGORITHMS (3 Semester Hours)
Design, comparison, and analysis of mathematical algorithms, including implementation and
testing using FORTRAN.
Prerequisite: CMSI 182 or 185 or MATH 282 or consent of the instructor.
MATH 598: SPECIAL STUDIES (1-3 Semester Hours)
Courses having a special syllabus and description not listed in the Bulletin.
MATH 599: INDEPENDENT STUDIES (1-3 Semester Hours)
Individualized study arranged by a student with a faculty member and approved by the Chairman
of the Department and the Dean.
194
SCIENCE COURSE OPTIONS FOR BREADTH REQUIREMENTS
Physics:
PHYS 101 INTRODUCTION TO MECHANICS- 4 Semester Hours
Vectors, equilibrium, Newton's laws of motion, work and energy, impulse and momentum,
harmonic motion, statics and dynamics.
Lecture, 3 hours. Corequisites: MATH 131 and PHYS 103.
PHYS 103 INTRODUCTION TO MECHANICS LAB- 0 Semester Hours
Laboratory experiments pertaining to mechanics. Measurements, projectile motion,
friction, Newton's laws, torque.
Laboratory, 3 hours. Corequisite: PHYS 101.
PHYS 201 INTRODUCTION TO ELECTRICITY AND MAGNETISM- 4 Semester Hours
Electrostatics. Current, resistance, and D.C. circuits. Magnetism. Induced
electromotive force. Electric and magnetic properties of matter. A.C. circuits.
Lecture, 3 hours. Prerequisites: PHYS 101 and 103. Corequisites: MATH 132 and
PHYS 203.
PHYS 203 INTRODUCTION TO ELECTRICITY & MAGNETISM LABORATORY- 0
Semester Hours
Laboratory experiments pertaining to electricity and magnetism. Coulomb's Law,
static electricity, electric field plotting, circuits, charge/mass ratio for electron.
Laboratory, 3 hours. Corequisite: PHYS 201.
Computer Science:
CMSI 182 INTRODUCTION TO COMPUTER SCIENCE (3 Semester Hours)
Great ideas in computer science, including some programming sing a high-level programming
language
CMSI 185 COMPUTER PROGRAMMING (3 Semester Hours)
Introduction to algorithms and computer programming using Java
CMSI 281 DATA STRUCTURES AND ALGORITHMS I(3 Semester Hours)
Introduction to data types, information structures, and algorithms. Topics include: collection
classes and interfaces for sets, lists, stacks, queues, and dictionaries; implementation techniques
such as arrays, linked lists, and efficient tree structures; introduction tot computational
complexity, elementary sorting; hashing.
Prerequisite: A grade of C (2.0) or better in CMSI 185
195
Appendix XIV:
LMU Mathematics Department Faculty Sketches and CVs.
196
RESUME
Dr. Lev Abolnikov
Professor of Mathematics
Loyola Marymount University
Los Angeles, CA 90045
(310)338-5103, e-mail: labolnik@popmail.lmu.edu
Home address: 5460 White Oak Ave., Apt. K-307
Encino, CA 91316
(818)990-0033, e-mail: lev@socal.rr.com
AREA OF EXPERTISE
-PROBABILITY THEORY AND STATISTICS
-STOCHASTIC PROCESSES
-MATHEMATICAL METHODS OF OPERATIONS RESEARCH
-QUEUEING THEORY AND INVENTORY CONTROL
EDUCATION
1967-1970

Ph.D. in Probability Theory, Statistics and Operations Research,
USSR Academy of Engineering, Management Sciences and Information, Leningrad,
USSR.
M.S. (diploma with honors) in Mathematics, Leningrad State University, Leningrad, USSR
EXPERIENCE
1981-
present Professor of Mathematics, Loyola Marymount University,
Los Angeles, CA
1980-1981
Visiting Professor of Mathematics, Claremont
Graduate School and Harvey Mudd College, Claremont, CA
1977-1980
Senior Statistician and System Analyst, Program Leader,
USSR Statistical Computer Center, Leningrad, USSR
1973-1977
Senior Research Scientist, Program Leader,
Scientific Research Institute for Automation, Manufacturing and Industrial Management,
Odessa, USSR
1963 –1973
Senior Research Statistician, Academy of Medical Sciences, Leningrad, USSR
PROFESSIONAL ORGANIZATIONS
“American Mathematical Society” (AMC)
“Operations Research Society of America” (ORSA)
“Journal of Mathematical Reviews” (Reviewer)
“Journal of Applied Mathematics and Stochastic Analysis” (JAMSA) (Associate Editor)
RECENT CONFERENCES AND SYMPOSIUMS PRESENTATIONS
1998
“Complex-analytic and matrix-analytic solutions for a queueing system controlled by queue
length increments”.
197
International Conference on Operations Research and Management Sciences, TelAviv, Israel.
“Bulk queueing systems with state-dependent parameters “. The 2-nd International
Symposium on Nonlinear Analysis, Athens, Greece.
1997
1997
“Queueing Processes and Optimization Problems
in Quality Control Systems”.
Annual Conference of the Institute of Operations Research and Management Sciences
(INFORMS), San Diego, California.
MOST RECENT PUBLICATIONS
1.
Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled
by arrivals. Journal of Appl. Math. And Stoch. Analysis, 13, (2000)
, 415-427
2.
Stochastic processes and optimization problems in quality control systems with a group-individual testing
procedure. Engineering Simulation, 16 (1999), 165-178.
3.
First passage processes in queueing system MX/GY/1 with service delay discipline. Internat. J. Math. and
Math. Sci. 17 (1994), no. 3, 571-586.
4.
Stochastic analysis of a controlled bulk queueing system with continuously operating server: continuous time
parameter queueing
process. Statist. Probab. Lett., 16 (1993), no. 2, 121-128.
5.
Semi-regenerative analysis of controlled bulk queueing systems with a bilevel service delay discipline and
some ergodic theorems. Comput.
Math. Appl., 25 (1993), no. 3, 107-116.
6.
A multilevel control bulk queueing system with vacationing server.
Oper. Res. Lett., 13 (1993), no. 3, 183-188
7.
Ergodicity conditions and invariant probability measure for an
imbedded Markov chain in a controlled bulk queueing system with a
bilevel service delay discipline. Part 1-2. Appl. Math. Lett., 5 (1992), no. 4, 25-27 and no. 5, 15–18.
8.
On a multilevel controlled bulk queueing system MX/GY(r, R)/1.
J. Appl. Math. Stochastic Anal., 5 (1992), no. 3, 237-260.
In addition to the above-mentioned papers, more than 40 other have been published in reviewed journals in the
USA and USSR. They deal with statistical and operations research methods in industrial and management
problems.
PERSONAL
Languages: English, Russian.
Interest:
sports, music, chess, literature.
198
CURRICULUM VITAE
Curtis Bennett
Associate Professor, Loyola Marymount University
665 1/2 W. Palm Ave.
El Segundo, CA 90245
(310) 615-0023
Citizenship: U.S.A.
email: cbennett@lmu.edu
web-page: http://myweb.lmu.edu/~cbennett/
Mathematics Department
Loyola Marymount University
One LMU Drive, Suite 2700
Los Angeles, CA 90045
(310) 338-5112
EDUCATION:
University of Chicago
Ph.D., Mathematics, Summer 1990
M.S., Mathematics, 1986
Colorado State University
Bachelor of Science - High Distinction, Spring 1985
AWARDS:
CASTL Fellowship, (Carnegie Academy for the Scholarship of Teachimg
and Learning), 2003-2004
National Security Agency Research Grant, 2000-2002
CASTL Fellowship, 2000-2001
National Science Foundation Postdoctoral Research Fellowship, 1992-1995
National Science Foundation Graduate Fellowship, 1985-1988
Century Fellow, University of Chicago, 1985-1988.
TEACHING AWARDS:
Mortar Board National Honor Society Excellence in Teaching Award
Bowling Green State University, 1998
Finalist, J. Sutherland Frame Teaching Award
Department of Mathematics, Michigan State University, 1992
HONORS:
Phi Betta Kappa, Colorado State University, 1984
Kappa Mu Epsilon, Colorado State University, 1983
President of Colorado Alpha Chapter of Kappa Mu Epsilon, 1984
Ph.D. THESES DIRECTED:
Lakshmi Evani, “Results on the BeSo order,” BGSU, Completed 2000
MASTER THESES DIRECTED:
Claudia Catalan, Loyola Marymount University, Completed 2003
PROFESSIONLA SOCIETIES:
American Mathematical Society
Mathematics Association of America
National Council for the Teachers of Mathematics
PROFESSIONAL EXPERIENCE:
Loyola Marymount University
Associate Professor, 2002-present
Bowling Green State University
Associate Professor, 1996-2003
Assistant Professor, 1993-1996
Michigan State University
199
Visiting Associate Professor, 2000-2001
Research Instructor, 1990-1992
Ohio State University
National Science Foundation Postdoctoral Fellow, 1992-1993
PROFESSIONL SERVICE:
AMS-MAA Committee on Disabilities, 2002-present
AMS Committee on Committees member, 2001-2003
Mathematics Advisory Panel, for Mathematics Teacher
MAA Ohio chapter, Committee on Student Mathematics,
member, 1995-1998, 1999-2002, Chair 2000-2001
AMS Committee on the Profession (COPROF), member,
1996-1999
AMS Subcommittee on Employment Issues (of COPROF),
member, 1993-present
AMS Subcommittee on Graduate Education (of Com. on Educ.),
member, 1995-1999
Greater Toledo Council for Teachers of Mathematics,
BGSU representative, 1997-1999
Editorial Staff of Young Mathematicians’ Network, 1993-1995,
(one of the founding editors)
Co-Organized AMS Special Session in Groups and Geometries
for AMS Regional Meeting in Kent State (1995)
Co-organized conference on groups and geometries at Bowling
Green State University in March, 1999
Organized Conference: New Directions in the Scholarship of Teaching and
Learning, Bowling Green State University, November 2001
Review articles for Math Reviews
Referee articles for various journals
UNIVERSITY SERVICE:
Loyola Marymount University
MASTeP Committee member (and co-chair) 2003-present
Assessment Committee for Mathematics Department, 2003-present
Undergraduate Advisor, 2003-present
University Committee on K-12 Education, 2002-present
Graduate Advisor (MAT - mathematics), 2002-present
Mathematics Dept. liaison to College of Education, 2002-present
Pi Mu Epsilon advisor, 2002-present
Bowling Green State University
Mathematics and Natural Sciences curriculum committee, 1999-2002
Undergraduate Council. 2002
Committee on Undergraduate Research, 2001-2002
Department Advisory Committee, 1996-2000, 2001-2002
Undergraduate Committee - member, 1995-1998, 1999-2002
Undergraduate Advisor, 1993-2002
William, Lowell Putnam Examination Coach, 1993-2002
Undergraduate Coordinator, 1999-2000
Math Problem Solver’s Committee - member, 1993-2000
Kappa Mu Epsilon, faculty advisor, 1995-1999
Education College search committee (mathematics education
position), 1997
Dept. of Mathematics and Statistics, Search Committee (algebra
position), 1997
Dept. of Mathematics and Statistics, Search Committee (mathematics
200
education position), 1996
Dept. of Mathematics and Statistics, Search Committee (mathematics
education position), 1995
Colloquium Committee - chair, 1995-1996
Colloquium, Committee - member, 1994
University Library Committee - member, 1994
PROFESSIONAL DEVELOPMENT:
Recent Workshops and Conferences Attended
MAA MATHFEST, Boulder, CO, August, 2003
Southern California Section of the MAA Spring Meeting, April 2003
AMS-MAA Joint Mathematics Meetings, Baltimore, January 2003
MAA PMET Workshop for teaching future mathematics teachers,
San Diego, January, 2002
INVITED MATHEMATICS RESEARCH LECTURES:
 T-Orders on the Coxeter Groups., Center for Cryptographic Research, San Diego, March 2003.
 A Paradoxical Coloring of Escher’s Angels and Devils, Claremont Colleges Mathematics Colloquium,
November 2002.
 Phan type theorems, Buildings in Geometric Group Theory, Würzburg, Germany, May 2002.
 1/19th of a generating function, MAA - Ohio Section meeting, April, 2002.
 A simple definition of the universal Grassmannian order, AMS-MAA Joint Mathematics Meetings, January
2002.
 A new proof of Phan’s theorem, AMS Central Section Meeting, September 2001.
 A Paradoxical Coloring of Escher’s Angels and Devils, Kansas State University, Mathematics Departmental
Colloquium, March 2000.
 Exponential of Infinite Dimensional Lie Algebras –– Kansas State University, Algebra Seminar, March 2000.
 Lights Out!, Fall Meeting, Ohio Section of the MAA, October, 1999.
 Affine -buildings and higher order buildings, Conference on (Moufang) n-gons and (Twin) Buildings, Ghent,
Belgium, June, 1999.
 Paradoxically coloring Escher’s Circle Limit IV, Ohio State University Group Theory and Graduate Student
Seminar, May, 1998.
 An Escher version of the Banach-Tarski paradox: Spring meeting of the Ohio chapter of the MAA: hour-long
invited address, 1998.
 Higher order buildings as Z Z-buildings, AMS Meeting, Central Section, March 1998.
 Extensions of Kac-Moody twin buildings using quasi-real roots –– AMS Meeting, Central Section, May 1997.
 Exponentiation of Infinite Dimensional Lie Algebras –– University of Toledo, June, 1996.
 Affine -buildings - Mathematisches Forschunginstitut, Oberwolfach, Germany, April, 1996.
 Exponentiation of Infinite Dimensional Lie Algebras –– University of Michigan (Group Theory and Lie Theory
Seminar), April, 1996.
 Exponentiation of Infinite Dimensional Lie Algebras –– Kent State University, February, 1996.
 Generalized -n-gons and Twin Trees –– preliminary report –– AMS Meeting, Central Section, Special Session
on Groups and Geometries –– Manhattan Kansas, March, 1994.
 Special Imaginary Roots of Kac-Moody Lie Algebras –– Bowling Green State University, March 1993.
 Special Imaginary Roots of Kac-Moody Lie Algebras –– University of Chicago Group Theory Seminar,
November 1902.
 When can a building be twinned? - preliminary report - AMS Meeting, Central Section, Special Session on
Groups and Geometries, Dayton, OH, Pctober 1992.
 Signed Dynkin Diagrams and Related Groups - XXIst Ohio State - Denison Conference, May 1992.
 Special Imaginary Roots of Kac-Moody Lie A;gebras - AMS Meeting, Baltimore, MD, January 1992.
 Generalized Spherical Buildings - Mathematisches Forschungsinstitut Oberwolfach, Special Session on Groups
and Geometries, July 1991.
 A Groupoid Approach to Buildings - Ohio State University, May 1991.
201
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Buildings and Groupoids - AMS Meeting, Southeastern Section, Special Session on Finite Groups and
Geometries - Tampa, FL, March 1991.
A Groupoid Approach to Buildings - Colorado State University, March 1991.
Affine -buildings - Yale University, October 1990.
Affine -buildings - AMS Meeting, Central Section, Special Session on Groups and Geometries - Mnhattan, KS,
March 1990.
Affine -buildings - Kansas State University, November 1989.
INVITED LECTURES, MATHEMATICS EDUCATION AND SCHOLARSHIP OF TEACHING AND
LEARNING
 The Story of Neal, University of Illinois conference on the Scholarship of Teaching and Learning, January 2003.
 Student Research Projects in a Mathematics Capstone Course for Secondary Teachers, AMS-MAA Joint
Mathematics Meetings, January 2003.
 Student Learning in a Mathematics Capstone Class for Future Teachers, Oxford College of Emory University,
Conference on the Scholarship of Teaching and Learning, November 2002.
 My Course Portfolio: A window on student learning and an entrance to further study, Disciplinary Styles in the
Scholarship of Teaching and Learning, Rockhurst, KS, April 2002.
 Poster presentation on course portfolio, AAHE National Conference on Higher Education, March 2002.
 Panelist AMS-MAA Joint Mathematics Meetings, 2002, Course Portfolios, January 2002.
 What is the Scholarship of Teaching and Learning, Conference on New Directions in the Scholarship of
Teaching and Learning, Bowling Green State University, November 2001.
 The effects of semester long research projects on a mathematics capstone class, Michigan State University,
Mathematics Education seminar, March 2001.
 Semester Long Mathematics Research Projects, Conference on the Scholarship of Teaching and Learning,
Youngstown State University, February 2001.
 An Example of a Capstone Course for Secondary Education Majors, AMS-MAA Joint Mathematics Meeting,
January 2001.
 The Many Discourses of the Scholarship of Teaching and Learning, Michigan State University, November 2000.
 Preparation of Mathematics Secondary Teachers - A Capstone Perspective, the MAA CRAFTY Conference,
Michigan State University, November 2000.
INVITED STUDENT (COLLEGIATE) PRESENTATIONS
5) Mathematical Pi(e), Grand Valley State College, Grand Valley, MI, October 1999, KME initiation talk.
6) Lights Out!, Xavier University, Cincinnati, OH, September 1999.
7) Lights Out!, John Carroll University, Cleveland, OH, April 1999.
INVITED SCHOOL PRESENTATIONS
1. Harry Potter Math, Conneaut Elementary School 5th grade class, November 1999.
OTHER INVITED PRESENTATION
 Panelist for Project NexT panel discussion on professional development, Joint Mathematics Meeting, 1998.
 Co-Organizer and moderator of AMS Panel Discussion: The Job Market for Mathematics Ph.D.s, Joint
Mathematics Meetings, January 1997.
 The Job Market for Mathematics Ph.D.s. - Michigan Sectional Meeting of the MAA, May 1996.
 Co-Organizer and Presenter of MAA Minicourse at the Joint Mathematics Meetings, January 1995. Title:
Learning About Today’s Job Market.
 Panelist - AMS-MAA-SIAM Committee on Employment Opportunities - Young Mathematicians Network Panel
Discussion, Joint Mathematics Meetings, January 1994.
PUBLICATION LIST
Mathematics Research Articles
 Automorphisms of Hyperelliptic Surfaces, Rocky Mountain Journal of Mathematics, 20(1), 1990, 31-37, (with
Rick Miranda).
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Imaginary Roots of a Kac-Moody Lie Algebera Whose Reflections Preserve Root Multiplicities, Journal of
Algebra, 158 (1), 1993, 2440267.
Signed Dynkin Diagrams and Related Groups, Group Theory, Proceedings of the Biennial Ohio State-Denison
Conference, 14-16 May, 1992, Edited by S. Sehgal and R. Solomon, World Scientific, New Jersey, 1993, pp. 3061.
Affine -Buildings I, Proceedings London Mathematical Society, 68 (3), 1994, pp. 541-576.
Partition Lattice q-Analogs Associated with q-Stirling Numbers, Journal of Algebraic Combinatorics, 3 (3),
1994, 261-284, (with B. Sagan and K. Dempsey).
A Generalization of Semimodular Supersolvable Lattices, Journal Combinatorial Theory, 72, 1995, 209-231,
(with B. Sagan).
Explicit Free Subgroups of Aut (R, ≤), Proceedings of the American Mathematical Society, 125 (1997), no. 5,
1305-1308.
Twin Trees and -gons, Transactions of the American Mathematical Society, 349, (1997), no. 5, 2069-2084.
Linear Forms in the Logarithms of Three Algebraic Numbers, Journal Théorie des Nombres Bordeaux, 9 (1997),
no. 1, 97-136. (With J. Blass, A. Glass, D. Meronk, and R. Steiner)
Enumerating A3(2) Blueprints and an Application, Journal of Experimental Mathematics, 7 (4), 1998, (with M.
Abramson).
Embeddings of Twin Trees, Geometriae Dedicata, 75, 209-215, 1999, (with M. Abramson).
Exponentiation of Infinite Dimensional Z-Graded Lie Algebras, 19 pp., Communications in Algebra, 28 (9),
2000, pp. 4013-4036.
Zero-Estimates for Polynomials in 3 and 4 Variables using Orbits and Stabilizers, in Hilbert’s Tenth Problem:
Relations with Arithmetic and Algebraic Geometry, AMS, 2000 (with Lisa K. Elderbrock and A.M.W. Glass).
A note on a theorem of Tits, Proceedings of the American Math Society, (with S. Shpectorov) posted on AMS
website for journal February, 2001.
A Topological Characterization of End Sets of a Twinning of a Tree, Europeam Journal of Combimatorics, 22
(1), pp. 27-35, 2001.
Phan-Curtis-Tits type theorems, (joint with R. Gramlich, C. Hoffman, and S. Shpectorov), Proceedings of the
Durham Symposium (2001), to appear.
A class of velocity fields with known Lagrangrian law, Journal of Statistical Physics, to appear.
A simple definition for the universal Grassmannian order, Journal of Combimatorial Theory, Series A, 102
(2003) 347-366, (joint with L. Evani and D. Grabiner)
Partial orders generalizing the weak order on Coxeter groups, Journal of Combinatorial Theory, Series A, 102
(2003) 331-346 (joint with R. Blok)
A new proof of a theorem of Phan, (joint with S. Shpectorov), accepted, Journal of Group Theory.
Fermat’s Last Theorem for Rational Exponents, American Mathematical Monthly, accepted.
Fibonacci decimals and generating functions, Math Horizons, provisionally accepted.
Mathematics Education and Scholarship of Teaching and Learning Articles
23. The scholarship of teaching and learning: a beginners view, Youngstown State University faculty newsletter,
May 2001, 2 pp.
24. Carnegie Project: Course Portfolio for a Mathematics Capstone Course for Mathematics Education majors,
http://www-math.bgsu.edu/cbennet/math417/PortfolioCover.htm, 2001.
25. My Course Portfolio: A window on student learning and an entrance to further study, Proceedings of Conference
on Disciplinary Styles in the Scholarship of Teaching and Learning, Rockhurst, KS, April 2002 (CD).
Expository Articles
26. A Paradoxical Decomposition of Escher’s Angels and Devils (Circle Limit IV), Mathematics Intelligencer,
Volume 22 (3), 2000, 39-46.
27. Topspin on the Symmetric Group, Math Horizons, 2000.
28. A Relationship Betweem Roots of a Polynomial and Roots of its Truncations, to appear in a Russian
compendium of research results, (with T. O’Brien)
Professional Development Articles
29. Another View of the Current Tough Job Market - Response, Notices of the American Mathematical Society, May
1995, pp. 570-571.
30. A Research Mentor is a Good Thing to Have, Starting Our Careers, AMS, 1999, 47-48.
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31.
32.
33.
34.
(Re)writing a thesis, and other mathematics for publication, Starting Our Careers, AMS, 1999, 61-63.
When at first you don’t succeed …, Starting Our Careers, AMS, 1999, 69-70.
Math Talks, Starting Our Careers, AMS, 1999, 105-107, (with F. Sottile)
Organizing a Special Session, Starting Our Careers, AMS, 1999, 108-110, (with F. Sottile)
Books
1. Starting Our Careers - A Collection of Essays and Advice on Professional Development from the Young
Mathematicians’ Network, editor, American Mathematical Society, 1999 (with A. Crannell)
Note: As is standard practice in pure mathematics, there is no first authorship for (almost all) papers. All papers
appear with authors’ names in alphabetical order. This does not imply any difference in effort or responsibility
for the results.
204
Michael C. Berg
Loyola Marymount University
Department of Mathematics
7900 Loyola Boulevard
Los Angeles, California 90045-8130
United States
Phone: (310) 338-5116
Email: mberg@lmu.edu
Education:
Ph.D., University of California, San Diego, Mathematics (Number Theory), 1985
Professional Experience:
Loyola Marymount University, College of Science and Engineering, Mathematics
Publications:
 Michael C. Berg. A Sufficient Condition for Generating Heisenberg Groups over Local
Fields. Far East Journal of Mathematics. 3(3): 371-384, 2001
 Michael C. Berg. On Local Objects Attached to Theta- and Zeta-Functions. Journal of Integral
Transforms and Special functions. 10(1): 13 - 24, Oct 2000
 Michael C. Berg. The Fourier-Analytic Proof of Quadratic Reciprocity /book: John Wiley &
Sons, Publishers. 2000
 Evert J. Post and Michael C. Berg. Mach's Principle in a Mixed Newton-Einstein Context.
Galilean Electrodynamics. 10(2): 36 - 40, 1999
 Michael C. Berg. A Relationship between Weil Indices and Local Constants. Western Number
Theory Conference, SFSU. 1998
 Michael C. Berg. On Heisenberg Groups and Low-dimensional Cohomology. 103-d Annual
Meeting of the AMS, MAA, SIAM, San Diego. 1997
 Evert J. Post, Michael C. Berg. Epistemics of Local and Global in Mathematics and Physics.
Proceedings of the Conference, Physical Interpretations of Relativity Theory (London). 1996
 Michael C. Berg. An Explicit Heisenberg Group and the Cauchy-Hecke-Weil-Kubota Proof of
Quadratic Reciprocity. Western Number Theory Conference, Asilomar. 1995
 Michael C. Berg. On Generalized Gauss-Hecke Sums and Theta Constants. Journal of Integral
Transforms and Special Functions. 3(1): 1-20, 1995
 Allen G. Thomas, Michael C. Berg. Medium PRF Set Selection: an Approach through
Combinatorics. IEE Proceedings, Radar, Sonar, Navigation. 141(6): 307-311, Dec 1994
 Michael C. Berg. On Certain Algebraic Aspects of the Analytic Proof of 2-Hilbert
Reciprocity. Presentation, Western Number Theory Conference, UCSD. 1994
 Michael C. Berg. On a Generalization of Hecke Theta Functions and the Analytic Proof of
Higher Reciprocity Laws. Journal of Number Theory. 44(1): 66-83, May 1993
 Michael C. Berg. The Analytic Proof of Higher Reciprocity Laws. 877-th Meeting of the
AMS, USC. 1992
Membership Information:
 American Mathematical Society
 Mathematical Association of America
 Pi Mu Epsilon
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Sigma Xi, The Scientific Research Society
United States Judo Association (Yudanshakai)
Language Skills:
Dutch: Reading fluent, Writing fluent, Speaking fluent
French: Reading functional, Writing functional, Speaking basic
German: Reading functional, Writing functional, Speaking functional
206
CURRICULUM VITAE
Jacqueline M. Dewar, Ph.D.
Professor of Mathematics, Loyola Marymount University
Department of Mathematics
Loyola Marymount University
One LMU Drive - Suite 2767
Los Angeles, CA 90045
(310) 338 - 5106 email: jdewar@lmu.edu
EDUCATION
University of Southern California, 1973
Saint Louis University, 1968
Ph.D. in Mathematics
B.S. in Mathematics
EMPLOYMENT
Loyola Marymount University
Professor of Mathematics, 1985-Present
Associate Professor of Mathematics, 1980-1985
Assistant Professor of Mathematics, 1973-1980
ADMINISTRATIVE EXPERIENCE
Loyola Marymount University
Campus Coordinator, Los Angeles Collaborative for Teacher Excellence, an NSF
Collaborative for Excellence in Teacher Preparation Program, 1995-2001
Director, Master of Arts in Teaching Mathematics Program, 1997-present
Coordinator, Single Subject Matter Program in Mathematics, 1994-present
Chairperson, Department of Mathematics,1983-1986
Director, Graduate Mathematics Program,1979-1981
PROFESSIONAL EXPERIENCE
Advisory Board Member
High School Mathematics from an Advanced Standpoint Project, funded by the Stuart
Foundation, 1999-2002
Member, LMU Math, Science and Engineering Consulting Team
Universidad Rafael Landivar, Guatemala City, Guatemala (August 1997)
Outside Evaluator
University of Redlands, Mathematics Department (1996)
CSU Dominguez Hills, Mathematics Department (1989)
Member, Women in Mathematics Delegation to China
People to People Citizen Ambassador Program (1990)
Math Science Interchange (Nonprofit founded to encourage girls in math and science)
Founding Member, 1978
Member, Planning Committee for annual Expanding Your Horizons Career Day in
Science and Engineering, 1978-present
Member, Board of Directors (since incorporation in 1983)
Vice President for Special Projects, 1990-present
Math,
HONORS AND AWARDS
2003-4 Carnegie Scholar, Carnegie Academy for the Study of Teaching and Learning
Faculty Recipient, 2002 Loyola Marymount University Women’s Herstory Award
Phi Beta Kappa, Saint Louis University, 1968
Pi Mu Epsilon, Saint Louis University, 1967
PUBLICATIONS (SINCE 1990)
207
A Future Teachers Conference - A Vehicle to Retain, Inform, and Inspire New and Prospective Teachers, co-authored
with J. Kasabian and L. Fathe, Journal of Mathematics and Science: Collaborative Explorations 5(1), Spring 2002.
Future Teachers Conference: A Planning Handbook, co-authored with J. Kasabian, published by the Los Angeles
Collaborative for Teacher Excellence, an NSF-funded CETP, 2000.
Meet the Teachers Roundtable – Connecting Future Teachers With Role Model Teachers: A Planning Handbook, coauthored with J. Kasabian, published by the Los Angeles Collaborative for Teacher Excellence, an NSF-funded
CETP, 2000.
"Addressing Gender Equity for Preservice Elementary Teachers," Undergraduate Mathematics Education Trends,
6(3) July, 1994.
"Using the Computer Language LOGO to Provide College Students With a Mathematical Experience," Collegiate
Microcomputer, IX(1) 59-61, February 1991.
"Recruitment and Retention of Students in Undergraduate Mathematics," co-authored with Pat Kenschaft, Miriam Cooney,
Vivian Kraines, Brenda Latka, and Barbara LiSanti, The College Mathematics Journal, 21(4) 294-301, September 1990.
Algebra and Trigonometry, 2nd edition, Algebra, 2nd edition, and Trigonometry, 2nd edition, co-authored with Dennis Zill,
McGraw-Hill, 1990.
Book reviews of:
 Africa Counts: Number and Pattern in African Culture appeared in Teaching Children Mathematics 6.9, May
2000, p. 200.
 Changing the Faces of Mathematics: Perspectives on Gender appeared in Teaching Children Mathematics 8.5,
January 2002, p. 299-300.
 Attaining Excellence: A TIMSS Resource Kit by Third International Mathematics and Science Study, U. S.
Department of Education, Office of Educational Research and Improvement appeared in Teaching Children
Mathematics 5.1 September 1998, pp. 59-60.
 Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity by Janet Trentacosta (ed.)
appeared in Teaching Children Mathematics 4.5 January 1998, p. 295.
 Finding the Connections: Linking Assessment, Instruction, and Curriculum in Elementary Mathematics, by Jean
Moon and Linda Schulman, appeared in Teaching Children Mathematics 3.8 April 1997, pp. 461-2.
 New Directions for Equity in Mathematics Education by Walter G. Secada, Elizabeth Fennema, and Lisa Byrd
Adajian appeared in Teaching Children Mathematics 2.7 March 1996, p. 440.
 The Scientist Within You: Experiments and Biographies of Distinguished Women in Science by appeared in
Teaching Children Mathematics 1.9 May 1995, p. 588.
 Addventures for Girls appeared in Teaching Children Mathematics 1.1 September, 1994.
208
PAPERS PRESENTED (SINCE 1990)
A Curriculum Development Odyssey, Co-presented with L. Kjeseth, M. Greenhalgh, and J. Kasabian at the Lilly
West Conference on College and University Teaching, Lake Arrowhead, CA, March, 2001.
Meet the Teachers Roundtable: Exemplary Teachers, Exemplary Lessons, Co-presented with J. Kasabian, F.
Manion and S. Tummers at the Lilly West Conference on College and University Teaching, Lake Arrowhead, CA,
March, 2001.
Cooperative Learning + Alternative Assessment: Adding Up to Make a Difference, Co-presented with L. Fathe, L.
Kjeseth, M. Greenhalgh, and J. Kasabian, Lilly West Conference on College and University Teaching, Lake
Arrowhead, CA, March, 2000.
Hands-on Activities Make a Difference, Co-presented with R. Vangor, Lilly West Conference on College and
University Teaching, Lake Arrowhead, CA, March, 2000.
From Portfolio in a Math Class, To Portfolio in Your Class, Lilly West Conference on College and University
Teaching, Lake Arrowhead, CA, March, 1999.
Teaming Up to Recruit and Prepare Future K-12 Math and Science Teachers, Co-presented with M. Greenhalgh and
J. Kasabian, Lilly West Conference on College and University Teaching, Lake Arrowhead, California. March, 1999.
Recruiting and Encouraging Future Teachers, Co-presented with J. Kasabian, Joint LACTE/MASTEP (NSF-funded
California Collaboratives for Teacher Excellence) Symposium, California State University, Los Angeles, CA,
October 16, 1998.
Women and Mathematics, Los Angeles City Teachers’ Mathematics Association Conference, California State
University, Dominguez Hills, California, March 14, 1998.
Building Skills, Confidence and Community in Freshman Mathematics Majors, Lilly West Conference on College
and University Teaching, Lake Arrowhead, CA, March, 1998.
Calculus Reform, Universidad Rafael Landivar, Guatemala City, Guatemala, August 1997.
Innovative Teaching Strategies in Mathematics, Universidad Rafael Landivar, Guatemala City, Guatemala, August
1997.
Building Skills and Confidence in Math Majors Through Problem Solving and Writing, Universidad Rafael Landivar,
Guatemala City, Guatemala, August 1997.
Teaching With Technology, Universidad Rafael Landivar, Guatemala City, Guatemala, August 1997.
Developing Skills, Confidence and Community in Freshman Mathematics Majors, "Celebrating Our Successes;
Understanding Our Challenges," a national conference for NSF-CETP programs, CSU Dominguez Hills, June, 1997.
Using E-mail Journals to Open Lines of Communication and Foster Self-reflection, Lilly West Conference on
College and University Teaching, Lake Arrowhead, California. March, 1997.
Using Portfolios in a Mathematical Proofs Course, Session on Alternative Assessment, Los Angeles Collaborative for
Teacher Excellence, November 1, 1996.
Mathematics: Contributions by Women, AMS/MAA Joint Mathematics Meeting, San Francisco, CA, January 1991.
Building Confidence in Problem Solving, 11th Annual Conference, California Mathematics Council of Community
Colleges South, Costa Mesa, CA. March 23, 1996.
Problem Solving, Writing Mathematics and Building Confidence, 35th Annual Fall Conference of the California
Mathematics Council Southern Section, Palm Springs, CA, November 5, 1994.
209
Women and Mathematics in China, LMU Faculty Seminar Series, February 20, 1992.
GREAT MATH: An Enrichment Program Staffed by Parent Volunteers, Seventh International Congress on
Mathematics Education, Quebec City, Quebec, Canada, August, 1992.
Using the Computer Language LOGO to Provide College Students with a Mathematical Experience, Fuzhow
University, Fuzhow, China, Women and Mathematics Delegation, People to People, July 1990.
CONFERENCES, WORKSHOPS, EVENTS ORGANIZED OR FACILITATED
“Quantitative Reasoning.” Facilitator, Discussion Session, SENCER (Science Education for New civic Engagement
and Responsibility) Summer Institute, Santa Clara University, Santa Clara, CA, August 2003
Annual Expanding Your Horizons in Math, Science and Engineering, Career Day for Girls in grades 6-10, Coorganizer, Loyola Marymount University, Los Angeles, CA,1978- present.
Annual Meet the Teachers Roundtable - Connecting Future K-12 Teachers with Role Model Teachers for Science
and Mathematics, Organizer, Loyola Marymount University, Los Angeles, CA, 1996-present.
Future Teachers Conference, Co-organizer, Santa Monica College, Santa Monica, CA, 1996 and 1997.
"Student Centered Learning: A View from the Trenches." Moderator, Panel discussion, LMU Chapter, California
Women in Higher Education, October, 1997.
Student Voices, What Are They Telling Us? Co-organizer and co-facilitator with J. Kasabian, Panel discussion,
NSF-CETP national conference. California State University Dominguez Hills, Carson, CA. June, 1997.
Reflection Session for LACTE Students, Co-organizer with J. Kasabian, Los Angeles County Teachers of
Mathematics Association Conference. California State University Dominguez Hills, Carson, CA. March 1997.
Using Cooperative Learning - Mathematics Roundtable Discussion, Facilitator, Regional Conference on College
Teaching: Using Cooperative Learning in Discipline Specific Settings, Occidental College, Los Angeles, CA.
February, 1997.
Students in LACTE, Organizer and chair, Student presentation, LACTE National Visiting Committee Meeting,
February, 1997.
Future Teachers Conference, Co-organizer, Santa Monica College, Santa Monica, CA, 1996 and 1997.
Math for Girls, Organizer and Teacher, Hands-on after-school course for 4th-8th grade girls, St. Jerome School, Los
Angeles, CA, 1993, 1996, 1997.
Great Math, Originator and coordinator, Math Enrichment Program for grades 5 to 8, St. Jerome School, Los
Angeles, CA, 1991-1996.
Family Math Night, Facilitator, Westchester Lutheran School and St. Jerome School, Los Angeles, CA, 1991-1993.
Math and Literature, Co-organizer and presenter, Teacher inservice, St. Jerome School, Los Angeles, CA, March and
May, 1992.
Problem Solving and Alternative Assessment, Organizer and presenter, Teacher inservice, St. Jerome School, Los
Angeles, CA, August 1991.
PROFESSIONAL MEMBERSHIPS
Association for Women in Mathematics
California Mathematics Council
210
Los Angeles County Teachers of Mathematics Association
Mathematical Association of America
Math/Science Interchange (Board of Directors)
National Council of Teachers of Mathematics
211
Ben G. Fitzpatrick
2916 Stanford Avenue
Marina del Rey, CA 90292
310-574-4993 (voice)
310-821-0373 (fax)
fitzpatrick@tempest-tech.com
Education:
Auburn University, B.S. (with highest honors), Applied Mathematics, 1981.
Auburn University, M.P.S. (master of probability and statistics), 1983.
Brown University, Sc. M., Applied Mathematics, 1986.
Brown University, Ph. D., Applied Mathematics, 1988.
Experience:
Loyola Marymount University.
Juanuary 2002--present
Los Angeles, California
Clarence J. Wallen, S. J., Professor of Mathematics: Teaching and research in applied
mathematics and statistics. Current efforts include the following:
 Instruction in probability and statistics to undergraduates in a variety of disciplines;
 Basic research in statistical and computational image processing, in modeling and analysis of
biological systems, and in transport and propagation in random media;
 Development of an applied industrial mathematics seminar, allowing students to work with
industrial partners to solve real-world problems; and
 Management of the departmental website.
Tempest Technologies LLC.
May 1998--present
Marina del Rey, California
Founding partner, president: Consulting in mathematical modeling, statistical analysis, and
software development in a variety of applications:
 Instrumentation products in oil recovery applications;
 Optical tracking through turbulence, in support of the Air Force Research Lab, Kirtland AFB,
and other corporate partners;
 Phase reconstruction and unwrapping in optical and radar applications;
 Adaptive noise cancellation in speech recognition applications;
 Biomedical devices for tissue classification;
 Test article recognition from limited vibration data; and
 Air-to-ground engagement analysis, simulation, and optimization.
Daniel H. Wagner Associates, Inc.
Aug 1997 – May 1998
Santa Clara, California
Associate: Consulting in mathematical and statistical analysis of complex signals and systems:
 Analyses of quadrupole ion trap mass spectrometry devices in support of patent infringement
litigation;
 Statistical analysis of DNA sequence data for the Department of Commerce and the Armed
Forces DNA Identification Lab, including exploratory data analysis and model development
for quality scoring of basecalls;
212


Modeling and data analysis for the design of an electromechanical probe to detect cartilage
degradation, supported by the National Institutes of Health, including project management and
technical direction, as well as numerical algorithm development, stochastic pattern
recognition and statistical data analysis;
Noise cancellation for speech recognition systems for enhanced tactical operations, supported
by the U. S. Army, including project management and technical direction, as well as time
series methods and wavelet decomposition ideas to adaptive filtering and processing of noisy
speech signals.
North Carolina State University
Aug 1992 – May 1998
Raleigh, North Carolina
Associate Professor of Mathematics (with tenure); Assistant Director of the Center for Research in
Scientific Computation: Teaching and research in applied mathematics, strategic and administrative
support of Center research and funding activities:
•
Developed nonparametric statistical models of hurricane frequency and severity
for the East Coast and Gulf Coast of the United States, for the purposes of studying
insurance risks, for the North Carolina Department of Insurance. Conducted data analysis
and directed teams of students in model development and assessment. Supported the State
in legal negotiations with insurance industry representatives.
•
Developed and analyzed new and innovative models for contaminant transport in
groundwater, under support from the Air Force Office of Scientific Research. Modeling
includes not only new stochastic models of soil property variability but also new
numerical models of the transport and biodegradation. Performed extensive data analysis
from Air Force field experiments in support of this model development.
•
Constructed distributed models of populations to understand individual variability
without the complexity of a fully individual based, Monte Carlo type of model. Through
the use of aggregate structured models coupled with distributions of individuals’
parameters, we developed computationally efficient simulation and estimation techniques
for comparing models to data supplied by UC Davis biological scientists.
•
Devised and implemented image reconstruction algorithms for applications to nonintrusive
flow sensing in ballistic range testing under support from the Air Force Office of Scientific
Research.
•
Initiated the development of nonlinear proliferation models to simulate kidney tumor growth,
and analyzed data to determine appropriate parameter values, under support from the
Chemical Industry Institute for Toxicology.
•
Analyzed and implemented time series forecasting methods for inventory management,
compared algorithms to client's data, devised improved methods, and he directed a team of
students in software development, for IBM.
•
Taught courses including calculus, stochastic processes, mathematical modeling, and real and
functional analysis.
•
Served as Assistant Professor August 1992 – August 1994.
213
The University of Tennessee
Aug 1989 – Aug 1992
Knoxville, Tennessee
Assistant Professor of Mathematics: Taught courses including calculus, numerical analysis,
mathematical modeling, and real and functional analysis, and control theory. Developed basic and
applied research programs in statistical and mathematical modeling.
TRW Defense Systems Group
July 1983 – August 1987
San Bernardino, CA
Systems Analyst: Developed statistical methods for classification of radar signals, devised optimization
methods for strategic defenses, and simulated missiles and projectiles in boost and free-fall phases of
flight.
Journal Publications:
1. “Statistical Tests for Model Comparison in Parameter Estimation Problems for Distributed Parameter
Systems,” by H. T. Banks and B. G. Fitzpatrick, J. Math. Bio., 28, 1990, pp. 501-527.
2. “Estimation of Growth Rate Distributions in Size Structured Population Modeling,” by H. T. Banks
and B. G. Fitzpatrick, Quart. of Applied Math., 49, no 2, 1991, pp. 215-235.
3. “Numerical Methods for an Optimal Investment-Consumption Model,” by B. G. Fitzpatrick and W. H.
Fleming, Math. of Oper. Res., 16, no. 2, 1991, pp. 823-841.
4. “Bayesian Analysis in Inverse Problems,” by Ben G. Fitzpatrick, Inverse Problems, 7, no 5, October
1991, pp. 675-702.
5. “Almost Sure Convergence in Distributed Parameter Identification Algorithms under Correlated
Noise,” by G. Yin and Ben G. Fitzpatrick, Appl. Math. Letters, 5, no. 4, 1992, pp. 41-44.
6. “Approximation and Control in Integral Equations of Nonlinear Vicsoelasticity,” by Ben G.
Fitzpatrick, J. Math. Systems, Estimation, and Control, 2, no. 4, 1992, pp. 483-501.
7. “A Fourth Order Scheme for Nonlinear Integral Equations of Viscoelasticity,” by Ben G. Fitzpatrick
and J. W. Gebbie, Appl. Math. Letters, 5, no. 3, 1992, pp. 63-67.
8. “On Invariance Principles for Distributed Parameter Identification Algorithms,” by G. Yin and Ben G.
Fitzpatrick, Informatica, 3, no. 1, 1992, pp. 98-118.
9. “Inverse Dynamics Paradigm: Adaptive Nonlinear Control and Identification of Large-Scale Power
Systems,” by R. C. Berkan, B. R. Upadhyaya, R. A. Kisner, and B. Fitzpatrick, Control--Theory and
Advanced Technology, 8, no 3, 1992, 465-477.
10. “Modeling and Estimation Problems for Structured Heterogeneous Populations,” by Ben G.
Fitzpatrick, J. Math. Anal. Appl., 172 no. 1, 1993, pp. 73-91.
11. “Parameter Estimation in Conservation Laws,” by Ben G. Fitzpatrick, J. Math. Systems, Estimation,
and Control, 3, no. 4, 1993, pp. 413-425.
12. “Empirical Distributions in Least Squares Estimation for Distributed Parameter Systems,” by Ben G.
Fitzpatrick and G. Yin, J. Math. Systems, Estimation, and Control, 5, no. 1, 1995, pp. 37-57.
13. “Statistical Tests of Fit in Estimation Problems for Structured Population Modeling,” by Ben G.
Fitzpatrick, Quart. Appl. Math., 53, no. 1, 1995, pp. 105-128.
14. “Approximation and Parameter Estimation Problems for Algal Aggregation Models,” by A. S. Ackleh,
B. G. Fitzpatrick, and T. G. Hallam, Math. Models and Methods in Appl. Sci., 3, no. 4, 1994, pp. 291311.
15. “Shape Matching with Smart Material Structures Using Piezoceramic Actuators,” by Ben G.
Fitzpatrick, J. Intell. Material Systems and Structures, 8 no. 10, 1997, pp. 876-882.
16. “Large Sample Behavior in Bayesian Analysis of Nonlinear Regression Models,” by Ben G.
Fitzpatrick and G. Yin, J. Math. Anal. Appl. 192, 1995, pp. 607-626.
17. “Modeling Aggregation and Growth Processes in an Algal Population: Analysis and Computations” by
A. S. Ackleh and B. G. Fitzpatrick, J. Math. Bio. 35, pp. 480-502, 1997.
214
18. “Analysis and Approximation for Inverse Problems in Contaminant Transport and Biodegradation
Models,” by Ben G. Fitzpatrick, J. Num. Func. Anal. Opt., 16, no. 7/8, 1995, pp. 847-866.
19. “On Continuous Dependence under Approximation for Groundwater Flow Models with Distributed
and Pointwise Observations,” by Ben G. Fitzpatrick and Michael A. Jeffris, Discrete and Continuous
Dynamical Systems, 2, no. 1, 1996, pp. 141-149.
20. “Estimation of Time Dependent Parameters in General Parabolic Evolution Systems,” by A.S. Ackleh
and Ben G. Fitzpatrick, J. Math. Anal. Appl. 203, 1996, pp. 464-480.
21. “Estimation of Discontinuous Parameters in General Nonautonomous Parabolic Systems,” by A.S.
Ackleh and Ben G. Fitzpatrick, Kybernetica 32 no. 6, 1996, pp. 543-556.
22. “An Adaptive Change Detection Scheme for a Nonlinear Beam Model,” by M. A. Demetriou and B.
G. Fitzpatrick, Kybernetica 32 no. 6, 1996, pp. 543-556.
23. “On Approximation in Total Variation Penalization for Image Reconstruction and Inverse Problems,”
by S. L. Keeling and B. G. Fitzpatrick, J. Num. Func. Anal. Opt. 18 , no. 9/10, 1997, pp. 941-958.
24. “Estimation of Groundwater Flow Parameters Using Least Squares.” by K. R. Bailey and B. G.
Fitzpatrick, Math. Comp. Modeling. 26 no. 11, pp. 117-127, 1997.
25. “Dispersion Modeling and Simulation in Subsurface Contaminant Transport,” by J. V. Butera, Ben G.
Fitzpatrick, and C. J. Wypasek, Math Models and Methods in Appl. Sci. 8 no. 8, 1998.
26. “Survival of the Fittest in a Generalized Logistic Model,” by A. S. Ackleh, D. Marshall, B. G.
Fitzpatrick, and H. Heatherly, Math Models and Methods in Appl. Sci. 9 pp. 1379-1391, 1999.
27. “Sampling Distribution of Approximate Errors for Least Squares Identification,” by G. Yin, Ben G.
Fitzpatrick, and K. Yin, Stoch. Anal. Appl. 17 no. 2, 1999, pp. 295-313.
28. “Estrogen Treatment Enhances Hereditary Renal Tumor Development in Eker Rats,” by D. Wolf, T.
Goldsworthy, E. Donner, R. Hardin, B. Fitzpatrick, and J. Everitt, submitted.
29. “Penalized Least Squares Methods for Imaging through Turbulence,” by B. G. Fitzpatrick and M. C.
Horton, in preparation.
30. “Modulated Boolean Point Processes for Modeling Hydraulic Conductivity,” by B. G. Fitzpatrick and
C. J. Wypasek, in preparation.
Refereed Proceedings Papers and Book Chapters
31. “Numerical Methods for Optimal Investment-Consumption Models,” by B. G. Fitzpatrick and W. H.
Fleming, Proc. 29th IEEE Conference on Decision and Control, 1990, Volume 4, pp. 2358-2361.
32. “Invariance Principles and Applications to Distributed Parameter Identification,” by G. Yin and B. G.
Fitzpatrick, Proc. 29th IEEE Conference on Decision and Control, 1990, Volume 6, pp. 3556-3557.
33. “Numerical Solution of a Control Problem for Optimal Cooling of Viscoelastic Films,” by Ben G.
Fitzpatrick, in Computation and Control II: Proceedings of the Second Bozeman Conference, K.
Bowers, J. Lund, Eds., Birkhauser, Boston, 1991, pp. 115-123.
34. “Parameter Estimation in Conservation Laws,” by Ben G. Fitzpatrick, Proc. 30th IEEE CDC,
Brighton, UK, Dec. 1991, 977--978.
35. “Homogenization of Von Karman Plate Equations,” by Ben G. Fitzpatrick and D. A. Rebnord, Proc.
31st IEEE CDC, Tucson, AZ, Dec. 16--18, 1992, 1160-1163.
36. “The Linear Regulator Problem for Systems with a Distribution of Parameters,” by M. Aczon, H. T.
Banks, and Ben G. Fitzpatrick, Proc. 31st IEEE CDC, Tucson, AZ, Dec. 16--18, 1992, 1168-1171.
37. “Bootstrap Methods for Inference in Least Squares Identification Problems,'' by Ben G. Fitzpatrick and
G. Yin, Chapter 4 of Identification and Control of Systems Governed by Partial Differential
Equations, H. T. Banks, R. Fabiano, K. Ito, Eds., SIAM, Philadelphia, 1993, pp. 45--58.
38. “Rate Distribution Modeling for Structured Heterogeneous Populations,” by Ben G. Fitzpatrick,
Proceedings of the International Conference on Control of Distributed Parameter Systems, W. Desch,
F. Kappel, K. Kunisch, eds., Birkhauser, Basel, 1994, pp. 131-142.
39. “Estimation of Distributed Individual Rates from Aggregate Population Data,” by H. T. Banks, B. G.
Fitzpatrick, and Y. Zhang, Proceedings of the International Conference on Differential Equations and
Applications to Biology and Industry, Claremont, 1994.
215
40. “A Comparison of Estimation Methods for Hydraulic Conductivity Functions from Field Data,” by
Ben G. Fitzpatrick and J. A. King, Computation and Control IV: Proceedings of the Fourth Bozeman
Conference, K. Bowers, J. Lund, eds., Birkhauser, Boston, 1995, pp. 155-168.
41. “Least Squares Estimation of Hydraulic Conductivity from Field Data,” by Kendall R. Bailey, Ben G.
Fitzpatrick, and M. A. Jeffris, Proceedings of the 1995 ASME 15th Conference on Mechanical
Vibration and Noise and Design Technical Conferences.
42. “A Bounded Variation Approach to Inverse Interferometry,” by Ben G. Fitzpatrick, Stephen L.
Keeling, and Stacey G. Rock, Proceedings of the 1995 ASME 15th Conference on Mechanical
Vibration and Noise and Design Technical Conferences.
43. “Convergence and Large Deviations in a Bayesian Approach to Functional Estimation Problems,” by
Ben G. Fitzpatrick and G. Yin, Proceedings of the 1995 ASME 15th Conference on Mechanical
Vibration and Noise and Design Technical Conferences.
44. “On Line Estimation of Stiffness in Nonlinear Beam Models with Piezoceramic Actuators,” by M. A.
Demetriou and B. G. Fitzpatrick, Proceedings of the 1995 ASME 15th Conference on Mechanical
Vibration and Noise and Design Technical Conferences.
45. “Monte Carlo Estimation of Diffusion Distributions at Intersampling Times,” by J. V. Butera, B. G.
Fitzpatrick, and C. J. Wypasek, Stochastic Analysis, Control, Optimization, and Applications: a
Volume in Honor of Wendell Fleming, W. McEneaney, G. Yin, and Q. Zhang, eds, Birkhauser,
Boston, 1999, 505-520.
46. “Estimation of Probability Distributions for Individual Parameters Using Aggregate Population Data,”
by H. T. Banks, B. G. Fitzpatrick, L.K. Potter, and Y. Zhang, Stochastic Analysis, Control,
Optimization, and Applications: a Volume in Honor of Wendell Fleming, W. McEneaney, G. Yin, and
Q. Zhang, eds, Birkhauser, Boston, 1999, 353-372.
47. “Control for UAV Operations under Imperfect Information,” by B. G. Fitzpatrick and W. McEneaney,
2002 AIAA Proceedings on Unmanned Air Vehicles, to appear.
48. “Mixed Initiative Planning and Control under Uncertainty,” M. Adams, W. Hall, M. Hanson, W.
McEneaney, and B. Fitzpatrick, 2002 AIAA Proceedings on Unmanned Air Vehicles, to appear
Non-Refereed Proceedings Papers and Technical Reports
49. “Inverse Problems for Distributed Systems: Statistical Tests and ANOVA,” by H.T Banks and B. G.
Fitzpatrick, in Proceedings of the International Symposium on Mathematical Approaches to
Environmental and Ecological Problems, Springer Lecture Notes in Biomathematics, 81, 1989, pp.
262-273.
50. “Statistical Methods for Parameter Identification and Model Selection in Distributed Systems,” by H.
T. Banks and B. G. Fitzpatrick, in Proceedings of the IFAC Fifth Symposium on Control of Distributed
Parameter Systems, June 26-29, 1989, Perpignan, France, pp. 191-193.
51. “Sample Distributions of Identification Algorithms for Distributed Parameter Systems,” by G. Yin and
Ben G. Fitzpatrick, Proceedings of the 1991 International Symposium on the Mathematical Theory of
Networks and Systems, Mita Press, Tokyo, 1992, pp. 569-574.
52. “Forecasting and Control Modeling for Inventory Planning Problems,” B. G. Fitzpatrick, H. T. Tran,
M. Brainard, S. Gray, M. C. Horton, M. H. Horton, and J. Schroeter, CRSC Technical Report TR9541.
53. “Automated DNA Sequence Quality Assessment for Diagnostics, Databases, and Forensics,” by J.
Sachs, A. Thatcher, C. Francis, and B. Fitzpatrick, D. H. Wagner Associates Report, 1998, 57 pages.
54. “Speech Recognition in Noisy Environments for Enhanced Tactical Performance,” by J. Sachs, M.
Grunert, B. Fitzpatrick, A. Christol, and C. Francis, D. H. Wagner Associates Report, 1998, 22
pages.
55. “Statistical Image-Based Approaches to Tracking Through Turbulence,” by B. Fitzpatrick, Tempest
Technologies Report, 1999, 35 pages.
56. “Phase Conjugation Control and Optimization Algorithms for Multiconjugate Adaptive Optics System
Design,” Tempest Technologies Report, 1999, 10 pages.
216
57. “Tracking Through Optical Turbulence,” by Ben G. Fitzpatrick, Tempest Technologies Report, 2000,
43 pages.
Invited and Contributed Conference Lectures
-
Contributed lecture at the Virginia Tech/ICAM conference on Numerical Solutions of Partial
Differential Equations, Blacksburg, VA, September 24-27, 1988.
Invited lecture at the Special Session on Mathematics in Population Biology, January 1989 Meeting of
the American Mathematical Society.
Contributed lecture at the Second International Conference on Mathematical Population Dynamics,
Rutgers, The State University of New Jersey, New Brunswick, New Jersey, May 17-20, 1989.
Invited Lecture at the 5th IFAC Symposium on Control of Distributed Parameter Systems, June 26-29,
1989, Perpignan, France.
Invited lecture at the Special Session on Control of Infinite Dimensional Systems, January 1990
Meeting of the American Mathematical Society.
Contributed lecture at the 1990 SIAM Annual Meeting, July 16-20, 1990.
Invited lecture at the 2nd Conference on Computation and Control, Montana State University,
Bozeman, Montana, August 1-7, 1990.
Invited lecture at 29th IEEE CDC, Honolulu, HI, December 5-7, 1990.
Contributed poster presentation at the Conference on Numerical Optimization Methods in Differential
Equations and Control, Raleigh, NC, July 15-17, 1991.
Contributed lecture at the Southeastern-Atlantic Regional Conference on Differential Equations,
October 25-26, 1991.
Contributed lecture at 30th IEEE CDC, Brighton, UK, December 11-13, 1991.
Contributed lecture at the AMS-SIAM-IMS Summer Research Conference, South Hadley, MA, July
11-17, 1992.
Invited lecture, Special Session on Control Theory in Economics, First World Congress of Nonlinear
Analysts, Tampa, FL, August 19-26, 1992
Invited lecture, Special Session on Structured Models in Ecology, First World Congress of Nonlinear
Analysts, Tampa, FL, August 19-26, 1992.
Invited lecture at 31st IEEE CDC, Tucson, AZ, December 16-18, 1992.
Invited lecture at the Special Session on Numerical Optimization, Southeastern Regional Meeting of
the AMS, Knoxville, TN, March 26-27, 1993.
Invited Lecture at the International Conference of Control and Estimation of Distributed Parameter
Systems, Vorau, Austria, July 18-26, 1993.
Invited Lecture at MTNS-93, Regensburg, Germany, August 2-8, 1993.
Invited Lecture at the Claremont Industrial Mathematics Workshop on Environmental Modeling,
Claremont, CA, August 20-21, 1993.
Problem presenter, Claremont Graduate Student Workshop in Mathematical Modeling, June 6-14,
1994.
Invited Lecture at IMACS World Congress, July 11-15, 1994.
Invited lecture at the 4th Conference on Computation and Control, Montana State University,
Bozeman, Montana, August 2-9, 1994.
Invited lecture at the Special Session on Environmental Mathematics, January 1995 Meeting of the
American Mathematical Society.
Invited lecture at 1995 ASME 15th Conference on Mechanical Vibration and Noise and Design
Technical Conferences, Boston, MA, September 16-20, 1995.
Contributed lecture at the SIAM Annual Meeting, Charlotte, NC, Oct. 23-27, 1995.
Contributed lecture at the SIAM Symposium on Geophysical Inverse Problems, Dec. 16-20, 1995,
Yosemite, CA.
Plenary Lecture, Southeastern Regional SIAM meeting, Clemson, SC, Mar. 28-30, 1996.
217
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Invited lecture at the 5th Conference on Computation and Control, Montana State University,
Bozeman, Montana, July 30-Aug 1, 1996.
Invited lecture at the Atmospheric Modeling and Laser Propagation Workshop, Phillips Lab, Kirtland
AFB, Albuquerque, NM, December 10-11, 1996.
Invited lecture at the 1997 Barrett Lectures, University of Tennessee, Knoxville, March 20-22, 1997.
Contributed lecture at the 1999 SIAM conference on Geosciences, San Antonio, TX, March 24-27,
1999.
Invited lecture at the 1999 SIAM workshop on Industrial Mathematics, Claremont, CA, June 16-19,
1999.
Invited lecture at the ABL Advanced Concepts Workshop, Albuquerque, NM, August 30-Sept 1, 1999.
Panelist in session on Industrial Careers, Southern California MAA meeting, March 17, 2001.
Invited Lecture, SIAM Annual meeting, July 11-14, 2001.
Invited Lecture, SIAM Conference on Imaging Science, March 4-6, 2002.
Invited Colloquia and Seminar Talks
-
Vanderbilt University, 1988.
University of Kentucky, 1989.
University of Southern California, 1990.
University of Tennessee Chemical Engineering Colloquium, 1991.
University of Kentucky, 1991.
Oak Ridge National Lab, 1991.
Clemson University, 1991.
University of Tennessee Engineering Science and Mechanics Colloquium, 1992.
North Carolina State University, 1992.
Wayne State University, 1993.
University of Graz, Austria, 1993.
University of New Hampshire, 1994.
Brown University, 1995.
Worcester Polytechnic Institute, 1995.
University of Graz, Austria, 1996.
Armstrong Laboratories, Biomathematics group, Brook AFB, 1996.
Worcester Polytechnic Institute, 1997.
University of Southern Louisiana, 1997.
D. H. Wagner Associates, 1997.
University of Southern California, 1998.
Claremont Graduate University, 1999.
Research and Educational Funding
-
``Numerical and Statistical Methods in Identification and Control,'' AFOSR, 10/1/90 through 12/31/92
(at UTK), $40,874.
1991-2 Tennessee Science Alliance Research Award, $3,600.
``Statistical Techniques for Identification and Robust Control in Distributed Parameter Systems,''
AFOSR, 3/1/93 through 2/28/95, $75,184.
``Modeling and Estimation Problems in Structured Population Dynamics,'' NCSU Faculty Research
and Professional Development award, 1/1/94 through 12/31/94, $5,000.
``Probabilistic Simulation Models for East Coast Hurricane Frequency and Severity,'' NC Department
of Insurance, 9/30/94 through 5/30/95, $32,400.
“Statistical Techniques for Modeling, Estimation, and Optimization in Distributed Parameter Systems,
AFOSR, 3/1/95 through 2/28/98, $187,835.
``Forecasting and Control Modeling for Inventory Planning Problems,'' IBM, 5/15/95 through
12/31/95, $30,025.
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``Industrial Mathematics Modeling Workshop for Graduate Students,'' NSA, 6/21/95 through 6/20/96,
$17,000.
``Tumor Modeling,'' The Chemical Industry Institute for Toxicology, 8/16/96 through 12/31/96,
$14,476.
Graduate Students Directed
1. Azmy Ackleh, Master of Science, Mathematics, UTK, 1990: completed Ph. D. in mathematical
ecology at UTK, 1993; currently an assistant professor of mathematics at the University of Southern
Louisiana.
2. John Wise, Master of Science, Mathematics, UTK, 1991: currently working for The 4D Company in
Nashville, TN.
3. John Trawick, Master of Science, Mathematics, UTK, 1992: currently working for a consulting firm
in Atlanta, GA.
4. Jay King, Master of Science, Mathematics, NCSU, August 1994, currently working at an actuarial
consulting firm in Atlanta, GA.
5. Michael Jeffris, Master of Science, Mathematics, NCSU, August, 1994, Ph. D. 1998, NCSU,
currently on the technical staff at MIT Lincoln Labs.
6. Frank Malizzo, Master of Science, Mathematics, NCSU, August 1994.
7. Jeff Barrows, Master of Science, Mathematics, NCSU, December 1994, currently a member of the
technical staff at Honeywell.
8. Jeff Butera, Ph. D. in Computational Mathematics, NCSU, 1997, currently an assistant professor at
High Point University.
9. Kendall Bailey, M.S. in Applied Mathematics, December, 1996, currently working for Schneider
Logistics, Wisconsin.
10. Michael Horton, Ph. D. in Applied Mathematics, NCSU, in progress, defended thesis in June 2000.
Currently a member of the technical staff, MIT Lincoln Labs.
Undergraduate Research Direction

In the NSF-sponsored Research Experience for Undergraduates program at UTK, I directed 4 students
over 3 summers. Two of these, Jeff Butera and Yue Zhang, subsequently enrolled in graduate school
at NCSU and received Ph.D. degrees, and one of them, Melissa Aczon, has received an NSF Graduate
Fellowship, with which she is completing a Ph. D. in numerical analysis at Stanford.

In industrially-sponsored projects at NCSU, I have directed research of two undergraduates, Chris
Karlof and Mike Brainard. Karlof is currently attending graduate school in applied mathematics,
while Brainard works in statistical data modeling at Capital One.
Courses Taught
Precalculus, Calculus (including Honors sections), Business Math, Mathematical Modeling
(at the undergraduate and graduate levels), Numerical Analysis (at the undergraduate and graduate levels),
Real and Functional Analysis, Stochastic Processes, Linear-Quadratic Control Theory, Inverse Problems
and Parameter Identification, Distributed Parameter Modeling in
Biological Systems, and a special topics course in Probabilistic Modeling of Hurricane Frequency and
Severity.
Professional Service Activity
-
Referee on papers for J. Math. Bio., J. Math. Anal. Appl., J. Opt. Theory Appl., SIAM J. Control
Opt., Dynamics and Control, Proceedings of the IEEE CDC, J. Population Bio., Automatica,
Computers and Mathematics.
219
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Referee for proposals to AFOSR, NSF, and the Arkansas Science and Technology Authority.
Organized Special Sessions at Meetings:
 ``Optimal Control and Applications,'' AMS Southeastern Regional Meeting, Knoxville, TN, March 2627, 1993.
 ``Parameter Identification,'' 1995 ASME 15th Conference on Mechanical Vibration and Noise and
Design Technical Conferences, Boston, MA, September 16-20, 1995.
-
Co-hosted 5th annual Graduate Student Industrial Mathematics Modeling Workshop, (sponsored in
part by the National Security Agency), NCSU, August 7-16, 1995.
-
Steering Committee, Southeastern-Atlantic Regional Conference on Differential Equations; Chairman
of the Organizing Committee, 1995 SEARCDE; held at NCSU, 13-14 October, 1995.
220
Michael D. Grady
NAME:
Michael D. Grady
ADDRESS:
(Home) 8400 Naylor Ave. Los Angeles, CA 90045
(Work) Mathematics Department, Loyola Marymount University
One LMU Drive, Suite 2700, Los Angeles., CA 90045
(email) mgrady@lmu.edu
PHONE:
(Home) 310-641-7687
(Work) 310-338-5107
PERSONAL DATA
BORN:
August 3, 1946
MARRIED:
December 22, 1973 to Mary-Margaret McGlone
CHILDREN: Sean Michael born October 24, 1978
Ryan Joseph born November 24, 1982
EDUCATION
St. Thomas Aquinas High School
Florissant, Missouri
Graduated June, 1964
University of Missouri, St. Louis
St. Louis, Missouri
Major: Mathematics
B.A., June, 1968
Southern Illinois University, Edwardsville
Edwardsville, Illinois
Major: Mathemastics
M.S., August, 1970
University of Utah
Salt Lake City, Utah
Major: Mathematics
Ph.D. June, 1975
JOB EXPERIENCE
Chairperson Dept. of Mathematics
Loyola Marymount University
Los Angeles, California
May 1993 - present
Professor of Mathematics
Loyola Marymoumt University
Los Angeles, California
September 1987 - present
Associate Professor of Mathematics
Loyola Marymoumt University
Los Angeles, California
September 1982 - August 1987
Assistant Professor of Mathematics
Loyola Marymoumt University
Los Angeles, California
September 1975 - August 1982
AEC Trainee
July 1974 - August 1974
221
Aerojet Nuclear Corporation
Idaho Falls, Idaho
Teaching Fellow September 1970 - June 1975
University of Utah
Salt Lake City, Utah
Systems Analyst Trainee
McDonnell-Douglas Corporation
St. Louis, Missouri
July 1969 - August 1970
Weights EngineerJuly 1968 - June 1969
McDonnell-Douglas Corporation
St. Louis, Missouri
PUBLICATIONS
“Sufficient Conditions for anm Operator-Valued Feynman-Kac Formula,” Transactions of the American Mathematics
Society.
Solutions Manuals for four precalculus level Mathematics texts.
Wadsworth Publishing Company, Belmont, California
Several editions of each of the following:
Precalculus (with Beckenbach and Wooton)
Wadsworth Publishing Company, Belmont, California
Modern College Algebra and Trigonometry (with Beckenbach and Drooyan)
Wadsworth Publishing Company, Belmont, California
Functions and Graphs (with Beckenbach and Drooyan)
Wadsworth Publishing Company, Belmont, California
College Algebra (witj Beckenbach, Drooyan, and Wooton)
Wadsworth Publishing Company, Belmont, California
SERVICE
Three years on the University Budget Committee
Two years on the Finance Committee of the Board of Trustees
One year as president of the faculty governance organization
Several (more than 3) years as a member of the faculty governance organization
Several ad hoc committees including:
one that wrote the constitution for the faculty senate
one that wrote a student evaluation of faculty that we used for 5 years
PROFESSIONAL MEMBERSHIPS
Mathematical Association of America
222
Lily S. Khadjavi
700 Westmount #108
West Hollywood, CA 90069
310-659-6525
khadjavi@post.harvard.edu
310-338-5969
Loyola Marymount University
Department of Mathematics
1 LMU Drive, Suite 2700
Los Angeles, CA 90045-8130
lkhadjavi@lmu.edu
EDUCATION
University of California, Berkeley, CA. Ph.D. in Mathematics, May 1999.
Dissertation advisor: Hendrik W. Lenstra, Jr.
University of Oslo, Norway. Honors grades in Norwegian. Spring 1991.
Harvard University, Cambridge, MA. A.B., June 1990. Cum Laude. Honors Thesis in number theory.
John Harvard Scholarship. Elizabeth C. Agassiz Certificate of Merit.
ACADEMIC EMPLOYMENT
Loyola Marymount University, Los Angeles, CA. Assistant Professor of Mathematics, tenure track.
Courses taught include calculus, abstract algebra, complex analysis, linear algebra, and a senior seminar in
cryptography. Core class in cryptography for non-math majors. Advised a senior thesis in quantum
computing. Spring 2000-present.
Mathematical Sciences Research Institute. Member, Fall 1999.
Institute for Advanced Studies. Teaching Assistant, IAS/Park City Math Institute. Ran problem
sessions for ``Elliptic curves, modular forms, and l-adic representations”. Summer 1999.
University of California, Berkeley. Research Assistant , Spring 1995, Fall 1997, Spring 1998, and Spring
1999. Graduate Student Instructor, various semesters, 1993-1998. Outstanding Graduate Student
Instructor Award, 1996-1997.
University of California, Berkeley and Mills College. Seminar Assistant for the Summer Math Institute,
a selective summer program for talented undergraduate math majors chosen nationally, one program
especially for women and the other for minority students. Assisted in both, in number theory and in
coding theory, guiding students with course work and research projects. 1993, 1994, 1995.
Harvard University. Instructor and Teaching Fellow. Taught in both the college and the Extension
School (adult students). Fall 1990, Summer 1988, 1989, 1991.
223
GRANTS AND RECOGNITION
NSF Grant, “Bridges to Science, Engineering and Technology”, Summer 2004, pending.
Faculty Summer Research Grant, ``Elliptic Curves and the ABC Conjecture”, LMU, Summer 2003.
Social Justice Research Grant, “Fighting inequity using a quantitative perspective”, LMU, Summer 2002.
Visiting Scholar, University of Southern California, November 2001-2002.
Faculty Summer Research Grant, “Sparse polynomials and complexity theory”, LMU, Summer 2001.
Advisor of the Year Award, Loyola Marymount University, Spring 2001.
Project NExT Fellow, Mathematical Association of America, 2000-2001.
Association for Women in Mathematics-National Science Foundation Travel Grant, July 2000.
New Faculty Research Grant, “Elliptic curves and Belyi maps”, LMU, Summer 2000.
Outstanding Graduate Student Instructor Award, U.C. Berkeley, 1996-1997.
US Department of Education National Need Fellowship, U.C. Berkeley, 1992-1993.
PAPERS
“An effective version of Belyi's Theorem,” Journal of Number Theory, 96 (2002), no. 1, 22-47.
“Belyi maps and elliptic curves,” with Victor Scharaschkin, submitted.
“Elliptic curves and the ABC Conjecture,” with Victor Scharaschkin, in preparation.
Current research projects:
Minimal degree Belyi maps and Belyi heights for algebraic curves.
Compositions of sparse polynomials and complexity questions.
Group write-ups from Mt. Holyoke R.E.U.:
“Sobolev Gradient in Steepest Descent”
“Coordinate-free Generalizations of the Gershgorin Circle Theorem.”
RESEARCH TALKS
Algebra and Combinatorics Seminar, University of Queensland, July 2003.
AMS-MAA Joint Meetings, Baltimore, January 2003.
Algebra Seminar, University of Southern California, April 2002.
Loyola Marymount Math Department Seminar, Los Angeles, CA, December 2001.
224
Claremont Colleges Math Colloquium, Pomona College, CA, November 2000.
Loyola Marymount Math Department Seminar, Los Angeles, CA, November 2000.
Five College Number Theory Seminar, Amherst College, Massachusetts, October 1999.
Number Theory Seminar, U.C. Berkeley, December 1998.
Second Annual Missouri Algebra Weekend, University of Missouri, October 1998.
Intercity Number Theory Seminar, Universiteit Leiden, Netherlands, March 1998.
West Coast Number Theory Conference, Asilomar, CA, December 1997.
Santa Clara University Math Colloquium, Santa Clara, CA, November 1997.
NSF Conference to Celebrate Women in Number Theory & Analysis, U.C. Berkeley, August 1997.
AWM/MSRI Julia Robinson Celebration of Women in Math, presented poster.
July 1996.
TEACHING TALKS
International Conference on the Teaching of Mathematics II, “Increasing retention of underrepresented
students through cooperative learning workshops'”, August 2002.
PDP, U.C. Berkeley, November 1999. Panel for undergraduates, “What can you do with a math major?”.
Math 300, U.C. Berkeley, November 1998. Panel in department course for training new teaching
assistants.
AMS/MAA Joint Meetings, San Francisco, January 1995. Panel organized by Deborah Haimo & Danny
Goroff on “The Chilly Classroom Climate,” a discussion of problems that can arise teaching.
PDP Orientation, U.C. Berkeley, Fall, 1994. Organized the training for new and returning teaching
assistants leading workshop sections.
225
PROFESSIONAL DEVELOPMENT
Project NExT Fellow, MAA, 2000-2001. Participant in workshops at MAA Mathfest, UCLA, August
2000; AMS-MAA Joint Meetings, January 2001; Mathfest, Madison, August 2001.
Hewlett Diversity Workshop, Center for Teaching Excellence. Participant, Loyola Marymount University,
Fall 2000.
SERVICE
Faculty Advisor. Advisor for three math major undergraduates, 2000-present.
Math Club/Pi Mu Epsilon Faculty Moderator, 2000-2001, 2002-2003.
Association for Gay and Lesbian Awareness/Gay Straight Aliance, 2000-present.
Advisor of the Year award, Loyola Marymount University, Spring 2001.
College Bound. Weekend program for African-American high school students interested in engineering.
Teach mathematics with engineering professor and undergraduate assistants. 2001-2002, 2003-present.
Science and Engineering Community Outreach Program, Loyola Marymount University. Summer
program for African-American, Latina/o, and Native American high school students to learn about
engineering. Students recruited from College Bound, Boyle Heights College Institute, and other local area
enrichment programs for at-risk students. Summer 2001, 2003.
Department of Mathematics Committees
Hiring Committee, 2001-2002.
Scholarship Committee, Fall 2002-2003.
Scheduling Committee, Fall 2002-2003.
Student Recruitment Committee, Spring 2002-present.
W.A.S.C. (assessment committee), Spring 2002.
University Committees
Faculty Senate, Fall 2003- present
Intercultural Advisory Committee, Spring 2002-present.
Sigma Xi Speakers Committee, Spring 2002-present.
Student Affairs Committee, Fall 2001-present.
Intercultural College Facilitator, College of Science and Engineering, Loyola Marymount University,
Fall 2002-present.
External service:
Lenstra Treuerfeest, Session Chair, Spring 2003.
Southern California Section Project NExT, Organizing Committee.
Helped with budget and grant application. Fall 2001-present.
Exanding Your Horizons, annual science day on campus for high schools girls, Loyola Marymount
University. Helped prepare for event. Spring 2000-present.
226
Project F.R.E.E. and Equality California (EQCA), civil rights related work, such as voter registration in
under-represented communities, domestic partner legislation, and other outreach and education, Los
Angeles and Compton. Spring 2000-present.
MISCELLANEOUS
I am a member of the AMS, AWM, MAA, and was a member of the Noetherian Ring as well as an officer
of the Math Graduate Student Association at Berkeley (1996-1997). I enjoy study abroad and spent
several months at the Universiteit Leiden in the Netherlands in the Spring of 1995 and 1998.
Languages include, in descending order, Norwegian (reading, writing, and speaking), French (reading and
basic speaking), Spanish (basic reading), Farsi (food knowledge), and Dutch (een piepklein beetje).
227
Suzanne Larson
Mathematics Department
Loyola Marymount University
Los Angeles, CA 90045
Education
Claremont Graduate School
St. Olaf College
Mathematics Ph.D.
Mathematics B.A.
January 1984
May 1979
Experience
Assistant Professor
Assistant Professor
Associate Professor
Full Professor
Marquette University
Loyola Marymount University
Loyola Marymount University
Loyola Marymount University
1983 - 1986
1986 - 1990
1990 - 1994
1995 -
Teaching Activities
Dr. Larson has taught a wide variety of courses and has worked to develop new courses at Loyola Marymount
University. She has taught courses such as a freshman Mathematics Workshop, Abstract Algebra, Real Analysis,
Topology, Discrete Mathematics, Geometry, Linear Algebra, Advanced Linear Algebra, Advanced Linear Algebra,
Probability and Statistics, Introduction to Axiomatic Systems, and Senior Mathematics Seminar.
Dr. Larosn has 16 years experience in working with Expanding Your Horizons programs which target
underrepresented female students in mathematics and science.
Publications
 Convexity Conditions on f-Rings, Canadian Journal of Mathematics, Volume 38 Number 1, 1986, pp. 48-64.
 Pseudoprime -Ideals in a Class of f-Rings, Proceedings of the American Mathematical Society, Volume 104
Number 3, 1988, pp. 685-692.
 Minimal Convex Extensions and Intersections of Primary -Ideals in f-Rings, Journal of Algebra, Volume 123
Number 1, 1989, pp. 99-110.
 Primary -Ideals in a Class of f-Rings, Ordered Algebraic Structures, Kluwer Academic Publishers, 1989, pp.
181-186.
 Sums of Semiprime, z and d -Ideals in a Class of f-Rings, Proceedings of the American Mathematical Society,
Volume 109 Number 4, 1990, pp. 895-901.
 When is Every Order Ideal a Ring Ideal?, coauthored with M. Henriksen and F. A. Smith, Commentationes
Mathematicae Universitatis Carolinae, Volume 32 Number 3, 1991, pp. 411-416.
 Primary -Ideals in a Class of f-Rings, Communications in Algebra, Volume 20 Number 7, 1992, pp. 20752094.
 Square Dominated -Ideals and -products and Sums of Semiprime -Ideal in f-Rings, Communications in
Algebra, Volume 20 Number 7, 1992, pp. 2095-2112.
 Semiprime f-Rings that are Subdirect Products of Valuation Domains, coauthored with M. Henrkisen, Ordered
Algebraic Structures, The Conrad Conference, Kluwer Academic Publishers, 1993, pp. 159-168.
 Lattice-Ordered Algebras that are Subdirect Products of Valuation Domains, coauthored with M. Henriksen, J.
Martinez and R.G. Woods, Transactions of the American Mathematical Society, to appear.







-Ideals of the Form I I , I :
2
I , Ideals Satisfying I  I(I : I) , and Primary
-Ideals in a Class of f-
Rings, Communications in Algebra, Volume 22 Number 8, 1994, pp. 3107-3131.
A Characterization of f-Rings in Which the Sum of Semipri,me -Ideals is Semiprime and its Consequences,
Communications in Algebra, Volume 23 Number 14, 1995, pp. 5461-5481.
The Intermediate Value Theorem for Polynomials over a Class of Lattice-Ordered Rings of Functions,
coauthored with M. Henriksen and J. Martinez, General Topology and Applications, Annals of the New York
Academy of Sciences, Volume 788, 1996, pp. 108-123.
Quasi-Normal f-Rings, Ordered Algebraic Structures, Kluwer Academic Publishers, 1996, pp. 146-158.
f-Rings in Which Every Maximal Ideal Contains Finitely Many Minimal Prime Ideals, Communications in
Algebra, Volume 25 Number 12, 1997, pp. 3859-3888.
The Intermediate Value Theorem in f-Rings, Communications in Algebra, Volume 30 Number 5, 2002, pp. 24692504.
Constructing Rings of Continuous Functions in Which There are Many Maximal Ideals with Nontrivial Rank,
Communications in Algebra, Volume 31 Number 5, 2003, pp. 2183-2206.
228
Offices Held in Professional Organizations
 Second Vice Chair of the Southern California Section of the Mathematical Association of America, 1991-92.
229
Herbert A. Medina
Department of Mathematics
Loyola Marymount University
One LMU Drive
Los Angeles, CA 90045
Tel: 310-338-5113; Fax: 310-338-3768
E-mail: hmedina@lmu.edu
Web Page: http://myweb.lmu.edu/medina
Education
 Ph.D. in Mathematics, University of California, Berkeley, 1992 Dissertation title: Hilbert Space Operators
Arising from Irrational Rotations on the Circle Group; Dissertation advisor: Henry Helson
 M.A. in Mathematics, University of California, Berkeley, 1987
 B.S. in Mathematics/Computer Sciemce, University of California Los Angeles, 1985
Academic Positions
8) Loyola Marymount University, Mathematics, Assistant Professor, 1992-1999; Associate Professor 1999-Present
Full-time, tenured faculty member.
 University of Puerto Rico - Humacao, Summer Institute in Mathematics for
Undergraduates (SIMU), Co-Director, Summer 1998-2002
Served as Co-Director and Co-Principal Investigator for SIMU, a six-week academic and research
program in mathematics for undergraduates from the United States and Puerto Rico
• Cornell University, Visiting Assistant Professor, Biometrics Unit, Summer 1996
Served as Summer Director for the Cornell-SACNAS Mathematical Sciences Summer Institute, a sixweek undergraduate program in mathematical biology
• Summer Mathematics Institute, U.C. Berkeley, Seminar Leader, Summer 1993, 1994
Taught a six-week course in continued fractions at the honors undergraduate level to a group of twelve
gifted minority undergraduates from across the country
• U.C. Berkeley, Minority Engineering Program, Calculus Instructor, 1986-1991
Lectured and organized an intensive two-week calculus course for approximately fifty students entering
the college of engineering
Consulting Work
• Godbe Research and Analysis, Half Moon Bay, California, 1996
Designed and analyzed a survey for Bay Area Rapid Transit (BART) that measured the effectiveness of a
BART mailer sent to new Bay Area residents
Mathematical Publications

Unitary Operators Arising from Irrational Rotations on the Circle Group, Michigan Mathematical
Journal, 41 (1994), 39-49.

On functions that are trivial cocycles for a s et of irrationals, II, with L.W. Baggett and K.D.
Merrill, Proceedings of the American Mathematical Society, 124 (1) (1996), 89-93.

Connections Between Additive Cocycles and Bishop Operators, Illinois Journal of Mathematics, 40
(3) (1996), 432-438.

Simultaneously symmetric functions, with L.W. Baggett and K.D. Merrill, American Mathematical
Monthly, 104 (6) (1997), 520-528.

Cohomology of polynomials under an irrational rotation, with L.W. Baggett and K.D. Merrill,
Proceedings of the American Mathematical Society, 126 (10) (1998), 2909-2918.

Generalized multiresolution analyses, and a construction procedure of all wavelet sets in Rn, with
L.W. Baggett and K.D. Merrill, Journal of Fourier Analysis, 5, (6), 1999, 563-573.

A sequence of Hermite interpolating-like polynomials for approximating inverse tangent, submitted
to the American Mathematical Monthly and available online.

The diagonalizable and nilpotent parts of a matrix, available online.

Apuntes de la Teoría de la Medida, in preparation and available online.
Other Publications
 Mid-SIMU Thoughts, with P.V. Negrón and I. Rubio, SACNAS News 2 (2) (1998), 13-14.
230
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



The Summer Institute in Mathematics for Undergraduates (SIMU): Aiming to Increase the Representation of
Latinos and Native Americans in Mathematics, with I. Rubio, Council on Undergraduate Research Quarterly, 20
(2) (1999), 64-71.
The Summer Institute in Mathematics for Undergraduate (SIMU) at the University of Puerto Rico - Humacao,
with I. Rubio, in Proceedings of the Conference on Summer Undergraduate Mathematics Research Programs, J.
Gallian, editor, American Mathematical Society, Providence, RI. 2000, 169-180.
How program design and implementation leads to achieving goals, in Proceedings of the Conference on Summer
Undergraduate Mathematics Research Programs, J. Gallian, editor, American Mathematical Society, Providence,
RI, 2000, 315-321.
How much should the Catholic university justice?, in The Just One Justices: The Role of Justice at the Heart of
Catholic Higher Education, M. McCullough, editor, Scranton Press, Tonawanda, NY, 2000, 81-88.
U.S. ethnic minorities earning mathematics degrees: A look at the numbers, in preparation.
Grants
 Cocycles of an Irrational Rotation, Loyola Marymount University, 1993 Summer Research Grant, $3,000.
 Continued Fractions, Loyola Marymount University, 1995 Summer Research Grant, $3,000.
 Cornell-SACNAS Mathematical Sciences Summer Institute, 1996, Co-Principal Investigator, National Security
Agency (NSA), $100,000; National Science Foundation (NSF), $55,000.
 Wavelets, Loyola Marymount University, 1997 Summer Research Grant, $3,000.
 Two Problems Arising from Irrational Rotations on the Circle, Loyola Marymount University, 1998 Summer
Research Grant, $3,000.
 Summer Institute in Mathematics for Undergraduates (SIMU) at the University of Puerto Rico - Humacao, CoPrincipal Investigator
a. 1998, National Security Agency (NSA), Grant No. MDA904-98-1-0526, $120,000;
National Aeronautics and Space Administration (NASA), $25,000; Alliance for Minority Participation
(AMP), $10,000.
b. 1999, National Security Agency (NSA), Grant No. MDA904-99-1-0041, $100,000; National Science
Foundation (NSF), Grant No. DMS-9907887, $47,000; National Aeronautics and Space Administration
(NASA), $12,000; Alliance for Minority Participation (AMP), $10,000.
c. 2000-2002, National Science Foundation (NSF) Research Experience for Undergraduates (REU), Grant No.
DMS-9987901, $360,000.
d. 2000, National Security Agency (NSA), Grant No. MDA904-00-1-0005, $120,000; Alliance for Minority
Participation (AMP), $10,000.
e. 2001, National Security Agency (NSA), Grant No. MDA904-01-1-0053, $120,000.
f. 2002, National Security Agency (NSA), Grant No. MDA904-02-1-0006, $120,000.
Mathematical Presentations
 Cocycles of an Irrational Rotation with One Point of Discontinuity, Kempner Colloquium, University of
Colorado, Boulder, March 16, 1992.
 A Glimpse at Continued Fractions, The Colorado College, December 10, 1992.
 A Result on Unbounded Additive Cocycles, The Colorado College, December 11, 1992.
 Teoría de la Medida, Universidad de El Salvador, a month-long series of talks on measure theory, July 1995.
 Symbolic and Numerical Study of Helixes in R3, University of Texas , El Paso, March 15, 1996.
 Cohomology of Polynomials Under an Irrational Rotation, Joint Mathematics Meetings, San, Diego, January 10,
1997.
 Simultaneously Symmetric Functions, XII Seminario Interuniversitario de Investigación Matématica, University
of Puerto Rico - Mayagüez, February 22, 1997.
 Estudio Simbólico y Numérico de hélices en R3, Seminario Matemático, University of Puerto Rico - Hum,acao,
April 15, 1997.
 A Procedure for Constructing all Wavelt Sets in Rn, XIII Seminario Interuniversitario de Investigación
Matématica, University of Puerto Rico - Humcao, February, 28, 1998.
 Using Mathematica to Illustrate Theorems from Advanced Linear Algebra, Joint Mathematics Meetings, New
Orleans, LA, January 12, 2001.
 A Sequence of Polynomials for Approximating Inverse Tangent, Tulane University, April 26, 2001.
 The Importance of Orthonormal Bases: An Introduction to Compressing Information, Annual Conference of the
Society for Advancement of Chicanos and Native Americans in Science (SACNAS), Anaheim, CA, September
27, 2002.
231

A Sequence of Hermite-Interpolating-Like Polynomials for Approximating Inverse Tangent, Los Angeles
Dynamics Seminar, California State University Long Beach, February 22, 2003.
Other Professional Presentations
 On Mathematics Summer Programs Aimed at Ethnic Minorities, The Mathematics Summer Experience: A
Working Conference on Mathematics Programs for Undergraduates, Crystal City, VA, October 2, 1999.
 Summer Programs’ Role in Cultivating Mathematical Talent within Groups Under-represented in Mathematics,
Models for Integrating Research into the Undergraduate Mathematics Curriculum, University of Arizona,
February 25, 2000.
 The Summer Institute in Mathematics for Undergraduates (SIMU): Opportunities and Challenges in Cultivating
Mathematical Talent, Joint Mathematics Meetings, Baltimore, MD, January 17, 2003.
Undergraduate Research Directed
1. Rebecca E. Pablo, A Geometric Look at the Continued Fractions of Quadratic Irrationals, a poster presentation
at the Annual Conference of the Society for the Advancement of Chicanos and Native Americans in Science
(SACNAS), El Paso, Texas, January 1995.
2. Rebecca E. Pablo, Symbolic and Numerical Study of Sexy Curves in 3-Dimensional Real Space, a poster
presentation at the meeting of the Southern California Section of the Mathematics Association of America, Los
Angeles, California, March 1995.
3. Albert J. Cortez, Transcendental Numbers with Bounded Partial Quotients: An Algorithm for Construction, an
oral presentation at the National Conference on Undergraduate Research (NCUR), Schenectady, New York,
April 1995.
4. Jessica A. Marzec, Fractal Dimension of Alfalfa Roots, co-directed with Gary Kuleck, a poster presentation at the
National Conference on Undergraduate Research (NCUR), Asheville, North Carolina, April 1996.
Other Academic Activities
1. Interactions Between Ergodic Theory and Number Theory, organized with L.W. Baggett and K.D. Merrill,
Special Session, Joint Mathematics Meetings, San Diego, January 1997.
2. Conference on Ergodic Theory and Dynamical Systems, invited participant, organized by the Technical
university of Wroclaw, Szklarska Poreba, Poland, September 7-13, 1997.
3. Junior Faculty Issues, panel participant, Society for Advancement of Chicanos and Native Americans in Science
(SACNAS) Annual Conference, October 10, 1997.
4. The Summer Mathematics Experience: A Working Conference on Summer Mathematics Programs for
Undergraduates, Organizing Committee Member, Crystal City, VA, September 30 - October 3, 1999.
Courses Taught at Loyola Marymount University
Real Analysis I, II; Methods of Applied Mathematics; Senior Seminar: Advanced Linear Algebra; Probability
and Statistics; Linear Algebra; Ordinary Differential Equations; Precalculus; Calculus I, II, III; Calculus for
Business Majors; Calculus for the Life Sciences II; Mathematics for Elementary School Teachers in Spanish;
A Look at Cryptography Throughout the Ages.
University Service
• College of Science and Engineering Committees
Sigma Xi Speaker Committee, 1993-1998
• University Committees
Student Affairs Committee, 1994-1997
Committee on Excellence in Teaching, 1995-1998
President’s Committee on Diversity, Co-Chair, 1995-1997
American Cultures Committee, 1995-Present
Student Development Services Assistant Dean Search Committee, Chair, 1997
Multicultural Development Committee
Committee on Ethnic Minority Faculty Affairs, 1993-96; Chair, 1995-96
University Planning Council 1999-Present
President’s Fritz B. Burns Distinguished Teaching Award Selection Committee,
2003
• Additional Service
Member of the LMU Faculty Senate, 1994-1997
LMU Latino Faculty Association, Co-Chair, 1996-1997, 1999-2000
232
Professional Memberships
American Mathematical Society (AMS)
Mathematical Association of America (MAA)
Society for Advancement of Chicanos and Native Americans in Science (SACNAS)
Other Memberships/Boards
• Board of Directors, Central American Resource Center (CARECEN), Chair, 19961999; Member 1999-2000.
• Western Interstate Commission for Higher Education (WICHE), California
Commissioner, 2000-Present.
Languages
Spanish: read, write and speak fluently
French, German: reading comprehension
233
Blake Mellor
Mailing Address:
Loyola Marymount University
University Hall
One LMU Drive, Suite 2700
Los Angeles, CA 90045-2659
Phone: (310) 338-5775
E-mail: bmellor@lmu.edu
URL: http://myweb.lmu.edu/bmellor
Positions Held
2002-present: Assistant Professor of Mathematics, Loyola Marymount University
1999-2002: Assistant Professor of Mathematics, Florida Atlantic University (Honors College)
1994-1999: Graduate Student Instructor, University of California, Berkeley
Education
1999: Ph.D. Mathematics, University of California, Berkeley
Dissertation: "Finite Type Link Homotopy Invariants", under Professor Robion Kirby
1993: B.A., Mathematics, magna cum laude, Harvard University
Thesis: "Heegaard Splittings and Casson's Invariant", under Professor Clifford Taubes
Papers
"A few weight systems arising from intersection graphs", preprint, May, 2002 (available at arXiv:math.GT/0004080),
to appear in the Michigan Mathematical Journal
"A geometric interpretation of Milnor's triple invariants", with Paul Melvin, Algebraic and Geometric Topology, vol.
3 (2003), paper no. 18, pp. 557-568
"On the existence of finite type link homotopy invariants", with Dylan Thurston, Journal of Knot Theory and its
Ramifications, vol. 10, no. 7, 2001, pp. 1025-1040
"Finite Type Link Homotopy Invariants II: Milnor's invariants", Journal of Knot Theory and its Ramifications, vol. 9,
no. 6, 2000, pp. 735-758
"Finite Type Link Concordance Invariants", Journal of Knot Theory and its Ramifications, vol. 9, no. 3, 2000, pp.
367-385
"The Intersection Graph Conjecture for Loop Diagrams", Journal of Knot Theory and its Ramifications, vol. 9, no. 2,
2000, pp. 187-211
"Finite Type Link Homotopy Invariants", Journal of Knot Theory and its Ramifications, vol. 8, no. 6, 1999, pp. 773787
Conferences Organized
Building a Community Partnership: Collaborations in Environmental Science, Education and Conservation, Honors
College, FAU, Jupiter, FL, July 2001 (Co-organizer)
Conference Sessions Organized
"Linking Mathematics with Other Disciplines", MAA Session, Joint Mathematics Meetings, Baltimore, January 18,
2003 (Co-organizer)
Panel on "Strategies for Mathematics for Liberal Arts", Project NExT, UCLA, August 3, 2000
Conference Presentations and Seminar/Colloquium Talks
"Discovery-based Science and Mathematics in an Environmental Context", MAA Poster Session on Projects
supported by the NSF DUE, Joint Mathematics Meetings, Baltimore, January 17, 2003 (with Stephanie
Fitchett, Honors College, Florida Atlantic University)
"To Have or Have Knot", Mathematics Colloquium, California Polytechnic University, San Luis Obispo, November
8, 2002 (Invited Talk)
"Finite type invariants and intersection graphs", Claremont Colleges Topology Seminar, October 29, 2002 (Invited
Talk)
"Seifert Surfaces and Milnor's Invariants", 965th meeting of the AMS, UNLV, Las Vegas, April 21-22, 2001
(Invited Talk)
234
"On the existence of finite type link homotopy invariants", Joint Mathematics Meetings, New Orleans, January 10-13,
2001
"A geometric interpretation of $\bar{\mu}(123)$", 959th meeting of the AMS, Columbia University, New York,
November 3-5, 2000
"Three weight systems arising from intersection graphs", AMS Mathematical Challenges of the 21 st Century, UCLA,
August 7-12, 2000
"Topological Psychology", MAA Mathfest, UCLA, August 3-5, 2000
"Intersection graphs and finite type invariants", Workshop on Low-Dimensional Topology, University of Warwick
(UK), July 10-21, 2000
"Finite Type Link Homotopy Invariants", 949th meeting of the AMS, UNC Charlotte, October 15-17, 1999
Conferences and Workshops Attended
ISAMA-BRIDGES, Granada, Spain, July 23-26 2003
MAA PMET workshop, SUNY Potsdam, June 8-19 2003
Southern California Topology Conference, Caltech, May 2003
Joint Mathematics Meetings, Baltimore, January 2003
Joint Mathematics Meetings, San Diego, January 2002
Building a Community Partnership: Collaborations in Environmental Science, Education and Conservation, Honors
College, FAU, Jupiter, FL, July 2001
Project Intermath Curriculum Workshop, Carroll College, Helena, MT, June 2001
Georgia International Topology Conference, University of Georgia, Athens, GA, May 2001
965th AMS meeting, UNLV, Las Vegas, April 2001
Joint Mathematics Meetings, New Orleans, January 2001
959th AMS meeting, Columbia University, New York, November 2000
AMS Mathematical Challenges of the 21st Century, UCLA, August 2000
MAA Mathfest, UCLA, August 2000
Workshop on Low-Dimensional Topology, University of Warwick, UK, July 2000
31st Southeastern International Conference on Combinatorics, Graph Theory and Computing, Florida Atlantic
University, March 2000
Joint AMS/MAA/SIAM Meeting, Washington, D.C., January 2000
949th AMS meeting, UNC Charlotte, October 1999
MAA Mathfest, Providence, R.I., July 1999
Joint AMS/MAA/SIAM Meeting, San Antonio, TX, January 1999
Kirbyfest, MSRI, UC Berkeley, June 1998
Joint AMS/MAA/SIAM Meeting, San Diego, CA, January 1998
Grants and Awards
LMU Summer Research Grant, 2003
NSF Grant #0088211, 2001-2003, Co-PI
"Discovery Based Science and Math in an Environmental Context"
Florida Atlantic University Research Initiation Award, #RIA-19, 2000
Academic Honors
Project NExT Fellow, 1999-2000
Department of Education National Needs Fellowship, 1993-1994
Harvard College Scholar, 1990-1991
National Merit Scholar
Teaching Experience
Loyola Marymount University
Differential Equations (Spring, 2003)
Discrete Methods (Spring, 2003)
Calculus I (Fall, 2002)
Honors College, Florida Atlantic University:
Symmetry (Spring, 2002)
Statistics (Fall, 2001)
Precalculus (Fall, 2000; Fall, 2001)
Calculus I (Spring, 2000; Spring, 2001)
235
Calculus II (Fall, 2000; Spring, 2002)
Calculus II for Physics (Spring, 2001)
Matrix Theory (Fall, 1999)
Discrete Mathematics (Spring, 2001)
Mathematical Reasoning (Spring, 2000; independent study)
Topology and Psychology (Spring, 2000; co-taught with Kevin Lanning)
Introduction to Programming in C (Fall, 1999)
Scientific Writing (Fall, 1999)
University of California, Berkeley:
Multivariable Calculus (Summer 1997)
Co-Instructor for undergraduate seminar on low-dimensional topology (Spring 1997)
Teaching Assistant for Calculus, Linear Algebra, Discrete Mathematics, Differentiable Manifolds (graduate
course)
Professional Service
Referee for Topology
Proposal reviewer for the National Science Foundation
University Service
LMU Math Dept. Curriculum Committee, 2002-2003
LMU Math Club Advisor, 2002-2003
HC Bylaws Committee (Chair), 2001-2002
HC Promotions and Tenure Guidelines Committee, 2001-02
Presiding Officer, Honors College Faculty Assembly, 2000-01
HC Academic Affairs and Student Life Committee, 2000-01
HC Promotions and Tenure Guidelines Committee, 2000-01
HC Screening Committee for Undergraduate Teaching Award (Chair), 2000-01
University Committee for Undergraduate Teaching Award, 2000-01
Search Committee for Mathematics, 2000-01
HC Faculty Committee (Chair), Fall 1999
HC Admissions Committee (Chair), 1999-2000
HC Promotions and Tenure Guidelines Committee, 1999-2000
HC Academic Affairs and Student Life Committee, 1999-2000
Search Committee for Mathematics (Chair), 1999-2000
Search Committee for Physics (Chair), 1999-2000
Search Committee for French, 1999-2000
University Faculty Council (alternate), 1999-2000
Languages: French, German.
Computer Languages: BASIC, Pascal, LISP, C, C++, Java, Unix
Professional Memberships
American Mathematical Society
Mathematical Association of America
Last Updated: 8/15/03
236
EDWARD C. MOSTEIG
Department of Mathematics, Loyola Marymount University
One LMU Drive Suite 2700, Los Angeles, CA 90045
Voice: 310-338-2381, Fax: 310-338-3768
emosteig@lmu.edu, http://myweb.lmu.edu/faculty/emosteig/
Education
Cornell University, Ph.D. in Mathematics, 2000
Cornell University, M.S. in Computer Science, 1999
University of Illinois, M.S. in Mathematics, 1996
University of Michigan, B.S. in Mathematics, 1993
Positions Held
Assistant Professor, Loyola Marymount University, August 2002 – present
Visiting Assistant Professor, Tulane University, August 2000 – July
2002
Publications
1. L. Fuchs, E. Mosteig. Additive Ideal Theory over Prüfer Domains,
Journal of Algebra, 252 (2002), 411–430.
2. E. Mosteig, M. Sweedler. Valuations and Filtrations, Journal of
Symbolic Computation, 34 (2002), 399–435.
3. E. Mosteig. Computing Leading Exponents of Noetherian Power
Series, Communications in Algebra, 30 (2002), 6055–6069.
4. I. H. Dinwoodie, E. Mosteig. Statistical Interference for Internal
Network Reliability with Spatial Dependence, SIAM Journal of
Discrete Math, to appear.
5. E. Mosteig, M. Sweedler. The Growth of Valuations on Rational
Function Fields, Proceedings of the American Mathematical Society,
to appear.
6. G. Boros, J. Little, V. Moll, E. Mosteig, R. Stanley. A Map on the
Space of Rational Functions, Rocky Mountain Journal of
Mathematics, to appear.
7. I. H. Dinwoodie, L. Matusevich, E. Mosteig. Transform Methods for
the Hypergeometric Distribution, submitted May 2002.
Teaching Experience
History of Mathematics
Department of Mathematics, Loyola Marymount University, Spring 2003
Taught course on the history of mathematics, emphasizing both European
and Non-European roots. Student research and classroom presentations
took place throughout the semester.
237
Calculus III
Department of Mathematics, Loyola Marymount University, Fall 2002,
Spring 2003
Instructed multidimensional calculus course that incorporated
collaborative learning. Combined lecture and workshop elements in a
class that did collaborative work on daily worksheets that ranged from
simple questions to small projects.
Euclidean and Non-Euclidean Geometry
Department of Mathematics, Loyola Marymount University, Fall 2002
Instructed a course in which the historical development of geometry was
discussed. An axiomatic approach was taken, developing Euclidean
geometry from the original axioms, gradually adding necessary axioms
and results as they were historically discovered, thus leading into
Non-Euclidean geometry.
Mathematics Instructor
Boeing Engineering Academy, Loyola Marymount University, 2002--2003
Instructed under-represented high school students on topics in
mathematics on weekends for the College Bound Outreach Program at
Loyola Marymount University. Introduced students to calculus themes
and its applications to engineering topics.
Gröbner Bases and Applications
Department of Mathematics, Tulane University Spring 2002
Taught course on applications of commutative algebra to various fields
of mathematics.
Linear Algebra
Department of Mathematics, Tulane University Fall 2001
Instructed honors undergraduate course that includes topics such as
linear transformations, eigenvalues, differential equations, and
applications. Computations, programs, and projects were written in
MATLAB.
Calculus III
Department of Mathematics, Tulane University Fall 2001
Instructed multidimensional calculus course that incorporated
collaborative learning. Combined lecture and workshop elements in a
class that did collaborative work on daily worksheets that ranged from
simple questions to small projects.
Combinatorics
Department of Mathematics, Tulane University Spring 2001
Instructed advanced undergraduate course that included topics such as
counting, generating functions, graph theory, constructing efficient
algorithms, and applications to algebra, number theory and
cryptography.
Graduate Algebra II
Department of Mathematics, Tulane University Spring 2001
238
Instructed course on introductory algebra including topics such as
Gröbner bases, categories, modules, and multilinear algebra.
Graduate Algebra I
Department of Mathematics, Tulane University Fall 2000
Instructed course on introductory algebra including group theory,
ring/field theory, and Galois theory.
Calculus I
Department of Mathematics, Tulane University Fall 2000
Instructed calculus course that made extensive use of graphing
calculators, wrote daily worksheets involving collaborative learning,
homework assignments, projects, and exams.
Gröbner Basis Seminar Associate
SIMU, University of Puerto Rico Summer 99
Instructed commutative and computational algebra to Latino and Native
American undergraduates at the Summer Institute of Mathematics for
Undergraduates. Duties consisted of monitoring and assisting groups
working on exercises in Gröbner Basis theory as well as running a
computer lab in which geometric computations were made using Maple.
Oversaw the groups working on research projects in areas such as
invariant theory, applications to chemistry, Padé approximations, and
linkages/robotics. Introduced some of the groups to software such as
CoCoA and Macaulay2 in the last half of the program.
Mathematics Instructor
Bridge Program, University of Illinois Summer 96, 97
Instructed intensive course for incoming minority undergraduate
students designed to improve arithmetical and elementary algebraic
skills to prepare them for their first year of college. Duties included
lecturing daily, developing curriculum, writing daily worksheets,
organizing group work sessions, and grading.
Math Coordinator
Skidmore College Summer 94, 95
Acted as liaison between the mathematics section of the local site at
Skidmore College and the CTY (Center for Talented Youth) headquarters
at Johns Hopkins University. Organized meetings, provided direction,
feedback, and aided in the development of curriculum for other math
instructors at the site. Instructed applied mathematics to gifted
middle school students in youth talent program.
Calculus Instructor
Department of Mathematics, Cornell University 8/96 – 5/99
Instructed calculus courses that made extensive use of graphing
calculators, wrote daily worksheets, homework assignments, quizzes, as
well as participated in weekly meetings that set the direction of the
course and developed the evening examinations and final. Worked solely
with two other graduate students on a project based calculus course in
which we developed the syllabus and coursework for the class. In
addition to lectures,
239
homework, quizzes, exams, and in-class projects, the students worked on
three major projects outside of class in which they met in groups with
the instructors to develop solutions to challenging problems. Students
later gave group presentations on a topic of their choice under the
direction of the instructors.
Mathematics Instructor
Department of Mathematics, University of Illinois 8/93 – 5/96
Instructed students and created curriculum for local version of
Harvard Calculus in spring 95 using the Hughes-Hallett, Gleason, et al.
textbook in conjunction with graphing calculators. Duties included
writing exams, in-class projects, daily worksheets, weekly assignments,
and evening projects. Developed curriculum and instructed college
algebra and precalculus courses in spring 94 and fall 94. Developed
in-class projects that made use of graphing calculators and taught
additional topics such as linearization via large-scale computations.
Acted as precalculus teaching assistant in fall 93, whose duties
include grading homework, quizzes, exams, running review sessions, and
heading a few sections of the course.
Automata and Complexity Theory Head Teaching Assistant
Department of Computer Science, Cornell University Fall 99
Gave occasional lectures, graded homeworks and exams, held office
hours, and ran review sessions for undergraduate honors course in
automata and complexity theory.
Coding Theory Teaching Assistant
Department of Electrical Engineering, Cornell University Fall 97
Graded homeworks and exams, held office hours, and ran review sessions
for graduate course in coding theory.
Merit Workshop Teaching Assistant
Department of Mathematics, University of Illinois Fall 95, 96
Organized group learning sessions for minority students in calculus
courses in which students are encouraged to work together to develop
problem solutions. Developed mini-course and research projects in which
students learned subjects outside of calculus such as number theory,
chaos, and discrete mathematics. Students met with instructors outside
of class to work on projects and develop a final presentation. Worked
with the
Calculus & Mathematica software developed at the University of
Illinois.
Students
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Kristina Altman, Tulane University, Master’s Thesis Director,
2002.
Michael Godzierez, Tulane University, Master’s Thesis Director,
2002.
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Selected Presentations
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Jan. 2003 CSU Poly-Pomona Colloquium, “Applications of Gröbner
Bases”
Apr. 2002 South Central Regional Weekend Algebra Conference, New
Orleans, LA. “The Growth of Valuations on Rational Function
Fields”
Mar. 2002 Colloquiumfest Conf. on Alg. Geo. and Computational
Algebra, Saskatoon, SK, Canada. “An Overview of Generalized
Gröbner Bases Via Valuations”
Jan. 2002 AMS-MAA Annual Joint Meetings, San Diego, CA. “Using
Valuations of Maximal Rank to Compute Gröbner Bases”
Sep. 2001 Grostat V Conference, New Orleans, LA. “Parameter
identifiability and estimation in a model of network reliability”
July 2001 University of Saskatchewan Math Colloquium, Saskatoon,
SK, Canada. “Applications of Gröbner Bases”
June 2001 IMACS Conference on Applications of Computer Algebra,
Albuquerque, NM. “Using Valuations to Compute Gröbner Bases”
April 2001 Algebra Weekend Conference, Columbia, MO. “An
Application of Puiseux’s Theorem”
Jan. 2001 AMS-MAA Annual Joint Meetings, New Orleans, LA. Panel
Member, “The Job Market”
Mar. 2000 Graduate Student Weekend, New Orleans, LA.
“Constructing Well-Ordered Maps”
Mar. 2000 Moravian College Math Colloquium, Bethlehem, PA.
“Computing Intersections of Surfaces”
Feb. 2000 Swarthmore College Math Colloquium, Swarthmore, PA. “An
Introduction to Gröbner Bases”
Feb. 2000 College of the Holy Cross Math Colloquium, Worcester,
MA. “Monomial Orders and Gröbner Bases”
Jan. 2000 AMS-MAA Annual Joint Meetings, Washington, D.C.
“Filtrations of Commutative Rings”
Oct. 1999 Route 81 Conference on Algebraic Geometry and
Commutative Algebra, Syracuse, NY. “Gröbner Bases Without Term
Orders”
Oct. 1999 1999 SACNAS National Conference, Portland, OR.
“Generalized Power Series and Valuations”
April 1999 NY Regional Graduate Math Conference, Syracuse, NY.
“Resolution of Planar Curve Singularities”
April 1997 Topics in Digital Communications, Ithaca, NY.
“Convolutional Codes and Inverses of Linear Sequential Circuits”
Feb. 1997 Seminar on Coding and Algebra, Ithaca, NY.
“Classification of Term Orders on Polynomial Rings”
April 1993 AMS Sectional Meeting, Washington, D.C. “Asymptotic
Cones of Elliptic Orbits in sp4(R)”
Oct. 1992 Undergraduate Symposium in Science, Argonne National
Laboratories, IL. “Adjoint Actions of Lie Groups”
Additional Experience
241
Grostat V, New Orleans, LA September 2001
Conference organizer for algebra session of Grostat V, the fifth annual
conference on applications of Gröbner bases and commutative algebra to
statistics.
Visiting Lecturer, University of Saskatchewan, Saskatoon, SA, Canada
July 2001
Gave a series of five lectures on applications of valuation theory to
computational algebra: (1) Term Orders and the Gröbner Basics, (2)
Constructing Well-Ordered Valuations, (3) The Digging Lemma and Rank
One Valuations, (4) A Proof of the Reverse-Well Orderedness of
Valuations Coming From Series with Negative
Support, and (5) Ideal Bases and Valuation Rings.
COCOA VII School on Computer Algebra, Kingston, ON, Canada Summer 2001
Studied applications of computational commutative algebra to automatic
theorem proving, industry, and algebraic geometry under the direction
of Tomas Recio, Laureano Gonzalez Vega, and Chris Peterson
COCOA VI School on Computer Algebra, Turin, Italy Spring 1999
Studied monomial ideals and ideals of points under the direction of
Tony Geramita, Lorenzo Robbiano, and Bernd Sturmfels.
Honors/Awards
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NSF Grant, Co-PI, Applications of Computational Algebra to HighDimensional Statistical Problems, 2002-2005.
Graduate Research Assistantship, Cornell University, 2000.
Graduate Teaching Assistantship, Cornell University, 1996 – 1999.
Nominated for Departmental Teaching Award, University of Illinois,
1994, 1995.
List of Excellent Teachers (Outstanding - Highest Ranking),
University of Illinois, 1994.
Graduate Teaching Assistantship, University of Illinois, 1993 –
1996.
High Honors in Mathematics, University of Michigan, 1993.
University of Michigan Alumni Association Scholarship, 1989 –
1990.
Department of Mathematics Scholarship, 1989 – 1990.
Committees and Memberships
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Hiring Committee, LMU, 2003-2004.
Scheduling Committee, 2003-2004.
Sophomore Scholarship Committee, LMU, 2002-2003.
Math Club Faculty Liaison, LMU, 2002-present.
Algebra Qualifying Exam, Wrote/Administered Tulane Graduate
Algebra Qualifying Exam, August 2001.
242
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Member, American Mathematical Society, 1993 – present.
Member, Mathematical Association of America, 1998 – present.
Member, Society for the Advancement of Chicanos and Native
Americans in Science, 1999 – present.
Member, Society for Industrial and Applied Mathematics, 1999 –
present.
Computer Committee Member, Center for Applied Mathematics, 1998 –
1999.
Bill Sears Colloquium Organizer, Center for Applied Mathematics,
1997.
Treasurer, Undergraduate Mathematics Organization, University of
Michigan, 1992 – 1993.
Additional Information
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Co-System Administrator, Center for Applied Mathematics, Summer
1998.
Programming: C, Java, Matlab, Maple, Mathematica, CoCoA,
Macaulay2, ML, HTML, LaTeX.
Platforms: LINUX, UNIX, Microsoft Windows.
Languages: French, Russian.
243
Patrick D. Shanahan
Department of Mathematics
Loyola Marymount University
One LMU Drive, Suite 2700
Los Angeles, CA 90045
Tel. (310) 337-7466
Fax. (310) 338-3768
E-mail pshanahan@lmu.edu
Education

Ph.D. Mathematics, University of California, Santa Barbara
Spring 1996
Dissertation: “Cyclic Dehn Surgery and the A-Polynomial of a Knot”
Advisor: Daryl Cooper
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M.A. Mathematics, University of California, Santa Barbara
Spring 1992
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B.A. Mathematics, California State University, Long Beach
Spring 1990
Academic Employment
9) Associate Professor, Loyola Marymount University
Fall 2003-Present
10) Assistant Professor, Loyola Marymount University
Fall 1996-Spring 2003
11) Teaching Associate, University of California, Santa Barbara
Fall 1994-Spring 1996
12) Teaching Assistant, University of California, Santa Barbara
Fall 1990-Spring 1994
Publications
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“A formula for the A-polynomial of twist knots.” Jim Hoste, co-author. To appear in Journal of Knot Theory
and Its Ramifications.
A First Course in Complex Analysis with Applications. Dennis Zill co-author. Jones and Bartlett Publishers.
Boston (2003).
“Trace fields of twist knots.” Jim Hoste, co-author. Journal of Knot Theory and Its Ramifications 10.4 (2001),
pp. 625-639.
“Cyclic Dehn surgery and the A-polynomial.” Topology and Its Applications 108 (2000), pp. 7-36.
“Three-dimensional representations of punctured torus bundles.” Brian Mangum, co-author. Journal of Knot
Theory and Its Ramifications 6.6 (1997), pp. 817-825.
Work in Preparation
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“A note on the commensurability classes of twist knots.” Jim Hoste, co-author/ Preprint.
“A proof of a theorem of Hodgson.” Jim Hoste, co-author. Unpublished manuscript.
“Strongly detected boundary slopes of the Whitehead link.” Alan Lash, co-author. Preprint.
Papers Presented at National Meetings
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“Trace Field of Twist Knots.” 107th Annual Joint AMS-MAA Mathematics Meeting. New Orleans, LA. January
2001.
“Degenerations of Representations and the Boundary Curve Space of the Whitehead Link.” 936th AMS
Meeting. Winston-Salem, NC. October 1998.
“Cyclic Dehn Surgery and the A-polynomial.” 104th Annual Joint AMS-MAA Mathematics Meeting.
Baltimore, MD. January 1998.
“Cyclic Dehn Surgery and the A-Polynomial of a Knot.” Wasatch Topology Conference. Park City, UT. June
1997.
244
Undergraduate Research Directed
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“Binomial Ideals from Graphs” by Elden Krause. Senior thesis. May 2002.
“A Differential Equation” by William Tsai. Math Horizons February 1999, p. 34. Solution acknowledged to
problem posed in September 1998 issue.
“Reducibility of Polynomials in Z4[x]” by Alysia Skilton. Solution submitted to problem posed in the College
Mathematics Journal February 1999 issue.
“A Tangle Model for Enzyme Action on DNA” by Patricia Cunningham. Seminar given at the Loyola
Marymount University Mathemtics Department Student/Faculty Colloquium. May 1997.
Academic Awards and Honors
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“Trace Fields of Hyperbolic Knots.” LMU Summer Research Grant, $4,000. June 2002.
“Hyberbolic Structures on Twist Knots.” LMU Summer Research Grant, $3,500. June 2000.
Invited Participant at the Conference in Honor of Robion Kirby. Mathematical Sciences Research Institute.
Berkeley, CA. June 1998.
MAA 1997-1998 Project NexT (New Experiences in Teaching) Fellow. Included grants from the Exxon
Education Foundation, $200, and the Los Angeles Collaborative for Teaching Excellence, $900.
“Three-Dimensional Representations of Knot Groups.” LMU Summer Research Grant, $3,000. June 1998.
Visiting Scholar, UCLA. Summer 1997.
“Detectable Surfaces in Two Component Link Complements. “ LMU Summer Research Grant, $3,000. June
1997.
Invited Participant at the Workshop on Computational and Algorithmic Methods in Three-Dimensional
Topology. Mathematical Sciences Research Institute. Berkeley, CA. March 1997. Included grant from the
National Science Foundation, $300.
Courses Taught
Quantitative Skills for the Modern World
Precalculus
Calculus I, II, and III
Methods of Proof
Real Variables I and II
Fundamental Concepts of Geometry
Senior Seminar: Calculus on Manifolds
Senior Seminar: Ideals, Varieties, and Algorithms
Elementary Statistics
Calculus for the Life Sciences
Ordinary Differential Equations
Linear Algebra
Abstract Algebra I and II
Complex Variables
Topology
Teaching Awards
Order of Omega Outstanding Professor Award. Spring 2001.
LMU Mathematics Society Professor of the Year. Spring 1999.
245
David M. Smith
Department of Mathematics
Loyola Marymount University
Los Angeles, CA 90045
310-338-5105
Personal:
U.S. Citizen, Married with two children
Education:
1971
1973
1978
Experience:
1978 - 1985
1979 - 1989
B.S.
M.S.
Ph.D.
1985 - 1989
1990 - 2003
Organizations:
Publications:
Oklahoma State University
Oklahoma State University
Oklahoma State University
Assistant Professor of Mathematics - LMU
Member of the Technical Staff, The Aerospace
Corp, Orbital Analysis Software Department
(Summer Employment)
Associate Professor of Mathematics - LMU
Professor of Mathematics - LMU
Association of Computing Machinery
Mathematical Association of America
A Program for Approximating Integrals, Byte (December, 1986, 113-112
Efficient Multiple-Precision Evaluation of Elementary Functions, Mathematics of Computation
(January, 1989) 131-134
A Fortran Package for Floating-Point Multiple-Precision Arithmetic, Transactions on
Mathematical Software (June, 1991) 273-283
Algorithm 693, Computer routines descriped in the 1991 TOMS article ACM Collected
Algorithms –– (June, 1991) www.netlib.org/toms/693
A Multiple-Precision Division Algorithm, Mathematics of Computation (January, 1996) 157163
Multiple Precision Complex Arithmetic and Functions, Transactions on Mathematical
Software (December, 1998) 359-367
Algorithm 786, Computer routines described in the 1998 TOMS article ACM Collected
Algorithms –– (December, 1998) www.netlib.org/toms/786
Multiple-Precision Gamma Function and Related Functions, Transactions on Mathematical
Software (December, 2001) 377-387
Algorithm 814, Computer routines described in the 2001 TOMS article ACM Collected
Algorithms –– (December 2001) www.netlib.org/toms/814
Using Multiple-Precision Arithmetic, Computing in Science and Engineering (July, 2003) 8893
Current research work:
I am currently working on a package containing multiple-precision routines for the exponential integral Ei(x),
generalized Exponential Integral En(x), Logarithmic Integral Li(x)., Sine Integral Si(x), Cosine Integral Ci(x),
Error functions Erf(x), Erfc(x), Bessel functions J n(x), Yn(x), and Fresnel Integrals C(x), S(x).
246
Warren S. Wright
Professor, Department of Mathematics
Loyola Marymount University
Los Angeles, CA 90045
swright@ lmu.edu
(310) 338 - 5114
Education
M.A. - 1965
University of Southern California
Los Angeles, California
Major: Mathematics
B.A. - 1963
Pomona College
Claremont, California
Major: Mathematics
Employment
1967 - present
Mathematics Faculty
Loyola Marymount University
Los Angeles, California
1965 - 67
Executive Officer and Commander
Headquarters Company
United States Army Air Defense School
Ft. Bliss, Texas
University Service
2002 – 2003
1997 - 2002
1993 - 1996
1989 - 1993
1989 - 1991
1987 - 1988
1986 - 1987
1978 - 1982
Chair, Senate Committee to Review Faculty Merit System
Chair, University Academic Planning and Review Committee
Chair, Rank and Tenure Committee
Chair, Mathematics Department
Chair, University Sabbatical Review Committee
President, Faculty Senate
Chair, University Academic Planning and Review Committee
Chair, Mathematics Department
247
Warren S. Wright
Publications - Textbooks
1995
Differential Equations with Computer Lab Experiments (with Zill) , PWS
Publishing Company
1995
Manual for Differential Equations with Computer Lab Experiments (with
Zill), PWS Publishing Company [A manual to accompany Differential
Equations with Computer Lab Experiments by Zill and Wright.]

Computer Lab Experiments in Differential Equations (with Zill), PWS Publishing Company
[A manual to accompany any beginning differential equations text.]
1988, 1983, 1978

Basic Mathematics for Calculus (with Zill and Dewar); Three Editions,
Wadsworth Publishing Co.
College Algebra and Trigonometry (with Zill and Dewar); Wadsworth Publishing Co.
Publications – Solutions Manuals
1999, 1992
Complete Solutions Manual and Student Solutions Manual to accompany
Zill and Cullen's Advanced Engineering Mathematics, Two Editions, PWSKent (first edition), Jones and Bartlett (second edition)
1998
Complete Solutions Manual and Student Solutions Manual to accompany
Zill's Differential Equations with Computer Lab Experiments (Second
Edition), PWS Publishing Company
2001, 1997, 1993
Differential
1989
Complete Solutions Manual to accompany Zill's A Frist Course in
Equations and Differential Equations with Boundary-Value Problems,
Three Editions, PWS-Kent Publishing Company (fourth and fifth editions),
Brooks-Cole (sixth and seventh editions)
2001, 1997, 1993,
Student Solutions Manual to accompany Zill's A First Course in
Differential Equations 1989, 1986, 1982
and Differential Equations with Boundary-Value
Problems, Six Editions, PWS-Kent
1979
Publishing Company (first five editions), Brooks-Cole (sixth and seventh
editions)
1995
Complete Solutions Manual and Student Solutions Manual to accompany
Zill and Wright's Differential Equations with Computer Lab Experiments,
PWS Publishing Company
248
1992, 1988, 1985
Complete Solutions Manual and Student Solutions Manual to accompany
Zill's Calculus, Three Editions; PWS-Kent Publishing Company
1990
Student Solutions Manual to accompany Zill and Dewar's Algebra and
Trigonometry, McGraw-Hill
1982
Student Solutions for The Fifth Edition of College Algebra to accompany
Beckenbach, Drooyan and Wooton’s College Algebra (Fifth Edition),
Wadsworth Publishing Co.
September 8, 2003
249
CONNIE J. WEEKS
Professor of Mathematics
Loyola Marymount Unviversity
One LMU Drive Suite 2700
Los Angeles, California 90045-8130
(310) 338-5108 or (310) 338-2774 (Sec.)
EDUCATION
Ph.D., Mathematics, January, 1978, University of Southern California
Dissertation: Estimation of States and Parameters in Continuous Nonlinear Systems with Discrete Observations.
Dissertation Advisor: Professor Alan Schumitsky
M.S., Mathematics, 1972, University of Southern California
B.S., Mathematics, 1970, Harvey Mudd College
ACADEMIC EXPERIENCE
Professor of Mathematics, Loyola Marymount University
September 1995 to Present
Associate Professor of Mathematics, Loyola Marymount University
September 1990 to June 1995
Assistant Professor of Mathematics, Loyola Marymount University
September 1986 to June 1990
Assistant Professor in the Department of Mechanical and Aerospace Engineering
Princeton University, July 1981 to June 1983
Assistant Professor of Mathematics, Loyola Marymount University
September 1977 to June 1979
INDUSTRY EXPERIENCE
Member of the Technical Staff, Jet Propulsion Laboratory, Future Mission Studies Group, Navigational Systems
section, July 1983 to September 1986. Performed orbit determination analysis for the Mariner Mark II Comet
Rendezvous/Asteroid Flyby Mission.
Member of the Technical Staff, Jet Propulsion Laboratory, Analysis and Simulations Group, Automated Systems
section, July 1979 to July 1981. Developed shape control and estimation algorithms for large space structures.
Member of the Technical Staff, The Aerospace Corporation, June 1978 to April 1979. Time Series Analysis and
Mathematical Modeling.
COURSES TAUGHT
250
Loyola Marymount University
MATH 685
MATH 620
MATH 598
MATH 598
MATH 598
MATH 357
MATH 355
MATH 321
MATH 322
MATH 245
MATH 234
MATH 131
MATH 130
MATH 122
MATH 121
MATH 112
MATH 111
Topics in Ordinary Differential Equations
Complex Analysis
Space Mathematics (Introduction to Orbit Determination)
Linear Optimization Theory (Graduate Course)
Topics in Optimization Theory (Senior Reading Course)
Complex Variables
Methods of Applied Mathematics
Real Variables I
Real Variables II
Ordinary Differential Equations
Calculus III (multivariable)
Calculus I
Pre-Calculus
Calculus for the Life Sciences
Pre-Calculus for the Life Sciences
Mathematical Analysis for Business II (Calculus)
Mathematical Analysis for Business I
California State University Northridge
Ma 450
Ma 245
Advanced Calculus
Ordinary Differential Equations
Princeton University
MAE 569
MAE 433
MAE 305
Linear Control and Estimation Theory
Automatic Control Systems
Mathematics in Engineering
My current MATH 575 class is a senior-graduate level introduction to orbit determination, taught from my own
notes. I have written chapters which utilize only the essentials of linear algebra, differential equations, functional
analysis, probability theory, estimation, and calculus of variations necessary to do the simplest problem. A computer
problem, a four-body problem of tracking a spacecraft orbiting an asteroid with Doppler data, completes the course.
The students were given the initial conditions for the Sun, Earth, asteroid and spacecraft. They were required to
integrate the equations of motion to obtain the positions of the Earth, asteroid and spacecraft with respect to the Sun,
compute the solution of the variational equations and the data partials, and to use the Kalman Filter to estimate the
position of the spacecraft.
RESEARCH GRANT
NASA Joint Venture (JOVE) Grant
June 1992-June 1995
AWARD: I received an award from the NASA/JOVE Program for the development of my Space Math class.
Departmental Committees
Curriculum, Scheduling, Hiring, Math Career, MAA Faculty Representative
251
UNIVERSITITY COMMITTEES
Loyola Marymount University (1986 to Present)
Rank and Tenure (Fall 2003-Spring 2009) Faculty Senate (1997-2003) President 2002-03, Vice President 2001-02
(Executive Committee, Committee on Committees, Bylaws, Chair of Elections Committee)
Research Committee (Fall 1995-Spring 1998)
Committee on the Status of Women (1996-7) Responsible for the portion of the Title 9 report concerning campus
policy on sexual harassment.
Faculty Advisor to the campus chapter of the Society of Women Engineers (SWE) for 1996-7.
Faculty Senate (Vice President 1994-95) (Secretary 1993-94)
University Committee on Committees (Fall 1993-Spring 1995) (Chairperson 1994-5)
Senate Elections Committee (1993-4)
University Library Committee (Fall 1993-Spring 1996)
Core Curriculum Committee (Science & Engineering 1991-2)
Representative to the Faculty Senate from Science & Engineering (Fall 1998-Spring 1991)
Princeton University
University Affirmative Action Committee
Departmental Graduate Committee
Loyola Marymount University (Fall 1977 to Spring 1979)
University Graduate Committee
Faculty Club Committee
Faculty Advisor to Pi Mu Epsilon
Faculty Advisor to Math Club
PROFESSOINAL AND HONORARY ORGANIZATIONS
Mathematics Association of America
American Astronautical Society
Sigma Xi, The Scientific Research Society
American Women in Mathematics
Pi Mu Epsilon
Conferences, Papers and Publications
Weeks, C.J. “A Test of Autonomous Navigation Using NEAR Laser Rangefinder Data” The Journal of the
Astronautical Sciences, Vol. 50, No. 3, 2002, pp. 325-337.
Weeks, C.J., Miller, J.K., Williams, B.G., “Calibration of Radiometric Data for General Relativity and Solar Plasma
During the Near-Earth Asteroid Rendezvous Spacecraft Solar Conjunction,” The Journal of the Astronautical
Sciences, Vol. 49, No. 4, October 2001, pp. 615-628.
Miller, J.K., Weeks, C.J., “Application of Tisserand’s Criterion to the Design of Gravity Assist Trajectories,” AIAA
2002-4717, Proceedings of the American Astronautical Society/American Institute of Aeronautics and Astronautics
Astrodynamics Specialist Conference, Monterey, California, August 2002.
“A Gravity Model for Navigation to Comets and Asteroids,” Paper AAS 02-140, AIAA/AAS Astrodynamics
Specialist Conference, San Antonio, Texas January 2002.
“A Test of Autonomous Navigation Using NEAR Laser Rangefinder Data,” AIAA/AAS Astrodynamics Specialist
Conference, Santa Barbara, CA., February, 2001, (Paper AAS 01-137).
252
“Calibration of Radiometric Data for General Relativity and Solar Plasma During the Near-Earth Asteroid
Rendezvous Spacecraft Solar Conjunction,” presented at the AAS/AIAA Spaceflight Mechanics Conference in
Girwood, Alaska, February , 1999, (Paper 99-440).
Weeks, Connie, Girardi, Anna, and Lynch, Jennifer, “Autonomous Navigation With Laser Altimetry,” Paper AAS
97-629, presented at the AAS/AIAA Astrodynamics Specialist Conference, August 1994, Sun Valley, Idaho.
Weeks, C.J., Bushelman, A., and Sabillo, R., “Direct Determination of a Spacecraft Orbit From the Doppler Data
Signature,” Paper AAS 95-144, presented at the AAS/AIAA Spaceflight Mechanics Meeting, February, 12-15, 1994,
Albuquerque, New Mexico.
Weeks, C.J., and Bowers, M.J., “Analytical Models of Doppler Data Signatures,” Journal of Guidance, Control and
Dynamics, Vol. 18, No. 6, November-December 1995, pp. 1287-1291.
Weeks, C.J., and Bowers, M.J., “Analytical Models of Doppler Data Signatures,” Paper AAS 94-178, AAS/AIAA
Spaceflight Mechanics Meeting, February 14-16, 1994, Cocoa Beach, Florida.
Weeks, C.J., “The Effect of Comet Outgassing and Dust Emission on the Navigation of an Orbiting Spacecraft,”
Paper AAS 93-624, AAS/AIAA Astrodynamics Specialist Conference in Victoria, B.C. on August 17, 1993.
Weeks, C.J., “The Effect of Comet Outgassing and Dust Emission on the Navigation of an Orbiting Spacecraft,” The
Journal of the Astronautical Sciences, Vol. 43, No. 3, 1995, pp. 327-343.
Miller, James K., Weeks, Connie J. and Wood, Lincoln D., “Orbit Determination Strategy and Accuracy for a Comet
Rendezvous Mission,” Journal of Guidance, Control and Dynamics, Vol. 13, No. 5, 1990, pp. 775-784.
Miller, James K., Weeks, Connie J., and Wood, Lincoln D.., “Orbit Determination of the Comet
Rendezvous/Asteroid Flyby Mission: Post Rendezvous,” AIAA Paper 89-0348, AIAA 27th Aerospace Sciences
Meeting, January 1989, Reno, Nevada.
Weeks, Connie J., “Orbit Determination for the Mariner Mark II Comet Rendezvous/Asteroid Flyby Mission: The
Orbiting Phase,” Advances in the Astronautical Sciences: Astrodynamics 1985, Vol. 58, Pt. II, Kaufman, B. et al., ed.,
Univelt, San Diego, 1986, pp. 1045-1064.
Weeks, Connie J., “Static Shape Determination and Control for Large Space Structures: I. The Flexible Beam,”
Journal of Dynamic Systems, Measurement and Control, Vol. 106, December 1984, pp. 261-266.
Weeks, Connie J., “Static Shape Determination and Control for Large Space Structures: II. A Large Space Antenna,”
Journal of Dynamic Systems, Measurement and Control, Vol. 106, December 1984, pp. 267-272.
Weeks, Connie J., “Static Shape Determination and Control for Large Space Structures, Jet Propulsion Laboratory
Report 81-71, October 1981. Pasadena: California Institute of Technology, 163, pp.
Weeks, Connie J., “Shape Control” Distributed Control of Large Space Structures, Jet Propulsion Laboratory Report
81-15, May 1, 1981, Pasadena: California Institute of Technology, pp. 37-59.
Weeks, Connie J., “Static Shape Determination and Control of Large Space Antenna,” Proceedings of the 20th
Annual Institute of Electrical and Electronics Engineers Conference on Decision and Control, December 1981, San
Diego (7 pp)
Weeks, Connie J., “Static Shape Determination and Control of Large Space Antenna,” Proceedings of the
International Symposium on Engineering Science and Mechanics, December 1981, Taiwan, Taiwan, National Cheng
Kung University Astronaughtic Society (15 pp).
Weeks, Connie J., “The Control and Estimation of Large Space Structures,” Proceedings of the Joint Automatic
Control Conference, August 1980, San Francisco, Vol. II (6 pp).
253
Weeks, Connie J., and Schumitzky, Alan, “Estimation of States and Parameters in Continuous Nonlinear Systems
with Discrete Observations,” Journal of Mathematical Analysis and Applications, Vol. 82, No. 1, July 1981, pp. 221254.
PRESENTATIONS AND SEMINARS
“Navigation of Missions to Comets and Asteroids,” presented to the Southern California Federation of Scientists,
September 10, 2002.
“Autonomous Navigation with LIDAR Altimetry,” Poster Presentation at the NASA PIDDP Workshop, Pasadena,
CA, June 10, 1997.
“Missions to Comets and Asteroids,” presented to the Kiwanis Club of Manhattan Beach, May 27, 1997.
“Mathematics in Space,” presented at the Expanding Your Horizons Conference at Loyola Marymount University,
April 24, 1994.
“The Effect of Comet Outgassing and Dust Emission on the Navigation of an Orbiting Spacecraft,” Poster
presentation at the NASA JOVE retreat July 10, 1993 in Galveston, Texas.
“Navigation Around Comets,” presented on Space Science Dauy at the University of Southern California, March
1993.
Keynote address on “The Exploration of the Solar System,” at the Expanding Your Horizons Conference at Loyola
Marymount University, April 27, 1991.
Keynote address to the Society of Women Engineers West Coast Conference at the Claremont Colleges, March 5,
1988.
“Orbit Determination for the Comet Rendezvous/Asteroid Flyby Mission,” Pi Mu Epsilon lecture at Loyola
Marymount University, March 1987.
Seminar: A series of eight lectures on Least Squares and Kalman Filtering, given at the Jet Propulsion Laboratory in
the Spring of 1986.
Colloquium: “The Application of Green’s Function Techniques to Problems in Control Theory and Estimation
Theory,” given to the Department of Mathematical Sciences, the University of Delaware, Newark, Delaware, March
1983.
Colloquium: “Static Shape Determination and Control for Large Space Structures,” Rensselaer Polytechnic Institute,
Troy, New York, April 15, 1982.
Chairman of Session on Tracking and Parameter Estimation, 1982 Conference on Information Sciences and Systems,
Princeton University, Princeton, New Jersey, March 17-19, 1982.
Seminar: “Shape Control and Determination for Large Space Structures,” Martin Marietta Corporation, Denver,
Colorado, July 13, 1981.
Colloquium: “Static Shape Control and Determination for Large Space Structures,” ta Harvey Mudd College,
Claremont, California, April 23, 1981.
Seminar: “The Space Shuttle and Large Space Structures,” given at Loyola Marymount University, Los Angeles,
California, November 19, 1980.
254
Thomas M. Zachariah
Department of Mathematics
Loyola Marymount University
Los Angeles, California 90045-2659
Telephone: (310)338-5109
e-mail: tzachari@lmu.edu
EDUCATION:
Claremont Graduate University
: Ph.D. (1984) in Mathematics (Geometric Probability).
: MA (1980) in Mathematics
TEACHING EXPERIENCE:
1994 - present:
1998 - 1994:
1986 - 1987
1984 - 1985
Associate Professor, Department of Mathematics, Loyola Marymount University, Los Angeles,
California 90045.
Assistant Professor, Department of Mathematics, Loyola Marymount University, Los Angeles,
California 90045.
Senior Lecturer. Applied Sciences Department at Guru Nanak Dev Engineering College, Bidar,
India.
Visiting Assistant Professor. Department of Mathematics, University of Alaska, Fairbanks,
Alaska.
SELECTED COURSES TAUGHT:
Calculus I, II, and III, Mathematics for Business I and II, Mathematics for Elementary School
Teachers I & II, Mathematical Modeling, Numerical Analysis, Ordinary Differential Equations,
Operations Research, Probability and Statistics, Statistics for Psychology Majors, and workshop
in Mathematics I & II.
CURRENT RESEARCH INTERESTS:
Mathematical modeling, geometrical probability and isoperimetric inequalities, integration of
computing in teaching mathematics.
SELECTED PUBLICATIONS AND PRESENTATIONS:
A Report on an Online Course for Non-science Majors, Proceedings of the Fifteenth International
Conference on Technology in Collegiate Mathematics, Vol. 15, 2003.
Developing Successful Math Majors: A two-Semester Course Sequence -Instructor’s Manual,
Jackie Dewar, Suzanne Larson, and Thomas Zachariah, manuscript completed, 2000.
Issues in Teaching an Undergraduate Math Modeling Course, Presentation at the. Mathematical
Association of America Southern California Section Meeting, fall 2000, Whittier College,
California.
255
Mathematical Modeling Using Mathematica, Presentation at the Second Biennial
Symposium on Mathematical Modeling in the Undergraduate Curriculum, University
of Wisconsin, La Crosse, June 1996.
Mathematics and Computing Technology, Presentation at the Seventh Annual
International Conference on Technology in Collegiate Mathematics held in November
1994.
An Introduction to Simulations in Modeling, COMAP modules, co-authored with
Robert Blatz, John Currano, KLD Gunawardena, Robert Nelson, and Dan Yates, July
1992.
External GRANTS:
Summer Research Experience for Community College Professors/K-12 Teachers, NSF,
summer 2003, involvement as a researcher.
Developing Successful Math Majors: A two-Semester Course Sequence -Instructor’s
Manual, LACTE, summer 2000, involvement as a course developer together with
Jackie Dewar and Suzanne Larson.
An Introduction to Probability and Statistics – Instructors Manual, LACTE, summer
2000, involvement as a course developer together with Suzanne Larson.
Mathematics Using Mathematica, NSF Instrumentation and Laboratory Improvement
Program, 2002 - 2004, involvement as the principal investigator.
Mathematical Modeling and Mathematica, Wolfram Research Inc., August 1992,
involvement as the principal investigator.
256
Dennis G. Zill, Ph.D.
Department of Mathematics
Loyola Marymount University
University Hall
One LMU Drive
Los Angeles, CA 90045
Email: dzill@lmu.edu
Office: (310) 338-5110
EDUCATION:
Ph.D. - 1967
Iowa State University
Major: Applied Mathematics
M.S. - 1964
Iowa State University
Major: Mathematics
B.A. - 1962
St. Mary's College (Minnesota)
Major: Mathematics
EMPLOYMENT:
1967-70,
Mathematics Faculty, Assistant Professor
Loras College
Dubuque, Iowa
1970-72,
Mathematics Faculty, Assistant Professor
California Polytechnic State University
San Luis Obispo, CA
1972 - present
Mathematics Faculty, Professor
Loyola Marymount University
Los Angeles, California
UNIVERSITY ADMINISTRATIVE POSITIONS:
1982
Chair, LMU Mathematics Department
1973-1977
Chair, LMU Mathematics Department
257
TEXTBOOK PUBLICATIONS
2002
A First Course in Complex Analysis (with P. Shanahan); Jones and
Bartlett Publishers
2001
A First Course in Differential Equations with Modeling Applications,
7th Edition; Brooks/Cole Publishing Co.
2000
Advanced Engineering Mathematics, 2nd Edition; Jones and Bartlett
Publishers
1998
Differential Equations with Computer Lab Experiments, 2nd Edition;
Brooks/Cole Publishing Co.
1995
Differential Equations with Computer Lab Experiments; PWS
Publishing Co.
1992
Advanced Engineering Mathematics (with M. Cullen); PWS-Kent
1992
Calculus, 3rd Edition; PWS-Kent
1990
Algebra and Trigonometry (with J. Dewar); McGraw-Hill Book Co.
1990
Trigonometry (with J. Dewar); McGraw-Hill Book Co.
1990
College Algebra (with J. Dewar); McGraw-Hill Book Co.
1984
Differential Equations with Boundary-Value Problems; PWS-Kent
1979
A First Course in Differential Equations with Applications; PWS-Kent
1979
College Algebra and Trigonometry (with W. Wright and J. Dewar);
Wadsworth Publishing Co.
1978
Basic Mathematics for Calculus (A pre-calculus text with W. Wright and
J. Dewar); Wadsworth Publishing Co.
1977
College Mathematics; Wadsworth Publishing Co.
1977
Introductory Calculus for Business, Economics and Social Science,
Wadsworth Publishing Co.
258