Advanced Mathematic s Educa tional S upport
Recommendations and Resources for Facilitating Collaboration Between Higher
Education Mathematics Faculty and Texas Public High Schools
First printing August 2003
Copyright 2003, The University of Texas at Austin. All rights reserved.
Permission is given to any person, group, or organization to copy and distribute this publication,
Advanced Mathematics Educational Support, for noncommercial educational purposes only, so long as the
appropriate credit is given. This permission is granted by the Charles A. Dana Center, a unit of the
College of Natural Sciences at The University of Texas at Austin.
About the Charles A. Dana Center’s Work in Mathematics and Science
The Charles A. Dana Center at The University of Texas at Austin works to support education leaders
and policymakers in strengthening Texas education. As a research unit of The University of Texas at
Austin’s College of Natural Sciences, the Dana Center maintains a special emphasis on mathematics
and science education. We offer professional development institutes and produce research-based
mathematics and science resources for educators to use in helping all students achieve academic
success. For more information, visit the Dana Center website at www.utdanacenter.org.
The development of Advanced Mathematics Educational Support was supported in part by the Sid. W.
Richardson Foundation and the Charles A. Dana Center at The University of Texas at Austin.
Additional funding was provided by an anonymous donor. Any opinions, findings, conclusions, or
recommendations expressed in this material are those of the authors and do not necessarily reflect
the views of the Sid W. Richardson Foundation or The University of Texas at Austin.
Author team and acknowledgments
Primary Authors
Ray Cannon, Department of Mathematics, Baylor University
Richard Parr, School Mathematics Project, Rice University
Ann Webb, College of Education, University of Texas at Tyler
Charles A. Dana Center Production Team
Bill Hopkins, AP Equity Initiative, Lead Editor
Kathi Cook, AP Equity Initiative, Editor
Susan Hudson Hull, Mathematics, Editor
Rachel Jenkins, Copy Editor
Amy Dolejs, Copy Editor
Phil Swann, Communications, Senior Designer
Advanced Mathematics Educational Support Advisory Team
Jasper Adams
Department of Mathematics, Stephen F. Austin State University
James Epperson
Department of Mathematics, University of Texas at Arlington
Gregg Fleisher
Advanced Placement Strategies, Inc., Dallas, Texas
Tony Hartman
Department of Mathematics, Texarkana College
Frank Hawkins
Department of Mathematics, Prairie View A&M University
Barbary Keith
Southwestern Regional Office, The College Board
Debbie Pace
Department of Mathematics, Stephen F. Austin State University
Spurgeon Parker
Department of Mathematics, Lee High School, Houston Independent
School District
Vince Schielack
Department of Mathematics, Texas A&M University
Uri Treisman
Charles A. Dana Center and Department of Mathematics,
The University of Texas at Austin
Gloria White
Eisenhower and Teacher Quality Grants, Texas Higher Education
Coordinating Board
Carol Williams
Department of Mathematics, Abilene Christian University
Some trademarked designations are used in this publication. Where we were aware of such a
designation, the trademarked term has been printed with initial capitalization. Registered trademarks
of the College Board used in this publication include Advanced Placement Program, AP, AP Vertical
Teams, College Board, Pre-AP, and SAT. (For more information on trademarked terms of The
College Board, see “The College Board Trademarks” at
www.collegeboard.com/html/trademark001.html.)
ACT is a trademark of ACT, Inc.
The Texas Academic Skills Program and TASP are trademarks of the Texas Higher Education
Coordinating Board and National Evaluation Systems, Inc.
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TEKS and TAKS Resources
The mathematics Texas Essential Knowledge and Skills (TEKS) were developed by the state of
Texas to clarify what all students should know and be able to do in mathematics in
kindergarten through grade 12. Districts are required to provide instruction that is aligned with
the mathematics TEKS, which were adopted by the State Board of Education in 1997 and
implemented statewide in 1998. The mathematics TEKS also constitute the objectives and
student expectations for the mathematics portion of the Texas Assessment of Knowledge and
Skills (TAKS).
The mathematics TEKS can be downloaded in printable format, free of charge, from the Texas
Education Agency website (www.tea.state.tx.us/teks). Bound versions of the mathematics and
science TEKS are available at cost from the Charles A. Dana Center at The University of Texas
at Austin (www.utdanacenter.org or 512-471-6190).
Resources for implementing the mathematics TEKS, including educator professional
development opportunities, are available through the Texas Education Agency and the Charles
A. Dana Center. Online resources can be found in the Mathematics TEKS Toolkit at
www.mathtekstoolkit.org.
Additional related products and services are available from the Dana Center via
www.utdanacenter.org.
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Table of Contents
Introduction.......................................................................................................................... 1
Why AMES Was Developed.............................................................................................................................................1
How AMES Was Developed.............................................................................................................................................2
Organization of AMES.........................................................................................................................................................2
A Shared Vision for Student Success: K–16 Mathematics in Texas ........................................ 3
The Texas High School Mathematics Program......................................................................................................3
K–12 Accountability in Texas—A Historical Perspective...............................................................................4
TAKS: Increased Accountability for High Schools.............................................................................................5
Transition to College............................................................................................................................................................6
Mathematics Education from the High School Perspective................................................... 7
Teacher Supply and Retention........................................................................................................................................7
The Situation.....................................................................................................................................................................7
Recommendations.........................................................................................................................................................7
Teacher Professional Development and Resources...........................................................................................8
The Situation.....................................................................................................................................................................8
Recommendations.........................................................................................................................................................9
Vertical Teaming....................................................................................................................................................................10
The Situation...................................................................................................................................................................10
Recommendations.......................................................................................................................................................10
Mathematics Education from the Higher Education Perspective.........................................12
Concurrent Enrollment....................................................................................................................................................12
The Situation...................................................................................................................................................................12
Recommendations.......................................................................................................................................................13
Standards for High School Calculus and Statistics.............................................................................................13
The Situation...................................................................................................................................................................13
Recommendations.......................................................................................................................................................14
Tenure and Promotion......................................................................................................................................................15
The Situation...................................................................................................................................................................15
Recommendations.......................................................................................................................................................15
Undergraduate Teacher Preparation..........................................................................................................................16
The Situation...................................................................................................................................................................16
Recommendations.......................................................................................................................................................16
Alternative Teacher Certification.................................................................................................................................17
The Situation...................................................................................................................................................................17
Recommendations.......................................................................................................................................................17
Promising Practices .............................................................................................................18
Master Teacher Summer Institutes..............................................................................................................................18
AP Summer Institute...........................................................................................................................................................18
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Advanced Mathematics Educational Support
The Rice University School Mathematics Project.............................................................................................19
Information Technology in Science Center for Teaching and Learning............................................19
UTeach .....................................................................................................................................................................................20
Systemic/Programmatic Administration Support..............................................................................................21
Lamar State College–Orange Mathematics Institutes........................................................................................22
Appendix A: Chapter 111. Texas Essential Knowledge and Skills for Mathematics .............. 24
Appendix B: Topic Outlines for Advanced Placement Mathematics.................................... 48
AP Calculus AB......................................................................................................................................................................48
AP Calculus BC......................................................................................................................................................................49
AP Statistics...............................................................................................................................................................................50
Appendix C: Vertical Teams: A Strategy for Building School Capacity..................................51
Higher Education Involvement in Vertical Teaming.......................................................................................51
Higher Education Leadership in Vertical Teaming..........................................................................................51
Vertical Teaming Not in Place..............................................................................................................................51
Vertical Teaming Already in Place.....................................................................................................................53
Vertical Alignment of the K–16 Curriculum.......................................................................................................54
References........................................................................................................................... 55
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Advanced Mathematics Educational Support
INTRODUCTION
Large-scale and well-replicated national studies (Adelman, 1999; Horn & Carroll, 2001) report
extremely high correlations between student participation in a rigorous high school curriculum
and their successful and timely completion of postsecondary education. Still, for far too many
students, academic preparation in mathematics is focused on passing state tests and completing
graduation requirements, rather than on taking challenging academic coursework that would
prepare them for success in postsecondary education. This document, Advanced Mathematics
Educational Support (AMES), was developed for higher education mathematics faculty to use as
they begin their work with schools to address this concern. It builds on the wisdom of
promising programs and practices from authors and advisors representing higher education and
high schools across Texas. AMES suggests strategies to improve communication and
collaboration between high school and higher education faculty, but it is not a synthesis of the
research. Instead, it is a collection of information, promising practices, and recommendations
designed to spark interest in working with public schools and help higher education faculty
begin a useful conversation with their public school counterparts about rigorous coursework
and student support.
Why AMES Was Developed
This document is intended for higher education faculty in mathematics. In it, we attempt to
capture the current situation of K–12 mathematics education from the view of public school
educators and higher education faculty, explore commonalities in the issues faced by both, and
propose some steps that higher education faculty might take to help public school educators
meet the goal of preparing all students for success in mathematics after high school.
Many college mathematics faculty report a wide variation in the mathematical preparation of
their first-year students. Roughly, these students fall into three groups. The first includes the
strongest students, those who have successfully completed four years of rigorous mathematics in
high school, including Advanced Placement (AP) courses, and who are well-prepared for
subsequent mathematics courses. The second group consists of students who have not taken
mathematics in their final year of high school and are ill-prepared to succeed in mathematics at
the collegiate level. The third group, consisting of students who have taken four years of
mathematics courses that were not at sufficient rigor to prepare them for success at the college
level, poses the most troublesome problems in placement. To increase the number of students in
the first group and decrease the number in the last two requires that college faculty become
more directly involved in students’ precollege mathematical preparation.
What would it take for public schools to provide a mathematics program so strong and inviting
that a large percentage of students—perhaps every student—could be prepared to successfully
complete challenging and rigorous mathematics courses throughout their four years in high
school? Precollege-level mathematics serves as a gateway to numerous postsecondary and career
opportunities for students. A strong mathematics background helps all students develop
analytical skills and knowledge that will be valuable in later life in a wide variety of settings.
Calculus, in particular, provides a useful background for work in many fields. Students majoring
in business, computer science, engineering, and the natural or life sciences in most four-year
institutions of higher education are required to successfully complete calculus. Those students
must enter college with a foundation in mathematics that at the very least prepares them for
calculus.
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Students, parents, high school counselors, teachers, and administrators must be made aware of
the fact that—while the state of Texas requires only three years of mathematics for high school
graduation—four years of substantive mathematics are needed to prepare for success at the
collegiate level. Many higher education mathematics faculty would suggest that more secondary
students should build a stronger foundation in algebra, trigonometry, and formal plane
geometry—rather than pushing to complete a calculus course in high school. The majority of
mathematics teachers on high school campuses would agree. In this document we explore ways
to foster dialogue and collaboration between higher education and K–12 faculty, with the
ultimate goal of increasing student participation and success in challenging and rigorous
mathematics courses throughout four years of high school.
How AMES Was Developed
AMES is the product of an advisory team working closely with three authors who have
experience in building collaborative relationships between their universities and public school
teachers. Each member of the advisory team was selected for his or her knowledge of the public
school situation or the higher education setting, or for experience in building bridges between
the two.
AMES is not a comprehensive inventory of strategies, nor is it a research study, or a survey of
the literature related to higher education–public school partnerships. Instead, it is a collection
of information, promising practices, and recommendations designed to spark interest in working
with public schools and help higher education faculty begin a useful conversation with their
public school counterparts.
Organization of AMES
AMES was developed to present views of Texas mathematics education from high school and
higher education perspectives. A shared understanding of both will be essential to improve
K–16 mathematics in Texas.
The section entitled Mathematics Education from the High School Perspective addresses issues
including the teaching force, professional development and resources, vertical teams, and
district concerns. The Texas accountability system sets the context for the high school
perspective. Mathematics Education from the Higher Education Perspective addresses issues of
concurrent enrollment, standards for high school calculus and statistics, tenure and promotion,
undergraduate teacher preparation, and alternative certification. Each of these sections
addresses the situation and includes general recommendations. Promising Practices includes
samples of model programs and approaches that address selected issues from the previous two
sections. The appendices include related information and resources.
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Advanced Mathematics Educational Support
A SHARED VISION FOR STUDENT SUCCESS: K–16 MATHEMATICS IN TEXAS
College mathematics faculty, college mathematics education faculty, and high school
mathematics educators share an ambitious vision for student achievement in mathematics. All
groups believe that all students should successfully complete high-quality mathematics courses
in high school that will enable them to enter higher education prepared to function on-level in
mathematics or to enter the workplace with the mathematical preparation necessary to succeed.
Thus, in practical terms, in Texas high schools, all students should successfully complete Algebra
I, Algebra II, and Geometry, and large percentages of students should successfully complete
Precalculus, AP Statistics, and/or AP Calculus.
Thus, all middle school, high school, and higher education mathematics educators should be
prepared in the mathematics content, pedagogy, and technology that will enable them to deliver
high-quality mathematics instruction to students. All public school and higher education
mathematics educators should have a good understanding of the state’s standards and vertical
alignment for mathematics content K–16, and should work closely together to promote student
success. While the state’s standards articulate the vertical alignment in the K–12 mathematics
curriculum, there is no such clear alignment between high school and college mathematics.
Before effective collaboration between college faculty and high school mathematics educators
can take place, both groups must have an understanding of the challenges that their colleagues
face daily.
The Texas High School Mathematics Program
For high school mathematics teachers in Texas, the challenges are many. These challenges
include
•
rigorous curriculum standards defined by the state,
•
state-mandated testing,
•
district-mandated curricular changes,
•
increased pressure to teach more students more mathematics,
•
changes in the format of the instructional day,
•
increased attention to meeting the affective needs of students, and
•
emphasis on educating all students, including those with special needs.
In Texas, the high school mathematics program has undergone substantial changes over a short
period. In 1997–98, the state adopted the Texas Essential Knowledge and Skills
(TEKS)—curriculum guidelines that establish what every student, from elementary school
through high school, must know and be able to do in core content areas. For all high school
mathematics courses, the TEKS call for increased emphasis on conceptual understanding
balanced with student mastery of algorithmic processes and skills. In addition the TEKS call for
the appropriate use of technology in all facets of mathematics instruction. Just as the nature of
the content areas that students are required to know has changed, so has the definition of what
Advanced Mathematics Educational Support
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those content areas include and the methods that are most effective in teaching those content
areas.
The TEKS define high school mathematics content for Algebra I, Geometry, Algebra II,
Precalculus, and Mathematical Models with Applications. In addition, students may earn state
mathematics credit for Advanced Placement Calculus (AB and BC), Advanced Placement
Statistics, International Baccalaureate courses, independent studies courses above the Algebra II
level, and concurrent enrollment1 courses. The state standards do not give high school credit for
courses below the Algebra I level—such as Pre-Algebra, Consumer Mathematics, and
Fundamentals of Mathematics—that had been offered in the past. Districts still have the option
of offering such mathematics courses for local credit. (Refer to Appendix A for the high school
Texas Essential Knowledge and Skills for Mathematics.)
To graduate, all Texas students must take three years of high school mathematics, including
Algebra I and Geometry. Students in the state’s Recommended High School Program are also
required to take Algebra II. Under the Distinguished Academic Program,2 the most rigorous
graduation program, students are encouraged, but not required, to take four years of
mathematics, including Precalculus. (Compare these mathematics requirements to four years of
English Language Arts and Reading required under the state’s minimum graduation plan with
options for taking AP English after completing English II.) Also, some schools are currently
requiring four years of high school mathematics, beginning with Algebra I, for high school
graduation. Beginning with the 2005–06 school year, with the statewide implementation of the
Recommended High School Program as mandatory, Texas students will be required to complete
three high school mathematics courses through Algebra II. While an improvement, in that it
will require students to advance further in their mathematical preparation than is currently
required under the minimum program, this new policy will not provide enough incentive to
increase enrollments in advanced courses beyond Algebra II. In addition, with the
implementation of this new graduation plan, superintendents will struggle even more to find
teachers with the content knowledge and pedagogical experience to teach Algebra II.3
K–12 Accountability in Texas—A Historical Perspective
Perhaps the issue that most affects classroom mathematics instruction in Texas high schools is
state-mandated assessments and their use as graduation requirements and as key components in
the Texas educational accountability system. Texas is in a period of transition in its assessment
system. Through the 2001–02 school year, the Texas Assessment of Academic Skills (TAAS)
was administered to all students in grades 3–8 and in grade 10. Passing both the English
Language Arts and Mathematics portions of the 10th-grade TAAS was a state-mandated
graduation requirement.
1
The practice of having high school students enroll in college courses while still in high school is known as
concurrent or dual enrollment. For more on concurrent enrollment, see “Concurrent Enrollment” in the section
headed Mathematics Education from the High School Perspective.
2
For more information on the Recommended High School Program and the Distinguished Academic Program, see
www.tea.state.tx.us/rules/tac.
3
Data provided in 2003 from the State Board for Educator Certification indicated that 27.8% of Texas Algebra II
teachers were either teaching out of field or were not fully certified (unpublished communication).
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Advanced Mathematics Educational Support
In 2003, the new Texas Assessment of Knowledge and Skills (TAKS) replaces TAAS. TAKS
assessments will be given in grades 3–11, with the 11th-grade exam serving as the new exitlevel graduation requirement.4 How students perform on these examinations is important to
schools and school districts because student achievement data is used in the Texas School
Accountability System. The total percentages of students, including disaggregated
subpopulations of students, at a school or district who pass each part of the exam, constitutes a
large part of a school or district’s accountability rating. Schools are rated as Exemplary,
Recognized, Acceptable, or Low Performing based in part on the passing rates of students on
these exams.5 The rating received carries serious implications for the district; repeated low
ratings can lead to state sanctions for low-performing districts. In addition, community pressure
for schools to reach the higher rating levels may be intense; this in turn leads to great pressure
on high school teachers to prepare students to do well on these exams.
In the past, the problem with preparing high school students for the state assessments has been
the lack of alignment between what was tested on the exit-level TAAS exam given at grade 10
(that is, mathematics through grade 8) and the high school mathematics curriculum. Many
schools responded to this pressure in inappropriate ways; for example, some schools conducted
intensive review of middle school mathematics during the sophomore mathematics courses,
while others instituted programs in which all students participated in TAAS drill and practice.
Much classroom time was also devoted to administering TAAS pretests and quizzes that were
intended to determine which objectives needed special attention. This meant that sufficient
classroom time was not devoted to high school–level mathematics.
TAKS: Increased Accountability for High Schools
Under the Texas Assessment of Knowledge and Skills system, students will be tested in
mathematics in grades 9 and 10 as well as on the exit-level 11th-grade test. The results of all
three assessments will be used as a factor in determining campus accountability ratings. More
importantly, the new TAKS exam will be better aligned with the high school mathematics
TEKS, assessing for the first time student success on Algebra I and Geometry content. In all
grades the TEKS and TAKS are more tightly aligned than were the TEKS and the Texas
Assessment of Academic Skills, or TAAS. Also, TAKS test items will reflect the increasing rigor
begun with the 2000 TAAS test, calling for students to reason and justify as well as
demonstrate knowledge and skills as required in the TEKS.6 This means that drill and practice
for the TAKS will be unlikely to ensure student success; what matters is that teachers teach
students to know and be able to do the mathematics as described in the TEKS. Another change
is that the high school mathematics TAKS will require that students have access to a graphing
calculator in order to answer questions analyzing multiple mathematics relationships and
representations.
4
The exit-level TAKS assesses student performance in mathematics, English language arts, social studies, and
science.
5
For a brief introduction to this accountability system, go to “Accountability,” at
www.tea.state.tx.us/accountability.html. For a more detailed explanation of the new system, see Texas Education
Agency, Department of Accountability Reporting and Research, 2003 Accountability Plan, available at
www.tea.state.tx.us/perfreport/account/2003/plan/index.html.
6
Samples of TAKS-like items are available from the Texas Education Agency’s Student Assessment Division in
“TAKS Information Booklets,” available at www.tea.state.tx.us/student.assessment/taks/booklets/index.html.
Beginning in summer 2003 released TAKS tests will be available at www.tea.state.tx.us/index.html.
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Transition to College
Since 1965 enrollment in public and private higher education institutions in Texas has increased
by 650,000 students. However, by 2015, Texas will need to add 500,000 more higher
education students to reflect national trends and estimated needs when compared to other
states.7 Thus, the level of preparation of incoming students is much more variable than was true
when smaller numbers of students went on to college. Some incoming college freshmen
demonstrate their academic proficiency in key subject areas by performing at acceptable levels
on the state-required Texas Higher Education Assessment (THEA)8 test or on an approved
alternative test. Others satisfy the college entry requirement with an acceptable performance on
the exit-level TAAS or TAKS exam or on the ACT or SAT.9 In addition, Texas high school
students who finish in the top 10 percent of their graduating class are guaranteed admission to
state colleges and universities, regardless of the course of study they completed in high school or
of their performance on THEA, ACT, or SAT.
Texas institutions of higher education are challenged to make appropriate placement decisions
for their incoming students. In many cases, higher education institutions have addressed any
gaps in student mathematical preparation by introducing noncredit developmental courses,
which too often require an inordinate amount of college mathematics department resources.
Many stakeholders, including state legislators, have expressed concern about the growth of
higher education developmental programs, including developmental mathematics courses, in
Texas public colleges and universities. The expansion of such programs serves to illustrate the
acute need for better alignment between high school mathematics programs and first-year
undergraduate mathematics courses.
7
Closing the Gaps: The Texas Higher Education Plan, Texas Higher Education Coordinating Board, 2001,
www.thecb.state.tx.us.
8
The THEA (formerly TASP) is an assessment program designed to ensure that students attending public
institutions of higher learning or educator preparation programs in Texas have the academic skills necessary to
perform effectively in college-level work. The THEA includes testing components designed to provide information
about the reading, mathematics, and writing skills of students entering higher education. Students whose THEA
performance shows they are not yet proficient in an academic area are required by their college or university to
participate in developmental education activities For more information, see the Texas Higher Education
Coordinating Board’s THEA website, www.thea.nesinc.com/.
9
For more information go to www.thecb.state.tx.us/CBRules.
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Advanced Mathematics Educational Support
MATHEMATICS EDUCATION FROM THE HIGH SCHOOL PERSPECTIVE
This section addresses issues including teacher supply and retention, professional development
and resources, and vertical teams. Each section describes an issue and includes general
recommendations and suggestions from the AMES advisory team to higher education faculty
wishing to address that issue. These are only a few of the actions a faculty member might take,
and these actions would be dependent on the local situation and district, campus, teacher, and
student needs.
Teacher Supply and Retention
The Situation
Perhaps the biggest issue facing Texas high schools, particularly urban schools, is finding and
retaining qualified and well-prepared mathematics teachers. Data collected by the Charles A.
Dana Center since 1997 indicate that students who are taught by instructors certified in their
field perform significantly better on statewide assessments; however, the same data revealed
that only 78.6% of high school Algebra I courses statewide were taught by certified teachers.
This percentage dropped to 68.1% in urban high schools. Recent research indicates that the
shortage of teachers certified to teach Algebra I and other high school mathematics courses in
Texas and in the nation will only grow worse.10 An increase in the state’s student population,
combined with an increase in the number of teachers reaching retirement age and an already
high teacher attrition rate, will leave Texas facing an even more critical shortage in future years.
Several school districts in the state already offer signing bonuses to certified high school
mathematics teachers who commit to teaching for a specified period of time. Dissatisfaction
with the job is given as the main reason that teachers leave the profession.11
Recommendations
•
Build partnerships among your institution of higher education’s mathematics
department, the college of education, and local school districts to design
mathematics teacher preparation programs that help prospective teachers make the
connection between the mathematics they learn at the university and the
mathematics they will teach in their classrooms.
•
Encourage strong mathematics students who might have a talent for teaching to
explore teaching as a profession. Both mathematics majors and minors should be
encouraged to teach at the high school, middle school, and elementary levels. In this
document’s Promising Practices section, see “UTeach” for an example of how this
might be implemented.
•
Use the expertise of local high school master classroom teachers to help deliver
components of the higher education teacher preparation program in order to
provide a consistent and clear public school context to the teacher preparation
program.
10
See, for example, Henke, R.R. & Zahn, L. (2001); State Board for Educator Certification (2002); Texas
Education Agency (1995); and Grissmer, D. & Kirby, S. (1987).
11
Ingersoll, R. M. (1999).
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•
Become familiar with the financial aid available to prospective teachers through
your institution as well through the Teach for Texas Conditional Grant Program and
the Teach for Texas Alternative Certification Grant Program.12
•
Create student organizations for cohorts of prospective mathematics teachers to
provide support and collegiality among these students.
Teacher Professional Development and Resources
The Situation
State policy requires that teachers certified since 1999 must participate in 150 hours of
continuing education over five years.13 In addition, all districts require their teachers to
participate in professional development on a yearly basis. To strengthen and support teachers’
content knowledge and instructional skills, K–12 schools provide teachers with professional
development opportunities and resources. Ongoing, meaningful professional development is key
to providing the support necessary for teachers to become more confident and successful in
teaching. Too often, however, professional development opportunities are not closely targeted
to the needs of the teachers or aligned with classroom instruction; consequently, the
professional development has little positive effect on improving student performance. Further,
classroom resources, including textbooks, may be poorly aligned to instruction. Even if the
professional development or resource does meet teacher needs, teachers frequently receive little
or no support in implementing either in their classrooms.14
Effective professional support should help teachers increase content knowledge and improve
pedagogical skills.15 Texas high school teachers have a variety of professional development
training opportunities available, offered from a variety of sources, including school districts,
education service centers,16 the state, and universities. Perhaps the most comprehensive and
widely available professional development in Texas is TEXTEAMS (Texas Teachers
Empowered for Achievement in Mathematics and Science). Developed and managed by the
Charles A. Dana Center with funding through the Texas Education Agency, TEXTEAMS
provides extended learning experiences in mathematics for teachers of all levels from
prekindergarten through Precalculus.17 TEXTEAMS institutes are correlated closely to the state
mathematics and science Texas Essential Knowledge and Skills and are delivered by skilled
mathematics and science educators who have received leadership training in the institutes.
Several TEXTEAMS institutes require higher education faculty as facilitators. TEXTEAMS
mathematics institutes provide teachers with mathematics instruction to deepen their
understanding of content, as well as activities that can be used with their students. In addition,
several universities around the state, through Eisenhower grant funding coordinated by the
12
Details are provided at www.collegefortexans.com.
Texas Administrative Code, Title 19, Part 7, Chapter 232, Subchapter R, Rule §232.850. This rule can be found
at www.sos.state.tx.us/tac/index.html.
14
Loucks-Horsley, Hewson, Love, and Stiles, 1998; Sowder, Philipp, Armstrong, and Schappelle, 1998.
15
Ball and Cohen, 1999; Briars, 2000; U.S. DOE, 2000; Kennedy, 1999.
16
Texas is divided into 20 geographic regions, each having an education service center. These centers coordinate
with the Texas Education Agency to provide educational services to schools within their region.
17
Since 1996 TEXTEAMS institutes have provided intensive, in-depth, content-based professional development
for more than 200,000 educators. These institutes are offered by more than 700 certified leaders throughout the
state. For more information, see the TEXTEAMS website at www.texteams.org.
13
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Texas Higher Education Coordinating Board, offer localized programs to help improve teacher
learning and student success. Local school districts and education service centers also develop
and provide professional development opportunities for teachers.
Recommendations
•
Work with local districts to determine teacher professional development needs.
•
Work with local districts and college of education colleagues to design, deliver, and
support ongoing, intensive professional development (face-to-face, online, and distance
learning) that meets teacher needs.
•
Become TEXTEAMS university leaders and use TEXTEAMS materials with preservice
and inservice teachers and graduate students.
•
Design college-credit summer courses for high school mathematics teachers that align
with teacher needs. Become familiar with grant funds available to college faculty for
this purpose through the Teacher Quality Higher Education Grants Program.18
•
Investigate the opportunity for your institution to offer College Board Advanced
Placement Summer Institutes that are designed to provide professional development for
teachers of designated AP courses.19
•
Become knowledgeable about high school curriculum resources, including textbooks,
assessments, technology, and supplemental materials, that support teaching the content
described in the TEKS. Offer to assist in resource development and evaluation when
appropriate.
•
Work with teachers to develop rich lessons that address multiple interrelated concepts
instead of surface lessons that address single concepts in isolation from each other.
•
Use existing master’s degree programs to attract teachers who need to learn or relearn
mathematics content.
•
Serve as a mentor or resource person for teachers and students. This might include
working with schools to build student support structures that directly improve student
achievement.
•
Make presentations on mathematics content and topics relevant to high school
mathematics teachers at state, local, and national professional conferences. For example,
the Conference for the Advancement of Mathematics Teaching (CAMT) and regional
and national conferences of the National Council of Teachers of Mathematics (NCTM)
encourage session proposals from university faculty.20
18
Details are provided on the Texas Higher Education Coordinating Board’s website at www.thecb.state.tx.us.
For information regarding how to become involved with AP Summer Institutes, contact the Southwestern
Regional Office of the College Board in Austin, Texas, at 512-891-8400 or 800-999-9139.
20
Details are provided at www.tenet.edu/camt and www.nctm.org.
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Advanced Mathematics Educational Support
9
Vertical Teaming
The Situation
In an effort to strengthen their mathematics programs, many districts have adopted the strategy
of using vertical teams to achieve a well-aligned, seamless mathematics curriculum. Such a
curriculum can be described as one that presents a thoughtful progression of concepts within
courses and from one course to another, avoiding redundancies, gaps, and jarring transitions.
Vertical team denotes a team made up of teachers in a given content area from a sequence of
grade levels who work together to build a strong instructional program, with a common goal of
ensuring success for all students. The College Board promotes vertical teaming as an Advanced
Placement strategy for teachers in a specific content area in grades 6–12 to foster the readiness
of junior- and senior-level high school students for the AP tests given for the purpose of
accruing college credits. In recent years, the concept has expanded to teachers in grades K–12
and has shown potential for K–16. Many districts have implemented “vertical teams” that
include only AP and Pre-AP teachers and do not share a common goal of ensuring success for
all students, or even address mathematics programmatic issues in substantive ways.
Unfortunately, many of these teams become dysfunctional and disband after a year or two. In
an unpublished pilot study conducted by the Dana Center between 1999 and 2001 the most
effective vertical teams included teachers from every level of the subject area—not just the
teachers of Pre-AP or AP students—who focused on programmatic issues and held high
expectations for all students.
Recommendations
•
Become familiar with various resources designed to build and support vertical
teams, especially the Advanced Placement Program Mathematics Vertical Teams
Toolkit (1998).21
•
If a local school district has implemented vertical teaming:
Learn about the district’s implementation experience and work with district
leadership to identify potential areas of concern.
Become part of the district’s vertical team and work to strengthen the district’s
implementation of vertical teaming (see Appendix C).
21
The Dana Center, in conjunction with the College Board, has developed the Advanced Placement Program
Mathematics Vertical Teams Toolkit, a resource guide for implementing the vertical teaming process to support strong
mathematics instruction. (This resource is available from the Dana Center and from the College Board.) Chapter 2
of the Toolkit proposes activities for initial team meetings that will help team members build trust with one
another, begin to explore curriculum issues, and set preliminary goals for the team. It is critical to the success of the
team to spend adequate time building trust among team members.
10
Advanced Mathematics Educational Support
•
If a local school district has not implemented vertical teaming:
Explore with district leadership the potential benefits of establishing a vertical
team.
Share information with the district leadership about the available resources
supporting vertical teams.
Help the district form a K–16 vertical team (see Appendix C).
Advanced Mathematics Educational Support
11
MATHEMATICS EDUCATION FROM THE HIGHER EDUCATION PERSPECTIVE
This section addresses issues of concurrent enrollment, standards for high school calculus and
statistics, tenure and promotion, undergraduate teacher preparation, and alternative teacher
certification. Each section briefly describes an issue and includes general recommendations from
the AMES advisory team.
Concurrent Enrollment
The Situation
The practice of having high school students enroll in college courses while still in high school is
known as concurrent or dual enrollment. Although these terms are often used interchangeably,
in the TEKS the term concurrent enrollment is used to describe the practice in which students
receive high school and college credit for the same course.22 Local implementation of
concurrent enrollment practices varies widely. For example, some students may wish to take an
advanced course that is not offered at their school, so they go to a nearby college to take the
course. In other cases, students may take the college course on their high school campus.
While concurrent enrollment appears to offer students enhanced educational opportunities, in
practice issues can arise. Identifying these issues can serve as a starting place for discussion
among high school educators, higher education faculty, parents, and other stakeholders. One
issue is that not all concurrent enrollment scenarios offer safeguards guaranteeing that the
course experience is truly at the collegiate level. Contributing to this problem are inconsistencies
in course content standards and faculty expertise. For example, if this course is taught by a high
school teacher without collaboration with university or college faculty, the content may not be
consistent with that of the same course taught at the college or university level. There may also
be a difference in the depth of mathematical content knowledge between high school and
higher education faculty.
Another issue is that the college-level course content that the student studies for concurrent
enrollment credit may not align with the content in the high school course for which the
student is given high school mathematics credit. These high school courses include Algebra II,
Precalculus, AP Statistics, AP Calculus AB, and AP Calculus BC. The content for these high
school courses is described by state law in the Texas Essential Knowledge and Skills (see
Appendix A). The TEKS also specify requirements for concurrent enrollment credit:
Section 111.60. Concurrent Enrollment in College Courses.
(a)
General requirements. Students shall be awarded one-half credit for each semester of successful
completion of a college course in which the student is concurrently enrolled while in high school.
(b)
Content requirements. In order for students to receive state graduation credit for concurrent
enrollment courses, content requirements must meet or exceed the essential knowledge and skills in a given course.
For example, if students complete College Algebra and receive concurrent enrollment credit for
Algebra II, the College Algebra course content must address all the TEKS in Algebra II.
22
The term dual enrollment would then be used to describe the practice in which high school students enroll in
college courses, but the credit for those college courses does not count toward high school graduation requirements.
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Advanced Mathematics Educational Support
Recommendations
•
Engage in dialogue with public school personnel about critical issues regarding
concurrent enrollment, such as course content, alignment with other courses, and
faculty expertise.
•
Collaborate with high school mathematics teachers to mutually ensure the rigor of
courses offered for concurrent credit. Build collegial relationships with these
teachers by building a network, meeting together on a regular basis, and attending
professional development and conferences together.
•
Work with public school personnel to develop and implement appropriate standards
and policies for concurrent enrollment.
•
Ensure that concurrent enrollment instructors are active members of both the high
school and the university or college mathematics departments. It is essential that
higher education faculty on a high school campus be engaged with high school
mathematics faculty. For example, a faculty member teaching College Algebra to
students receiving concurrent credit should communicate with teachers of pertinent
courses on the high school campus.
•
Collaborate with public school personnel to ensure that concurrent enrollment
courses meet the requirements described in the Texas Essential Knowledge and
Skills before awarding state graduation credit.
Standards for High School Calculus and Statistics
The Situation
Any student who takes calculus in secondary school and performs satisfactorily in that course
should place out of the comparable college calculus course. Calculus courses that meet the level
of rigor necessary for this to occur must be preceded by four full years of mathematical
preparation in algebra, geometry, trigonometry, analytic geometry, and elementary functions
(College Board, 2002). Students who enroll in a high school statistics course should have
successfully completed three full years of mathematical preparation in algebra and geometry,
including successful completion of a rigorous Algebra II course. As a response to this
recommendation, Texas designated AP Calculus AB and AP Calculus BC as the only high school
calculus courses approved for graduation credit. Similarly, AP Statistics is the only statistics
course approved for graduation credit. However, contrary to the state’s intent, some schools
offer a non-AP calculus or statistics course and give credit for Independent Study or award a
local credit.
According to the AMES advisory team, many of their university colleagues do not support the
AP program because of their negative experiences with many students who took Advanced
Placement courses in high school. For example, many students claim, “I had AP calculus in high
school,” yet these students did not take the AP exam and frequently can demonstrate only some
rudimentary calculus techniques. It is very probable that such students did not successfully
complete a true AP course; the standards for AP courses are extremely high. For example, an
AP calculus course is built around a course description reflective of a college calculus course.
Advanced Mathematics Educational Support
13
There are actually two AP calculus courses, entitled Calculus AB and Calculus BC. (For a list of
topics for these courses, see Appendix B.) The AB course and exam cover roughly the first
college semester of calculus, through integration by substitution and separation of variables; the
BC course and exam cover the first two semesters of calculus, including numeric and Taylor
series, and the calculus of parametric, polar, and vector functions of one variable.
Like AP Calculus, AP Statistics is built around a course description and examination designed
to reflect the content of a typical introductory college course in statistics. The purpose of the
AP course in statistics is to introduce students to major concepts and tools for collecting,
analyzing, and drawing conclusions from data. The AP course is organized around four broad
topics: data exploration, experimental design, modeling, and inference. This course provides
students with an experience equivalent to that of a one-semester, introductory non-calculusbased college course in statistics. (For a list of course topics, see Appendix B.)
All three AP mathematics courses are designed to culminate in students taking an AP exam in
May. These exams are the same throughout the country and are graded by a select gathering of
highly qualified college and high school calculus or statistics teachers. The existence of these
independently administered and graded exams, together with the high standards of the grading
process, safeguard the quality of the program.
Recommendations
23
24
•
Become knowledgeable about the public school curriculum to help teachers
maintain high standards in all mathematics courses so that all students are
adequately prepared for their next mathematics course. (See Appendix A for the
high school Texas Essential Knowledge and Skills for Mathematics.)
•
Collaborate with local school districts to educate high school seniors, parents, and
counselors regarding the importance of mathematics in students’ options for a
college major. Work with local school districts to implement policies that ensure
that students take a rigorous mathematics course their senior year.
•
Work with local school districts to implement policies that ensure that local high
schools that offer calculus and statistics courses offer only AP Calculus and/or AP
Statistics.
•
Encourage school districts to develop a policy requiring every student enrolled in an
AP course to take the corresponding AP exam.
•
Become an easily accessible resource for local AP teachers. Support the development
of AP teachers by becoming involved in College Board AP summer institutes and
providing ongoing professional development and support.23 Encourage school
districts to send their teachers to AP training.
•
Become a reader for the AP exams.24
For information about how to become involved with AP summer institutes, go to www.collegeboard.org.
For information on how to become a reader for the AP exams, go to www.collegeboard.org.
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Advanced Mathematics Educational Support
Tenure and Promotion
The Situation
Time constraints are a barrier to effective collaboration between higher education and public
schools. For many college faculty, time spent on research is crucial, because tenure, promotion,
and merit raises rely heavily on research productivity. Although this problem is not easily
solved, major research-funding organizations, such as the National Science Foundation, require
K–12 educational outreach components for many of the science and mathematics research
grants awarded.
A university culture that supports and welcomes coordination between mathematics faculty and
education faculty and that desires strong ties with K–12 educators will make it easier for
educators with an interest in bridging the gap between secondary and college instruction to act
on their inclinations and develop strong connections that help both the secondary classroom
teacher and the university professoriate. Several universities who see teacher inservice and
preservice preparation as an important component of their mission have begun to value faculty
work with K–12 educators for tenure and promotion. For example, in this document’s
Promising Practices section, see “Systemic/Programmatic Administration Support.”
Recommendations
•
Work to establish policies in your department and college that ensure that
meaningful involvement with K–12 educators is valued toward tenure. (In this
document’s Promising Practices section, see “Systemic/Programmatic Administration
Support.”)
•
Become an active participant in a statewide network that is considering issues
around revision of tenure policies. For example, the annual October Preservice
Conference25 provides a forum for discussion of these issues.
•
Build meaningful collaborations with public school and college of education
counterparts that will support teachers and lead to student success. If a measure of
student success can be attributed to mathematics faculty involvement, this
strengthens the argument for meaningful involvement in high school counting
toward tenure. (In this document’s Promising Practices section, see
“Systemic/Programmatic Administration Support.”)
•
Advance the knowledge base for improving student performance in mathematics by
valuing and rewarding faculty research on secondary mathematics instructional
practices.
25
The Charles A. Dana Center has hosted the annual October Preservice Conference for mathematics, science, and
education faculty for the past eight years, with participation of faculty from across the state. For more information,
see www.utdanacenter.org.
Advanced Mathematics Educational Support
15
Undergraduate Teacher Preparation
The Situation
A number of reports (Kilpatrick, Swafford, and Findell, 2001; Ma, 1999; U.S. DOE, 2002)
underscore the importance of strong content knowledge for all mathematics teachers and
specifically emphasize the need for a focus on the mathematics that they will be expected to
teach. State and national mandates urge greater involvement by more university mathematicians
in the mathematical preparation of teachers.26 The Texas State Board for Educator
Certification in April 2000 responded to these challenges by adopting new certification
standards for Texas teachers in each of the major subject areas. These new certification
standards move higher education from a credit-based to a standards-based system for teacher
preparation, and include a high-stakes accountability system for colleges and universities that
prepare teachers.
These new certification standards delineate, for the first time, not how many courses beginning
educators have to take, but rather what knowledge and skills they must acquire. These
standards also specify content knowledge and teaching skills for teachers of specific grade
levels—that is, early childhood–grade 4, grades 4–8, and grades 8–12. These teacher
certification standards are connected to the TEKS, the state’s curriculum guidelines for what
public school students must know and be able to do. New teacher certification
examinations—the Texas Examinations of Educator Standards (TExES)—are aligned with the
new standards. An institution that prepares teachers is accredited based on how well its
prospective teachers perform on this test.27
This standards-based program allows for greater flexibility and creativity in the design of
teacher preparation programs, while ensuring greater uniformity and quality in teacher
preparation. It is also forcing a reexamination of the mathematics preparation of teachers across
the state and creating a need for greater collaboration between colleges of education and
mathematics departments in designing high-quality programs that meet state and national
standards.28
Recommendations
•
Department of mathematics faculty should examine current curriculum and
programs in relationship to the SBEC beginning teacher certification standards and
ensure that the mathematics content required for certification is infused within the
required mathematics courses.
26
For example, the NSF mathematics and science partnership grants require substantial involvement of university
mathematics faculty in improving K–16 education. In addition numerous reports recommend the deep involvement
of mathematics faculty, both with the preparation of prospective teachers and with K–12 schools (American
Mathematical Society, 2001; U.S. DOE, 2000).
27
For additional information, see the State Board for Educator Certification website, www.sbec.state.tx.us.
28
A project from the Charles A. Dana Center currently underway, “Strengthening and Supporting Standards-Based
Mathematics Teacher Preparation (S3MTP),” will provide opportunities for faculty collaboration, professional
recognition, and leadership, and develop resources for implementing standards-based teacher preparation and
certification. Details are provided at www.utdanacenter.org/mathematics/highered/projects.html.
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Advanced Mathematics Educational Support
•
Network with mathematics faculty on your campus and at other institutions who
are involved with teacher education to build a solid mathematics program to
prepare teachers.
•
Support faculty members in your institution who choose to invest their time and
apply their expertise to the preparation of mathematics teachers.
Alternative Teacher Certification
The Situation
In an effort to alleviate the severe shortage of mathematics teachers in Texas, a variety of
alternative certification programs have been developed and offered by education service
centers, school districts, colleges and universities, and other entities. Alternative certification is
available to people who wish to become teachers and who hold a baccalaureate degree but do
not have a teaching certificate. Because of the No Child Left Behind Act of 2001, all alternative
certification programs will need to be reviewed to meet the requirements for producing highly
qualified teachers.
Recommendations
•
Become informed about and involved in the alternative certification programs in
your region.29
•
Become familiar with the financial aid available to alternative certification
candidates through your institution as well as through the Teach for Texas
Alternative Certification Grant Program.30
29
For more information regarding alternative certification, go to the State Board for Educator Certification website
at www.sbec.state.tx.us.
30
Details are provided at www.collegefortexans.com.
Advanced Mathematics Educational Support
17
PROMISING PRACTICES
This section includes samples of model programs and promising practices that address selected
issues from the previous two sections. It includes brief descriptions of the programs with
weblinks that offer additional information. This section is not intended to include all programs
available, nor are we suggesting that programs can be easily transported from one institution to
another. However, it does offer a snapshot of responses to concerns about teacher preparation
and teacher quality that can be used to spur discussion and generate ideas that can lead to
improved teacher preparedness and connections between high schools and universities.
Master Teacher Summer Institutes
The University of Texas at Austin
www.mtsi.utexas.org
The Master Teacher Summer Institutes at The University of Texas at Austin are designed for
teachers of Advanced Placement courses to work with university content faculty and master
teachers. The goal of these month-long professional development experiences is to increase the
diversity and academic skills of Texas public school students enrolled in pre-AP and AP courses
and to improve the content, technological, and content-specific pedagogical knowledge of the
teacher participants. Through these institutes, high school teachers are able to work with
university faculty in a way that increases teacher knowledge of the content they teach, offers
suggestions for how to use that new knowledge in the classroom, and helps them build their
own program of professional development that will lead to continued improvement of their
teaching. The Master Teacher Summer Institutes are offered through the College of Natural
Sciences and the College of Liberal Arts.
AP Summer Institute
Baylor University
Since 1995 Baylor University has hosted an annual summer Advanced Placement Institute for
high school teachers. This AP Institute provides an excellent opportunity for teachers and
professors to learn from each other about contemporary students, curriculum, and academic
expectations. The high school teachers benefit from having access to university professors who
are experts in their fields, to relevant research and tools in the university library, and to special
events designed specifically for them. Baylor benefits from the contact with these teachers of
the academically gifted young people who are among its prospective students. To get the
institute started, Baylor first contacted the Southwestern Regional Office of the College Board
in Austin. This office provided not only guidance on the format for the institute, but also a list
of potential consultants to conduct the workshops. The College Board also helped with
publicity for the institute, provided resource materials and certificates of attendance for the
participants, and shared information about state reimbursements for the cost of the institute to
qualified participants.
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Advanced Mathematics Educational Support
The Rice University School Mathematics Project
rusmp.rice.edu
The Rice University School Mathematics Project was created in 1987 with funding from a
three-year National Science Foundation grant. It was initially designed to include 48 middle and
high school teachers from a diverse group of local schools and school districts. In the first year
these teachers participated in an intensive six-week summer instructional program on the Rice
University campus. The goals of the program from its onset were:
•
active involvement of research mathematicians, mathematics educators, and
classroom teachers in the development of the mathematical content taught in the
program,
•
collaboration of mathematics educators and master teachers to ensure that models
of content and instructional delivery were appropriate for the secondary classroom,
•
use of an inquiry and problemsolving approach whenever possible,
•
development of participants to serve as “lead teachers” at their respective campuses
so that the training could be disseminated more broadly,
•
development of content-based curricular materials by participants,
•
appropriate use of technology, and
•
interaction between secondary teachers and research mathematicians.
The Rice University School Mathematics Project continued with funding from private and local
foundations. It includes creating academic-year courses for teachers in mathematics content and
pedagogy, planning and writing curriculum and supplementary materials for local school
districts, and developing innovative ways to bridge the gap between teacher training and
classroom implementation. One of these programs, the RUSMP Urban Program, combines
teacher training with summer school and academic year instruction and has been nationally
acclaimed as a leading staff development model.
The project provides training through its various programs to about 600 teachers annually,
including 120 each year in the Summer Campus Program, which is now in its 17th year. As the
project has expanded, it has evolved from its initial beginnings as part of the Mathematics
Department into a center within the School of Natural Sciences.
Information Technology in Science Center for Teaching and Learning
Texas A&M University
its.tamu.edu
The Information Technology in Science Center for Teaching and Learning, a partnership
between the College of Science and the College of Education at Texas A&M University, is an
interdisciplinary graduate program that seeks to replenish the nation’s supply of education
specialists in science, mathematics, and technology through learner-centered opportunities
Advanced Mathematics Educational Support
19
involving scientists, mathematicians, education researchers. and education practitioners. The
Center’s three main goals are (1) production of education specialists, (2) creation of new
knowledge through research on the impact of information technology in the form of modeling,
visualization, and interaction with complex data sets on learning and teaching science and
mathematics, and (3) development and dissemination of quality professional development
experiences structured around the impact of information technology on learning and teaching
science and mathematics in grades 7–12.
The ITS Center’s innovative approach combines faculty, graduate students, and master teachers
who work together as a science-based team to develop key questions, design and conduct
research, and provide professional development experiences while gaining valuable
technological skills.
Participants are part of a research program based on integration, coordination, and application.
In addition to integrated science, education, and technology experiences, participants are
involved in research experiences coordinated by the center to ensure that each participant’s
individual project becomes part of a coherent whole. The ITS program offers direct applications
and connections to classrooms across the state and nation by transferring current science,
mathematics, and engineering research into grades 7–12 classrooms to assist teachers in
integrating technology into the science curriculum.
UTeach
The University of Texas at Austin
www.uteach.utexas.edu
The UTeach program at The University of Texas at Austin is a teacher-preparation program at a
traditional four-year institution. UTeach stands as a model because of the intensity with which
the Colleges of Natural Science and Education and the Austin Independent School District
have collaborated to create a teacher-preparation program that is attractive to students and that
prepares them to teach with knowledge of research-based practices.
UTeach was developed in consultation with a group of master high school teachers and the
Texas State Board for Educator Certification, according to the new guidelines for teacher
certification and new national and state standards for K–12 education in mathematics and
science. All UTeach degree plans can be completed in four years. Teaching techniques, field
experience, the study of mathematics and science, and certification are fully integrated. The
courses were codeveloped for this program by the College of Natural Sciences and the College
of Education.
Hallmarks of the UTeach program include:31
•
31
Active recruitment and support of natural science undergraduates who are
interested in careers in secondary math and science education. Support includes
tuition reimbursement, placement with small cohorts of students, paid internships,
and guidance by master teachers.
Material that follows is from the UTeach website (www.uteach.utexas.edu).
20
Advanced Mathematics Educational Support
•
Emphasis on preparing teachers who will be knowledgeable of their discipline,
experienced with involving students in scientific inquiry, and practiced in employing
new technologies to enhance student learning.
•
A revised, streamlined professional education sequence drawing on research on
learning, standards-based curricula, multiple forms of assessment, and proven
strategies for achieving equity and integrating technology into math and science
education.
•
Integrated preservice and content experiences which prepare UTeach students to
teach all levels of material—from the core curriculum to Advanced Placement
courses—to students of diverse cultural and socioeconomic backgrounds.
•
Program flexibility with multiple entry points (from freshman to postbaccalaureate), integrated degree plans, and proficiency-based assessment, including
the development of individual teaching portfolios.
In addition to natural sciences and education faculty, UTeach employs master teachers in the
program as instructors, advisers, and field supervisors. Since it began in 1997, UTeach has
grown to over 250 students, and is already graduating over 50 teachers per year. Student
retention as undergraduates is much higher than College of Natural Sciences averages, and
nearly 40% of the first teaching graduates have been minority students.
Systemic/Programmatic Administration Support
Stephen F. Austin State University
www.sfasu.edu/math
One major obstacle university mathematics faculty face when becoming involved in public
school education is the criteria for merit and tenure at the university level. (See the section
“Tenure and Promotion” in this document’s section on Mathematics Education from the Higher
Education Perspective.) In the Stephen F. Austin State University Department of Mathematics
and Statistics, active engagement with K–12 schools by mathematics faculty is counted toward
merit and tenure, as is professional development or service. Department policy supported by the
university states that funded grants, including those related to public school teaching, are
counted the same as a refereed publication in a professional journal. With a funded grant
project, consideration is given to the number of people affected by the project, the money that
is brought to the university, and the publicity and goodwill associated with the grant project.
Efforts of the Department of Mathematics and Statistics at SFASU to improve teacher
preparation and quality receive the full support of the upper administration (deans, vice
presidents, president, and governing board). The mathematics and statistics department
chairman actively promotes the idea that the university’s upper administration will support any
efforts that improve the success rate in mathematics. Most upper administrators are interested
in the university’s students being successful in their chosen program, which in many instances
includes a significant amount of mathematics. In part because of this work to improve K–12
education, students who enter the university will be better prepared. Thus, retention rates will
rise.
Advanced Mathematics Educational Support
21
As of 2002 SFASU had seven graduate mathematics courses and six undergraduate courses
designed specifically to meet the needs of preservice and inservice teachers. One incentive at
SFASU to encourage faculty members to become involved in mathematics education is release
time given for faculty to develop new mathematics education courses.
Even at SFASU, with its favorable merit and tenure policies, attracting mathematics faculty
members who become involved in the preparation and/or professional development of K–12
teachers is difficult. Encouraging faculty after they come to the university is the easy part, and it
helps to have an administration that supports such faculty members’ efforts.
Lamar State College–Orange Mathematics Institutes
Lamar State College–Orange, Texas
www.orange.lamar.edu/Academics/Academic_Eisenhower.htm
Providing high-quality professional development to PK–12 teachers is a challenge. For those
college or university personnel with an Eisenhower Higher Education Grant32 project, a special
challenge has been the length of the training that this grant requires be provided, which is
approximately 100 hours. Since reviewers from throughout the country judge Eisenhower
proposals, it is also important that the content provided in training funded by Eisenhower
grants be of high quality.
Beginning in 1999, the mathematical content of the Lamar State College–Orange Eisenhowerfunded mathematics institutes was built around TEXTEAMS (Texas Teachers Empowered for
Achievement in Mathematics and Science) institutes. (Refer to “Teacher Professional
Development and Resources” in the Mathematics Education from the High School Perspective section
of this document.) TEXTEAMS materials are designed to help educators understand and
implement the mathematics and science Texas Essential Knowledge and Skills (TEKS) and to
support student success on TEKS-based assessments, such as the Texas Assessment of
Knowledge and Skills (TAKS). Each TEXTEAMS institute is developed around unifying
themes and concepts articulated in the TEKS. Additionally, TEXTEAMS institutes focus on
helping educators think deeply about the “big ideas” found in the mathematics and science
curriculum, such as proportionality in grades 6–8.
The TEXTEAMS institutes provide the foundation upon which teachers can discuss, explore,
and experience mathematical content in a hands-on environment, using appropriate technology.
Participating teachers are given time to thoroughly explore the mathematics content. For
example, the TEXTEAMS institute Algebra I: 2000 and Beyond was designed as five-day, 30hour professional development. In the Lamar State College–Orange Mathematics Institute,
teachers spend 40 hours during the summer and 21 hours in the fall semester exploring the
content of Algebra I. The academic year provides LSCO institute participants the opportunity
to implement with their students the content learned in the summer institutes. After teaching
the content in their classrooms, LSCO institute participants have the opportunity to revisit the
32
Since 1985–86 the Texas Higher Education Coordinating Board has provided grants through the federal
Eisenhower Professional Development Grants Program in mathematics and science for K–12 school personnel.
Beginning with the 2003–04 academic year, the Texas Higher Education Coordinating Board is funding the Teacher
Quality Professional Development Grants Program to replace Eisenhower grants. Grants are awarded to higher education
institutions in April for implementation from May 1 of the funding start year to September 30 of the following year.
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Advanced Mathematics Educational Support
institute experiences in a structured setting and discuss successes, problems, and/or suggested
modifications to their classroom instruction. This same principle has been applied to other
LSCO institutes targeted at both middle school and high school programs.
One major advantage of using TEXTEAMS materials as the foundation for LSCO Eisenhower
institutes is the hours of work saved by institute personnel in the development of content for
the courses. The advisory teams, authors, and presenters of the TEXTEAMS institutes have
already completed much work when the institutes are ready for the training sessions. The higher
education instructors have the opportunity to build upon their work rather than developing
content from scratch.
Other advantages of using TEXTEAMS materials within the LSCO mathematics institutes
include:
•
TEKS-based activities are used that are already teacher-tested and student-tested.
•
Some of the institutes include sample student work, which further enhances the
content.
•
The institutes provide an opportunity for higher education personnel to observe
teachers discussing the content, how it will work with their students, and other
issues related to implementation in the classroom.
For higher education faculty who have never received or conducted an Eisenhower-funded
professional development project, the use of the TEXTEAMS materials provides a blueprint to
begin their involvement with the professional development for PK–12 school teachers. The
higher education faculty can supplement or modify the content to whatever extent they choose.
Advanced Mathematics Educational Support
23
APPENDIX A
CHAPTER 111. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS
Subchapter C. High School33
Statutory Authority: The provisions of this Subchapter C issued under the Texas Education Code, §28.002,
unless otherwise noted.
§111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12.
The provisions of this subchapter shall be implemented beginning September 1, 1998, and at that time,
shall supersede §75.63(e)-(g) of this title (relating to Mathematics).
Source: The provisions of this §111.31 adopted to be effective September 1, 1996, 21 TexReg 7371.
§111.32. Algebra I (One Credit).
(a)
Basic understandings.
(1)
Foundation concepts for high school mathematics. As presented in Grades K-8, the basic
understandings of number, operation, and quantitative reasoning; patterns, relationships, and
algebraic thinking; geometry; measurement; and probability and statistics are essential
foundations for all work in high school mathematics. Students will continue to build on this
foundation as they expand their understanding through other mathematical experiences.
(2)
Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra;
symbols provide powerful ways to represent mathematical situations and to express
generalizations. Students use symbols in a variety of ways to study relationships among
quantities.
(3)
Function concepts. Functions represent the systematic dependence of one quantity on another.
Students use functions to represent and model problem situations and to analyze and interpret
relationships.
(4)
Relationship between equations and functions. Equations arise as a way of asking and
answering questions involving functional relationships. Students work in many situations to
set up equations and use a variety of methods to solve these equations.
(5)
Tools for algebraic thinking. Techniques for working with functions and equations are essential
in understanding underlying relationships. Students use a variety of representations (concrete,
numerical, algorithmic, graphical), tools, and technology, including, but not limited to,
powerful and accessible hand-held calculators and computers with graphing capabilities and
model mathematical situations to solve meaningful problems.
(6)
Underlying mathematical processes. Many processes underlie all content areas in mathematics.
As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics,
and reasoning, as well as multiple representations, applications and modeling, and justification
and proof.
33
The high school mathematics TEKS can be found on the Texas Education Agency website at
http://www.tea.state.tx.us/rules/tac/chapter111/ch111c.html.
24
Advanced Mathematics Educational Support
(b)
Foundations for functions: knowledge and skills and performance descriptions.
(1)
(2)
(3)
The student understands that a
function represents a dependence of
one quantity on another and can be
described in a variety of ways.
The student uses the properties and
attributes of functions.
The student understands how
algebra can be used to express
generalizations and recognizes and
uses the power of symbols to
represent situations.
Advanced Mathematics Educational Support
Following are performance descriptions.
(A)
The student describes independent and
dependent quantities in functional
relationships.
(B)
The student gathers and records data, or
uses data sets, to determine functional
(systematic) relationships between
quantities.
(C)
The student describes functional
relationships for given problem
situations and writes equations or
inequalities to answer questions arising
from the situations.
(D)
The student represents relationships
among quantities using concrete models,
tables, graphs, diagrams, verbal
descriptions, equations, and inequalities.
(E)
The student interprets and makes
inferences from functional relationships.
Following are performance descriptions.
(A)
The student identifies and sketches the
general forms of linear (y = x) and
quadratic (y = x2) parent functions.
(B)
For a variety of situations, the student
identifies the mathematical domains and
ranges and determines reasonable domain
and range values for given situations.
(C)
The student interprets situations in terms
of given graphs or creates situations that
fit given graphs.
(D)
In solving problems, the student collects
and organizes data, makes and interprets
scatterplots, and models, predicts, and
makes decisions and critical judgments.
Following are performance descriptions.
(A)
The student uses symbols to represent
unknowns and variables.
(B)
Given situations, the student looks for
patterns and represents generalizations
algebraically.
25
(4)
(c)
Following are performance descriptions.
(A)
The student finds specific function
values, simplifies polynomial
expressions, transforms and solves
equations, and factors as necessary in
problem situations.
(B)
The student uses the commutative,
associative, and distributive properties to
simplify algebraic expressions.
Linear functions: knowledge and skills and performance descriptions.
(1)
(2)
26
The student understands the
importance of the skills required to
manipulate symbols in order to
solve problems and uses the
necessary algebraic skills required
to simplify algebraic expressions
and solve equations and
inequalities in problem situations.
The student understands that linear
functions can be represented in
different ways and translates among
their various representations.
The student understands the
meaning of the slope and intercepts
of linear functions and interprets
and describes the effects of changes
in parameters of linear functions in
real-world and mathematical
situations.
Following are performance descriptions.
(A)
The student determines whether or not
given situations can be represented by
linear functions.
(B)
The student determines the domain and
range values for which linear functions
make sense for given situations.
(C)
The student translates among and uses
algebraic, tabular, graphical, or verbal
descriptions of linear functions.
Following are performance descriptions.
(A)
The student develops the concept of
slope as rate of change and determines
slopes from graphs, tables, and algebraic
representations.
(B)
The student interprets the meaning of
slope and intercepts in situations using
data, symbolic representations, or
graphs.
(C)
The student investigates, describes, and
predicts the effects of changes in m and b
on the graph of y = mx + b.
(D)
The student graphs and writes equations
of lines given characteristics such as two
points, a point and a slope, or a slope
and y-intercept.
(E)
The student determines the intercepts of
linear functions from graphs, tables, and
algebraic representations.
(F)
The student interprets and predicts the
effects of changing slope and y-intercept
in applied situations.
Advanced Mathematics Educational Support
(G)
(3)
(4)
The student formulates equations
and inequalities based on linear
functions, uses a variety of
methods to solve them, and
analyzes the solutions in terms of
the situation.
The student formulates systems of
linear equations from problem
situations, uses a variety of
methods to solve them, and
analyzes the solutions in terms of
the situation.
The student relates direct variation to
linear functions and solves problems
involving proportional change.
Following are performance descriptions.
(A)
The student analyzes situations involving
linear functions and formulates linear
equations or inequalities to solve
problems.
(B)
The student investigates methods for
solving linear equations and inequalities
using concrete models, graphs, and the
properties of equality, selects a method,
and solves the equations and
inequalities.
(C)
For given contexts, the student interprets
and determines the reasonableness of
solutions to linear equations and
inequalities.
Following are performance descriptions.
(A)
The student analyzes situations and
formulates systems of linear equations to
solve problems.
(B)
The student solves systems of linear
equations using concrete models, graphs,
tables, and algebraic methods.
(C)
For given contexts, the student interprets
and determines the reasonableness of
solutions to systems of linear equations.
(d)
Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.
(1)
The student understands that the graphs of
quadratic functions are affected by the
parameters of the function and can interpret
and describe the effects of changes in the
parameters of quadratic functions.
Advanced Mathematics Educational Support
Following are performance descriptions.
(A)
The student determines the domain and
range values for which quadratic
functions make sense for given
situations.
(B)
The student investigates, describes, and
predicts the effects of changes in a on the
graph of y = ax2.
(C)
The student investigates, describes, and
predicts the effects of changes in c on the
graph of y = x2 + c.
(D)
For problem situations, the student
analyzes graphs of quadratic functions
and draws conclusions.
27
(2)
(3)
The student understands there is more than
one way to solve a quadratic equation and
solves them using appropriate methods.
The student understands there are situations
modeled by functions that are neither linear
nor quadratic and models the situations.
Following are performance descriptions.
(A)
The student solves quadratic equations
using concrete models, tables, graphs,
and algebraic methods.
(B)
The student relates the solutions of
quadratic equations to the roots of their
functions.
Following are performance descriptions.
(A)
The student uses patterns to generate the
laws of exponents and applies them in
problem-solving situations.
(B)
The student analyzes data and represents
situations involving inverse variation
using concrete models, tables, graphs, or
algebraic methods.
(C)
The student analyzes data and represents
situations involving exponential growth
and decay using concrete models, tables,
graphs, or algebraic methods.
Source: The provisions of this §111.32 adopted to be effective September 1, 1996, 21 TexReg 7371.
28
Advanced Mathematics Educational Support
§111.33. Algebra II (One-Half to One Credit).
(a)
(b)
Basic understandings.
(1)
Foundation concepts for high school mathematics. As presented in Grades K-8, the basic
understandings of number, operation, and quantitative reasoning; patterns, relationships, and
algebraic thinking; geometry; measurement; and probability and statistics are essential
foundations for all work in high school mathematics. Students continue to build on this
foundation as they expand their understanding through other mathematical experiences.
(2)
Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra;
symbols provide powerful ways to represent mathematical situations and to express
generalizations. Students study algebraic concepts and the relationships among them to better
understand the structure of algebra.
(3)
Functions, equations, and their relationship. The study of functions, equations, and their
relationship is central to all of mathematics. Students perceive functions and equations as
means for analyzing and understanding a broad variety of relationships and as a useful tool for
expressing generalizations.
(4)
Relationship between algebra and geometry. Equations and functions are algebraic tools that can
be used to represent geometric curves and figures; similarly, geometric figures can illustrate
algebraic relationships. Students perceive the connections between algebra and geometry and use
the tools of one to help solve problems in the other.
(5)
Tools for algebraic thinking. Techniques for working with functions and equations are essential
in understanding underlying relationships. Students use a variety of representations (concrete,
numerical, algorithmic, graphical), tools, and technology, including, but not limited to,
powerful and accessible hand-held calculators and computers with graphing capabilities and
model mathematical situations to solve meaningful problems.
(6)
Underlying mathematical processes. Many processes underlie all content areas in mathematics.
As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics,
and reasoning, as well as multiple representations, applications and modeling, and justification
and proof.
Foundations for functions: knowledge and skills and performance descriptions.
(1)
The student uses properties and
attributes of functions and applies
functions to problem situations.
Advanced Mathematics Educational Support
Following are performance descriptions.
(A)
For a variety of situations, the student
identifies the mathematical domains and
ranges and determines reasonable domain
and range values for given situations.
(B)
In solving problems, the student collects
data and records results, organizes the
data, makes scatterplots, fits the curves
to the appropriate parent function,
interprets the results, and proceeds to
model, predict, and make decisions and
critical judgments.
29
(2)
(3)
(c)
The student formulates systems of
equations and inequalities from
problem situations, uses a variety
of methods to solve them, and
analyzes the solutions in terms of
the situations.
Following are performance descriptions.
(A)
The student uses tools including
matrices, factoring, and properties of
exponents to simplify expressions and
transform and solve equations.
(B)
The student uses complex numbers to
describe the solutions of quadratic
equations.
(C)
The student connects the function
notation of y = and f(x) =.
Following are performance descriptions.
(A)
The student analyzes situations and
formulates systems of equations or
inequalities in two or more unknowns to
solve problems.
(B)
The student uses algebraic methods,
graphs, tables, or matrices, to solve
systems of equations or inequalities.
(C)
For given contexts, the student interprets
and determines the reasonableness of
solutions to systems of equations or
inequalities.
Algebra and geometry: knowledge and skills and performance descriptions.
(1)
(2)
30
The student understands the
importance of the skills required to
manipulate symbols in order to
solve problems and uses the
necessary algebraic skills required
to simplify algebraic expressions
and solve equations and
inequalities in problem situations.
The student connects algebraic and
geometric representations of
functions.
The student knows the relationship
between the geometric and
algebraic descriptions of conic
sections.
Following are performance descriptions.
(A)
The student identifies and sketches
graphs of parent functions, including
linear (y = x), quadratic (y = x2), square
root (y = √x), inverse (y = 1/x),
exponential (y = ax), and logarithmic
(y = loga x) functions.
(B)
The student extends parent functions
with parameters such as m in y = mx
and describes parameter changes on the
graph of parent functions.
(C)
The student recognizes inverse
relationships between various functions.
Following are performance descriptions.
(A)
The student describes a conic section as
the intersection of a plane and a cone.
Advanced Mathematics Educational Support
.
(d)
(B)
In order to sketch graphs of conic
sections, the student relates simple
parameter changes in the equation to
corresponding changes in the graph.
(C)
The student identifies symmetries from
graphs of conic sections.
(D)
The student identifies the conic section
from a given equation.
(E)
The student uses the method of
completing the square.
Quadratic and square root functions: knowledge and skills and performance descriptions.
(1)
(2)
(3)
The student understands that
quadratic functions can be
represented in different ways and
translates among their various
representations.
The student interprets and describes
the effects of changes in the
parameters of quadratic functions in
applied and mathematical
situations.
The student formulates equations
and inequalities based on quadratic
functions, uses a variety of
methods to solve them, and
analyzes the solutions in terms of
the situation.
Advanced Mathematics Educational Support
Following are performance descriptions.
(A)
For given contexts, the student
determines the reasonable domain and
range values of quadratic functions, as
well as interprets and determines the
reasonableness of solutions to quadratic
equations and inequalities.
(B)
The student relates representations of
quadratic functions, such as algebraic,
tabular, graphical, and verbal
descriptions.
(C)
The student determines a quadratic
function from its roots or a graph.
Following are performance descriptions
(A)
The student uses characteristics of the
quadratic parent function to sketch the
related graphs and connects between the
y = ax2 + bx + c and the
y = a(x - h)2 + k symbolic
representations of quadratic functions.
(B)
The student uses the parent function to
investigate, describe, and predict the
effects of changes in a, h, and k on the
graphs of y = a(x - h)2 + k form of a
function in applied and purely
mathematical situations.
Following are performance descriptions.
(A)
The student analyzes situations involving
quadratic functions and formulates
quadratic equations or inequalities to
solve problems.
(B)
The student analyzes and interprets the
solutions of quadratic equations using
discriminants and solves quadratic
equations using the quadratic formula.
31
(4)
(e)
The student formulates equations
and inequalities based on square
root functions, uses a variety of
methods to solve them, and
analyzes the solutions in terms of
the situation.
The student compares and translates
between algebraic and graphical solutions
of quadratic equations.
(D)
The student solves quadratic equations
and inequalities.
Following are performance descriptions.
(A)
The student uses the parent function to
investigate, describe, and predict the
effects of parameter changes on the
graphs of square root functions and
describes limitations on the domains and
ranges.
(B)
The student relates representations of
square root functions, such as algebraic,
tabular, graphical, and verbal
descriptions.
(C)
For given contexts, the student
determines the reasonable domain and
range values of square root functions, as
well as interprets and determines the
reasonableness of solutions to square
root equations and inequalities.
(D)
The student solves square root equations
and inequalities using graphs, tables, and
algebraic methods.
(E)
The student analyzes situations modeled
by square root functions, formulates
equations or inequalities, selects a
method, and solves problems.
(F)
The student expresses inverses of
quadratic functions using square root
functions.
Rational functions: knowledge and skills and performance descriptions.
The student formulates equations and
inequalities based on rational functions,
uses a variety of methods to solve them,
and analyzes the solutions in terms of the
situation.
32
(C)
Following are performance descriptions.
(1)
The student uses quotients to describe
the graphs of rational functions,
describes limitations on the domains and
ranges, and examines asymptotic
behavior.
(2)
The student analyzes various
representations of rational functions with
respect to problem situations.
Advanced Mathematics Educational Support
(f)
(3)
For given contexts, the student
determines the reasonable domain and
range values of rational functions, as
well as interprets and determines the
reasonableness of solutions to rational
equations and inequalities.
(4)
The student solves rational equations and
inequalities using graphs, tables, and
algebraic methods.
(5)
The student analyzes a situation modeled
by a rational function, formulates an
equation or inequality composed of a
linear or quadratic function, and solves
the problem.
(6)
The student uses direct and inverse
variation functions as models to make
predictions in problem situations.
Exponential and logarithmic functions: knowledge and skills and performance descriptions.
The student formulates equations and
inequalities based on exponential and
logarithmic functions, uses a variety of
methods to solve them, and analyzes the
solutions in terms of the situation.
Advanced Mathematics Educational Support
Following are performance descriptions.
(1)
The student develops the definition of
logarithms by exploring and describing
the relationship between exponential
functions and their inverses.
(2)
The student uses the parent functions to
investigate, describe, and predict the
effects of parameter changes on the
graphs of exponential and logarithmic
functions, describes limitations on the
domains and ranges, and examines
asymptotic behavior.
(3)
For given contexts, the student
determines the reasonable domain and
range values of exponential and
logarithmic functions, as well as
interprets and determines the
reasonableness of solutions to
exponential and logarithmic equations
and inequalities.
(4)
The student solves exponential and
logarithmic equations and inequalities
using graphs, tables, and algebraic
methods.
(5)
The student analyzes a situation modeled
by an exponential function, formulates
an equation or inequality, and solves the
problem.
33
§111.34. Geometry (One Credit).
(a)
(b)
Basic understandings.
(1)
Foundation concepts for high school mathematics. As presented in Grades K-8, the basic
understandings of number, operation, and quantitative reasoning; patterns, relationships, and
algebraic thinking; geometry; measurement; and probability and statistics are essential
foundations for all work in high school mathematics. Students continue to build on this
foundation as they expand their understanding through other mathematical experiences.
(2)
Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry;
shapes and figures provide powerful ways to represent mathematical situations and to express
generalizations about space and spatial relationships. Students use geometric thinking to
understand mathematical concepts and the relationships among them.
(3)
Geometric figures and their properties. Geometry consists of the study of geometric figures of
zero, one, two, and three dimensions and the relationships among them. Students study
properties and relationships having to do with size, shape, location, direction, and orientation
of these figures.
(4)
The relationship between geometry, other mathematics, and other disciplines. Geometry can be
used to model and represent many mathematical and real-world situations. Students perceive the
connection between geometry and the real and mathematical worlds and use geometric ideas,
relationships, and properties to solve problems.
(5)
Tools for geometric thinking. Techniques for working with spatial figures and their properties
are essential in understanding underlying relationships. Students use a variety of representations
(concrete, pictorial, algebraic, and coordinate), tools, and technology, including, but not limited
to, powerful and accessible hand-held calculators and computers with graphing capabilities to
solve meaningful problems by representing figures, transforming figures, analyzing
relationships, and proving things about them.
(6)
Underlying mathematical processes. Many processes underlie all content areas in mathematics.
As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics,
and reasoning, as well as multiple representations, applications and modeling, and justification
and proof.
Geometric structure: knowledge and skills and performance descriptions.
(1)
34
The student understands the
structure of, and relationships
within, an axiomatic system.
Following are performance descriptions.
(A)
The student develops an awareness of the
structure of a mathematical system,
connecting definitions, postulates,
logical reasoning, and theorems.
(B)
Through the historical development of
geometric systems, the student
recognizes that mathematics is developed
for a variety of purposes.
(C)
The student compares and contrasts the
structures and implications of Euclidean
and non-Euclidean geometries.
Advanced Mathematics Educational Support
(2)
(3)
(4)
(c)
The student analyzes geometric
relationships in order to make
and verify conjectures.
The student understands the
importance of logical reasoning,
justification, and proof in
mathematics.
The student uses a variety of
representations to describe
geometric relationships and solve
problems.
Following are performance descriptions.
(A)
The student uses constructions to explore
attributes of geometric figures and to
make conjectures about geometric
relationships.
(B)
The student makes and verifies
conjectures about angles, lines,
polygons, circles, and three-dimensional
figures, choosing from a variety of
approaches such as coordinate,
transformational, or axiomatic.
Following are performance descriptions.
(A)
The student determines if the converse of
a conditional statement is true or false.
(B)
The student constructs and justifies
statements about geometric figures and
their properties.
(C)
The student demonstrates what it means
to prove mathematically that statements
are true.
(D)
The student uses inductive reasoning to
formulate a conjecture.
(E)
The student uses deductive reasoning to
prove a statement.
Following is a performance description.
The student selects an appropriate
representation (concrete, pictorial, graphical,
verbal, or symbolic) in order to solve
problems.
Geometric patterns: knowledge and skills and performance descriptions.
The student identifies, analyzes, and
describes patterns that emerge from twoand three-dimensional geometric figures.
Advanced Mathematics Educational Support
Following are performance descriptions.
(1)
The student uses numeric and geometric
patterns to make generalizations about
geometric properties, including
properties of polygons, ratios in similar
figures and solids, and angle
relationships in polygons and circles.
(2)
The student uses properties of
transformations and their compositions
to make connections between
mathematics and the real world in
applications such as tessellations or
fractals.
35
(3)
(d)
Dimensionality and the geometry of location: knowledge and skills and performance descriptions.
(1)
(2)
(e)
The student analyzes the
relationship between threedimensional objects and related
two-dimensional representations
and uses these representations to
solve problems.
The student understands that
coordinate systems provide
convenient and efficient ways of
representing geometric figures and
uses them accordingly.
Following are performance descriptions.
(A)
The student describes, and draws cross
sections and other slices of threedimensional objects.
(B)
The student uses nets to represent and
construct three-dimensional objects.
(C)
The student uses top, front, side, and
corner views of three-dimensional objects
to create accurate and complete
representations and solve problems.
Following are performance descriptions.
(A)
The student uses one- and twodimensional coordinate systems to
represent points, lines, line segments,
and figures.
(B)
The student uses slopes and equations of
lines to investigate geometric
relationships, including parallel lines,
perpendicular lines, and special segments
of triangles and other polygons.
(C)
The student develops and uses formulas
including distance and midpoint.
Congruence and the geometry of size: knowledge and skills and performance descriptions.
(1)
36
The student identifies and applies
patterns from right triangles to solve
problems, including special right
triangles (45-45-90 and 30-60-90) and
triangles whose sides are Pythagorean
triples.
The student extends measurement
concepts to find area, perimeter,
and volume in problem situations.
Following are performance descriptions.
(A)
The student finds areas of regular
polygons and composite figures.
(B)
The student finds areas of sectors and arc
lengths of circles using proportional
reasoning.
(C)
The student develops, extends, and uses
the Pythagorean Theorem.
(D)
The student finds surface areas and
volumes of prisms, pyramids, spheres,
cones, and cylinders in problem
situations.
Advanced Mathematics Educational Support
(2)
(3)
(f)
The student analyzes properties and
describes relationships in geometric
figures.
The student applies the concept of
congruence to justify properties of
figures and solve problems.
Following are performance descriptions.
(A)
Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties of
parallel and perpendicular lines.
(B)
Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties and
attributes of polygons and their
component parts.
(C)
Based on explorations and using concrete
models, the student formulates and tests
conjectures about the properties and
attributes of circles and the lines that
intersect them.
(D)
The student analyzes the characteristics of
three-dimensional figures and their
component parts.
Following are performance descriptions.
(A)
The student uses congruence
transformations to make conjectures and
justify properties of geometric figures.
(B)
The student justifies and applies triangle
congruence relationships.
Similarity and the geometry of shape: knowledge and skills and performance descriptions.
The student applies the concepts of
similarity to justify properties of figures
and solve problems.
Following are performance descriptions.
(1)
The student uses similarity properties
and transformations to explore and
justify conjectures about geometric
figures.
(2)
The student uses ratios to solve problems
involving similar figures.
(3)
In a variety of ways, the student
develops, applies, and justifies triangle
similarity relationships, such as right
triangle ratios, trigonometric ratios, and
Pythagorean triples.
(4)
The student describes the effect on
perimeter, area, and volume when length,
width, or height of a three-dimensional
solid is changed and applies this idea in
solving problems.
Source: The provisions of this §111.34 adopted to be effective September 1, 1996, 21 TexReg 7371.
Advanced Mathematics Educational Support
37
§111.35. Precalculus (One-Half to One Credit).
(a)
General requirements. The provisions of this section shall be implemented beginning September 1,
1998, and at that time shall supersede §75.63(bb) of this title (relating to Mathematics). Students can
be awarded one-half to one credit for successful completion of this course. Recommended prerequisites:
Algebra II, Geometry.
(b)
Introduction.
(c)
(1)
In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry
foundations as they expand their understanding through other mathematical experiences.
Students use symbolic reasoning and analytical methods to represent mathematical situations,
to express generalizations, and to study mathematical concepts and the relationships among
them. Students use functions, equations, and limits as useful tools for expressing
generalizations and as means for analyzing and understanding a broad variety of mathematical
relationships. Students also use functions as well as symbolic reasoning to represent and
connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model
physical situations. Students use a variety of representations (concrete, numerical, algorithmic,
graphical), tools, and technology to model functions and equations and solve real-life problems.
(2)
As students do mathematics, they continually use problem-solving, language and
communication, connections within and outside mathematics, and reasoning. Students also use
multiple representations, applications and modeling, justification and proof, and computation
in problem-solving contexts.
Knowledge and skills.
(1)
38
The student defines functions,
describes characteristics of
functions, and translates among
verbal, numerical, graphical, and
symbolic representations of
functions, including polynomial,
rational, radical, exponential,
logarithmic, trigonometric, and
piecewise-defined functions.
The student is expected to:
(A)
describe parent functions symbolically
n
and graphically, including y = x ,
x
y = ln x, y = loga x, y = 1/x, y = e ,
x
y = a , y = sin x, etc.;
(B)
determine the domain and range of
functions using graphs, tables, and
symbols;
(C)
describe symmetry of graphs of even and
odd functions;
(D)
recognize and use connections among
significant points of a function (roots,
maximum points, and minimum points),
the graph of a function, and the symbolic
representation of a function; and
(E)
investigate continuity, end behavior,
vertical and horizontal asymptotes, and
limits and connect these characteristics to
the graph of a function.
Advanced Mathematics Educational Support
(2)
(3)
(4)
The student interprets the meaning
of the symbolic representations of
functions and operations on
functions within a context.
The student uses functions and
their properties to model and solve
real-life problems.
The student uses sequences and
series to represent, analyze, and
solve real-life problems.
Advanced Mathematics Educational Support
The student is expected to:
(A)
apply basic transformations, including
a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|,
f(|x|), to the parent functions;
B)
perform operations including
composition on functions, find inverses,
and describe these procedures and results
verbally, numerically, symbolically, and
graphically; and
(C)
investigate identities graphically and
verify them symbolically, including
logarithmic properties, trigonometric
identities, and exponential properties.
The student is expected to:
(A)
use functions such as logarithmic,
exponential, trigonometric, polynomial,
etc. to model real-life data;
(B)
use regression to determine a function to
model real-life data;
(C)
use properties of functions to analyze and
solve problems and make predictions;
and
(D)
solve problems from physical situations
using trigonometry, including the use of
Law of Sines, Law of Cosines, and area
formulas.
The student is expected to:
(A)
represent patterns using arithmetic and
geometric sequences and series;
(B)
use arithmetic, geometric, and other
sequences and series to solve real-life
problems;
(C)
describe limits of sequences and apply
their properties to investigate convergent
and divergent series; and
(D)
apply sequences and series to solve
problems including sums and binomial
expansion.
39
(5)
(6)
The student uses conic sections,
their properties, and parametric
representations to model physical
situations.
The student uses vectors to model
physical situations.
The student is expected to:
(A)
use conic sections to model motion, such
as the graph of velocity vs. position of a
pendulum and motions of planets;
(B)
use properties of conic sections to
describe physical phenomena such as the
reflective properties of light and sound;
(C)
convert between parametric and
rectangular forms of functions and
equations to graph them; and
(D)
use parametric functions to simulate
problems involving motion.
The student is expected to:
(A)
use the concept of vectors to model
situations defined by magnitude and
direction; and
(B)
analyze and solve vector problems
generated by real-life situations.
Source: The provisions of this §111.35 adopted to be effective September 1, 1998, 22 TexReg 7623.
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Advanced Mathematics Educational Support
§111.36. Mathematical Models with Applications (One-Half to One Credit).
(a)
General requirements. The provisions of this section shall be implemented beginning September 1,
1998. Students can be awarded one-half to one credit for successful completion of this course.
Recommended prerequisite: Algebra I.
(b)
Introduction.
(c)
(1)
In Mathematical Models with Applications, students continue to build on the K-8 and Algebra I
foundations as they expand their understanding through other mathematical experiences.
Students use algebraic, graphical, and geometric reasoning to recognize patterns and structure,
to model information, and to solve problems from various disciplines. Students use
mathematical methods to model and solve real-life applied problems involving money, data,
chance, patterns, music, design, and science. Students use mathematical models from algebra,
geometry, probability, and statistics and connections among these to solve problems from a
wide variety of advanced applications in both mathematical and nonmathematical situations.
Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools,
and technology to link modeling techniques and purely mathematical concepts and to solve
applied problems.
(2)
As students do mathematics, they continually use problem-solving, language and
communication, connections within and outside mathematics, and reasoning. Students also use
multiple representations, applications and modeling, justification and proof, and computation
in problem-solving contexts.
Knowledge and skills.
(1)
(2)
The student uses a variety of
strategies and approaches to solve
both routine and non-routine
problems.
The student uses graphical and
numerical techniques to study
patterns and analyze data.
Advanced Mathematics Educational Support
The student is expected to:
(A)
compare and analyze various methods for
solving a real-life problem;
(B)
use multiple approaches (algebraic,
graphical, and geometric methods) to
solve problems from a variety of
disciplines; and
(C)
select a method to solve a problem,
defend the method, and justify the
reasonableness of the results.
The student is expected to:
(A)
interpret information from various
graphs, including line graphs, bar
graphs, circle graphs, histograms, and
scatterplots to draw conclusions from the
data;
(B)
analyze numerical data using measures of
central tendency, variability, and
correlation in order to make inferences;
(C)
analyze graphs from journals,
newspapers, and other sources to
determine the validity of stated
arguments; and
41
(D)
(3)
(4)
(5)
(6)
42
The student develops and
implements a plan for collecting
and analyzing data in order to make
decisions.
The student uses probability
models to describe everyday
situations involving chance.
The student uses functional
relationships to solve problems
related to personal income.
The student uses algebraic
formulas, graphs, and amortization
models to solve problems
involving credit.
use regression methods available through
technology to describe various models
for data such as linear, quadratic,
exponential, etc., select the most
appropriate model, and use the model to
interpret information.
The student is expected to:
(A)
formulate a meaningful question,
determine the data needed to answer the
question, gather the appropriate data,
analyze the data, and draw reasonable
conclusions;
(B)
communicate methods used, analysis
conducted, and conclusions drawn for a
data-analysis project by written report,
visual display, oral report, or multimedia presentation; and
(C)
determine the appropriateness of a model
for making predictions from a given set
of data.
The student is expected to:
(A)
compare theoretical and empirical
probability; and
(B)
use experiments to determine the
reasonableness of a theoretical model
such as binomial, geometric, etc.
The student is expected to:
(A)
use rates, linear functions, and direct
variation to solve problems involving
personal finance and budgeting,
including compensations and deductions;
(B)
solve problems involving personal taxes;
and
(C)
analyze data to make decisions about
banking.
The student is expected to:
(A)
analyze methods of payment available in
retail purchasing and compare relative
advantages and disadvantages of each
option;
(B)
use amortization models to investigate
home financing and compare buying and
renting a home; and
Advanced Mathematics Educational Support
(C)
(7)
(8)
(9)
The student uses algebraic
formulas, numerical techniques,
and graphs to solve problems
related to financial planning.
The student uses algebraic and
geometric models to describe
situations and solve problems.
The student uses algebraic and
geometric models to represent
patterns and structures.
use amortization models to investigate
automobile financing and compare
buying and leasing a vehicle.
The student is expected to:
(A)
analyze types of savings options
involving simple and compound interest
and compare relative advantages of these
options;
(B)
analyze and compare coverage options
and rates in insurance; and
(C)
investigate and compare investment
options including stocks, bonds,
annuities, and retirement plans.
The student is expected to:
(A)
use geometric models available through
technology to model growth and decay
in areas such as population, biology, and
ecology;
(B)
use trigonometric ratios and functions
available through technology to calculate
distances and model periodic motion;
and
(C)
use direct and inverse variation to
describe physical laws such as Hook's,
Newton's, and Boyle's laws.
The student is expected to:
(A)
use geometric transformations,
symmetry, and perspective drawings to
describe mathematical patterns and
structure in art and architecture; and
(B)
use geometric transformations,
proportions, and periodic motion to
describe mathematical patterns and
structure in music.
Source: The provisions of this §111.36 adopted to be effective September 1, 1998, 22 TexReg 7623.
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Subchapter D. Other High School Mathematics Courses
Statutory Authority: The provisions of this Subchapter D issued under the Texas Education Code, §28.002,
unless otherwise noted.
§111.51. Implementation of Texas Essential Knowledge and Skills for Mathematics, Other High School
Mathematics Courses.
The provisions of this subchapter shall be implemented by school districts beginning September 1, 1998,
and at that time shall supersede §75.63(o), (q)-(u), and (cc) of this title (relating to Mathematics).
Source: The provisions of this §111.51 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.52. Independent Study in Mathematics (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of
Independent Study in Mathematics. Required prerequisites: Algebra II, Geometry. Students may repeat
this course with different course content for a second credit.
(b)
Content requirements. Students will extend their mathematical understanding beyond the Algebra II
level in a specific area or areas of mathematics, such as theory of equations, number theory, nonEuclidean geometry, advanced survey of mathematics, or history of mathematics. The requirements for
each course must be approved by the local district before the course begins.
(c)
If this course is being used to satisfy requirements for the Distinguished Achievement Program, student
research/products must be presented before a panel of professionals or approved by the student's mentor.
Source: The provisions of this §111.52 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.53. Advanced Placement (AP) Statistics (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of this
course. Recommended prerequisites: Algebra II, Geometry.
(b)
Content requirements. Content requirements for Advanced Placement (AP) Statistics are prescribed in
the College Board Publication Advanced Placement Course Description: Statistics, published by The
College Board. This publication may be obtained from the College Board Advanced Placement
Program.
Source: The provisions of this §111.53 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.54. Advanced Placement (AP) Calculus AB (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of this
course. Recommended prerequisite: Precalculus.
(b)
Content requirements. Content requirements for Advanced Placement (AP) Calculus AB are prescribed
in the College Board Publication Advanced Placement Course Description Mathematics: Calculus AB,
Calculus BC, published by The College Board. This publication may be obtained from the College
Board Advanced Placement Program.
Source: The provisions of this §111.54 adopted to be effective September 1, 1998, 22 TexReg 7623.
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§111.55. Advanced Placement (AP) Calculus BC (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of this
course. Recommended prerequisite: Precalculus.
(b)
Content requirements. Content requirements for Advanced Placement (AP) Calculus BC are prescribed
in the College Board Publication Advanced Placement Course Description: Calculus AB, Calculus BC,
published by The College Board. This publication may be obtained from the College Board Advanced
Placement Program.
Source: The provisions of this §111.55 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.56. IB Mathematical Studies Subsidiary Level (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of IB
Mathematical Studies Subsidiary Level. To offer this course, the district must meet all requirements of
the International Baccalaureate Organization, including teacher training/certification and IB assessment.
Recommended prerequisites: Algebra II, Geometry.
(b)
Content requirements. Content requirements for IB Mathematical Studies Subsidiary Level are
prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from
International Baccalaureate of North America.
Source: The provisions of this §111.56 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.57. IB Mathematical Methods Subsidiary Level (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of IB
Mathematical Methods Subsidiary Level. To offer this course, the district must meet all requirements
of the International Baccalaureate Organization, including teacher training/certification and IB
assessment. Recommended prerequisites: Algebra II, Geometry.
(b)
Content requirements. Content requirements for IB Mathematical Methods Subsidiary Level are
prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from
International Baccalaureate of North America.
Source: The provisions of this §111.57 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.58. IB Mathematics Higher Level (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of IB
Mathematics Higher Level. To offer this course, the district must meet all requirements of the
International Baccalaureate Organization, including teacher training/certification and IB assessment.
Recommended prerequisite: IB Mathematical Studies Subsidiary Level or IB Mathematical Methods
Subsidiary Level.
(b)
Content requirements. Content requirements for IB Mathematics Higher Level are prescribed by the
International Baccalaureate Organization. Curriculum guides may be obtained from International
Baccalaureate of North America.
Source: The provisions of this §111.58 adopted to be effective September 1, 1998, 22 TexReg 7623.
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Advanced Mathematics Educational Support
§111.59. IB Advanced Mathematics Subsidiary Level (One-Half to One Credit).
(a)
General requirements. Students can be awarded one-half to one credit for successful completion of IB
Advanced Mathematics Subsidiary Level. To offer this course, the district must meet all requirements
of the International Baccalaureate Organization, including teacher training/certification and IB
assessment. Recommended prerequisite: IB Mathematics Higher Level.
(b)
Content requirements. Content requirements for IB Advanced Mathematics Subsidiary Level are
prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from
International Baccalaureate of North America.
Source: The provisions of this §111.59 adopted to be effective September 1, 1998, 22 TexReg 7623.
§111.60. Concurrent Enrollment in College Courses.
(a)
General requirements. Students shall be awarded one-half credit for each semester of successful
completion of a college course in which the student is concurrently enrolled while in high school.
(b)
Content requirements. In order for students to receive state graduation credit for concurrent enrollment
courses, content requirements must meet or exceed the essential knowledge and skills in a given course.
Source: The provisions of this §111.60 adopted to be effective September 1, 1998, 22 TexReg 7623.
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APPENDIX B
TOPIC OUTLINES FOR ADVANCED PLACEMENT MATHEMATICS
The following general topic outlines for AP Calculus AB, AP Calculus BC, and AP Statistics are
adapted from the College Board course outlines for each of the courses. More detailed course
descriptions are available from the College Board.34
AP Calculus AB
Functions, Graphs and Limits
•
•
•
•
•
•
•
Analysis of graphs
Understanding the concept of a limit
Calculating limits
Limits involving infinity
Asymptotic and unbounded behavior
Continuity
Implications of continuity
Differentiation
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Concept of the derivative
Derivatives of basic functions
Product and quotient rules
Chain rule
Implicit differentiation
Mean value theorem
Corresponding characteristics of f and f'
Geometric interpretation of the second derivative
Corresponding characteristics among f, f', and f"
Tangent lines as approximations
Optimization
Analysis of curves given in Cartesian form
Modeling rates of change
Related rates
Integration
•
•
•
•
•
•
•
The Riemann Sum
The definite integral
Fundamental Theorem of Calculus
Antidifferentiation techniques
Properties of integrals
Accumulation of rate of change
Numerical approximations of definite integrals
34
To download the Advanced Placement course descriptions, go to
apcentral.collegeboard.com/courses/descriptions and choose from the table of AP courses at the bottom of the
page.
48
Advanced Mathematics Educational Support
•
•
•
•
Applications of the integral
Separable differential equations
Initial value problems
Exponential growth and decay
AP Calculus BC
Functions, Graphs and Limits
•
•
•
•
•
•
•
Analysis of graphs
Understanding the concept of a limit
Calculating limits
Limits involving infinity
Asymptotic and unbounded behavior
Continuity
Implications of continuity
Differentiation
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Concept of the derivative
Derivatives of basic functions
Derivatives of parametric, polar, and vector functions
Product and quotient rules
Chain rule
Implicit differentiation
Mean value theorem
Corresponding characteristics of f and f'
Geometric interpretation of the second derivative
Corresponding characteristics among f, f', and f"
Tangent lines as approximations
Optimization
Analysis of curves given in Cartesian form
Analysis of curves given in parametric, polar, or vector forms
Modeling rates of change
Related rates
L’Hopital’s Rule
Integration
•
•
•
•
•
•
•
•
•
•
•
The Riemann Sum
The definite integral
Fundamental Theorem of Calculus
Antidifferentiation techniques
Properties of integrals
Accumulation of rate of change
Numerical approximations of definite integrals
Applications of the integral
Separable differential equations
Logistic differential equations
Initial value problems
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•
•
•
•
Slope fields
Euler’s method
Exponential growth and decay
Improper integrals
Sequences and Series
•
•
•
•
Concept of series
Special series
Tests for convergence
Taylor series
AP Statistics
Exploring Data
•
•
•
•
•
•
•
•
•
•
Describing distributions with graphs
Numerical summaries of distributions
Boxplots
Comparing distributions
Measures of relative position
Correlation
Linear regression
Model quality
Transformations to achieve linearity
Analyzing two-way tables
Planning a Study
•
•
•
Simple random sampling
Stratified, cluster and systematic sampling
Experimental design
Anticipating Patterns
•
•
•
Random variables
Normal distribution
Sampling distributions
Statistical Inference: Confirming Models
•
•
•
•
•
•
•
•
•
50
Confidence intervals
Estimating population parameters
Tests of significance
Type I/II errors and power
Difference in means
Difference in proportions
Chi-Square test for goodness of fit
Inference for two-way tables
Inference for slope of the least squares line
Advanced Mathematics Educational Support
APPENDIX C
VERTICAL TEAMS: A STRATEGY FOR BUILDING SCHOOL CAPACITY
Higher Education Involvement in Vertical Teaming
Vertical teaming is a vehicle through which higher education mathematics faculty can
understand and influence the secondary school mathematics experience of students. Ideally, the
gateway to the full range of mathematics courses should remain open for every student as long
as possible. The courses students take as early as the sixth grade often determine the sequence of
mathematics courses available to them throughout high school. In fact, the mathematics
curriculum that students take beginning in kindergarten determines the mathematical content
they can pursue in secondary school as well as at the collegiate level.
Although not every student will go on to pursue a mathematics-based field, all students will be
stronger for the experiences of a challenging and rigorous four-year secondary mathematics
program. The mathematics and analytical skills they learn will be useful to them as citizens and
in whatever fields they choose to pursue. Moreover, higher expectations for themselves in
mathematics can enhance student self-esteem and may translate to higher expectations in other
aspects of their lives as well. To succeed in giving all students access to a challenging and
rigorous secondary mathematics program, all teachers K–16 must communicate and work with
one another. A key strategy to foster such collaboration is the formation of K–16 vertical
teams. The primary goal of the vertical team strategy is to enhance all students’ achievement by
increasing communication and cooperation among the members of the teaching team about the
mathematics program at their schools. Through such communication and cooperation, vertical
teams can facilitate the implementation of academic changes and support structures necessary to
make high achievement in mathematics by all students a reality.
Higher Education Leadership in Vertical Teaming
Since vertical teams are already working in some Texas districts, the first item of business is for
higher education mathematics faculty to determine whether a vertical team is already in place in
the district with which they wish to work.
Vertical Teaming Not in Place
If vertical teaming is not in place, then the higher education faculty member can work to
initiate interest. In other words, the higher education faculty member must sell the local
decision-makers on the need and benefits of vertical teaming. A suggested sequence of events
follows.
Step 1: Contact the local district superintendent. The superintendent should understand the
benefits of vertical teaming for the faculty and students in the district, including the
impact on students, teachers, and the districts’ accountability rating. If the district has
not been involved in vertical teaming, then initial involvement in one content area is
strongly suggested, with mathematics as this area.
Step 2: Ask the superintendent for permission to visit with the high school administration
about the concept of vertical teaming. Again, the campus administrators will need to see
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the benefits for implementing vertical teaming on their campus. The benefits for the
campus will include the same benefits as for the district.
Step 3: Ask the campus administrators for permission to visit with the mathematics faculty
about implementing vertical teams on the campus. Ideally, the campus administrators
should attend this meeting. The benefits for the teachers and their students are discussed
at this meeting. One strong benefit for the teachers is that they will receive students
who are better prepared to pursue challenging and rigorous mathematics.
Step 4: Visit feeder schools35 to discuss the vertical team concept with the feeder schools’
campus administrators and mathematics faculty. Hopefully, the vertical team concept
can be backed down to the kindergarten level; thus, the team initially may include
teachers from grades K–12 and the higher education mathematics faculty members.
Step 5: Work with district leadership to select members of the vertical team. The vertical team
should involve interested teachers from each grade level and/or mathematics course on
each campus. Past experience has indicated that the number of members on a successful
vertical team should be between 10 and 20. Each individual affected by the vertical
team can provide input to the team’s work, but a vehicle should be established for such
input, rather than have an unwieldy number of members on the vertical team. In
addition to mathematics teachers, members of the vertical team should include the
following:
•
At least one member who has some degree of expertise related to the use of
technology in mathematics education.
•
At least one member who has a deep, extensive knowledge of college-level
mathematics. Efforts to create a strong mathematics program must include serious
attention to the mathematics itself, and if students are to be adequately prepared to
succeed in postsecondary mathematics, the team will need a picture of what
postsecondary mathematics looks like.
•
Administrators from the schools represented on the vertical team, particularly in
discussions of inclusiveness and student support structures, access to advanced
mathematics courses, professional development, and local action planning.
•
Counselors, especially in discussions of inclusiveness and student support structures,
access to advanced mathematics courses, and local action planning.
To be maximally effective, the mathematics vertical team will need support from campus
administrators at each of the schools represented by the team. Critical roles played by
administrators include the following:
•
Making the establishment of a strong, inclusive mathematics program a top priority
for the school;
•
Assisting with shaping the team’s goals and vision of where their efforts are headed;
35
A feeder school is a school whose students attend the next grade level at another campus. For example, a
certain middle school may “feed into” one designated high school.
52
Advanced Mathematics Educational Support
•
Communicating with district administration, teachers in other departments, the
school board, parents, and community members about the goals of the mathematics
vertical team; explaining why the team’s efforts are important; helping others to
value the team’s efforts; and defusing any concerns about the team and their efforts;
•
Examining how the team’s work will benefit the entire school and building the
team’s work into a schoolwide improvement plan or strategic plan;
•
Providing strong visible support for the team’s efforts.
Administrators need to help with a number of logistical issues. Administrators should not
expect vertical teams to meet on the teachers’ own time without compensation; not only is
such an expectation unfair, it also defeats the purpose of the formalization of communication
structures enabled by vertical teaming. Many teachers do not wish to be away from their classes
during the school day, even if the team meets only 4 to 5 times a year. Additionally, since the
vertical team will typically involve several campuses, travel time must be considered in
scheduling time for team meetings. If a viable time to meet cannot be identified during the
school day, then the higher education faculty can help in identifying grants to fund teachers’
extra hours. Higher education faculty can also be a valuable resource in writing proposals for
funding projects. If school district or grant funds are not available, then a local sponsor might
be identified for the vertical team efforts. The local sponsor may be an individual or business
partnership with an interest in mathematics preparation of the local youth.
After members of the vertical team are identified and time to meet is identified, the team must
develop into a cohesive unit rather than a group of individuals. This development of a cohesive
unit is much the same as that done by a coach as he or she takes a group of individual athletes
and builds them into a team. Each member must develop the attitude that the team’s interest is
of greater value than any one individual’s interest.
Vertical Teaming Already in Place
If a functioning vertical team is already in place in a district or campus, higher education
mathematics faculty face different challenges. One major issue can be that the higher education
mathematics faculty member has to integrate him- or herself into an already established team.
However, this is an issue not only with the addition of a higher education faculty member to
the team, but also as any new members are added to the vertical team as a result of retirement,
resignation, or personal reasons. Periodically, the vertical team will need to experience
additional team-building activities to orient new members.
If a team is already in place and the higher education faculty member desires to become
involved, then the faculty member should go through a process similar to that described above.
However, if the team is already functioning, then the faculty member will not need to “sell” the
district on the concept but instead will need to share with the various administrators and
teachers the benefits that can occur from the involvement of higher education faculty, such as
improved communication between higher education and the local school system and better
preparation for students who leave the school system to attend a college or university.
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Vertical Alignment of the K–16 Curriculum
The vertical alignment of the curriculum process K–16 operates similarly to the process in
K–12. To include higher education, one of the first steps for vertical team is to decide what
higher education courses the team should focus on. That is, what higher education courses
should the team connect vertically with high school mathematics courses, by identifying the
mathematical content needed for success in those higher education courses? The higher
education faculty can identify the content necessary for success in the higher education courses.
They can also guide the secondary mathematics teachers regarding what content needs to be
covered as a prerequisite for success in those higher education courses.
Some vertical teams may wish to consider a combination of College Algebra and Trigonometry,
while others may consider the calculus sequence. After selecting the courses, the higher
education mathematics faculty identifies critical skills and knowledge that all students need to
have to succeed in that higher education course. Next, the team identifies where that content is
taught and learned in the secondary curriculum. Then, the highest-level relevant mathematics
course in the high school curriculum is investigated. The high school teachers should identify
the skills and knowledge that the students need to be successful in those high school courses.
After this content is identified, then the team identifies where that content is taught and
learned in the secondary curriculum as described in the TEKS. The high school faculty can
provide information on why some of the content may or may not be appropriate for the high
school course. Working together as a team, the high school and higher education faculty can
reach consensus on the appropriate content at each level and the best tools for teaching that
content for the success of all students.
As collaborations develop that promote coordination of precollege and postsecondary
mathematics teaching, one outcome will be enhanced capacity at the high school level to
prepare more students to immediately enroll in the more challenging mathematics courses that
they need for success.
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Charles A. Dana Center. (1998). Advanced Placement Program mathematics vertical team toolkit. New
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