Longwood University
Department of Mathematics & Computer Science
MATH 309, Sections 04 and 05 (Fall 2011)
Numeration Systems
Professor:
Office Location:
Phone/Voicemail:
Math Office:
E-mail:
Office Hours:
Dr. Maria Timmerman
339 Ruffner
434.395.2890 (w), 434.978.7184 (h)
434.395.2194 (Gale Moss)
Class Times:
T, R
timmermanma@longwood.edu
4:00 – 5:00 pm, Tuesday; 1:30 – 2:30 pm Wednesday;
10:00 – 12 noon, Thursday; and by appointment
12:30– 1:45; 2:00 – 3:15, 350 Ruffner
(Be sure to check the time of your assigned section.)
Texts:
Parker, T.H. & Baldridge, S.J. (2004). (Second Edition) Elementary Mathematics for Teachers.
Okemos, Michigan: Sefton-Ash Publishing.
Primary Mathematics 3A Textbook. (2008). Singapore: Marshall Cavendish International.
Primary Mathematics 4A Textbook. (2008). Singapore: Marshall Cavendish International.
Primary Mathematics 5A Textbook. (2008). Singapore: Marshall Cavendish International.
Primary Mathematics 5A Workbook. (2008). Singapore: Marshall Cavendish International.
Primary Mathematics 6A Textbook. (2008). Singapore: Marshall Cavendish International.
Other Required or Suggested Materials:
 Notebook/sprial notebook for class notes & homework problem sets (assignments will be
collected at random), calculator (any type), and colored pencils (optional)
 CDs or jump-drives to save course material & computer files
 BRING the main textbook to each class session
 Bookmark the Virginia Department of Education (VDOE) website to become familiar with and use
the new 2009 K-8 Mathematics Virginia Standards of Learning, which are being implemented this fall,
for the 2011-2012 school year.
(see: http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml)
Additional Information: Students are responsible for checking the ANNOUNCEMENTS and
ASSIGNMENTS in Blackboard in advance of each class period. See http://blackboard.longwood.edu.
Also, students are responsible for downloading the syllabus and all other needed course documents
from Blackboard, printing them if hardcopies are desired, and knowing the information contained in
these documents. All academic regulations in the current Longwood Catalog will be followed.
Turn cell phones to Vibrate during class. As part of our community of learners, to be fully
engaged in our class sessions, please do not use your cell phone, including text
messages, during class. In addition, cell phones may NOT be used as a calculator during
tests/quizzes.
Course Description:
Examines various numeration systems past and present, computational algorithms, basic sets of
operations, elementary number theory, concepts of rational numbers, and proportional reasoning in a
problem-solving environment through student investigations and hands-on activities. 3 credits.
MATH 309 - Fall 2011
1
M. Timmerman
Course Objectives:
This course is designed to prepare students who plan to pursue PK-6 and/or PK-8 teacher licensure through
the liberal studies major. A major goal of the course is to re-examine the mathematics content
knowledge you will teach, as well as develop pedagogical ideas for teaching PK-6 and/or PK-8
mathematics, and how children learn mathematics developmentally. Given that mathematical reasoning
develops as a result of problem solving, the course will provide you with many opportunities to solve
problems, and examine how students develop ways to reason mathematically as a result of problem solving.
Although you already know PK-8 mathematics at a certain level, it is very likely that what you know is how to
do some mathematical procedures, but you may not understand why those procedures work. This course will
challenge you to think beyond the how to the why. This kind of knowledge is crucial for the depth and
flexibility of understanding that you will need in order to teach PK-6 and/or PK-8 mathematics for conceptual
understanding. In addition, a major objective of PK-8 mathematics is to develop students’ number sense,
which is the foundation for understanding all other mathematical concepts and ideas.
Course Competencies:
By the completion of this Math 309 course, students should be able to:
 deepen your understanding of PK-8 mathematics content knowledge.
 be familiar with the new 2009 K-8 Mathematics Virginia Standards of Learning, including released test
items used for assessment, which are being implemented this fall, for the 2011-2012 school year.
 communicate mathematical ideas orally and in writing.
 gain experience and use a variety of strategies in problem solving.
 solve problems/tasks/activities related to PK-8 mathematics content in multiple ways.
 use hands-on activities and tools to solve mathematics problems.
 know and incorporate various models to represent specific concepts and skills related to number and
operation sense through the four operations of addition, subtraction, multiplication, and division for
whole numbers and rational numbers.
 develop their professional competence, confidence, and enthusiasm for learning and teaching PK-6
and/or PK-8 mathematics.
Course Content Outline and Methodology:
The course will cover mathematics topics in Chapters 1, 2, 3, 5, and 6 from the text. Class sessions
will engage you in student-centered learning environments with hands-on explorations, collaborative
small-group work and discussion, pairs engaged in problem-solving tasks and communication, and
whole-class discussions that model the instructional practices that are advocated in the Curriculum
and Evaluation Standards (NCTM, 1989), the Professional Standards for Teaching Mathematics
(NCTM, 1991), the Assessment Standards for School Mathematics (NCTM, 1995), the Principles and
Standards for School Mathematics (NCTM, 2000), and the Curriculum Focal Points (2006). These
reform documents support a constructivist perspective for the teaching and learning of mathematics.
***Outside of class assignments include chapter readings, online journal articles, reflective writing
assignments, and homework problem sets from the Primary Mathematics grade-level texts.
Mathematics teaching and learning are problem-solving activities. You should spend at a
minimum 2 to 3 hours preparing for each class session – reading, studying, and completing
assignments. To succeed as a mathematics student and teacher it is essential to know mathematics
vocabulary. As you read each section, you may want to write each mathematics word on a 3x5 note
card, along with its definition, symbol (if any) and examples. Not only will this make studying for tests
and quizzes much easier, it will also be a useful resource in your future teaching career.
MATH 309 - Fall 2011
2
M. Timmerman
Inclement Weather Policy
Information concerning cancellation of classes due to inclement weather is available at www.longwood.edu,
on the campus radio, WMLU 91.3 FM, or by calling 434.395.2000. In addition, I will post an announcement
on Blackboard if the weather prevents travel to Longwood. The website is http://blackboard.longwood.edu.
Statement of Compliance with Americans with Disabilities
Any student who feels s/he may need an accommodation based on the impact of a physical,
psychological, medical, or learning disability should contact the instructor privately. If you have not
already done so, please contact the Office for Disability Services (103 Graham Building, 395-2391) to
register for services.
Course Requirements and Evaluation:
 (1) Mathematics Autobiography, goals,
learning theory paper – 5 points
 (2) Problem Sets (unannounced) – 10 points
 (1) Assessment Project – 20 points
 (3) Short Quizzes – 60 points
 (2) In-Class Tests – 90 points
FINAL GRADE – based on a total of 270 points
 Assignments, Project, Quizzes, Tests, Total – 185 points
 Final Exam – 85 points
Course Grade Assignment:
92-100%
A
90-91
A87-89
B+
82-86
B
80-81
B77-79
C+
72-76
C
70-71
C67-69
D+
62-66
D
60-61
D0-59
F
FINAL GRADE = percentage of your points based on
a total of 270 points (check Blackboard gradebook)
Example – A total of 189 points out of 270 points is a
70%, which is a “C-”
Letter grades will signify the following:
A Excellent performance in all areas.
B Satisfactory performance in all areas and excellent performance in some areas.
C Satisfactory performance in all areas.
D Less than satisfactory performance in some areas.
F Less than satisfactory performance in all areas.
Non-graded Assignments: (but still very important to your success!)
 In-Class Activities – participation will help you learn concepts for the course
 READ Chapter materials regularly to assist your learning and to formulate questions
MATH 309 - Fall 2011
3
M. Timmerman
Mathematics Content Assessments
In-Class Quizzes & Tests – “paper & pencil” tests based on in-class activities, homework problems,
and assignments; these will consist of content as listed on the course schedule including mathematics
problems and discussion information based on in-class activities, powerpoints, and class notes
Final Exam – will consist of mathematics content from the entire semester including mathematics
problems and discussion information based on in-class activities, powerpoints, and class notes
Assessment Project – Using various websites to access the NCTM Principles and Standards, Virginia
Standards of Learning, released VA SOL test items, and TIMSS international test items, you will have
an opportunity to compare and contrast international and state assessments, and describe how test
items aligned with standards may be used to inform instruction. Further guidelines will be provided
early in the semester. Due: Tuesday, October 18th.
Homework Problem Sets – These are assigned at the end of each section of the Parker and
Baldridge textbook and Primary Mathematics grade-level textbooks. Complete these problems after
we have discussed the section material in class through class activities and power points. (Or, you
may work ahead of schedule, too.) Although the author may say to use ‘mental math,’ be sure to
show all strategies used to solve problems, not just ‘final answers’ for full credit. You may use your
textbook, other reference books, and work in small groups to complete problem sets. At least 2 of the
homework problem sets will be collected unannounced during the semester.
Homework Problem Set Rubric – 5 points each
Assigned homework sets of problems should be completed on a regular basis solely for your benefit.
The assigned problems are a minimum; you may complete more of the end-of-section problems. The
completion of daily reading assignments will enable you to effectively participate in class and learn
from others. Homework Problem Sets will be completed outside of class and collected unannounced
during the semester.
5
Student is in attendance; ALL assigned problems attempted (on-time) and illustrates
meaningful consideration of mathematical ideas and problem-solving strategies. In
most cases, you will need to include some explanations of your reasoning. Full credit
is NOT given for work showing ONLY final answers. The focus is on the process
of problem solving and describing your mathematical thinking.
3
Student is in attendance; MOST (more than 80%) assigned problems attempted (ontime) and illustrates some consideration of mathematical ideas and problem-solving
strategies.
2
Problem sets are incomplete in some way (e.g., only shows a final solution or very
minimal work with lack of clarity in procedures) or turned in late due to lack of
attendance (even if it is an excused absence.
0
Problem sets are unsatisfactory and/or student does not complete and turn in
assignment.
MATH 309 - Fall 2011
4
M. Timmerman
Class Attendance and Contribution
Students are expected to attend all classes. Please be sure to arrive on time for each class session.
You are responsible for doing the assigned readings before class and incorporating your thinking in
class discussions. Class attendance is expected, both for you to learn and so that others may benefit
from your input.
Though it often feels safer to not let anyone know what you are thinking, only active participation in the ongoing articulation of unformulated ideas can promote your learning. Being a risk taker and willing to share
your evolving understandings with others are important abilities for future teachers to develop and
demonstrate. In short, from day one you are evaluated as a (becoming) teacher - no longer as a student!
You are expected to participate in ALL group activities and class discussions. The level of participation
will affect the quality and quantity of your learning.
The following attendance policies are in effect for this course:
 A grade of “0” or “F” may be given on work missed because of unexcused absences.
 After 3 absences (excused or unexcused), absences result in a loss of 3 points per
absence and are subtracted from your total points at the end of the course.
 If you are late for class 3 times, it is the equivalent of one absence.
 A course grade of “F” may be assigned when the student has missed a total (excused and
unexcused) of 25 percent of the scheduled class meeting times (more than 7 absences)
 There are NO MAKEUPS for in-class activities (except as noted in descriptions above) as it is
impossible to recreate the discussion that occurred during class.
 Acceptable excused absences (as listed in the Longwood catalog) will be taken into
consideration for missed quizzes or tests; however, it is to your advantage to make PRIOR
arrangements.
EXTRA CREDIT
To pass this course, you will need to meet all of the course requirements. In addition, up to 5 bonus points
will be awarded for attendance at the PCTM mathematics conference, held on the 2nd day of Reading Days,
Tuesday, October 11th, here at Longwood. The conference is held at Ruffner Hall, 9-3 pm. In order to receive
extra credit, you will need to attend at least 3 sessions (each session is one-half hour) and write a short
paper based on the presentations you attended. Further guidelines for the paper will be given prior to the
conference. No other extra credit is available for this course.
Longwood’s Honor System
A strong tradition of honor is fundamental to the quality of living and learning in the Longwood
community. The Honor System was founded in 1910, and its purpose is to create and sustain a
community in which all persons are treated with trust, respect, and dignity. Longwood affirms the
value and necessity of integrity in all intellectual community endeavors. Students are expected to
assume full responsibility for their actions and to refrain from lying, cheating, stealing, and plagiarism.
The Longwood Honor Code applies to all work for the course as follows:
 Any out-of-class practice work or hand-in assignment can include using text information,
discussion with other class members, and/or discussion with professor. However, the final

product submitted for the grade must be the student’s own work.
Any in-class activities that involve teamwork allows for discussion within your team as directed
by the professor.
MATH 309 - Fall 2011
5
M. Timmerman


ALL QUIZZES & TESTS ARE TO BE COMPLETED INDIVIDUALLY. Please write and sign the
honor code on all exams indicating that: “I have neither given nor received help on this work,
nor am I aware of any infraction of the Honor Code.”
Any student that violates the Honor Code will receive a zero on the assignment and
will be reported to the Longwood University Honor Board.
Tentative Course Schedule – may be changed as needed
Week of
Text Chapters
August 23 – August 25
Introduction to
Chapter 1
Chapter 1
Numeration Systems
NCTM and Virginia
Content and Process
Standards
August 30 – September 1
Class Activity & Assignment Due Dates
Introductions
Community of Learners
Focus: mathematics, how children learn
mathematics, and teaching mathematics
Before Thursday class session: Reading –
Preface pp. iii-x and Chapter 1, Section 1.1
Due: Mathematics autobiography, goals, and
theory of how students learn mathematics
paper—Thursday, August 25t
Number Sense
Early Counting, Set
and Measurement
Models
Before Tuesday class session: Complete Homework
Set 1, p. 6. #1, 3, 4, 5, 6 (for #5, make sure to use
Primary Math text #3A);
Chapter 1
Before Tuesday class session: Complete Homework
Set 2, p. 13, #1, 3, and 4. NOTE: Although the
Before Thursday class session: Reading – Chapter 1,
Four Basic
Section 1.2; AND Download & Bring Ten-Frame
Number Relationships posted on Blackboard
September 6 – September 8
Place Value
XMANIA
Mental Math
Addition/Subtraction
September 13 – September 15
Chapter 1
More Addition and
Subtraction
Open Number Lines
Multiplication of
Whole Numbers
MATH 309 - Fall 2011
author states to solve problems mentally, write down
your strategies or thinking, not just the final
answers.
Before Thursday class session: No new reading or
written assignment
Before Tuesday class session: Reading – Chapter 1,
Section 1.3 AND Complete Homework Set #3, pp.
18-19, #1, 2, 3, 4, 6, 7; Be sure to record your work
for the ‘mental math’ strategies.
Short Quiz 1– Thursday, September 15th
Before Thursday class session: Reading – Chapter 1,
Section 1.4 AND Complete Homework Set #4, p. 24,
#1, 4, 5, 7, 8; Be sure to record your work for
‘mental math’ strategies; not just final answers.
6
M. Timmerman
September 20 – September 22
Chapter 1
More on
Multiplication
Division of Whole
Numbers
September 27 – September 29
Grade Estimates Due by noon,
September 26th (D and F)
Chapter 2
Classroom Practice
Polya’s 4 Phases
Problem Solving
Chapter 2
CGI Word Problems
October 4 – October 6
Chapter 2
Art of Word
Problems: Using Bar
Diagrams
October 11 – October 13
No class Tuesday,
October 11th
Reading Day and
PCTM Conference
October 18 – October 20
Before Tuesday class session: Reading – Chapter 1,
Section 1.5
Before Thursday class session: Complete Homework
Set #5, pp. 30-31, #1, 4, and 7
Before Tuesday class session: Reading – Chapter 1,
Section 1.6 AND Complete Homework Set #6, pp.
36-37, #1, 2, and 4; Reading – Chapter 1, Section
1.7 (Read with a ‘critical eye’ – Take notes; Be able
to discuss points of disagreement with author’s
views)
Before Thursday class session: Reading – Chapter 2,
Section 2.1 AND Complete Homework Set #7, p. 47,
#1a, 1b, 2a, 2b, 3a, 3b, 5a – 5d; IMPORTANT:
Record your work for ‘mental math’ strategies and
describe your thinking.
In-Class Test 1 – Tuesday, October 4th
Before Thursday class session: Reading – Chapter 2,
Section 2.2 AND Complete Homework Set #8, p. 51,
#1, 3, 4
No class Tuesday,
October 11th
Chapter 3
Before Thursday class session: Reading – Chapter 2,
Section 2.3, AND complete Homework Set #9, p. 56,
#1, 2, 4, ‘Teacher’s Solutions’ means to solve the
problem using a Bar Diagram
Chapter 3
Due: Assessment Project, Tues., October 18th
Before Tuesday class session: Read posting on BB,
‘Did you know? The utility of other bases’ AND solve
the 8 Homework problems stated with the posting
Changing Bases
Strands of
Mathematical
Proficiency
Addition and
Subtraction Invented
Algorithms (review)
Multiplication
Invented Algorithms
Before Thursday class session: Reading – Chapter 3,
Section 3.1 and 3.2; Complete Homework Set #10,
p. 62, #4a and 4b (IMPORTANT: use base 10 blocks
for models, instead of chip models AND write the
partial sums method, like we did on the 3 little pigs
class activity), and #5; Complete Homework Set
#11, p. 65, #4, 5a, 5b (again: use base 10 blocks
for models, instead of chip models AND write the
partial differences method), #6 (only read this part
in the Primary 3A text, #7 (use Bar Diagrams for
Teacher’s Solutions), and #8
MATH 309 - Fall 2011
7
M. Timmerman
October 25 – October 27
Chapter 3
Short Quiz 2 – Tuesday, October 25th
Chapter 5
Before Thursday class session: Reading – Chapter 3,
Sections 3.4 and 3.5 AND Complete Homework Set
#13, p. 76, #1, 4 (revised directions: draw using
base ten blocks and record method), #5, and #7:
use partial quotients or scaffold method; AND
Complete Homework Set #14, p. 81, #1, 2
OMIT Section 3.6
Chapter 5
Before Tuesday class session: Reading – Chapter 5,
Section 5.1 AND Complete Homework Set #19, p.
112, #4
Division
Invented Algorithms
Number Theory
Odd and Even
Numbers
November 1 – November 3
Checks for Divisibility
Prime and Composite
Numbers
GCF and LCM
November 8 -November 10
Chapter 6
Fraction Concepts
Before Tuesday class session: Reading – Chapter 3,
Section 3.3 AND Complete Homework Set #12, p.
70-71, #1, 2, 3 (NEW directions: use partial products
and draw rectangular arrays for #3), #6 (use Bar
Diagrams for Teacher’s Solutions), #7 (again, use
rectangular arrays and partial products), and #8
Before Thursday class session: Reading – Chapter 5,
Sections 5.2 and 5.3 AND Complete Homework Set
#20, p. 117, #1-4; AND Homework Set #21, p. 121,
#1a, 1c, 2c, 2d, 2e, 2h, 3d, 3e, 3h
Short Quiz 3 – Tuesday, November 8th
Download & bring to class fraction activity sheets
(may share with a partner)
Before Tuesday class session: Reading – Chapter 5,
Sections 5.4 and 5.5; Complete Homework Set #22,
p. 124, #1-3 (IMPORTANT: Be sure to show work to
determine if a number is prime or not prime for #1,
and ‘test’ more numbers than those listed on the
chart for 3c) AND Homework Set #23, p. 130, #1, 4,
6a, (New directions: use any methods for finding
GCF and LCM)
November 15-November 17
MATH 309 - Fall 2011
More on Fraction
Concepts
Before Thursday class session: Reading – Chapter 6,
Section 6.1; AND Complete Homework Set #24, p.
137, #2, 3, 6c, 6d
Chapter 6
Before Tuesday class session: Reading – Chapter 6,
Section 6.2; AND Complete Homework Set #25, pp.
142-143, #1, 3a, 6, 8; AND in Primary Textbook 5A:
Addition/Subtraction
of Fractions
p. 40, Practice 3B, #5, 6, and 7 AND p. 43, Practice
3C, #5, 7, and 9
In-Class Test 2 – Thursday, November 17th
8
M. Timmerman
November 22-November 24
No class Thursday,
November 24th
Thanksgiving
November 29 – December 1
Last Day of Classes Friday
Wednesday, December 7th
Chapter 6
Before Tuesday class session: Reading – Chapter 6,
Section 6.3; AND Complete Homework Set #26, p.
149, #5 (revise directions: please use
rectangles/diagrams for problems 6-10 (omit #5) of
Practice E in Primary Textbook 5A) and #9: do not
use the area model, instead decompose and multiply
using the distributive property
Chapter 6
Before Tuesday class session: Reading – Chapter 6,
Sections 6.4 and 6.5; AND Homework Set #27, p.
154, #1 using any strategy; Homework Set #28, pp.
158-159, #1 and #4
Review
Before Thursday class session: Questions for exam?
Multiplication of
Fractions
Division of Fractions
Final Exams
(In-Class)
12:30 – 1:45 class: Weds, 8:00 – 10:30 am
2:00 – 3:15 class: Weds, 11:30 – 2:00 pm
Description of 1st written paper:
Mathematics Autobiography, Individual Goals, and Theory of Students’ Learning of
Mathematics – 5 points
DUE: Thursday, August 25th
At the beginning of the course, in a two-to-three page word-processed essay (double spaced, ~ 1
inch margins, and 12-10 point font), you are asked to write a mathematics autobiography describing
your past experiences in learning mathematics. Please print a hard copy and turn in Thursday.
FIRST: As you describe your experiences, you may find it useful to answer some (not all) of the
following questions:
• What topics in mathematics did you like, and which did you dislike?
• Who were the people who played a positive role in your mathematical life, and why?
• Who played a negative role, and why?
• Describe your good mathematical experiences and the poor experiences.
• In what environments do you learn best?
• What environments hinder your learning?
SECOND: In a separate paragraph or using bullet statements, identify the individual goals you plan
to pursue during this course. For example, respond to some (not all) of the following questions: Why
are you taking this course? (do not state that it is degree requirement) What do you want to learn?
What do you hope to accomplish? What would you like to work on in the course?
THIRD: In a separate statement or paragraph, respond to the following question: What is
mathematics? Then, in a separate paragraph(s), describe your theory of how children learn
mathematics in 2nd grade. This will probably be different for each of us (and that’s OK as we begin
learning with each other). Depending on your different experiences, some of you may have a lot or
just a little to say for your beginning mathematical theories.
MATH 309 - Fall 2011
9
M. Timmerman