MATH 653

Longwood University
Department of Mathematics & Computer Science
MATH 653 - 94 (Summer 2010)
Rational Numbers and Proportional Reasoning for K-8 Teachers
Professor: Maria Timmerman
UVa Doctoral Student: Kateri Thunder
Instructor: Sarah Minervino
Instructor: DeAnna Moreau
Office Hours:
Kateri: Tuesday, 1:30 – 2:30 pm
Sara: Thursday, 1:30 – 2:30 pm
Contact Info: , 434.962.4579
Contact Info: , 434.409.6785
Contact Info: , 202.352.2365
Contact Info: , 804.347.2437
(To contact the teaching team - Email is preferred.)
Dr. Timmerman:
Tuesday, 7 – 8 pm
DeAnna: Thursday, 7 - 8 pm (June 17 & July 29, 7:30 – 8:30 pm)
Contact Sara Minervino with technology difficulties.
Monday & Wednesday Synchronous class sessions: 6:30 – 8:00 pm; Log on to Longwood Blackboard
at Please log on ~10 minutes before class begins – Wimba Classroom.
Fosnot, C.T. & Dolk M. (2002). Young mathematicians at work: Constructing fractions, decimals, and
percents. Portsmouth, NH: Heinemann.
Lamon, S.J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and
instructional strategies for teachers (2nd Edition). Mahwah, NJ: Lawrence Erlbaum Associates.
Smith, M.S., Silver, E.A., & Stein, M.K. (2005). Improving instruction in rational numbers and proportionality :
Using cases to transform mathematics teaching and learning. New York, NY : Teachers College Press.
Other Required Materials:
 Notebook/sprial notebook for notes, calculator (any type), graph paper, and colored pencils (optional)
 Headsets, Computer disks, CDs or jump-drives to save course material & computer files
 DigiMemo tool (provided by the grant - cohort group only)
 Turn cell phones to Vibrate during face-to-face class sessions.
Other Information: Students are responsible for checking the ANNOUNCEMENTS and ASSIGNMENTS in
Blackboard in advance of each class session. The website is Also, students are
responsible for downloading all needed course documents from Blackboard, printing them if hardcopies are
desired, and knowing the information contained in these documents. NOTE: All students should receive a
letter from Longwood stating your user ID and directions for logging on to Blackboard. Please contact the Help
Desk (434.395.4357) if you need instructions for logging on, or have other IT questions.
All papers should be written using Microsoft Word and submitted through the course Blackboard site to be
graded. For grant cohort students: All assignments will be completed using a DigiMemo and the DigiMemo
Handwriting eBook files will be submitted through the course Blackboard site to be graded. For non-grant
students: Written reflections must be word-processed, and HW problem sets may be hand-written or wordprocessed, scanned, and submitted through the course Blackboard site to be graded. All students should write
HW problem sets clearly, and large enough to be able to see their work when posted in class powerpoints.
Course Description: 3 Graduate Credit Hours
Rational Numbers and Proportional Reasoning for K-8 Teachers is one of the core mathematics courses
for the K – 8 Mathematics Specialist Programs developed through a cooperative arrangement of Longwood
University, Norfolk State University, University of Virginia, and Virginia Commonwealth University. The
course will focus on the content and processes that support the Virginia computation and estimation strand
MATH 653 – Summer 2010
M. Timmerman
of the Virginia K-8 Mathematics Standards of Learning, specifically as it applies to rational numbers and
proportional reasoning. The five NCTM process standards—problem solving, reasoning and proof,
connections, communication, and representation—will be emphasized throughout the course.
Course Objectives:
This course is designed to engage participants in the following: (a) constructing a deeper understanding of the
essential content of the K-8 strands of rational numbers and proportional reasoning, (b) examining students’ ways
of reasoning mathematically about rational numbers and proportional reasoning, and (c) developing pedagogical
content knowledge of rational numbers and proportional reasoning (appropriate for K-8 Mathematics Teacher
Specialists.) Given that proportional reasoning develops as a result of solving problems, a major focus of the
course will provide participants with opportunities to solve many problems, and examine how students develop
proportional reasoning as a result of problem solving.
In particular, participants will explore a wide variety of rational number problems and activities through structured
tasks designed to develop participants’ depth and flexible understanding of the multiple roles that rational
numbers and proportional reasoning play in mathematics and to consider the following questions: (a) What are
the big ideas related to rational numbers in a given situation? (b) What strategies can be used to solve problems?
and (c) What models can be used to represent problems?
Course Competencies:
By the completion of this Math 653 course, students should be able to:
Solve rational number problems/tasks in multiple ways without using standard algorithms or rules.
Know, understand and apply the process of mathematical problem solving in the context of rational
number and proportional reasoning problems.
[Demonstrate an] Understanding of the knowledge, skills, and processes of the Virginia Mathematics
Standards of Learning [related to rational numbers] and how curriculum may be organized to teach
these standards to learners (VA Mathematics Specialist Endorsement Requirement).
Communicate their mathematical thinking orally and in writing to peers and teaching team.
[Demonstrate an] Understanding of and the ability to use the five processes—becoming
mathematical problem solvers, reasoning mathematically, communicating mathematically, making
mathematical connections, and using mathematical representations—at different levels of complexity
(VA Mathematics Specialist Endorsement Requirement).
[Demonstrate an] Understanding of the history of mathematics, including the contributions of
different individuals and cultures toward the development of mathematics and the role of
mathematics in culture and society (VA Mathematics Specialist Endorsement Requirement).
[Demonstrate an] Understanding of the sequential nature of mathematics and the mathematical
structures inherent in the content strands [related to rational numbers and proportional reasoning]
(VA Mathematics Specialist Endorsement Requirement).
[Demonstrate an] Understanding of and the ability to use strategies for managing, assessing, and
monitoring student learning, including diagnosing student errors [related to rational numbers and
proportional reasoning] (VA Mathematics Specialist Endorsement Requirement).
Use a variety of representations and strategies to model problem situations dealing with rational
numbers and proportional reasoning in order to support and deepen their own and their students’
mathematical understanding.
MATH 653 – Summer 2010
M. Timmerman
Demonstrate computational proficiency, including a conceptual understanding of rational numbers,
ways of representing number, relationships among number and number systems, and the meaning
of operations related to rational numbers and proportional reasoning.
[Demonstrate an] Understanding of and proficiency in grammar, usage, and mechanics and their
integration in writing (VA Mathematics Specialist Endorsement Requirement).
Course Requirements and Evaluation:
 Whole-group participation and attendance – 10%
 Small-group participation (Blog postings & responses) – 10%
 Homework Problem Sets and Reading Reflections – 20%
 Mid-term Exam – 10%
 Interview Project – 20%
 Course Portfolio (includes history of mathematics paper and final reflection paper) – 15%
 Final Exam – 15%
Course Grade Scale
93 – 100%
86 – 92%
78 - 85%
70 – 77%
below 70%
Whole-group Participation and Attendance: Class activities require communication, interactions, and
discussions with other class members, and these cannot be reproduced. Class attendance is expected
(synchronous and face-to-face sessions), both for you to learn and so that others may benefit from your input.
You will be graded on participation and attendance. Attendance includes being on time for class and returning
from breaks in a timely manner.
At most one excused online absence is allowed. Additionally, you must get permission in advance from the
course instructors, if you must be absent due to an illness, family emergency, or other extenuating
circumstance. You will be expected to complete all work you will miss and to complete a make-up assignment.
In some weeks we meet twice for online sessions; therefore, the make-up assignment is due the 2nd session
after a missed class. You must assume full responsibility for all material covered during your absence.
Any unexcused absences will result in your final grade being reduced a letter grade for each absence. An
archive of all Wimba class sessions will be posted on our course Blackboard site. If you miss class or would like
to review something discussed in class, you can access that archive at your convenience.
Session chapter readings and online articles: Readings should be completed in preparation for both
synchronous sessions and asynchronous blog entries and responses. You are expected to analyze and reflect
on all readings and be prepared to contribute to ‘live discussion,’ and your small group blog. The quality of the
class will depend on the extent of your participation. Each participant should be prepared to lead class
discussions of the readings and homework set problems.
Problem Sets: Each class session there will be problems assigned often using different problem-solving
strategies related to the course readings. When requested, solutions must be detailed with work shown using
Polya’s problem-solving process structure. You may use your book, other references, small-group blogs, and
discussion with other class members when completing these problems. However, the final product
submitted for a grade must be the student’s own work. Problems will be graded on the problem-solving
process, effort, and the use of alternative strategies. Problems will not be graded entirely for accuracy of the
final result. These problems may be shared and discussed the next class session, and problem sets will be
turned in on a random basis throughout the semester.
MATH 653 – Summer 2010
M. Timmerman
Reflective Writings: During the course, you will be asked to reflect on focus questions related to assigned
chapter readings and posted articles on Blackboard. Take-away statements and specific quotes will often
provide the framework for these reflections posted on your small-group blog.
Small-group Participation: Using the Blog tool, you will have an opportunity to post some homework
problem sets, and reflect on students’ thinking and teaching practices found in the chapter readings and cases
in the Smith et al. course text. The Blog is only accessible to you, your small group, and the teaching team.
You will be graded on your small-group participation when using the Blog tool.
After a Wednesday synchronous class session, you will need to post selected problem strategies and solutions
from the homework problem set, and any requested reflections from the chapter readings by Saturday, at 6
pm. Often, reflections on the chapter readings, will take the form of writing ‘take away’ statements. By a takeaway, we mean something you found interesting in the chapter that you would like to discuss using your
small-group blog, or ask a question to be discussed in a future whole-class session.
Next, by Monday, at 6 pm, read and respond using the Blog at least once to another person’s homework
mathematics problem and at least once to a reading reflection in your small group.
After a Monday synchronous class session, you will need to read and respond using the Blog at least once to a
mathematics problem and at least once to a reading reflection by Wednesday, at 6 pm. This may be related to
a new problem shared ‘synchronously’ in the Monday night session, or a problem from the previous week.
Student Interview Project: Conduct a rational number-based interview with two different K-8 students,
preferably at the same grade level. If possible, tape the interviews so it easier to revisit the interviews, and
analyze each student’s thinking. Further details will be discussed early in the course. Interview Project is
due Fri. July 30th at 1 pm, face-to-face session in Roanoke.
Course Portfolio: Each participant will complete a course portfolio that will serve as one assessment of
individual growth and understanding of mathematical ideas, student learning, and teaching practices over the
course. Further details will be provided early in the course. Portfolio due Mon. Aug. 9th by 6 pm.
Mid-term Exam and Final Exam: Consists of K-8 mathematics content, how students think about and learn
mathematics, and pedagogy examined in the course including mathematical concepts, processes, and
discussion information based on assignments, class activities, powerpoints, and notes. You must complete
each exam within a 3-hour block of time. Mid-term Exam is due Fri. July 9th by 6 pm.
Any use of resources or collaboration with people in or out of the class will be considered a violation of the
Longwood Honor System. For grant cohort students : You will complete all test problems in a DigiMemo
Handwriting eBook file and submit the file through the course Blackboard page. For non-grant students: You
may use word-processing and drawings, and scan your work, submitted through the course Blackboard site to
be graded. Final Exam is due Fri. Aug. 13th by 6 pm.
Additional Information
Students are expected to purchase all required materials, attend all class sessions, complete all assignments as
given, and participate in all class activities. All academic regulations in the current 2009 - 2010 Longwood
Catalog will be followed. (see )
Graduate Credit: For each hour you attend class, you should plan to spend 2 to 3 hours on
coursework. Thus, for a 3-credit graduate course, for each class session (90-minute synchronous and
90-minute asynchronous), you will need to spend a minimum of 6 to 9 hours preparing for each class
meeting – reading, studying, and completing assignments. Major assignments will require more time.
MATH 653 – Summer 2010
M. Timmerman
Assignments and Projects: Assignments should be completed and submitted on time. Late work will
result in significant grade reductions according to the following scale: 1 day late, 10 percent reduction;
2 days late, 25 percent reduction; 3 days late, 40 percent reduction; 4 days late, 60 percent reduction;
more than 4 days late, grade of zero will be assigned. Students who have unique extenuating
circumstances may be allowed extensions on a case-by-case basis, but such extensions must be
requested prior to the date the assignment or project is due.
Notebook: Each participant should keep a notebook that contains activities, ideas for discussion,
assignments, and class notes to encourage your development of mathematical and pedagogical ideas.
The notebook is NON-GRADED but will help you with several assignments.
Suggested items in Notebook:
Daily notes related to mathematics problems and discussions (both during class and while you
are completing out-of-class chapter readings and problem sets)
Copies of class activities
Individual ideas that can help develop the basis for final reflection in the course portfolio
Copies of reflective writings
Longwood’s Honor System
A strong tradition of honor is fundamental to the quality of living and learning in the Longwood community.
Longwood affirms the value and necessity of integrity in all intellectual community endeavors. The 2009 2010 Longwood Honor System policy statement and purpose is located at:
Upon enrollment in this course, all students are presumed to have acquainted themselves with and have an
understanding of the Honor System. Therefore, it is a student's responsibility to ask the teaching team to
clarify expectations for each assignment in order to be in compliance with the Honor System.
ALL EXAMS ARE TO BE COMPLETED INDIVIDUALLY. Please write out and sign your name to the
Longwood Academic Pledge 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 graded assignments and
will be reported to the Longwood University Honor Board.
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.
Tentative Course Schedule – schedule may be changed as needed
Session 1, Online:
W June 9, 6:30-8pm
Session 2, Online:
M June 14, 6:30-8pm
Session 3, Live:
F June 18, 1:00-6pm
MATH 653 – Summer 2010
Overview of the Course
What is Mathematizing?
Fair Shares (Submarine Sandwich Problem)
Reasoning about Rational Numbers
Relative and Absolute Terms
Additive and Multiplicative Thinking
Thinking About the Whole
Qualitative Comparisons
M. Timmerman
Session 4, Live:
S June 19, 8am-2pm
Session 5, Online:
W June 23, 6:30-8pm
Session 6, Online:
M June 28, 6:30-8pm
Session 7, Online:
W June 30, 6:30-8pm
Session 8, Online:
M July 5, 6:30-8pm
Session 9, Online:
W July 7, 6:30-8pm
Session 10, Online:
M July 26, 6:30-8pm
Session 11, Online:
W July 28, 6:30-8pm
Session 12, Live:
F July 30, 1:00-6pm
Session 13, Live:
S July 31, 8am-2pm
Session 14, Online:
W August 4, 6:30-8pm
Session 15, Online:
M August 9, 6:30-8pm
Session 16, Online:
W August 11, 6:30-8pm
Estimating with Benchmarks
Equivalent Fractions
Comparing Fractions
Connecting Fractions, Decimals, and Percents (Case of Randy Harris)
Ratio Tables and Unitizing (Cat Food Problem)
Estimating Sums and Differences with Rational Numbers
Models for Addition and Subtraction of Rational Numbers
Addition and Subtraction of Rational Numbers
Choosing a Common Whole
Addition and Subtraction of Rational Numbers
Models for Multiplication and Division of Rational Numbers
Midterm Exam due Fri. July 9th
Multiplication and Division of Rational Numbers
Multiplication and Division of Rational Numbers
Multiplication and Division of Rational Numbers
Student Interviews due Fri. July 30th
Ratios (Case of Marie Hanson)
Graphs of Direct Proportions and Proportional Functions
Rates (Mixtures)
Ratios (Capture/Recapture)
More Proportional Reasoning
Portfolio due Mon. Aug. 9th
More Proportional Reasoning
Final Exam due Fri. Aug. 13th
Teachers as Mathematicians
Reference Books of Interest:
National Council of Teachers of Mathematics (2002). Making sense of fractions, ratios, and proportions. B.
Litwillwer, & G. Bright (Eds.). NCTM Yearbook. Reston, VA: Author.
National Council of Teachers of Mathematics. (2003). A research companion to the principles and standards for
school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through
grade 8 mathematics: A quest for coherence. Reston, VA: Author.
National Research Council (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford,
and B. Findell (Eds.). Washington, DC: National Academies Press.
Sowder, J.T., Philipp, R.A., Armstrong, B.E., & Schappelle, B.P. (1998). Middle-grade teachers’ mathematical
knowledge and its relationship to instruction: A research monograph. Albany, NY: State University of
New York Press.
MATH 653 – Summer 2010
M. Timmerman