ED 262a - Teaching Mathematics in Elementary Classrooms
Instructor: Joan D. Martin, Ph.D.
E-mail: [email protected]
Course Description Teaching Mathematics in Elementary Classrooms focuses mainly on the
number domains that prospective elementary mathematics teachers are expected to teach, from
a more advanced perspective. Participants will explore underlying concepts and properties of
our base-ten system including integers and rational numbers while examining patterns; number
theory; operations; and various addition, subtraction, multiplication, and division algorithms.
The domains of geometry and measurement will be studied; the classroom learning community
and students’ work will be discussed and examined. Participants will immerse themselves in the
doing of mathematics and solving problems. The course will emphasize the practices of the
Common Core State Standards for Mathematics, and it will model appropriate pedagogy for
teaching mathematics in the elementary grades.
Course Texts:
• Van de Walle, John A., & Karp, Karen, & Bay-Williams, Jennifer M. (2016). Elementary and
middle school mathematics: Teaching developmentally. Boston, MA: Pearson [ISBN 0134046951]
• Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of
fundamental mathematics in China and the United States. Mahwah, N.J: Lawrence Erlbaum
Associates.
Additional Readings (provided in class or on Latte):
• Bass, Hyman. “Computational Fluency, Algorithms, and Mathematical Proficiency: On
Mathematician’s Perspective.” Teaching Children Mathematics (TCM). NCTM, February 2003.
• Lampert, M. (2001). Teaching problems and the problems of teaching. New Haven: Yale
University Press.
• Boaler, Jo. (2016). Mathematical mindsets: Unleashing students' potential through creative math,
inspiring messages, and innovative teaching. San Francisco, CA: Jossey-Bass & Pfeiffer Imprints.
Learning Goals
This course is designed for participants to:
 Enhance their knowledge of the mathematics taught in elementary school and expand their
comfort zones with mathematics content;
 Broaden their conceptions of mathematics teaching and learning in the elementary grades, with
attention to equity;
 Embrace an environment of reflection, risk-taking, and deep engagement with mathematics;
 Solve a variety of problems using multiple problem-solving techniques;
 Enjoy the teaching and learning of mathematics and become more confident in their ability to
do mathematics.
Evaluation/Course Requirements
Class participation (27%)
Attendance will be taken at each session. Mathematical understandings develop as participants
work through problems and discuss their thinking. Participation is essential and will be an
important part of your grade. Participation is based on in-class activities and solving problems,
as well as the discussion of these and any assigned readings. If you are unable to attend a
particular class session, please inform me beforehand (if possible). You will be responsible for
contacting someone in the class to find out what transpired in your absence. You then will be
required to write up a description of the concepts covered in class and what occurred. This
write-up will be due along with the assignment from the missed class.
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
Joan D. Martin, Ph.D.
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Checking for messages and sending text messages is not appropriate during class time. Also,
using a computer during class time for reasons other than class materials is not appropriate.
Homework (28%)
Participants will be required to complete assignments given for each class. Late assignments
will not receive full credit. To earn full credit on assignments that involve solving problems,
you must do the following:
1. Show all work used to solve the problem.
2. If you are unable to reach a solution, please show your attempts to solve the
problem and articulate the difficulty and issues you encountered. You will earn
credit for conscientious effort to complete the assignment.
On reading assignments that involve a write-up you must do the following:
1. Identify two points that resonate with you and your teaching.
2. Elaborate why you feel these ideas are important.
3. Write one short paragraph for each point.
Project Outline (5%)
Participants will choose an elementary math concept and develop a teaching outline (with
supporting evidence from readings) for a lesson that addresses: 1. The goals of the lesson; 2.
The student-outcome expectations; 3. Problem(s) for students to solve in the learning of the
concept; 4. Pitfalls that students may encounter; and 5. How the pitfalls could be addressed or
possibly avoided. This project outline will be further developed in ED267a, which you will be
taking this Fall.
Final Exam (40%)
Participants will take an in-class exam to assess their knowledge of the content of this course.
Methodology
Instructional approaches include problem-solving, readings, written assignments, and in-class
mathematical activities, games, and discussions. Participation will include small and whole group
discourse and analysis of problem solutions. At the end of each class an informal assessment will
be taken in the form of an “exit ticket.”
Academic Accommodations
If you are a student who needs academic accommodations because of a documented disability you
should contact me, and present your letter of accommodation, as soon as possible. If you have
questions about documenting a disability or requesting academic accommodations you should
contact Jessica Basile <[email protected]> Director of Graduate Student Affairs. Letters of
accommodations should be presented as the start of the semester to ensure provision of
accommodations. Accommodations cannot be granted retroactively.
Academic Integrity
Academic integrity is central to the mission of educational excellence at Brandeis University. Each
student is expected to turn in work completed independently, except when assignments specifically
authorize collaborative effort. It is not acceptable to use the words or ideas of another person be it a
world-class philosopher or fellow student without proper acknowledgement of the source. This
means that you must use footnotes and quotation marks to indicate the source of any phrases,
sentences, paragraphs, or ideas in published volumes, on the Internet, or created by another student.
Violations of University policies on academic integrity, described in Section 3 of Rights and
Responsibilities, may result in failure in the course or on the assignment, and could end in
suspension from the University. If you are in doubt about the instructions for any assignment in this
course, you must ask for clarification. Also see http://brandeis.edu/coures/instruction/academicintegrity/integrity/index.html.
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
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Session Descriptions
Session #1 Tuesday, 6/21/16
Mathematical Problem Solving
The goal of this session is to develop mathematical thinking tools that encompass problem solving
skills, representation skills, and reasoning skills in a community of learners. Students will focus on the
kind of mathematical thinking they are using in solving problems. Non-routine ways of looking at
problem solving will be emphasized as a way of engaging participants in their own style of
approaching mathematics. Classic problems, such as those involving the number of handshakes,
squares on a chessboard, and Pascal’s triangle, will be investigated as well as lesser known ones.
Strategies for solving problems will be discussed.
Assignment --- Due Thursday 6/23:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al:
Chapter 1, Teaching Mathematics in the 21st Century, pp. 1-12; Chapter 2, Exploring What It Means to
Know and Do Mathematics, pp. 13- 32; and Chapter 3, Teaching through Problem Solving, pp. 33- 56.
• Complete the Interview Questionnaire: Teaching Mathematics in Elementary Classrooms ED 262a
using the template provided.
• Solve Problem Sheet #1 given out in today’s class.
Session #2 Thursday, 6/23/16
Number Systems
The goal of this session is for students to gain an appreciation for the advantages and sophistication of
our base 10 number system primarily through comparison of properties of other number systems. They
will represent and construct numbers in different number systems such as Roman, Babylonian, and
Egyptian, and understand the function of place value, and the role of zero. Participants will interpret
the value of numbers in base systems other than base ten (e.g., base 2 and base 5), add and subtract in
these bases, and translate number values from one base system to another, in order to build a strong
understanding of number systems in general.
Assignment --- Due Tuesday 6/28/16:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al:
Chapter 8, Developing Early Number Concepts and Number Sense, pp. 142 – 166; Chapter 11,
Developing Whole-Number Place-value Concepts, pp. 222 – 246; and Chapter 14, Algebraic Thinking,
Equations, and Functions, pp. 299 – 338.
• Solve problems #5 and #39 from MTEL 53 Practice Test. The website is:
<http://www.mtel.nesinc.com>.
• Solve Problem Sheet #2 given out in today’s class.
Session #3 Tuesday, 6/28/16
Number Theory and Numeration
Participants will explore patterns and relationships in the counting numbers, as well as number line
relationships and the different sets of numbers in the real number system (i.e., counting, natural,
integers, rational and irrational). They will use the Sieve of Eratosthenes to determine prime and
composite numbers. Other topics will be: prime factorization and its use (e.g., common factor and least
common multiple); properties of operations; order of operations; divisibility rules; square, cube and
figurate numbers. The underlying concepts and relationships of algebra that build bridges from
arithmetic to algebra will be included in the session’s work.
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
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Assignment --- Due Thursday 6/30/16:
• Solve problems #6, #7, and #13 from MTEL 53 Practice Test. The website is:
<http://www.mtel.nesinc.com>.
• Read Teaching Problems and the Problems of Teaching, by Magdalene Lampert,
Chapter 4, pp. 51-100. Identify two points that resonate with you and your teaching and elaborate why
you feel these ideas are important. Write one short paragraph for each point.
• Read Knowing and Teaching Elementary Mathematics by Liping Ma: Chapter 1, Subtraction with
Regrouping: Approaches to Teaching a Topic pp. 1-27; and Chapter 2, Multidigit Number
Multiplication: Dealing with Students’ Mistakes, pp. 28-54. Identify two points (one from each
chapter) that resonate with you and your teaching and elaborate why you feel these ideas are important.
Write one short paragraph for each point.
• Solve Problem Sheet #3 given out in today’s class.
Session # 4 Thursday, 6/30/16
Classroom Culture and Discourse (Formative Assessment)
Participants will discuss the effect of classroom discourse on mathematics learning. This discussion
will take place in the larger context of establishing a classroom culture that is conducive to children
revealing their mathematical understandings and misunderstandings.
Assignment --- Due Tuesday 7/5/15:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al,
Chapter 12, pp. 247-276.
• Solve problems #1 and #9 from MTEL 53 Practice Test. The website is: <http://www.mtel.nesinc.com>.
• Read article, “Computational Fluency, Algorithms, and Mathematical Proficiency: One
Mathematician’s Perspective,” by Hyman Bass taken from Teaching Children Mathematics.
Identify two points that resonate with you and your teaching and elaborate why you feel these ideas
are important. Write one short paragraph for each point.
• Solve Problem Sheet #4 given out in today’s class.
Session #5 Tuesday, 7/5/16
Addition and Subtraction
Participants will be introduced to the broad use of basic number and operation knowledge in
computing. They will focus on strategies used to perform the inverse operations of addition and
subtraction, including children’s invented procedures, as well as procedures used in other cultures. The
importance of understanding place value will be addressed. Participants will use manipulatives to
represent numbers and operations, and consider the effectiveness of using these materials to teach
concepts. The various types of addition and subtraction problems and difficulties students may
encounter will be explored. Use of the open number line as a problem-solving tool for young students
will be used. A discussion on Liping Ma’s terminology of composing and decomposing units will
focus on the use of language to describe the operations and the inherent inadequacy of terms such as
“borrowing” when describing the standard algorithm for subtraction.
Assignment ---- Due Thursday 7/7/16:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al,
Chapter 13, Developing Strategies for Multiplication and Division, pp. 277-298.
• Solve Problem Sheet #5 given out in today’s class.
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
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Session #6 Thursday, 7/7/16 Multiplication/Division
Participants will focus on the operations of multiplication and division. They will consider the types of
units that result when two quantities are multiplied, and consider three contexts for multiplication:
repeated addition, area model, and Cartesian product model. They will also examine various
multiplication algorithms such as partial products and lattice multiplication and investigate the
advantages and disadvantages of different algorithms. They will discuss the arguments for
computational fluency in the learning of traditional algorithms. Participants will also examine division
models (partitioning and repeated subtraction) as they analyze problems involving division. They will
explore the division techniques of partial quotients and column division, and compare them to the
traditional long division algorithm, and determine how to interpret remainders.
Assignment ---- Due Tuesday 7/12/16:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al:
Chapter 15, Developing Fraction Concepts, pp. 339-370; Chapter 17, Developing Concepts of
Decimals and Percents, pp. 403-428; and Chapter 18, Ratio, Proportions, and Proportional
Reasoning, pp. 403-452.
• Solve problem #14 from MTEL 53 Practice Test. The website is: <http://www.mtel.nesinc.com>.
• Solve Problem Sheet #6 given out in today’s class.
Session #7 Tuesday, 7/12/16
Rational Numbers
Participants will engage in a variety of problems that require a deep understanding of the development
of rational number concepts. They will examine the different conceptions of rational numbers
(measure, quotient, operator, ratio, location on number line) as well as the concepts of units and
unitizing. They will look at ways to interpret, model, and work with rational numbers, and discover
methods of developing fraction sense through: working with unit fractions; ordering of fractions
without necessarily finding the common denominator; and working with fraction bars, pattern blocks,
the number line, and dot paper as tools to visualize the rational numbers.
Assignment ----Due Thursday 7/14/16:
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al,
Chapter 16, Developing Fraction Operations, pp. 371-402.
• Solve Problem Sheet #7 given out in today’s class.
Session #8 Thursday, 7/14/16
Operations with Fractions and Decimals
Participants will explore ways of understanding the traditional algorithms for multiplying and dividing
fractions and decimals. They will study area models for multiplying fractions as a method for
understanding the “whys” behind multiplication. They will discover methods to prove and explain the
phrase, “invert and multiply,” as they find ways to perform division of fractions that have meaning for
students. They will work with alternative algorithms for multiplying and dividing decimals that
concentrate on developing meaning for the operations.
Assignment --- Due Tuesday 7/19/16
• Read Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle et al:
Chapter 19, Developing Measurement Concepts, pp. 453 – 487; Chapter 20, Geometric Thinking and
Geometric Concepts, pp. 488 – 525.
• Solve Problem Sheet #8 given out in today’s class.
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
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Session #9 Tuesday, 7/19/16
Geometry and Measurement
The goal of this session is to continue the mathematical thinking and problem solving in the realm of
geometric and measurement topics. The van Heile Levels of Geometric Thinking will be explained and
participants will construct a geometric model with given parameters. The focus will be on: two-and
three-dimensional shapes; perimeter; area; surface area; and volume.
Assignment --- Study for Final Exam --- Thursday, 7/21/16
Submit outline - teaching/learning project to be further developed in Fall semester course ED267a
Session #10 Thursday, 7/21/15
Final Exam
ED 262a - Teaching Mathematics in Elementary Classrooms – summer, 2016
Joan D. Martin, Ph.D.
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