COMMUNICATION AS A PROCESS FOR STUDENTS TO LEARN
MATHEMATICAL
Akihiko Takahashi
DePaul University
The Principals and Standards for School Mathematics (NCTM, 2000) suggest that
communication is one of the important processes for teachers to facilitate during
mathematics lessons. Although the Principals and Standards emphasize that theprocess of learning mathematical contents is as important as the acquisition ofconcepts and procedures, we still do not see many mathematics classrooms thatemphasize the process of learning mathematics in the U.S. In order to provide
students with opportunities to learn mathematics through communication, this paper
will discuss the use of the "open-ended approach" to encourage students tocommunicate with each other as they learn mathematics.
INSTRUCTION AS INTERACTION
Rather than passively listening to teachers talk, students are expected to be actively
involved in mathematics (Brown, 1994). It is expected that teaching and learning
mathematics be viewed as the product of interaction among teachers, students, andthe mathematics (National Research Council, 2001). In other words, students shouldbe able to learn the contents of mathematics actively through interaction withteachers and peers. Thus, facilitating communication should be an integral part of
supporting students in their learning of mathematics through interaction.
In order for students to be able to communicate each other, it is important to have alearning structure that stimulates communication among the students. If all thestudents are told to use the same approach to finding an answer they may not need tocommunicate with each other. They would focus on listening to what the teacher says
and to the mathematical contents presented by the teacher. On the other side, if
several different ideas are presented by the students during the class, they areencouraged to explain each idea and to justify each idea as reasonable. Especially, if
several different approaches/solutions are presented and the teacher expects students
to figure out which approach/solution is reasonable, students are required tounderstand these ideas and to make the decision through communication.
Principles and Standards for School Mathematics (National Council of Teachers ofMathematics, 1989, 2000) includes a discussion of the necessity for learning
mathematical content through the processes of problem solving, reasoning and proof,
communication, connections, and representation. This means that it is important forteachers to carefully plan their focus on not only what the students need to learn but
also how the students acquire the mathematical contents such as knowledge andprocedures during the class.

- Akihiko Takahashi 1 -USE OF THE OPEN-ENDED APPROACH
In order to have meaningful communication during mathematics lessons, teachers
need to provide rich mathematical situations for their students. It is expected that themathematical situation will encourage students to learn something new bycommunicating with teachers and peers. One of the ways to provide students withsuch opportunities is to employ the open-ended approach (Becker & Shimada, 1997).
The open-ended approach is an instructional approach using an open-ended problem,
which has multiple solutions or multiple approaches to a solution. The lesson
proceeds by using several students’ responses to the given problem to provide
experience in finding something new in the process. The open-ended approach is an
outcome of the developmental research project on methods of evaluating
higher-order-thinking skills in mathematics education between 1971 and 1976.
Although the project was aimed at improving the evaluation of students' thinking
skills, the researchers became aware that lessons based on solving open-ended
problems have a rich potential for improving teaching and learning. The researcherssummarize the advantages of using the open-ended problem in mathematics lessons
as follows.
• Students participate more actively in the lesson and express their ideas more
frequently.
• Students have more opportunities to make comprehensive used of theirmathematical knowledge and skills.
• Even low-achieving students can respond to the problem in some significant
ways of their own.
• Students are intrinsically motivated to give proofs.
• Students have rich experiences in the pleasure of discovery and receive theapproval of fellow students.
(Becker & Shimada, 1997)
Since the open-ended approach was introduced, US teachers have started using theopen-ended approach in their classrooms. Teachers who are practicing lesson study,
especially, often use open-ended problems when they plan lessons for lesson study.
Although having open-ended problems is a new idea and it may not easy for teachers
to use them in their lesson, these teachers have been working hard to encourage
students to interact each other to learn mathematic actively. Several lesson plans havebeen developed for US students based on popular open-ended problems in Japan. The
attached lesson plan shows an example of the lesson plan for a research lesson thatwas conducted in Fresno, California (Appendix I).
By employing the open-ended approach, the lesson was designed to provide students
an opportunity to develop their competence in using mathematical expressions and- Akihiko Takahashi 2 -equations. The lesson began with an introductory activity to let students understandhow to use an equation to describe an arrangement of dots. The teachers thenpresented an open-ended problem where students compared and discussed a varietyof ways to describe the arrangement of 25 dots by using equations. During theprocess of comparing and discussing a variety of mathematical expressions andequations, students communicated with each other to understand how other students
described the same arrangement of 25 dots.

Find the number of dots and describe your way to find thenumber of dots by using a mathematical expression/equation.
If you can, find and describe several different ways to countthe number of dots.

RETHINKING DEVELOPING LESSON PLAN
When lesson design focuses not only on what students need to learn but also on how
students acquire mathematical knowledge, experience, and skills, teachers arerequired to develop lesson plans differently. Commonly used lesson plans mainly
focus on what students need to learn and do not focus much on the process of student
learning during the lesson. Lesson Study provides teachers the opportunity to thinkdeeply about the students' learning process. Because the major purpose of observing
a research lesson is to examine students' learning process for the sake of thepost-lesson discussion, lesson planning requires teachers to anticipate students’variety of approaches to the solutions and to plan for facilitating discussion with thestudents. These processes of planning lessons help teachers think carefully to design
the lesson by including students learning process.
Since lesson study provide teachers such learning opportunity, designing lesson
through lesson study is an ideal way to help teachers provide learning opportunity for
their students through communication.
References
Becker, J. P., & Shimada, S. (1997). The open-ended approach: A new proposal forteaching mathematics. Reston, Virginia: National Council of Teachers ofMathematics.
Brown, A. (1994). The advance of learning. Educational Researcher, 23(8), 4-12.
National Council of Teachers of Mathematics. (1989). Curriculm and evaluation standards
for school mathematics. Reston, Virginia: National Council of Teachers ofMathematics.

- Akihiko Takahashi 3 -National Council of Teachers of Mathematics. (2000). Principles and standards for schoolmathematics. Reston, Virginia: National Council of Teachers of Mathematics.
National Research Council. (2001). Adding it up: Helping children learn mathematics.
Washington, DC: National Academy Press.

- Akihiko Takahashi 4 -Mathematics Lesson Plan for Fifth GradeFor the lesson on Friday, July 22
At the Homan Elementary School, Fresno, CA
Instructor: Akihiko Takahashi
1) Title of the Lesson:
How do you find the number of the dots?
2) Goal of the Lesson:a) Students will begin to recognize equations as a way to represent quantitative relations mathematically.
b) Students will begin to be able to communicate with other students by:i) expressing their thinking processes using equations.
ii) inferring possible thinking processes of other students from the equations they present.
3) Instruction of the Lesson
Learning algebra is a key for students to further their proficiency that they have developed by learning numberand operations. The NCTM standers expect students to be able to 1) understand patterns, relations, and functions, 2)
represent and analyze mathematical situations and structures using algebraic symbols, 3) use mathematical models
to represent and understand quantitative relationships, 4) analyze change in various contexts. Among those
expectations, this lesson is designed to provide students with opportunity to experience representational activities,
which is the focus of the second expectation.
Representational activities of algebra in elementary and early middle school grades involve translating verbal
expressions into mathematical (symbolic) expressions and equations. These activities are the essentials for students
to be able to represent mathematical situations and structures by using mathematical expressions and equations inorder to analyzing those situations and structures to understand quantitative relationships. These representational
activities should focus not only on representing mathematical situations and structures but also interpreting
mathematical expressions and equations. However, teaching algebra in elementary grades has been mainly focusedon recognizing patterns and relations and not much on representational activities. As a result many students are
struggling to express patterns and relations by using mathematical expressions and equations, although they are able
to describe them by using verbal expression, tables, and models such as diagram and figures. Recognizing patterns
and relations is important but we should not underrate the importance of the meaningful acquisition of algebraic
notation because it is one of the most difficult aspects of algebra for students (Edwards, 2000).
In order for students to be able to translate their verbal expressions and other informal expressions such as using
diagrams and tables into mathematical expressions and equations, it is important for students to recognize thatmathematical expressions and equations as the representation of relations. Unfortunately many students view
mathematical expressions and equations as a signal to compute because elementary school mathematics tends to beheavily answer oriented (Thompson, Philipp, Thompson, and Boyd 1994). Thus, students should have experiences
to adjust their view of mathematical expressions and equations in order to use them to analyze mathematicalsituations and structures. One of the ways to provide students such an experience might be posing an open-ended
problem that requires each student express the mathematical situation in various ways by using equations andcommunicate with other students by interpreting their equations.
By employing open-ended approach, this lesson is designed to provide students an opportunity to develop theircompetence to use mathematical expressions and equations. The lesson begins with an introductory activity to let
students understand how to use an equation to describe an arrangement of dots, then presents following open-ended
problem to compare and discuss variety of ways to describe the arrangement of 25 dots by using equations. This
type of open-ended problem is known as a problem with multiple solutions, because the major focus of this problem
solving activities is to represent the arrangement of the dots by using an equation and not to find the number of thedots.
Find the number of dots and describe your way to findthe number of dots by using a mathematicalexpression/equation.
If you can, find and describe several different ways tocount the number of dots.

- Akihiko Takahashi 5 -Because of the nature of open-ended approach, the main concern for the teacher during this lesson is to facilitate
discussion meaningfully by including all the students in the class. In order to do so, the instructor will carefully
examine students’ equations during students’ individual/group work and plan the discussion immediately following
students’ individual/group work.
4) Flow of the lesson
Steps, Learning Activities
Teacher’s Questions and Expected Student
Reactionsa. Introduction
How many dots are there? Tell me what yousee.
Anticipated students’ responsesi.
one in the center andfour in the outside
ii. two, one, and twoiii. three and two
Anticipated students’ responses
three and three
two, two, and two

Teacher’s Support

Let students see the major focus oftoday’s activities
Help students realize that they can use
equations to describe the arrangementof dots by translating students’ verbal
expression into an equation.
(a)i) 1+4=5ii) 2+1+2 =5, 2x2+1=5iii) 3+2=5(b)i) 3+3=6, 2x3=6ii) 2+2+2=6, 3x2=6

Points of Evaluation

•••

2.

Do students realizethat there are
multiple ways todescribe thearrangement ofdots?
Do students
understand thatthey can describe
an arrangement ofdots by using
equations?
Are students
motivated to finddifferent ways todescribe thearrangement ofdots by using
different
equations?

Posing Problem

Find the number of dots and describe your way to findthe number of dots by using a mathematicalexpression/equation.
If you can, find and describe several different ways tocount the number of dots.

Students should use worksheets to write theirequations so that they can use them during thewhole class discussion to share and describe
their equations.

Provide students worksheets with alarge picture of the dots for theproblem.
Let students write an equation that
describes their counting method oneach worksheet. Students will beable to have as many worksheets asthey want.
Students might work with theirpartners.

- Akihiko Takahashi 6 -Do students understandhow to use theworksheet?

Do students work
comfortably with theirpartners?

3.

Solving Problem

Anticipated students’ responses
Does each student findthe number of dots, andis he/she able todescribe his/her way byusing the diagram on
the worksheet?

1+3+5+7+5+3+1=25
2x1+2x3+2x5+7=25
7+2(1+3+5)=25

3x3+4x4=25
4x4+9=25

4x4 +3x3=25
4+4+4+4+3+3+3=25

5x5=25
4x5+5=25

3x5+2x4+2x1=25
3x5+2x3+4x1=25

Does each student write
an equation to describe
his/her counting
methods.

5x5=25

4. Sharing their equations
Share your equations to the class but do not tellother where the equation comes from.
<Show but don’t tell>
Other students: Let’s figure out where theequation comes from.

5. Summing up
Reflect what we learned by looking at the board
writing and recognize mathematicalexpression/equation can be a way to express
mathematical situation.

Facilitate opportunities for students tointerpret an equation. Encourage all
the students to be engaged in therepresenting and interpreting activity
because it is possible that someequations might come from different
ways of representing the arrangementof the dots. The unique solutions
among the students’ solutions will bediscussed later.

Do students see
relationships between adiagram and anequation?

Let students write their learning
experience as a journal reflection.

5) Evaluationa) Did the students represent quantitative relations by using equations?
b) Did the students communicate with other students by expressing their thinking processes using equations, andinferring possible thinking processes of other students from the equations they present?

- Akihiko Takahashi 7 -