Process Standard

The goals of the NCTM's communication process standard are that "in grades 9–12, the mathematics curriculum should include the continued development of language and symbolism to communicate ideas so that all students can—

• reflect upon and clarify their thinking about mathematical ideas and relationships;
• formulate mathematical definitions and express generalizations discovered through investigations;
• express mathematical ideas orally and in writing;
• read written presentations of mathematics with understanding;
• ask clarifying and extending questions related to mathematics they have read or heard about;
• appreciate the economy, power, and elegance of mathematical notation and its role in the development of mathematical idea."
  (NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 140)

Video Overview

Excerpts of seven classrooms demonstrate communication about mathematical ideas in a range of formats, including whole–class discussions, small groups, writing, and student presentations. Students use paper and pencil, graphing calculators, and a Calculator–Based Laboratory^TM as they talk and work together on mathematical problems. Specifically, students use communication to tackle the following problems:

• graphing trigonometric functions
• writing functions to model real-world phenomena
• finding derivatives of exponential functions
• using matrices in statistical analysis
• determining methods to solve equations
• solving problems using exponents
• creating graphs and equations from collected data

---

A Pre–Viewing Exploration for Teacher Workshops

Discovering Different Views of a Geometric Proof

Objective: To use communication to create different methods for developing a geometric proof.
Materials: One piece of chart paper and one marker for each small group.

Activity
Solve this problem in small groups.
Prove that the diagonals of a parallelogram bisect each other.
Discuss different solution methods in your small group. Decide on one solution, write it on the chart paper, and present it to the entire group. In your presentation, explain how and why you chose the solution.

Exploring Content

Discuss the following questions in your small group and share your conclusions with the entire group.

1. Describe the learning preferences of teachers who used each solution method. Are their preferences analytical? holistic? inductive? deductive?
2. Make a list of criteria that make a proof valid. Consult the proofs provided in the answers and the proofs presented by other teachers. Do you think each proof is valid? If students had created them, would you give full credit for each solution? Explain your answer.

Exploring Teaching Issues

1. How would you encourage communication between students doing a similar activity?
2. How would you encourage students to use varied methods to solve problems in geometry, algebra, trigonometry, and calculus?
3. What assessment issues might be affected by having students consider multiple approaches to solving a problem?

---

Exploration Answers

Activity Possible proofs:

1. Use geometry. Assume that BD does not bisect AC. This means AO > or < OC. Assume AO > OC. Then OA/OC is not equal to 1/1. Since ABCD is a parallelogram, Angle 1 is congruent with Angle 2 and Angle 3 is congruent with Angle 4 (opposite sides are parallel and alternate interior angles are congruent). Therefore, BOA and COD are congruent, because if two angles in one triangle are congruent to two angles in another triangle, the third angles are congruent. Thus, triangles AOB and DOC are similar. That means that the sides are in proportion and AB/DC is equal to 1/1 because the opposite sides of a parallelogram are congruent. Therefore, the ratio of OA/OC must be 1/1. Contradiction: BD must bisect AC, therefore the diagonals of a parallelogram bisect each other.

2. Use algebra.
   A = (–3,2)
   B = (3,2)
   C = (1,–2)
   D = (–5,–2)
   Draw AC and DB. Use the midpoint formula to determine the midpoints of each line.
   Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2) = ((–3 + 1) / 2, (2 – 2) / 2)) = (–1, 0) The midpoint of AC is O(–1,0)
   Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2) = ((–5 + 3) / 2, (–2 + 2) / 2)) = (–1, 0) The midpoint of DB is O(–1,0).
   The lines intersect in the same point, therefore they must bisect each other.

3. Use geometry.

   1 Parallelogram ABCD	1 Given
   2 AB||DC	2 Definition of a parallelogram
   3 Angle ABO is congruent with Angle CDO	3 Alternate interior angle of parallel lines are congruent
   4 AB is congruent with DC	4 Opposite sides of parallelogram are congruent
   5 Angle BOA is congruent with Angle COD	5 Vertical angles are congruent
   6 Triangle ABO is congruent with Triangle CDO	6 AAS
   7 AO is congruent with OC	7 CPCTC
   8 DO is congruent with OB	8 CPCTC
   9 The diagonals bisect each other	9 By definition of line bisector

4. Use a measuring device, such as a ruler or other technology, to measure AO, OC, OB, and OD.

Exploring Content

1. Answers will vary depending on individual teachers. In analytical thinking, an answer is revealed by examining the individual parts; in holistic thinking, an answer is found by looking at a pattern; in inductive thinking, a rule is established after a pattern is found; and in deductive thinking, a rule is applied to show a condition is true.
2. Possible answers are identifying and defending basic assumptions and verifying that there are no inconsistencies with the basic assumptions or with other statements in the proof.

---

Topics for Discussion

Means of Communication

• Cite the conditions that encouraged student communication.
• What evidence did you observe of students' confidence in their ability to communicate mathematical understanding?
• How did students use mathematical symbols to express their understanding?
• How did communication help students to solve problems and develop their reasoning skills?
• How did teachers respond to students to encourage deductive and inductive reasoning?
• In what ways do graphing calculators affect communication?
• What is the value of having students make presentations?

Classroom Organization

• How did the organization of the different classrooms facilitate communication?
• How do students explore problems differently when they work in groups?
• What reasons did students give for the value of working in groups?

Assessment

• How did students clarify their thinking?
• In what ways did students take responsibility for their learning?
• Cite the opportunities for teachers to assess their students.
• What student work could the teachers use to inform future lessons? Explain.
• How does connecting mathematical symbols to a context or another representation of a problem help students and teachers check student understanding?

Home | Catalog | About Us | Search | Contact Us | Site Map © 1997-2009 Annenberg Media. All rights reserved. Legal Policy