Reasoning

Process Standard

The goals of the NCTM's reasoning process standard are that "in grades 9–12, the mathematics curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills so that all students can—

• make and test conjectures;
• formulate counterexamples;
• follow logical arguments;
• judge the validity of arguments;
• construct simple valid arguments;
  and so that, in addition, college–intending students can—
• construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction."
  (NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 143)

Video Overview

Excerpts of seven classrooms show students reasoning about mathematical ideas in whole–class discussions, small group work, and class presentations. In some lessons, students use graphing calculators to explore their conjectures. In other lessons, they use manipulatives—such as paper squares, paper triangles, and a bungee–jump model—to investigate concepts. As students listen to one another's ideas, make conjectures, and develop problem–solving strategies, they strive to find and explain answers. Specifically, students use reasoning in:

• exploring congruence properties
• determining how to collect data
• using different approaches to find a derivative
• developing a geometric proof
• writing trigonometric functions
• investigating patterns to develop a formula
• making conjectures about geometric properties

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A Pre–Viewing Exploration for Teacher Workshops

Using Strategies to Build Towers

Objective: To use inductive and deductive reasoning to investigate patterns and functions.
Materials: Three cardboard disks and three straws constructed according to the diagram at right and one set of construction paper disks (six different sizes) for each small group.

Activity
Do this activity in small groups.
Place the disks on one straw. Use the disks to determine the function that represents the minimum number of moves necessary to transfer all of the disks from one straw to another. You can use all three straws for the transfer. The disks must always be placed in increasing order of size from the largest disk on the bottom to the smallest disk on the top.

1. Describe the process in which you moved the disks from one straw to another. Did you develop a strategy? If so, describe the strategy and explain why you think it worked to solve the problem.
2. After completing the moves for 2, 3, 4, 5, and 6 disks, complete the following chart for those disks.
   

   Number of disks in tower	Minimum number of moves
   2
   3
   4
   5
   6

   After determining the minimum number of moves necessary to transfer all of the disks to another straw, compare your group's answers with those of other groups. Discuss any differences in approaches and answers.
3. Make a conjecture about the minimum number of moves for n disks. Explain the reasoning you used to find the rule for this function.
4. If you had 64 disks, how many moves would you have to make to transfer the disks? How long would it take? What kind of reasoning did you use to find the answer?
5. What are the similarities and differences between the reasoning you used to solve questions #3 and #4?

Exploring Content

1. How could you alter this investigation to further promote reasoning and to introduce the concept of variable?

Exploring Teaching Issues

1. How would you evaluate students in this type of activity?

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Exploration Answers

Activity

1. Straws are named A, B, C. The disks start off on straw C. In the case of three disks, the disks are numbered 1 to 3 from smallest to largest. The process involves the following seven moves: 1 to A; 2 to B; 1 to B; 3 to A; 1 to C; 2 to A; 1 to A.

2. 2 disks	3 disks	4 disks	5 disks	6 disks
   3	7	15	31	63

3. The number of moves increases as the number of disks increases. This increase does not appear to be linear or quadratic. The numbers that represent the number of moves are close to powers of 2 (4, 8, 16, 32 . . . ). In fact, they are one less than the powers of 2. Therefore, the pattern can be described as one less than 2 raised to the power of the number of disks (2ⁿ–1).
4. Calculating the number would take too much time, but it is represented by 2⁶⁴–1. If you take a move every second, it would take nearly 6 billion centuries to finish moving all 64 disks. Deductive reasoning.
5. They are similar in that they are logical reasoning. They are different because inductive reasoning builds the rule from the data and deductive reasoning applies the rule to determine the solution.

Exploring Content

1. Answers will vary. One possible answer is you could change the number of straws, which would allow students to see that change in one value results in different outcomes. The problem allows you to generalize in terms of a constant value, in this case n.

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Topics for Discussion

Inductive and Deductive Reasoning

• Cite examples of inductive and deductive reasoning in the video.
• Discuss how students might use both inductive and deductive reasoning to establish the validity of an argument in a two–column proof.
• How could students use inductive and deductive reasoning together in one of your lessons?
• Describe how you teach your students to make and test conjectures, formulate counterexamples, follow logical arguments, judge the validity of arguments, and construct simple valid arguments.

Lesson Structure

• How do you structure lessons differently to encourage reasoning for students working in groups and for students working individually?
• Discuss how different contexts of communication, such as group work, writing, and presentations, promote reasoning.
• Why is it important to ask students to make generalizations and develop a rule?
• How does using multiple approaches and multiple representations promote student reasoning?
• Describe how you help students reason when there is more than one possible answer to a problem.

Classroom Environment

• Give examples of how students in the video developed their understanding by asking and responding to questions.
• How do students benefit from knowing there are often different ways to approach a problem?
• What are your questioning techniques for encouraging reasoning?
• How have you or could you use technology to promote reasoning in one of your lessons?

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