Using Conjectures to Teach Students the Role of Proof

Kristy Thacker

Sced 473

September 28, 2005

Dr. Groth

Using Conjectures to Teach Students the Role of Proof

Cox, Rhonda L. “Using Conjectures to Teach Students the Role of Proof.” Mathematics Teacher. Vol 97. pg 48-52.

Summary

The most difficult part of a geometry teacher’s job is teaching students to write proofs. Only a select few of them really ever get it, and they greatly dislike the process and did not understand the need to learn how to write a proof. Therefore Rhonda Cox designed a unit that would improve students’ ability to write proofs but also show them the role of proof writing in mathematics and actually involve them in the activity of being a mathematician. She feels that if you teach proofs so that students can understand how mathematics is done, similar to how science teaches students, they’ll understand how to write proofs. A proof is only one step in a process of learning and discovering new mathematics. The steps include making a conjecture based on observation, testing the conjecture, work to justify the conjecture through proof, critique, and final acceptance. Before students are able to write proofs they must be mentally ready and have a certain level of understanding. By the time Miss Cox starts the unit her class has studied the basic parts of a proof, how to write “given” and “prove” statements from a conditional statement.

The purpose of this unit is for the students to make and test their own conjectures. For this reason Miss Cox takes the textbook from her students so they can not consult them. She uses groups since students benefit from discussion with other students in both the conjecturing process and the proof-writing process. The group work provides a strong support system for the stronger students to reach out to the weaker students. The teacher has several roles in this unit including a facilitator and model of behavior. Miss Cox usually asks questions to steer her students into the right direction. The hardest role for the teacher is giving up control of the learning environment to the students, because this unit is completely student-generated. It has been found that students find usually all of the concepts that are in the textbook and several more. This accomplishment shows a maturing ability to reason many steps ahead and to anticipate the kinds of things that they will be able to prove from properties. It is important to remember that each group may proof the same conjectures differently. When reviewing the proof, a copy of the proof is placed on the overhead and the entire class critiques each part of the proof. When the class accepts the proof, the statement is accepted as a theorem. Then the class names the theorem and adds it to the class list. The students feel this method of learning proofs is not only fun buy enjoyable. Approaching proofs in this matter improves self-esteem, responsibility, the ability to defend one’s ideas, persistence, logical reasoning, creativity, teamwork, and curiosity.

Commentary

This article not only showed how the teacher taught proofs, but it showed why she decided to create a different method instead of keeping the traditional method. I thought it was impressive she took a survey to find out how her students felt about learning this way. I didn’t like that she didn’t comment on the problems she expressed with this approach of teaching. There had to be something that doesn’t run as smoothly as you hope, but Rhonda Cox makes it seem like this unit has no problems.

Discussion Questions

1. Would you feel that allowing students to explore and create their own conjectures would make them better proof writers? (Would you use this method?)
2. What are some problems you can foresee in this method of teaching?
3. What do you feel the strengths are in the traditional method of teaching proofs?