Students and their own Conjectures

I recently read an article called "Using Conjectures to Teach Students the Role of Proof" by Rhonda Cox. It talks about how she prefers to teach proofs to her students. She taught students proofs different from the way it was traditionally taught. Traditionally, the teacher gives the students a statement and asks the students to prove it. Since the teacher asked the students to prove the statement, they assume it must be true. Rhonda did not find that to be an effective teaching strategy. Instead, she allowed her students to come up with their own conjectures and prove those. She said that most most students understood proofs better even though they still did not like doing them.

I believe that this teaching style for proofs can be used for all kinds of topics in mathematics. It allows students to explore and not be spoon fed answers. I believe it can me more enjoyable for students. I would imagine there are not a lot of us who enjoy sitting around taking notes while the teacher or professor lectures. This idea of exploration can be very powerful because it prevent memorization. If a student is simply told something they will have to try and memorize it. If they figure it out themselves, they will retain the information better because they will know how to find it and understand it better.

What happens when you try and memorize

A perfect opportunity to use this teaching style was missed by the teacher I am currently observing at Jenison High School. They were learning about the rules for combining exponents. This was a perfect chance to have students explore how exponents combine.

Some of the rules taught were:
You can only combine exponents when they have the same base

(x^a)(x^b) = (x^(a+b))

(x^a)/(x^b) = (x^(a-b))

x^(-a) = 1/x^a

(x^a)^b = x^(ab)

(x/y)^a = (x^a/y^a)

After they were taught all of these rules, they were given a worksheet and practice these rules. It seemed almost too simple. They were given a sheet that had all of the rules on it. While doing the practice worksheet, they used their rules sheet and basically copied the rules over.


I believe if the students were given some time they could have figured these rules out themselves. As a teacher I would have prompted them by saying "Keep the base the same, and what rules you can find about exponents. Adding? Subtracting? Multiplying? Others?" This gives them an idea of what to look for but forces them to find the answer. I believe students would remember that activity better then me simply standing at the front of the class and saying these rules work because I said so.

I think this idea can work with every concept in math. As teachers we always talk about how we don't want to simply tell student the answer. We want them to think critically and not simply memorize what we teach them. I believe that using this technique of having the students come up with their own conjectures eliminates memorization to some degree. Students will remember something better if they themselves came up with the answer in the first place. As Rhonda Cox found out, students may not like learning the material, but they will understand it better.