Last week my students saw and practiced their first method of proof in my freshman geometry class: Proof by Indirect Reasoning. I knew this would seem a strange concept to them. "If I think something is false, why would I say it is true?" was a question I knew I would be asked. In reality, I looked out at a sea of blank stares after showing the first example. Teaching how to write proofs takes time, though, and after a bit of practice it is finally starting to click.
For my initial lesson, I came up with an analogy to present the concept of indirect proof that I felt might help the students relate with this method of reasoning, however backwards it may seem. My advice to my students was this: Think like a lawyer.
Here is how our analogy went:
Me: Alright, who in the class looks especially guilty of a crime today?
After many-a-suggestion, I decided to pick a student that would really play up the part so we could have some fun with it.
Me: What crime did Johnny* commit?
*Name changed for student's privacy.
Johnny: I robbed a bank.
Said a little too quickly, I thought. ;)
Me: Okay. I'm a prosecutor in the courtroom. I think that Johnny robbed the bank, but he is pleading not guilty. This is how I will begin my proof: Let's assume that Johnny is innocent. Johnny, what is your alibi?
Johnny: I was on vacation.
Me: Were you on vacation the entire week of the crime?
Johnny: Yes.
Me: Okay, Johnny. But here's the thing. I have a security tape that shows you shopping here in town the night before the crime took place. You say you were on vacation. Unless you can be in two places at once, I'd say this is a contradiction!
We than had a small sidebar conversation about the definition of a contradiction, which they had already written in their notes.
Me: This means my original assumption - that you were innocent - must be false. Therefore, you are guilty!
This was a fun way to introduce the topic of indirect proof so that my students could make a connection. Everyone has seen this method of reasoning used by a lawyer on TV, right? While it didn't make my students immediately become expert proof-writers, I did feel that it helped them to gain some insight and understanding into how and why this method of proof is used. It also supports the stance of Eric Knuth in his article, Proof as a Tool for Learning Mathematics. He asserts that proofs should not only show that something is true, but also why it is true. This will enable them to better comprehend the underlying mathematics. After all, if students don't understand the why, have we really done our job?
If anyone else has some analogies or strategies for teaching a particular method of proof, I'd love to hear about it in the comments section!
For my initial lesson, I came up with an analogy to present the concept of indirect proof that I felt might help the students relate with this method of reasoning, however backwards it may seem. My advice to my students was this: Think like a lawyer.
Here is how our analogy went:
Me: Alright, who in the class looks especially guilty of a crime today?
After many-a-suggestion, I decided to pick a student that would really play up the part so we could have some fun with it.
Me: What crime did Johnny* commit?
*Name changed for student's privacy.
Johnny: I robbed a bank.
Said a little too quickly, I thought. ;)
Me: Okay. I'm a prosecutor in the courtroom. I think that Johnny robbed the bank, but he is pleading not guilty. This is how I will begin my proof: Let's assume that Johnny is innocent. Johnny, what is your alibi?
Johnny: I was on vacation.
Me: Were you on vacation the entire week of the crime?
Johnny: Yes.
Me: Okay, Johnny. But here's the thing. I have a security tape that shows you shopping here in town the night before the crime took place. You say you were on vacation. Unless you can be in two places at once, I'd say this is a contradiction!
We than had a small sidebar conversation about the definition of a contradiction, which they had already written in their notes.
Me: This means my original assumption - that you were innocent - must be false. Therefore, you are guilty!
This was a fun way to introduce the topic of indirect proof so that my students could make a connection. Everyone has seen this method of reasoning used by a lawyer on TV, right? While it didn't make my students immediately become expert proof-writers, I did feel that it helped them to gain some insight and understanding into how and why this method of proof is used. It also supports the stance of Eric Knuth in his article, Proof as a Tool for Learning Mathematics. He asserts that proofs should not only show that something is true, but also why it is true. This will enable them to better comprehend the underlying mathematics. After all, if students don't understand the why, have we really done our job?
If anyone else has some analogies or strategies for teaching a particular method of proof, I'd love to hear about it in the comments section!
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