Day 30: Proofs?

I have a sub tomorrow. Last year it would have been no big deal. Probably because I was so focused on surviving. Now I am kinda disappointed I don’t get the spend the day with my students and frustrated at myself for giving them all the same worksheet when they really need to keep moving forward.

Today in Algebra I was planning on taking the easy road and giving students a day to practice. I wasn’t feeling great about it because that would mean back-to-back days of practice on a concept they have almost mastered already. What is awesome about my students is they would have done this without any thought. But just because they are willing to work for me without question, doesn’t mean I should take advantage of it.

Instead I did a sort of speed dating activity. We rearranged the room so there were rows of desks facing each other. Students had two minutes to work on a problem with their partner then one partner rotated. I threw other questions in there like state flags and capitals of states; they loved it.

Afterwards I asked them for feedback; didn’t want to change much, just maybe a little more than 2 minutes on some problems (that doesn’t mean there isn’t any room for structuring it a little better on my end though!).

In Geometry we started proofs. (Actually, first there was an awesome visual pattern found here)

And I was worried about how today would go.

At a workshop this summer with Dan he talked about how you can always add more, but after all the mathematical structure has been added you can’t take it away.

I went with it and put this up on the board:

Day30.2

I asked students what they could conclude from this.

There wasn’t any complaining or distractions or groaning. They jumped right on it.

Some threw numbers into the mix;

Day30.1

And eventually we were able to reach the conclusion. I talked about how this was a specific case and how 6.1 and 83.9 is another. If we want this to be true for every case, we would be spending a lot of time plugging in numbers making sure they work.

After I asked students for to write out a game plan for how they could use the provided information to reach the conclusion we wanted.

They did it.

And it was awesome.

I guess all my emphasis on how the process is so much more important than the solution is paying off. Those whose game plan fell short didn’t care; they learned from it and hearing about other’s approaches. I wrapped the discussion up by writing a formal proof using THEIR rules.

For the rest of the period students worked on Justin’s Formal System Proofs. (Provided by a co-worker) This is a great way to introduce proofs and reinforce logic in general.

Day30Here is the direct link to the goods.

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Featured comment:

Joshua provides two great suggestions for fostering student interest:

…encourage your students to have opinions about proofs. Which ones do they like, which ones do they find most convincing, which ones do they find easiest to understand, for two proofs of the same result, how do they compare, etc…

Prove or Disprove and Salvage if Possible (=PODASIP, inspired by the PROMYS Program in Boston.) Give them at least a couple cases where they’ve been asked to prove something that isn’t true. This ambiguity is common for working mathematicians and many non-mathematical situations…

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3 thoughts on “Day 30: Proofs?

  1. Two suggestions

    The first is taken from ideas in Paul Lockhart’s Measurement: encourage your students to have opinions about proofs. Which ones do they like, which ones do they find most convincing, which ones do they find easiest to understand, for two proofs of the same result, how do they compare, etc.

    Two reasons to do this:
    (1) the concept of a proof often lends support to the meme that math is all about right and wrong. Aesthetic judgement clearly doesn’t fit that meme, nor do the deeper mathematical habits that come from thinking about whether a proof seems to really illuminate the truth of the theorem, whether it has extensions or further implications, etc.

    (2) These questions provide another great window into what the students are actually thinking. For example, do they really understand the theorem as stated? Do they actually believe it is true? Do they have an intuition about why it might be true?

    As an illustration, I came in with a collection of “proofs” of the pythagorean theorem (some weren’t proofs) and was really surprised by the discussion around which were more illuminating, which were more convincing, which were the most “beautiful.”

    Second suggestion: Prove or Disprove and Salvage if Possible (=PODASIP, inspired by the PROMYS Program in Boston.) Give them at least a couple cases where they’ve been asked to prove something that isn’t true. This ambiguity is common for working mathematicians and many non-mathematical situations, so that’s one benefit.

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    1. These are awesome suggestions. I shared these with my department as well, we are excited to give them a try! I am learning there is so much more to proofs than just a statement and reason.

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