## Carl's Teaching Blog

#### A place to talk about teaching and learning

My observation was scheduled for today, but it didn’t happen because I had to attend a meeting so I won’t post about that today.  Since I am probably more concerned about keeping this #MTBoS30 streak alive than I should be, I am going to post a link to something else instead.  Below is one of my “Mathematical Thinking” worksheets from last cycle and I want to see if anyone could help me encourage students to write proofs on some of these tasks.

#### Generalizing Problems

In this worksheet I wanted students to work on proving ideas, by that I mean taking something that works in some situation, and showing that it works in many, or infinite situations.  I wanted it to appeal to a broad range of student content levels so all of the proofs are based on number concepts, not really algebra or geometry.

Students struggled in writing proofs, although most were able to make appropriate guess as to what the proof would show.  Students would largely show an example of any of these statements with different numbers, but wouldn’t know the first steps to actually come up with a proof.

For the first page, lots of students would say “7+8+9=24, 3*8=24, It works” and have nothing else to say.

I need help figuring out how to get students to write a statement describing how the phenomena present in their examples can be applied to the rest of the number system.

10/30 #MTBoS30

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## 8 Comments

1. #### Kate Nowak

So, I have a few ideas of things to try. Maybe you have already thought of these things and have good reasons for not doing them. I dunno.

1. Let them notice the pattern and make the conjecture. It might capture their interest better than stating the thing to prove. So for the first question, say, hey, what’s the sum of 3, 4, and 5? Whats the sum of 11,12, and 13? How about 20, 21, and 22? Do you notice anything that always seems to be true? If not keep adding sets of three consecutive numbers until you notice something that seems like it is always true.

2. Have they seen examples of what a proof looks like? Reading and understanding proofs, and copying techniques you’ve seen in other proofs, are important tools. It might be good for them to see a proof done a few different ways, like with algebra and with a diagram.

3. An entree to this kind of thinking that’s a little gentler might be pile problems. There are a million of them at visualpatterns.org.

4. For some reason, the words “prove” and “proof” are scary. I’ve had better luck with phrases like “convince ourselves that this always has to be true.”

Hope that helps!

• #### Carl Oliver

Thanks for the feedback Kate.

1. I see what you’re saying, I have away the conjecture in an attempt to scaffold. Kids could have been really proud of themselves for noticing and may have led to more confidence for the rest of the day.
2. After the class I have out this https://docs.google.com/a/cityas.org/document/d/1RSM6WBJUVxiCHaNqUnZ2Xo0cLRaVv5vTn2L2XZ4NTW4/edit?usp=docslis
The thinking here was they would work in groups to see which of either “Brantonio”‘s or “Andorra”‘s work seemed not really well explained. Goal was to see a couple different problem solving styles too. This was after the fact, we hadn’t done any pros before.

3. I’ll check that out.

4. I’m totally stealing those

Thank you!

2. #### Patrick Honner

There are some nice problems included in your set that are both substantial and accessible. I agree with Kate that you might consider building into the expoloration an opportunity for the students to make the conjectures themselves.

Another idea is to ask the students to pose a simpler, but related question. For example, in the three consecutive numbers problem, could you somehow lead students to think about a simliar result for two consecutive numbers? Might they be more able to articulate a “proof” of that result?

And the “proof by example” approach is a common student response. in my experience, the best way to counteract it is to let students play with patterns that break on a regular basis in order to foster their mathematical skepticism.

• #### Carl Oliver

Thanks for the feedback Patrick.

Itsounds like you’re saying I could ask students to pose a similar question after we have seen some examples? And you’re saying everyone can come up with their own question? Maybe we could even generate a list of related questions at the board and the goal is for the class to see how many we can see when or not.

I really need to do more of that “break the pattern” with my kids to develop more of the mathematical skepticism.

Thanks for the comments!

3. #### Blair Miller

I’ve taught this topic recently with my Grade 11 students and have found them really struggling. I’d agree with Kate that students need to see examples of proofs to start to get comfortable with how to provide the reasoning.

The other thing I found helpful was scaffolding the generalisation. Students didn’t know where to start with even generalising thinks like an even number (2n), an odd number (2n + 1), consecutive numbers (n, n+1, n+2), fractions (a/b). It might be helpful to have them come up with generalisations for these concepts first.before trying to generalise more complex rules.

I agree with Patrick that looking for counterexamples to break patterns is helpful in showing that not all patterns continue indefinitely (sum of the digits in the multiples of 3 greater than 12 where a pattern starts to emerge, but then changes). Regardless, students still found this topic extremely difficult and often responded with inductive reasoning/proof by example even when asked for deductive reasoning.

• #### Carl Oliver

I think having kids represent their thinking about the pattern was really lacking (probably because I was in a hurry). I talked about the n-1, n, n+1 representation on the board, along with an example with blocks, but “I” was doing it, so the students viewed it as something to copy in their notes, not use on their own.

It is really hard to do proofs for everyone. I know a lot of people don’t posture math majors because of not understanding how to prove. Hopefully I will be able to make some progress by finding ways to get students to think about problems differently, and deeply.

Thanks for your comment!

4. #### Max Ray (@maxmathforum)

Whenever I think of proof I always think of DeVilliers: http://mzone.mweb.co.za/residents/profmd/proofa.pdf. Reading his work (especially the book Rethinking Proof with Geometer’s Sketchpad) pushed me to be more conscious of creating an intellectual need for proof, not just proof-as-exercise.

Things like:
* Making sure that not every conjecture offered to students was true — otherwise the social context is proof enough!
* Finding problems where the proof was an explanation, appealed to students’ sense that “Okay, this seems to be true, but _why_?” (example: visual proof that the sums of triangular numbers are squares, or that the sums of consecutive odd numbers are squares, or that the sum 1/2 + 1/4 + 1/8 + … approaches 1)
* Finding problems where the proof led to other conjectures, or at least useful results… like your problem about young Karl being forced to add up the number from 1 to 100 and figuring out he could easily add up the numbers from 1 to 10000 or 1 to 1000000000. Kids get a kick out of those!
* Supporting students to be the ones to generate conjectures through visual patterns, games that require generalization, etc. (Games that require generalization being things like finding a winning strategy for a game even if you don’t know if you’ll go first, how many will be playing, etc.)

Did you see David Cox’s blog on Fostering the Wreck-It Mindset?

Okay, last thing: Sometimes the way you structure the human-to-human interaction can really change how the same worksheet goes. With the idea that proof is a social/communication tool, what if you paired students up and one student was given the evens and the other the odds (on a sheet where some conjectures were true and others broke down eventually). Their job was to somehow convince their partner that their conjecture was always, always true… and their partner’s job was to show that the conjecture broke down somehow. The partnership gets points for correctly ferreting out and convincing you that a conjecture is sometimes, always, or never true. Bonus points if either partner uses a drawing, and double bonus points if either partner uses an algebraic representation. I wonder if just making explicit the two “sides” in every proof — the convincer and the wrecker — would help to motivate students bringing strategies other than trying one example to the table.

• #### Carl Oliver

Thanks Max for such a thoughtful and detailed comment.

I think I will include a number of those ideas when I reteach this unit next year. I especially want to look in to more of the “Wreck-it” kind of things as I heard people throwing the term around while I was at NCTM, but I didn’t know where it originated from. I had not heard of David Foster’s post, but it looks insightful. I won’t have time to work in too many more activities as our school is about to reach a fever pitch of “am I going to graduate.” Thanks again for your help.

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