This week I had the pleasure to see Dan Meyer, Cathy Fosnot and Graham Fletcher at OAME’s Leadership conference.

Each of the sessions were inspiring and informative… but halfway through the conference I noticed a common message that the first 2 keynote speakers were suggesting:

Dan Meyer showed us several examples of what mathematical surprise looks like in mathematics class (so students will be interested in making sense of what they are learning), while Cathy Fosnot shared with us how important it is for students to be puzzled in the process of developing as young mathematicians. Both messages revolved around what I would consider the most important Process Expectation in the Ontario curriculum – Reasoning and Proving.

###### Reasoning and Proving

While some see Reasoning and Proving as being about how well an answer is constructed for a given problem – how well communicated/justified a solution is – this is not at all how I see it. Reasoning is about sense-making… it’s about generalizing why things work… it’s about knowing if something will always, sometimes or never be true…it is about the “that’s why it works” kinds of experiences we want our students engaged in. Reasoning is really what mathematics is all about. It’s the pursuit of trying to help our students think mathematically (hence the name of my blog site).

###### A Non-Example of Reasoning and Proving

In the Ontario curriculum, students in grade 7 are expected to be able to:

**identify, through investigation**, the minimum side and angle information (i.e.,side-side-side; side-angle-side; angle-side-angle) needed to describe a unique triangle

Many textbooks take an expectation like this and remove the need for reasoning. Take a look:

As you can see, the textbook here shares that there are 3 “conditions for congruence”. It shares the objective at the top of the page. Really there is nothing left to figure out, just a few questions to complete. You might also notice, that the phrase “explain your reasoning” is used here… but isn’t used in the sense-making way suggested earlier… it is used as a synonym for “show your work”. This isn’t reasoning! And there is no “identifying through investigation” here at all – as the verbs in our expectation indicate!

A Example of Reasoning and Proving

Instead of starting with a description of which sets of information are possible minimal information for triangle congruence, we started with this prompt:

Given a few minutes, each student created their own triangles, measured the side lengths and angles, then thought of which 3 pieces of information (out of the 6 measurements they measured) they would share. We noticed that each successful student either shared 2 angles, with a side length in between the angles (ASA), or 2 side lengths with the angle in between the sides (SAS). We could have let the lesson end there, but we decided to ask if any of the other possible sets of 3 pieces of information could work:

While most textbooks share that there are 3 possible sets of minimal information, 2 of which our students easily figured out, we wondered if any of the other sets listed above will be enough information to create a unique triangle. Asking the original question didn’t offer puzzlement or surprise because everyone answered the problem without much struggle. As math teachers we might be sure about ASA, SAS and SSS, but I want you to try the other possible pieces of information yourself:

Create triangle ABC where AB=8cm, BC=6cm, ∠BCA=60°

Create triangle FGH where ∠FGH=45°, ∠GHF=100°, HF=12cm

Create triangle JKL where ∠JKL=30°, ∠KLJ=70°, ∠LJK=80°

If you were given the information above, could you guarantee that everyone would create the exact same triangles? What if I suggested that if you were to provide ANY 4 pieces of information, you would definitely be able to create a unique triangle… would that be true? Is it possible to supply only 2 pieces of information and have someone create a unique triangle? **You might be surprised here… but that requires you to do the math yourself:)**

###### Final Thoughts

Graham Fletcher in his closing remarks asked us a few important questions:

- Are you the kind or teacher who teaches the content, then offers problems (like the textbook page in the beginning)? Or are you the kind of teacher who uses a problem to help your students learn?
- How are you using surprise or puzzlement in your classroom? Where do you look for ideas?
- If you find yourself covering information, instead of helping your students learn to think mathematically, you might want to take a look at resources that aim to help you teach THROUGH problem solving (I got the problem used here in Marian Small’s new Open Questions resource). Where else might you look?
- What does Day 1 look like when learning a new concept?
- Do you see Reasoning and Proving as a way to have students to show their work (like the textbook might suggest) or do you see Reasoning and Proving as a process of sense-making (as Marian Small shares)?
- Do your students experience moments of cognitive disequilibrium… followed by time for them to struggle independently or with a partner? Are they regularly engaged in sense-making opportunities, sharing their thinking, debating…?
- The example I shared here isn’t the most flashy example of surprise, but I used it purposefully because I wanted to illustrate that any topic can be turned into an opportunity for students to do the thinking. I would love to discuss a topic that you feel students can’t reason through… Let’s think together about if it’s possible to create an experience where students can experience mathematical surprise… or puzzlement… or be engaged in sense-making… Let’s think together about how we can make Reasoning and Proving a focus for you and your students!

I’d love to continue the conversation. Write a response, or send me a message on Twitter ( @markchubb3 ).

I think the example problem would produce more discussion if the question is to determine the fewest number of information needed to draw a congruent triangle. There is a problem like that in Japanese 5th grade textbook.

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I agree Tad. Thanks for the suggestion. How would you word your problem then?

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Here is the sequence of questions in the translated textbook:

[4] Think about how to draw triangles DEF so it will be congruent to triangle ABC on the right. (picture of triangle ABC is given with angle A, 80 degrees, angle B, 35 degrees, angle C, 65 degrees, AB, 5.5 cm, BC 6cm, and CA 3.5 cm.

First sub-question: By drawing a segment EF the same length as side BC, the vertices E and F are fixed. How can we decide where the last vertex, D, should be?

Main investigation question: Let’s think about which of the following information is needed to determine the position of vertex D: the lengths of sides AB, and AC, and the measures of angles A, B and C.

Since this is a Grade 5 textbook, they do not actually try to prove formally methods to draw congruent triangles. Expectation is that through discussion, students understand that they only need 2 additional measurements (3 altogether) are needed.

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Mark,

Thank you so much for posting – this is the area I’ve always found the most problematic in getting students engaged in Geometry. We also found most approaches to the triangle congruence theorems too teacher focused and not engaging at all in developing mathematical thinking and reasoning. We’ve adopted something slightly different from your lesson that also added in situations when the usual congruence theorems (SSS, ASA, etc.) don’t work (triangle inequality, two obtuse angles, etc.) to make sure the students were thinking of conditions for the triangles as well. We’ve found that patty paper allows students to create, share and compare their work (especially for the SSA case). We also liked that the lessons are slightly tactile and group oriented. I can share them with you if you want.

Kevin

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Thanks for the feedback Kevin. I’d love to see what you have. Always great to push thinking and share ideas:)

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