**Introduction**

Last week I shared how my grade 3 students were preparing to prove their conjecture about subtraction: If you add +1 to both numbers in subtraction, the difference will not change. I really appreciate all the feedback and conversations that happened around the topic on twitter. It was also interesting to read blog posts from Jenna Laib , Simon Gregg and Sarah Caban about their experience working with proofs and representations.

My class finally finally got to our proof lesson last Monday, it was an extra long lesson, students came up with many great models and their teacher ended up with many questions. All this made for an extra long post.

**Towards Lesson 4**

We ended last week with building and sharing different ways to represent subtraction. I promised that on Monday we will work on **Proving the Claim**. I admit, I was nervous regarding the level of frustration and confusion the task could cause, and I was not certain how to prepare my students for it. So I did the only thing that came to mind; I scared them.

**Lesson 4: Proof and Evidence**

I started with warning my students that this is going to be the hardest part of the whole process. They came to the point where they need to prove their conjecture for all positive integers. The only time my students attempted proving was in the beginning of the year when we used snap cubes to reason about sums of odd and negative numbers (the lesson inspired by this post in TCM blog).

I reminded my students about this experience, pointed at the selection of tools and manipulatives and sent them off to work. I sensed an air getting heavy with confusion just as I anticipated. Before it ascended to panic, I made a suggestion to rebuild the models that students used last time and then see if they could tweak them to prove our conjecture. I tried really hard to not impose my thinking on my students, so I was mainly wondering around asking students how their models work, recording their explanations and asking for clarifications. The sense of direction was all that was needed, and I will leave the rest of this post to my student’s work. I wish I could post the videos, but my school board is very particular about not sharing students’ voices or pictures online so I will have to stick to the transcripts.

**Students’ Proofs**

The next one got me really excited.

There was also a trio of “story models” that showed that if you subtract 1, the conjecture will also work. I really enjoyed the narrative part.

**Students’ Reflections:**

The lesson took longer than I expected, so I asked for a more open reflection with our general prompts: questions, challenges, observations, discoveries.

**My Notices and Wonders:**

I wish I could say that all of my students experienced the same degree of success with these lessons, developed multiple models for proof and expanded their range of subtraction strategies and their understanding of the operation. I am afraid it is not the case. When I was analyzing my lesson, I got back to thinking about my goals. I wanted my students to

- explore different representations and contexts for subtraction
- gain deeper understanding of subtraction and its properties
- attempt and experience proving
- construct effective arguments
- explain our ideas in words, models, diagrams and mathematical symbols

Even students who were not able to come up with a working model on their own, had a chance to discuss and review other models. They had a chance to reason, to try and to model. They might not have gotten the result but it doesn’t mean that they didn’t get the experience. I wonder if they count it as their success.

Thanks for sharing your work with students. I enjoyed reading the captions that accompanied the student work. It was interested to see what they said about the representations. Don’t worry if not every student ended at the same level of understanding. I think the experiences they had working toward your goals does count as success.

In my experience, not every student is ready to make the giant leap to proof. This is partly due to the fact that some students are solid enough with the conjecture to move beyond testing it a bunch of times. For these students, repeated reasoning (MP8) is really beneficial. The fact they test out a bunch of examples to prove or disprove the claim provides much needed practice and experience playing with the operations.

That being said, I believe all students benefit from these experiences with representations because they help build a deeper understanding of how the operations and numbers behave. For students who notice and can express the regularity in the form of a claim, working toward proving the claim through representations and contexts will help them understand the structure behind the claim. For students that aren’t ready to think in general terms, the work on representation helps them develop visual models for subtraction.

Having done this kind of work with second graders, I can say that it takes time and not every student gets to the same finishing spot in terms of understanding the general claim. However, over the course of the year, the more we did these explorations, the stronger students became when looking for and making use of structure this way.

There is a lot of powerful thinking in your class and some interesting connections between the representations and the contexts. It will be interesting to see what students do with this idea and if some are able to extend the claim to include changing the minuend and subtrahend by any amount.

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Thank you for your feedback, Mike. I agree, moving towards generalizations won’t happen at the same time for everyone but I plan to continue looking for similar experiences. I have been collecting our questions and claims which I am thinking to re-organize into somethink like a claims wall that we can work with throughout the year. I found more ideas in chapter 7 of your book which made me wonder what claims might students in grades 3-6 make.

Regarding applications and extension of the claim, about a half of the students discovered that not just 1, but a larger number will work too and some found that you can subtract the number as well as add it. That was another positive thing about this lesson structure; regardless of the level, all the students were challenged and engaged.

What I was surprised with, is that when I put a number string on the board that would call for making friendly numbers, many students did not see the connection. They seemed to have difficulty moving to practical applications. I plan to continue working with claim’s extensions and doing number talks to support students in making these connections.

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