Proof and Evidence: Reasoning About Subtraction Part 2


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


“Music is actually a lot like math”.


“How do I know that it always works? Because it never ends if we just keep adding by 2s, even, even, even”.

The next one got me really excited.

“All of the bottom is white. I added this way as a handle or to add more of these squares. Green is the difference and the red is the subtraction amount. Red and green are the main number. The black adds one more to the total number and subtracts one more.” -“Why do you add one cube? In our conjecture you add +1 to the first number and +1 to the second, and you just add 1?” – “Because this one cube adds 1 to the main number and the other number, like the minuend AND the subtrahend.”

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.

“This is 10 – 6. The white is 6. I made it like this because white is like extinguished water and that’s why I also have blue which is the remaining water. And that equals 4. The original number is 10”.
“I made this like cheese. This is like an entire square of cheese. Five of these got stolen by mice cause they like cheese and now there is only 4 bits of cheese left. 4+5 equals 9, 9 is the original number”.
“This is the leaf, burning. This is an entire leaf of 8. This one works like 8-4=4”.

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.


I knew this one was coming!

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.


The Best Intentions..

For the “We All Fall Down” edition of the blogging initiative! I skipped the last two prompts with the valid excuse of report cards writing, but I couldn’t skip this one.

The Fail

I finally get my students to calm down from all the excitement and ask, “So… do you see the problem? How can we solve it?” I look around and to my horror realize that no, they don’t, because there is no problem.

The Premise

I really bought into Dan Meyer’s “If math is the aspirin then how do you create a headache?” idea.  I always felt that teaching is close to storytelling, to creating an engaging narrative. Then after watching some of Dan’s videos and reading some blog posts, articles and discussions around, it appeared very intuitive. Of course, I won’t be interested in solving problems that I don’t have. I won’t spend my mental energy or waste my memory capacity on that. If I don’t care, I won’t learn.

All this seemed so obvious and easy that the real challenge got obscured: How do you plant and grow the problem? What seed do you throw in? What notices and wonders do you nurture?

The Lesson Plan

In early September, I planned a lesson to introduce standard units of measurement. The idea was to let kids experience how standard units allow us to have common language to communicate our measurements. Students were asked to measure different items in the classroom with… whatever they felt like. Then we were to discuss the length of their desks and to realize that we can’t come to a consensus because we all used different units.

The Lesson

Students were very enthusiastic and measured a lot of different items around the classroom, shared their measurements and observations, recorded them. Then the moment came to discuss our problem and to see how we all need to use same units… “What problem?” When I looked around I saw twenty-five happy eight year olds who have been having time of their life measuring the classroom with everything from pencils to their heads. They didn’t have a problem, they had a blast. They did not care that we all had different measurements for our desk. It was exhilarating to find out how many different ways we could measure our desk! I wanted my kids to need metric system. Well, after my lesson they didn’t want anything to do with metric system.

The time for the lesson was almost up. No way to salvage the lesson materialized in the last 10 minutes of the class. I spent this time mumbling about standard units of measurement who nobody cared about.

I started typing in how I fixed the lesson next day building on my failed one, and then decided I should leave it out. I’ll share if someone is interested. The most important impact of this fail was that it made me more sensitive to my assumptions, made me question the questions I ask my kids and the prompts I give to them. It made me experience that I can’t force a problem on my students, it has to come from them.

The Question

What would you have done to save the lesson?

Proof and Evidence: Reasoning About Subtraction Part 1


It’s the third year that I spend some intentional time in my classroom working on a “making friendly numbers” subtraction strategy: 234 – 19 = 235 – 20. I was not happy about my previous two attempts. Some kids I think just bought into it, some kept confusing it with addition (take from one number and add to another); in any case, I felt like students were not able to figure out why this works, and they either took my word for it or tossed it away. In year one, I gave it to them. In year two, I made a lesson with Cuisenaire rods. I offered my students a great model they didn’t really care for at that moment.

A few students brought this strategy up this year during our number talks. “Why does it work?” – “I don’t know, my teacher told me it always does.” It was time to try again for the third time.

Original plan was a bit blurry: get the number string on, get students to notice the rule. Then model it with the snap cubes in two towers; let students work with partners to change the number of cubes in each tower without changing the difference.

Intermission 1

The new idea came after Mike Flynn’s session about exploring multiple representations, a first session in Effective Practices for Advancing the Teaching and Learning of Mathematics course in Mount Holyoke College. I tossed my original plans and decided to try and follow the example that was shared in the session; let students notice the pattern, come up with the claim and then prove the claim using their own representations.

Lesson 1: Developing the Claim

First lesson was a brief number talk. 3-1, 4-2, 5-3. Students noticed that if they add “1” to both subtraction numbers, it seems like the difference stays always the same. I also started recording the steps of mathematical thinking that we will follow. My worries during the lesson were that a few students did not seem to notice a pattern. I hoped that the Testing lesson will help.

Lesson 2: Testing the Claim

The plan was to ask students to spend 15 minutes testing the claim with as many examples as possible. Then things went sideways. “Can I try negative numbers?” We don’t teach integers operations until grade seven. I teach grade three. I panicked then said, “Go for it.” Then there were groups of kids engaging in some interesting discussions.

  • It doesn’t work! (-5) – 3 is not the same as (-4) – 2!
  • You are not subtracting -3, you are subtracting positive 3.
  • If (-5 )– 3 = (-8), then what is -5 – (-3)?

Not sure what I was thinking when I briefly introduced integers earlier this year. You can’t give eight year old kids a new math toy and expect that they won’t play with it. I’m bracing myself for multiplication and division of integers in the spring.

One student discovered that if you subtract a number from itself, you will always get zero. From that moment on, he forgot about original claim and was testing his new discovery. I think we will be proving it next. It was all fun if not quite what I anticipated. At the end of 15 minutes we all agreed that the rule seems to always work.

Intermission 2

The plan was to now ask kids to use any models to prove this claim. And that’s when I realized that I didn’t allow my students to play with different tools and to develop their own subtraction models. We built numbers, we did not build “subtraction.” I felt we are not ready. I decided to let kids to explore different representations for subtraction first.

Lesson 3

“Today we will be testing the tools that we will later use to prove our claim.” Students worked in 4 groups to build a model, draw a diagram and write a story for 9-5=4. I had a selection of different items ready for the students to pick; from ten frames and balances to dominoes and snap cubes. First ten minutes were terrifying. Students used manipulatives to build symbols 9,5 and 4. I told that with their manipulatives they are trying to explain it to someone who might not anything about subtraction or symbols for numbers, but can count.

After all groups seemed to have come up with some working representations, students interviewed each other to find a representation that was different from their own and reflected on it in their math journals.


Closing Thoughts and Plans

These few lessons made me realize that when I ask kids to explain their thinking, I often give them my tools and they end up modeling my thinking rather than developing and explaining their own. They need to find their own tools and develop their own reasoning and I need to give them more opportunities to do it. On Monday we will move on to our proving.

Here is the template I made to plan my future Mathematical Reasoning Lessons.


How do you lead students from drawing the picture to drawing the diagram? From building numerical symbols to building a model?

My Favorite: From Sharing to Discussion

My grade 3 students love sharing: stories, artifacts, ideas. They are much less interested in listening – to what others have to share. Which kind of defeats the purpose. Math time is not an exception.

I believe that discussions play important role in elementary mathematics classroom and all students need to become comfortable with constructing arguments, defending their reasoning, questioning and proving. I try to design my lessons and routines so that they would include these skills as their natural component. And yet sharing doesn’t turn into discussing as often as I hope it would.

The background:

There were some things that worked in the past for me: Circle counting (I learned about this routine from Sadie Estrella here ) encouraged students to listen to each other and WODB (from Christopher Danielson and Mary Bourassa) seemed like a scavenger hunt to my students who were interested in “who found what else and how it can help me to find something else”.

But I am always on a lookout for more ways to engage more students into productive and attentive mathematical discussion, and my favorite “aha” this week was this “one new way”.

About a month ago in December, I joined a Global Math Department webinar lead by Andrew Stadel (@mr_stadel) , Clothesline Math: The Master Number Sense Maker.

The lesson:

So as January came around, I decided to start my first day of school in grade 3 classroom with Clothesline lesson: looking at numbers, addition and subtraction equations and arranging them on the number line; first in the right order and then in the right spot relative to each other.


Students worked in groups first to put the numbers in order, then each group moved one number on the large clothesline at the front. Everyone agreed on the order but more precise locations were already causing a lot of disagreements. Usually, I would ask students to come up one by one to move one number and justify their reasoning and then have some discussion routines to roll. The same kids usually want to come up. The same kids usually want to have a discussion. The same kids want to argue about making the location more perfect. And all these kids do not constitute a majority of my class.

Using some webinar ideas, this time I asked a student to come up, move one number and remain silent. Other students had to try and “read her mind” to figure out the reasoning. Whatever the reason, this shift worked. Kids who are usually not comfortable presenting did not hesitate to come up, to move the numbers, and then to listen to others try and figure out their reasoning. “Not quite”, “On the right track”, “That was what I was trying to do”, “There is more to it” – the list of responses developed on the go. At the end, they found it easier to summarize their explanation after listening to all these other people trying to “read their minds”.

The “favorite” stuff:

The best conversations happened around some solutions that most of the class quickly identified as incorrect. They still had to try and figure out what was the reasoning behind it instead of jumping straight to disagreeing.

Student 1 comes up and puts the card 5+7 between 0 and 10-5.

Me: “What do you think is the reasoning?”

Student 2: “Maybe 5+7 is close to 0 because 5 and 7 together are smaller than 10 and 5?”

Student 3: “Maybe it’s smaller because 7-5 is 2 and it’s very close to 0. But one is minusing and another is adding, so it might be important.”

Student 1: “Can I revise my thinking?” Moves 5+7 past 10-5 and 20-10. I did not think that I need to add.”

What worked well:

-New students joined the discussions.

-More conversations happened without my interference.

-Students listened to each and “revised their thinking” based on each others feedback.

-Students who find it challenging to explain their reasoning felt more confident when they had the “help” of others working not to challenge their ideas but to understand them.

My Questions now:

  • How can I adapt similar routine to other activities?
  • What other strategies and routines other teachers use to encourage authentic discussion?

I appreciate your ideas and suggestions!