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!

The first post is the first post

I suppose this is an embodiment of my one and only New Year’s resolution.

Multiple times during this school year I had an urge to share the project or the question and I’ve reached the point where I want to create the space to do it. I am not certain how often I will be able to update; maintaining the required class blog generally drains my blogging energy. But I have a few projects coming that I am excited about and I hope some of them will end up here.

Happy Holidays!