Category Archives: Generalizations

Math Reasoning Stages


Tracy Zager (@TracyZager) asked for some thoughts/pushback on these stages of math reasoning imagined as a flow, so here are my thoughts based on my experience in the classroom…

Pattern Sniffing –  After students see a pattern I find they continue, using that pattern, for a while before thinking about a generalization. So, maybe “Extending Pattern Using the Pattern” comes after this in my mind?

Wondering – When they wonder, they definitely look at relationships, but I am not sure they wonder if it will always be true at this point? Now that I just wrote that, I am thinking maybe “Extending Pattern Using the Pattern” comes after this one?

Articulating – “Can I communicate what I am seeing happening in a precise way?” I don’t know if they are thinking too much about it at this point but more seeing it happening? Could

I don’t know where this fits necessarily, if it is embedded in one of them, or if it really fits at all:), but there is a point where mathematically students prove a generalization works with certain number and not others because “the numbers have to work that way” (structure) without the conceptual proof of why that is. For example, “Even dimensions of a rectangle will give you an even area.” Students can make the statement that it has to work every time because when you multiply even numbers it is always an even product….true, but isn’t there proof to that. So, it is like a string of proof by depth?

Investigating and Explaining Why – I feel the relationships and patterns question to themselves comes back up here too.

I love thinking about this process for students and the teacher implications between each step. What questions and/or feedback do we give as students go through this that isn’t too helpful or leading, but not too vague that leave them in one spot spinning wheels? Paging @MPershan…

*Chatting with Tracy after I wrote this, she was focused on mathematicians, not students. I find some holds true in both cases to different sophistications. 

Hope that helps a bit Tracy! Hopping on a plane but as soon as I have wifi I will add a couple more question I have to think about around this!


Developing Claims – Rectangles With Equal Perimeters

Yesterday’s math lesson launched based on this student activity book page in Investigations…


Students read the introduction and I first asked them, “If we were to build these in Minecraft, was feet an appropriate unit of measure?”  Some thought that feet seemed too small for a garden and instead wanted to use yards, that was, until one student schooled us all on Minecraft. Come to find out, Minecraft uses the metric system with each block representing one cubic meter. We then changed the unit to meters and were on our way.

I gave them 10 minutes, either with a partner or individually, to build as many different rectangular garden designs as they could with 30 meters of fencing. As I walked around, it was interesting to see how students were designing their gardens. Some started building random dimensions and adding/subtracting fence links to eventually hit 30 while others had thought of one rectangle to start and adjusted from there by subtracting/adding from sides. It was fun to see the ones that discovered a pattern in their building and sped through the rest of the rectangles. Some beautiful patterns emerged….

IMG_8854Photo Jan 14, 1 23 24 PM

Photo Jan 14, 1 22 35 PMAfter 10 minutes, I asked them fill in the table on their activity page and chat with their table to see if anyone had any dimensions they didn’t have. As a table, they came up with some noticings based on their table, Minecraft builds, and process of building.

IMG_8862 IMG_8863 IMG_8865 IMG_8866 IMG_8867 IMG_8868 IMG_8869 IMG_8870 IMG_8871 IMG_8872 IMG_8873The class ended after this group work. Talking with another colleague about the lesson at the end of the day, we chatted about what claims could come out of this lesson. It was so interesting to me to think about not only the geometric claims, in terms of rectangles and dimensions, but also the numeric claims that can evolve from these conversations as well. And to think that these number claims are grounded in a visual connection is pretty awesome to me!

I was so I started today by pairing up every two tables to share with one another. One table read their noticings while the job of the other table was to ask clarifying questions. That gave me time to circulate, listen, and choose the claims for our class share out. After the table shares, we reconvened as a whole class and chatted about a few of the claims as a group.

IMG_8853I asked them to think about similarities and if we could combine some of them to form one claim based on our work.

Class ended with a journal entry in which they worked independently to begin to form a claim or the beginnings of one. Here is where I will start tomorrow…

IMG_8855 IMG_8856 IMG_8857 IMG_8858 IMG_8859 How fun!

As a follow up on this post, this is the assessment on their work with perimeter and area. During the class period, only one student started working on finding the square that would give them the largest area with a perimeter of 30. After this idea came out in our class discussion, using fractional dimensions was something that others were thinking about during the assessment.  Here is a student’s (different than the initial student) work that I thought was pretty fantastic:

Photo Jan 16, 11 51 34 AM





Articulating Claims in Math

This summer I was fortunate to hear Virginia Bastable keynote about the work in her book Connecting Arithmetic to Algebra. If you have not read this book, it is a must! It explores the process by which we have students notice regularities, articulate claims, create arguments and representations, and make generalizations.

9780325041919It is something, as elementary school teachers, we need to really be thinking about more in our math classes. Are we creating environments that encourage students to think about the math behind the strategies and make generalizations based on the properties of operations? I have taken this recent reading and made it a priority in my classroom.

I always have students notice and discuss patterns and regularities but I don’t often have them create generalizations for us to revisit as we move through the year. For example if their claim works for whole numbers, shouldn’t I revisit that as we work with fractions? Does your claim still hold true?

As a class routine, I posted this on the board and asked students to fill in the blanks to make it true:

12 x 4 = ___ x ____

Quickly, students wrote down answers, had their hands up, and one student blurted, “This is easy, you don’t even have to solve it!” Typically blurting out answers before others are done thinking drives me a bit crazy, but this time I was thinking…Yes! I asked who else thought the same thing. I had at least one hand up at each table so I asked them to discuss with their table how that is possible. We came back together and each table said they could double/half to fill in the blanks. I took answers on the board and got the expected: 6 x 8, 24 x 2, 48 x 1, 3 x 16 and then I even had a 96 x 1/2 and 36 x 1 1/3! I asked about the 48 x 1…did you get that by double/halving from the original problem? What is happening there? They noticed that it was x 4, ÷ 4, and then the same with 3 x 16. I asked them to take some individual time to see if they thought their strategy would always work and could prove it with a representation. They then talked at their tables and I asked each table to write a claim, something they think is true about this work.

I got some who kept solving problems to prove it works:


I had a couple try out the representation (exclaiming how hard it was to draw what is happening:)


Here are some of their claims:



This is the class list from my second period class. I especially liked that one of them said it only worked with multiplication. How fun to revisit!

IMG_8107Instead of losing these, I started a Claim Wall to post and have students add to and revisit throughout the year. I am trying to think through how to have students comment on them, possibly agree/disagree post its?


If you would like more information about Virginia’s work, there are courses available here: Check it out, great stuff!


Making Mathematical Connections

Each day I start the class with a Number Talk. I thought to continue building our multiplication strategies and make connections to our volume work, I would do Dot Quick Images. This is one of the images that I did yesterday:


In this image I hoped to bring out the commutative and associative properties (not by name, but the idea of what is happening in each) within their solutions as well as the use of the 3 x 3 array to get the number of groups in the picture and 3 x 4 array to get the number of dots in each group.  This would be the moment when I wish I took a picture of the board with their responses, but in the flow of the lesson, forgot. Many said they did 12 x 9 to get 108. I especially loved that some said they said they didn’t know 12 x 9, so did 12 x 10 and took a group of 12 away:) I said that when I read 12 x 9, I think of 12 groups of 9, trying to elicit the commutative property. I had them talk to their neighbor and we agreed this picture looked like 9 groups of 12, but there was a way to make it look like 12 x 9. I wrote both on the board agreed the amount of dots didn’t change, just the way we looked at it did. This went on into another image and we began our first lesson on volume. I blogged about that work here.

So, after chatting with a colleague after the lesson, we thought it would be interesting after the work yesterday, to reflect back to that number talk. Today I put the same image back up and I did a much better job of pulling out the (3 x 4) x 9 through better questioning and because they were solid in the answer, they could reason about it a bit deeper. I also told them to be thinking about our lesson yesterday to see if they could see any connection between the two. I finished the number talk and gave them 2 minutes to reflect in their journal about any connections they saw. Here is what we shared as a class…

IMG_7753 IMG_7754 IMG_7755 IMG_7762 IMG_7763 IMG_7764 IMG_7765 IMG_7766 IMG_7767 IMG_7769

So much to love here…..I loved the idea of layering the dot arrays to make a box. I loved the connection to the properties in each….