When a Drawing is Not Just a Drawing

One of my biggest learnings in my role as a board co-ordinator has been around mental math in elementary schools. To be honest, when I first heard the term, I assumed mental math had to do with memorization and just learning multiplication facts. 

I now know it to be a procedure very rich in strategy, promoting flexibility of number, and the very important mathematical process of representing a problem. Being able to see the math not only contributes to the understanding of the question at hand, but also to assessing the reasonableness of the answer.

Visualization and representation of mathematics does not come easily to me - I was the student who became good at math by memorizing procedures. I still have to put a lot of effort into picturing multiplication in an array or area model, or picturing factorization by splitting items into groups, or picturing what happens to fractions as they are operated upon.

This past week, I had the opportunity to model the annotation process in mental math mini-lessons in a grade 8 class (on the right, below), as well as a grade 10 locally developed class (on the left, below), while a second teacher led the discussion. 

This is still very new to me, and it throws me quite far out of my comfort zone. Here are a few of the things I'm learning about annotating student thinking:

Choose the representation you want to use ahead of time

Will you use a number line to represent students' thinking in this number talk? A ratio table? An array? Or will you annotate entirely with symbolic notation? Are you choosing this representation because of students' familiarity (or lack of familiarity) with it? Or because it better illustrates the idea you wanted to get across? How does the representation relate to your questions?

With the grade 10 class, we wanted to link multiplication and division to something very hands-on and practical, like building a deck, so we used an area model. With the grade 8 class, they had already done a lot of work on representing multiplication with arrays, so we built on that to annotate how division is connected with multiplication by continuing to use arrays. 

If the goal is to help students visualize the math, then we need to choose the representation with great intention.

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Practice annotating in advance

Once you've selected the representation you'll use, work through the number talk ahead of time, annotating the answers you think students will give. Will you use different colours? How large will you make the annotation? Will everything be drawn vertically, or will you have some annotations below the original question, and some to the side?

Last week, I accidentally flipped one of the arrays in the grade 10 class so that it didn't match the array in the previous number sentence, potentially causing some confusion. In the grade 8 class, I made an array WAY too large so that when I partitioned it, the sizes of the partitions didn't make sense (making proportional drawings on the fly is definitely a struggle for me!). The more you can practice drawing what you want the students to see, the more natural it will be in the moment. 

I'm learning that as much as I'd like to think that I can get up and draw area models (they're just rectangles!) based on student thinking, to do it properly I have to be much more deliberate.

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Consider how to draw other ways of thinking 

Not all students think like we do. Some students may give incorrect answers, or come at a question from a completely different angle than how we would approach it. How will you acknowledge and annotate those strategies? 

To help prepare for the lessons last week, I ran my string by a non-math friend and annotated her responses as practice. I also tried to anticipate some very different answers, and practiced annotating those solutions. While you can't prepare for every possibility (sure enough, in the grade 10 class we worked through a strategy neither of us had anticipated beforehand), I felt more comfortable going "off-script" and the anticipation helped me think on my feet. 

A good annotation can also help uncover student misconceptions, and help them realize where they might have gone wrong. Or, it might lead to a completely different way of thinking about a problem. In either case, that visualization can be quite powerful.

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My go-to books, as I learn more about mental math and number talks, are Making Number Talks Matter by Humphreys & Parker (2015), and Building Powerful Numeracy for Middle and High School Students by Harris (2011).