4 moves to make math visible with kids, using counters


Let’s say you’re a parent helping a Grade 1 child with their math, and they’re subtracting eight from 17, using small items — counters — like Smarties, multicoloured Rocket candies or Lego pieces.

The child counts out 17 items. Then, they count eight of those items to take away. Finally, they start counting the remaining items.

Here is where parents who haven’t revisited math instruction for decades get confused and want to show their child what they believe is a faster or better way: by picking up a pencil and paper to stack the 17 on top of the eight.

Don’t be surprised if your child brings home a few unfamiliar strategies. Many classrooms today embrace “number talks” — what math educator Sherry Parish defines as discussions about computation problems “designed to elicit specific strategies that focus on number relationships and number theory.”

Making math thinking visible

In the example above that involves subtracting eight from 17, parents may be thinking the next step in teaching a child is relaying the rule that you never take a bigger number from a smaller one.

They may want to tell the child: “You can’t take eight away from seven, so you borrow a 10 from the tens column …” That’s when the parent and child both realize they are exactly where they started, subtracting eight from 17. Yikes!

Instead of falling into the trap of showing and telling your preferred method for subtracting, mathematics educators recommend listening to learners and talking about what they already know.

What the child says and does can be a resource for understanding subtraction.

Here are ways parents or teachers could support a learner’s strategy by helping make their thinking visible.

Move No. 1: Make arrangements

Talking about numbers involves understanding “how muchness” — for instance: how 17 is 20 less three, or how it’s 15 and two more. Math educators talk about the importance of learning to “subitze” — learning to quickly see how many without counting, through quick visual recognition.

Using counters helps students see quantities. This can be done by making specific arrangements with counters, like grouping counters into circles, rows or clusters of threes, fours and fives. For example, if the number 17 is arranged into subitized images of five, a child might say they see three fives and two remaining single units.

Seventeen seen as five, five, five and two. How would you subtract eight? (Marc Husband)

Prompting them to subtract eight might provoke a strategy where the child takes away five, and then another three. Depending on where the child starts taking away (from left to right versus right to left), they could be left with different subitized images: for example, two, five and two; or five and four.

Such processes of “decomposing” numbers — breaking down the eight into five, two and one — builds what’s called number sense (understanding how numbers are related).

Move No. 2: Colour code

Using different colours can help kids see numbers within numbers. (Marc Husband)

Using two different colour counters can enhance a learner’s ability to see activities that are happening in their minds.

Supposing the child says, in subtracting eight from 17, they want to subtract seven first. You can then lay out seven green counters and 10 blue ones, and then remove seven green and one blue. Coding numbers with colours helps students to see numbers within numbers (that seven and one make eight).

Move No. 3: Show the action

When learners subtract, they’re doing a mental action. Seeing this action with counters supports understanding the concept of “taking away.”

Let’s continue with our example of 17 minus eight with the 17 organized into subitized images of five and two. If the child says that they want to subtract the seven first, a parent or teacher can illustrate this by pulling (in a downwards motion) the arrangements of seven, followed by one from the remaining 10.

Moving the counters demonstrates what thinking processes are happening. (Marc Husband)

Emphasizing the mental activities by doing and recording is beneficial for all learners. It may in particular also be a way for educators to seek to engage Indigenous learners. As math researcher and educator Lisa Lunney Borden writes, in a case study of Mi’kmaw students in Atlantic Canada learning math, most Indigenous languages in Canada have verb-based origins, and language generates world views and cultural ways of knowing. From this perspective, emphasizing process and action in math may help engage Indigenous learners and affirm their identity formation whether or not they currently speak their ancestral language.

Move No. 4: Check-in repeatedly

While you’re arranging counters, colour coding and showing actions to make ideas visible, you can support a two-way conversation by checking in multiple times with kids. Consider the following questions or prompts:

“Can you say it again?”: If you’re unsure what they did, don’t be afraid to ask them to repeat it. Taking time to figure out what they did values their thinking and can be engaging for everyone, including the parent or educator.

Gesture: When you move counters to make a learner’s thinking visible, gesture by making a circle with a finger over the counters they say they’re seeing. While colour coding, you might ask: “What numbers do you see now? Show me the 10. Where’s the seven? The one?” Take turns gesturing over the counters to seek agreement on what you’re recording.

Go slowly: Students’ thinking can happen fast, so slowing down what’s happening in their head is a good thing. To support slowing down, ask questions like: “What did you do first?”

Let the learner take the lead: If the learner says their strategy is: “I subtracted seven first, then one more,” pull seven counters down and ask them which counter (from the remaining 10) should be pulled down to make the eight. The learner might choose a counter different than you. There is no correct answer here — pulling the middle counter down, for example, might make it easier for them to see the remaining nine.

We would love to hear how these moves support talking about math!

Author Bios: Marc Husband is Assistant Professor, School of Education at St. Francis Xavier University and Heather Bourrie is a PhD Candidate, Mathematics Education at York University, Canada