Through visual representations, mathematical language and opportunities to question and rethink, students developed greater confidence in communicating their thinking and approaching unfamiliar problems.
“Mathematics is a subject that is all about creativity, making connections, and sense-making.”
— Jo Boaler
When students can see the maths, they can talk about it.
This became the core idea behind our Grade 2 Math Talk journey. It started with simple but intriguing mathematical images—dot patterns, tens frames, number lines and visual models—and invited students to look closely before looking for an answer. We wanted them to notice the mathematics, make connections and find a way to communicate what they were seeing.
In mathematics, the final answer can sometimes hide the thinking that produced it. Two students can arrive at the same answer while seeing the mathematics in completely different ways. One might count, another might group, another might partition, while another might recognise a relationship immediately. We wanted Math Talk to open up those different ways of seeing.
We wanted students to experience the jobs of mathematicians—to explain, justify, convince, reason and share. The focus was not simply on whether students could solve a problem, but whether they could communicate the mathematical thinking behind their solution.
The representations became particularly powerful because they helped make mathematical relationships visible. Students could see quantities, structures and connections that might otherwise have been difficult to hold in their minds.
Instead of carrying several relationships mentally, they had something in front of them that they could return to, manipulate and compare. This helped reduce some of the cognitive load involved in working with multiple pieces of mathematical information at once.
The visual was not there to make the mathematics more attractive.
It was there to make the mathematics easier to see and think about.
Once students had something they could see and work with, we needed to give them the language to communicate their thinking. We intentionally introduced mathematical vocabulary such as partition, compare, represent, strategy, justify, reason and convince.
These words were not introduced simply for students to remember. They became part of the way students communicated mathematical relationships and explained their choices. Their explanations began to move beyond what they had done towards why they had done it.
This connected strongly with Jo Boaler’s emphasis on mathematical reasoning. Explaining mathematical work is not something that happens after the mathematics; it is part of the mathematics. A correct answer could tell us where a student had arrived. Their explanation helped us understand how they had got there.
As Math Talk developed, we began listening beyond the final answer. Could students explain a relationship? Could they justify a strategy? Could they use evidence to support their thinking? Could they compare different approaches and explain what they noticed?
The jobs of mathematicians began to become part of the mathematical work itself. Students were explaining, justifying, convincing, reasoning and sharing ideas that could be examined. An answer could become the beginning of another question rather than the end of the task. A strategy could be challenged. A representation could reveal something unexpected. Students could return to the mathematics and ask:
“What makes this make sense?”
What Happens When the Answer Doesn't Work?
Some of the richest moments came when students were unsure. A strategy might lead to an unexpected answer, a representation might reveal something they had not anticipated, or a problem might reach a point where they did not know what to try next.
Instead of immediately providing another procedure, we gave students time to stay with the mathematics. They could return to the representation, rearrange the objects, redraw the model, adjust the number line or try representing the problem in another way. Gradually, students became more willing to remain in that uncertainty. They tried another possibility, reconsidered an earlier idea and continued working when their first approach did not succeed.
This changed the way we looked at mistakes too. An unexpected answer could reveal something about how a student was thinking. Rather than treating the mistake only as something to correct, we could use the representation to investigate the reasoning behind it. Jo Boaler’s idea that “Every time a student makes a mistake in math, they grow a synapse” resonated with this approach. The value was not in making a mistake, but in what the mistake allowed us to investigate.
Another idea from Boaler resonated strongly with our experience: some mathematicians are not particularly fast with numbers because they “think deeply and carefully about mathematics.” This challenged the assumption that speed is a measure of mathematical ability. Math Talk created space for students to pause, look again, represent an idea and reason before moving forward. Sometimes, slowing down was what allowed students to see something they had missed the first time.
As students became familiar with different representations, they began to recognise that the same mathematical idea could look different depending on how it was represented. A tens frame might reveal composition. An open number line might make jumps and relationships visible. A drawing might expose the structure of a problem. An equation might express the same relationship symbolically.
This led to an important question:
What does each representation help us see?
Students began to understand that representations were not simply different ways of recording an answer. Each could reveal something different about mathematics.
The next question was whether these ways of thinking would transfer when the mathematical context changed. We began noticing students using mathematical practices in situations that did not look exactly like the examples they had encountered before. They could represent an unfamiliar problem, look for relationships and draw on reasoning even when a familiar procedure was not immediately available.
Transfer was not simply remembering a strategy. It was knowing when mathematical thinking could help.
This was also where we began to see students making more decisions within mathematics. They could consider which representation might help, how they could show their thinking and which strategy might make sense for the problem in front of them.
They were not simply waiting for the next instruction.
They were beginning to decide how to enter mathematics.
Looking back, the shift was not simply about students talking more. It was about giving students more ways to see, represent and reason about mathematics. The visual representation gave mathematical thinking a form. Language helped students articulate it. Math Talk gave them a space to examine it. Difficult moments gave them opportunities to revisit and rethink. Different representations opened possibilities for mathematical choice, and these practices began to travel into new situations.
What began with mathematical provocations and visual representations gradually became a different way of approaching mathematics. Students were not only looking for an answer; they were looking for relationships, making sense of what they saw, explaining their choices and deciding how they wanted to approach a problem.
Ultimately, the question is not simply, “Can the child get the answer?”
It is: “Can the child see mathematics, make sense of it, represent it, explain it, justify it, convince others and reason about it?”
Because when students can see the maths, they can talk about it. And when they can talk about it, they can begin to do the jobs of mathematicians.
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