We’ve argued for a long time that the real experience of mathematics is inextricably tied to play. But if you’re a parent or teacher, you’ve seen kids play in mathematically irrelevant ways. How do we hit that sweet spot of mathematical play?

One way is to recognize mathematical questions and ideas when they arise from the play itself. Another is to subtly introduce them, without sacrificing the play when you do. So it isn’t, “Stop playing so we can do these flashcards.” Instead, it’s “I wonder how many Jenga blocks you could stack on a single block without it falling.”

We’ve found that there are four central questions that can help uncover or motivate mathematical play.

1. How many?
2. How much?
3. What kind?
4. What if?

Each of these questions highlights a topic from mathematics, and arguably these same questions are the ones that continue to motivate modern mathematical research. Roughly, the correlation goes:

1. How many = questions of number
2. How much = questions of measurement
3. What kind = questions of classification & geometry
4. What if = questions of logic & imagination

For younger kids, the first two questions can often be simplified into a “which is bigger” or “which is more” kind of question, which can avoid counting if they’re not ready to do it.

Once you have these four questions at your disposal, opportunities for mathematical play start popping up everywhere. A walk around the block can be a chance to play the game of estimating steps and then checking whether you were right. (“How many steps do you think it will take to reach that tree?”)

Or an argument over who has more juice can become an experiment to see which cup actually holds more liquid. (How much juice can it hold? Which cup holds more? How can we figure that out?”)

Little kids will automatically classify and sort. What’s fascinating to me is how central this question is throughout mathematics. It seems like every field begins with a question of what kinds of objects are possible. What kinds of pentagons tessellate? What kinds of of polygons can you make by putting squares next to each other?

As for What If questions, cartoonist Randall Munroe has practically built a career out of them. The freedom to assume that the rules or the setup is different is one of the keys to owning your mathematical experience.

If you’ve got a favorite question you ask to find that sweet spot on the Venn diagram, let us know!

A last thought: perhaps the Venn diagram above isn’t to scale. Maybe it should be more like this: