A growing debate over the “science of math” is bringing renewed attention to structured instruction, foundational skills, screening, and remediation. These are important conversations. But amid the focus on what students need to know and the skills they are missing, I worry that we are paying too little attention to another question: What mathematics are we actually showing our children?
After more than two decades of teaching mathematics, I have come to believe that the mathematics children encounter in school bears little resemblance to what draws mathematicians to the subject in the first place.
Mathematicians tinker. We try examples, draw pictures, look for patterns, and make conjectures. We ask what happens if an assumption changes. We follow ideas that sometimes go nowhere. We become interested in a problem because something about it seems intriguing, unexpected, or beautiful. This kind of play is part of how mathematics is done.
Children should encounter that mathematics, too.
Yet this larger math world is often saved for later, after students have mastered enough of the curriculum to be considered ready for it. “Enrichment” experiences are often reserved for students who finish first or are considered more advanced, while students who struggle may encounter even less of that mathematical landscape as their attention is increasingly directed toward the skills they are missing.
We would find this strange in reading. Children learning the alphabet, phonics, and grammar are also surrounded by stories and books they cannot yet read independently. We do not wait for them to become proficient readers before reading wonderful literature aloud to them. They experience the richness of language while they are still learning the mechanics of reading. It is part of what gives children a reason to want to read.
Why should mathematics be different?
Children do not need to understand everything about an idea to be captivated by it. We can let them encounter exciting mathematics before they have all the tools to explain it and allow their understanding to deepen as they grow. The entry point does not need to be complicated. An everyday object, a piece of paper, or a familiar tradition can open onto surprisingly deep mathematics.
Last month, Jewish families celebrated Rosh Hashanah by dipping apples in honey for a sweet new year. A honeycomb offers a mathematical question accessible to a child: Why hexagons?
Regular triangles, squares, and hexagons can each tile the Euclidean plane without gaps or overlaps. Yet, honeybees build with hexagons. A young child can explore which shapes fit together. An older student can compare areas and perimeters or investigate why only certain regular polygons tile the plane. Either activity can be an accessible entry point into the more advanced understanding that the regular hexagonal honeycomb requires the smallest total perimeter out of the possible partitions of a plane into regions of equal area. There is something delightful about discovering that bees are remarkably good at optimization.
Or give students paper cut into regular heptagons and ask them to arrange three around a common vertex. These seven-sided polygons cannot lie flat when attached without overlapping because their three interior angles total more than the 360 degrees available around a point in the Euclidean plane. The paper begins to buckle.
Instead of stopping there, ask: Is there a geometry in which three regular heptagons can meet around every vertex?
There is. In the hyperbolic plane, where the rules of non-Euclidean geometry allow polygons to have smaller interior angles than their Euclidean counterparts, such a tessellation is possible. A handful of paper heptagons has opened a doorway into a geometry beyond the Euclidean geometry students typically encounter.
Or take two strips of paper. Give each a half twist, in opposite directions, and tape the ends together to form two Möbius bands. Tape the two bands together at right angles. Before cutting along their center lines, ask students what they think will happen. The result is two interlocked hearts.
These experiences are playful, but the mathematics is substantial: optimization, non-Euclidean geometry, and topology. More importantly, each begins with something to notice and a question worth pursuing. Why do bees use hexagons? Why won’t these heptagons lie flat? What will happen if I cut this?
That is much closer to the way mathematicians experience mathematics.
And it matters especially for children who struggle. When a child falls behind, focusing more intensely on missing skills and prerequisites is understandable. But if that child’s experience of mathematics becomes primarily one of remediation, we risk giving the students who most need a reason to remain engaged the narrowest view of the subject.
A child who struggles with multiplication can still think about infinity. A student with gaps in algebra can still be intrigued by a Möbius band. Rich mathematics should not be a reward reserved for students who have already mastered foundational skills. Experiencing the depth and beauty of the subject can itself give students a reason to develop those skills.
None of this means that children do not need arithmetic fluency, algebraic reasoning, mathematical language, and sustained practice. They do. The choice is not between foundational skills and mathematical exploration. Students need both. Learning the mechanics of mathematics should not require waiting to encounter its ideas, just as learning the mechanics of reading does not require waiting to encounter good literature.
That has a practical implication for classrooms. Do not save the interesting mathematics for Friday afternoon, for the students who finish early, or for an enrichment program. Put something interesting in front of everyone. Let students notice, build, cut, draw, and conjecture. Let them make a prediction and discover something unexpected. Then, give them the mathematical tools to push their questions further.
As we decide how best to strengthen mathematics instruction, our job is not only to help children learn mathematics. It is to make sure they have the chance to see the enormous mathematical world waiting for them.
