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I want to begin this text with a quote from Miguel de Guzmán, a Spanish professor of Mathematical Analysis, a respected mathematician, teacher, and communicator. As president of the ICMI (International Commission on Mathematical Instruction), he had a deep influence on math education in Spain and around the world, and he was a key figure in its popularization.
«Play and beauty are at the origin of a large part of mathematics. If mathematicians throughout history have had so much fun playing and contemplating their play and their science, why not try to learn it and communicate it through play and beauty?»
— Miguel de Guzmán
With this reflection, Miguel de Guzmán captures something often overlooked: math is not simply a set of rigid rules or abstract symbols, but a field born of curiosity, imagination, and the joy of discovery.
If mathematicians themselves describe their work as playful, why do we hesitate to put play at the center of children’s math experiences? So, let us talk about play.
In many classrooms, math and play are two separate worlds: the first, serious and rigorous; the second, spontaneous and fun. The challenge—but also the opportunity—is to build bridges between them. Play-based learning has been widely studied in early childhood education; today, we know that play supports social-emotional development, improves self-regulation, and builds the foundation for complex cognitive skills.
In math, research shows that rich math learning and high-quality play complement each other rather than compete.
A teacher who asks intentional questions or introduces math language during play does not interrupt the experience; rather it deepens it. Likewise, children who take on math challenges through games or participate in guided play situations often return to their free play with richer strategies and more complex ideas (Clements & Sarama, 2005; Van Oers, 1996; Vogt et al., 2020).
The false dichotomy between “play” and “rigor” disappears when we recognize that math grows in environments where curiosity and structure coexist.
Of course, not all play is the same, and each type of play opens different mathematical opportunities. In early childhood classrooms, play-based learning can be understood as a continuum:
Sources: California UPK, Van Oers (1996)
“Math can be integrated into children’s play and ongoing activities, but this often requires a curriculum and a knowledgeable adult who creates a supportive environment and provides challenges, suggestions, tasks, and language. Combining free play with intentional teaching, and promoting play with math objects and math ideas is pedagogically very powerful.”
(Clements & Sarama, 2005a)
At one end, free play, inquiry (questioning), and collaborative play offer opportunities to mathematize play situations. These are child-led contexts in which the teacher helps math learning emerge by introducing questions, vocabulary, or new materials.
For example, imagine a group of children playing freely and building with blocks. At first, they stack for fun, but even in this playful activity, they are naturally using different math concepts and processes as they explore these materials. For instance, they will discover that some blocks cannot be stacked because they have curved faces, while others, with all flat faces, can be placed in any position, allowing them to build a tower.
Later, students begin comparing the heights of their towers, and at that moment, the teacher comes over and asks, “How do you know which tower is the tallest?” This question immediately adds a new dimension to the play. Children begin comparing their towers not only by height, but also by the number of blocks each one has. They begin counting blocks and discussing whether “tall” refers to physical height or to the number of blocks used. They start asking questions such as “What happens if the blocks are different from each other?” “How can we compare heights?” and so on.
In that moment, the teacher interprets which math skills are in play. Children are practicing cardinality by counting how many blocks are in each tower and perhaps connecting that count to the idea of height. It is not just about comparing two towers, but about understanding that cardinality (the number of items in a set) and measurement (height or length) are related concepts. The teacher could even help connect these ideas by saying something like: “How could we measure the height of the towers with a tape measure to check?” This intervention supports the development of number sense, measurement, and comparison, while keeping play as the context.
At the other end of the continuum, we find playful learning and educational games. In these cases, the starting point is a specific math goal that is integrated into game-format activities that sustain interest and participation. Both approaches (mathematizing play and making math playful) are valid and necessary, and the art of teaching lies in knowing when each one is most appropriate.
Now imagine that later that same week, the teacher introduces a card-matching game (Memory) with numeral cards and quantity cards.
Children flip over two cards each time, trying to find the match between the numeral and the corresponding number of dots. Laughter and excitement fill the classroom when they shout, “I found it!” In this case, the math content is intentional: the game is designed to practice cardinality, numeral recognition, and matching numerals to quantities.
The teacher observes and deepens the experience with questions such as: “How did you know these two cards went together?” or “Which numbers are still missing?” In this way, math is directly integrated into the activity, and the learning goals are clear and structured.
The card game example contrasts with the block-building example in that the former is explicitly designed to develop specific math skills. In this case, the goals (cardinality, numeral recognition, and quantity-to-numeral correspondence) are clear and integrated into the activity.
In contrast, in tower building, the play is also mathematically rich, but it is not initially structured around a specific content or process; instead, they emerge as students explore and interact with the materials and with each other. The teacher interprets and, in some cases, expands mathematical thinking as it emerges.
In this context, the teacher’s role is to guide attention toward different connections and make the mathematics visible.
What is clear is that the teacher’s role is neither passive nor directive, but dynamic. Sometimes they observe and step back; other times they join the play to ask a question at the right moment; other times they offer new materials or game rules.
The teacher acts as a bridge between play and math. The challenge is not deciding whether to include play in the math classroom, but how to respect its spontaneity while also guiding it toward rich math learning.
Although the two examples show how play and math can coexist, in many classrooms, play and mathematical rigor are often seen as opposites. A key challenge for many teachers is reconciling the two, and the belief persists that only structured activities, such as worksheets, ensure “serious” or “rigorous” math learning.
This idea creates a noticeable tension, especially when teachers feel they must choose between free play and highly structured instruction, losing the middle space where play and rigor can coexist.
In an age 5 group, they do these two activities. In which one do you think they are learning more math?
The problem arises when we do not fully understand the complexity of early math learning. Identifying and drawing out math from playful situations requires a deep understanding of how young children’s mathematical thinking develops.
For example, when a teacher observes children comparing towers during free play, they must recognize an opportunity to work on measurement and cardinality, even if the child’s focus is “building” rather than “counting.” Teachers need to learn to notice these moments and understand how they connect to broader math concepts.
However, building this bridge is not easy. Designing playful experiences that promote specific math learning requires stepping out of your comfort zone.
Supporting free play centered on open exploration is not the same as designing an experience where that intentionally integrates math without losing the joy and curiosity of play. This means giving up total control, which is difficult for many teachers, especially when they are used to the extremes of free play or direct instruction.
This tension between play and rigor is a topic we will explore in more depth in future posts. We will talk about how to create classroom spaces where both can coexist. For now, think about how to begin identifying the math present in children’s play and how to start building this bridge in your teaching practice.
Remember: if play and beauty are truly at the origin of mathematics, then early-years classrooms should reflect that origin every day—not as a concession to fun, but as the most natural and powerful way to learn.
References
California Universal Prekindergarten. (n.d.). Retrieved January, 2026, from https://cauniversalprek.org/
Clements, D. H., & Sarama, J. (2005a). Math play. Parent & Child, 12(4), 36–45.
Clements, D. H., & Sarama, J. (2005b). Math play: How young children approach math. Early Childhood Today, 19(4), 50–57.
Fábrega, J., Edo, M. y Torregrosa, A. (2025). Play and mathematics in early childhood education: Classification and analysis of play typologies. Edma 0-6: Educación Matemática en la Infancia, 14(2), 1-28. DOI: https://doi.org/10.24197/d4jj4c68
Van Oers, B. (1996). Are you sure? The promotion of mathematical thinking in the play activities of young children. European Early Childhood Education Research Journal, 4(1), 71-89.
Vogt, F., Hauser, B., & Stebler, R. (2020). Learning through play – pedagogy and learning outcomes in early childhood mathematics. Journal of Early Childhood Research, 18(2), 167-183. https://doi.org/10.1177/1476718X20912747
