Play, Mathematics, and False Dichotomies

NIEER is hosting a blog forum on play-based learning in early childhood education, including posts from national experts in the field. Learn more about the forum here. Some worry that the push for quality education even partially driven by a desire to improve achievement may deprive children of important childhood experiences. Others worry that unstructured play without teacher engagement does little to develop children’s minds, particularly for children at high risk of academic failure. 

By Douglas H. Clements & Julie Sarama, University of Denver

Let’s stop the cycle of “abuse”—or at least confusion—that stems from false dichotomies in early education. “Play vs. academics” is arguably the main one. Of course children should play. But this does not mean they should not learn, and even play, with mathematics. Consider the following.

  1. In their free play, children naturally engage in mathematics.[i] Observations of preschoolers show that when they play, they engage in mathematical thinking at least once in almost half of each minute of play. Almost 9 out of 10 of children engage in at one or more math activities during play episodes.[ii]
  2. This mathematical play reveals intuitive knowledge of many concepts that most people think young children can’t understand, from arithmetic, to proportions, to parallelism and right angles. Unfortunately, the same children may not “understand” these concepts when they arrive in middle school. If they are not helped to mathematize (reflect on, give language to) their early “theorems in action,”[iii] the ideas do not become theorems in thought.
  3. Many adults, including early educators, believe that sequenced, intentional instruction will harm children’s play. These concerns are misplaced. Math and literacy instruction increase the quality of young children’s play. Children in classrooms with a stronger emphasis on literacy or math are more likely to engage in a higher quality of social-dramatic play.[iv] The new ideas energize high-level play activity. Thus, high-quality instruction in math and high-quality free play do not have to “compete” for time in the classroom. Doing both makes each richer. Unfortunately, many adults believe that “open-ended free play” is good and “lessons” in math are not.[v]  They don’t believe that preschoolers need specific math teaching.[vi] They don’t realize that they are depriving children not only of the joy and fascination of mathematics, but of higher-quality free play as well.
  4. If children play with mathematical objects before they’re asked to solve problems with them, they are more successful and more creative.[vii]
  5. These and other examples bring us to a fascinating type of play: mathematical play. Here we do not mean play that involves mathematics. We mean playing with mathematics itself.
    building blocks

What does this look like in action? Just after her fourth birthday, Abby was playing with three of the five identical toy train engines her father had brought home. Passing by, her mother asked, “Where are the other trains?” Although her mother was out of sight, Abby spoke to herself. “Oh, I have five. Ummm…[pointing to each engine]you are one, two three. I’m missing ‘four’ and ‘five’—two are missing! [She played with the trains for another minute.] No, I changed my mind…I have ‘one,’ ‘three,’ and ‘five.’ I’m missing ‘two’ and ‘four.’ I gotta find them two.”

When Abby first figured out how many she was missing, she was using math in her play. But when she decided that she would renumber the three engines she had with her ‘one,’ ‘three,’ and ‘five’ and the missing engines ‘two’ and ‘four’ she was playing with the notion that the assignment of numbers to a collection of objects is arbitrary. She was counting not just objects, but also words. She counted the words “four, five” to see there were two missing, and then figured that counting the renumbered counting words “two” and “four” also yielded the result of “two.” She was playing with the idea that counting words themselves could be counted.

What does all this mean regarding children’s development and learning? Free play experiences form the intuitive, implicit conceptual foundation for later mathematics. Later, children represent and elaborate these ideas—creating models of an everyday activity with mathematical objects, such as numbers and shapes; mathematical actions, such as counting or transforming shapes; and their structural relationships. This is the process of mathematization.[viii]. Recognizing the difference between foundational and mathematized experiences is necessary to avoid confusion about the type of activity in which children are engaged.[ix] They need both.

Unfortunately, adults often do not provide the mathematics experiences.[x] Our own work with teachers on curricula has been stonewalled many times by an administrator saying: “Our philosophy is that we are play-based.” Not only does this statement ignore all the evidence on play and learning,[xi] it is based on a pernicious false dichotomy that harms the children in their care. 

In summary, young children engage in significant mathematical thinking and reasoning in their play, especially if

(a) they have knowledge about the materials they are using (e.g., building blocks or other manipulatives or toys),

(b) the task is understandable and motivating, and

(c) the context is familiar and comfortable.[xii]

Math can be integrated with children’s ongoing play and activities…but this usually 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 mathematical objects and mathematical ideas is pedagogically powerful.[xiii]


[i] van Oers, B. (1996). Are you sure? Stimulating mathematical thinking during young children’s play. European Early Childhood Education Research Journal, 4, 71-87.

[ii] Seo, K.-H., & Ginsburg, H. P. (2004). What is developmentally appropriate in early childhood mathematics education? In D. H. Clements, J. Sarama & A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 91-104). Mahwah, NJ: Erlbaum.

[iii] Vergnaud, G. (1978). The acquisition of arithmetical concepts. In E. Cohors-Fresenborg & I. Wachsmuth (Eds.), Proceedings of the 2nd Conference of the International Group for the Psychology of Mathematics Education (pp. 344-355). Osnabruck, Germany.

[iv] Aydogan, C., Plummer, C., Kang, S. J., Bilbrey, C., Farran, D. C., & Lipsey, M. W. (2005). An investigation of prekindergarten curricula: Influences on classroom characteristics and child engagement. Paper presented at the NAEYC, Washington, DC.

[v] Sarama, J. (2002). Listening to teachers: Planning for professional development. Teaching Children Mathematics, 9, 36-39.

Sarama, J., & DiBiase, A.-M. (2004). The professional development challenge in preschool mathematics. In D. H. Clements, J. Sarama & A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 415-446). Mahwah, NJ: Erlbaum.

[vi] Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.

[vii] Bruner, J. (1985). On teaching thinking:  An afterthought. In S. F. Chipman, J. W. Segal & R. Glaser (Eds.), Thinking and learning skills.  Volume 2:  Research and open questions (Vol. 2, pp. 597-608). Hillsdale, NJ: Erlbaum.

Holton, D., Ahmed, A., Williams, H., & Hill, C. (2001). On the importance of mathematical play. International Journal of Mathematical Education in Science and Technology, 32, 401-415.

[viii] Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York, NY: Routledge.

[ix] Kronholz, J. (2000, May 16). See Johnny jump! Hey, isn’t it math he’s really doing?, The Wall Street Journal, p. A1; A12.

[x] Tudge, J. R. H., & Doucet, F. (2004). Early mathematical experiences: Observing young Black and White children’s everyday activities. Early Childhood Research Quarterly, 19, 21-39.

[xi] Sarama, J., & Clements, D. H. (2009). Building blocks and cognitive building blocks: Playing to know the world mathematically. American Journal of Play, 1, 313-337.

[xii] Alexander, P. A., White, C. S., & Daugherty, M. (1997). Analogical reasoning and early mathematics learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 117-147). Mahwah, NJ: Erlbaum.

[xiii] 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.

Clements, D. H., & Sarama, J. (in press). Learning and teaching early math: The learning trajectories approach (2nd ed.). New York, NY: Routledge.


  1. Douglas & Julie, Can you share your thoughts, and research, on the use of worksheets for instructional purposes in an early childhood classroom?

  2. Christyn Dundorf on

    I love the idea of powerful play – that is, play that is intentionally and sensitively monitored and supported by skilled adults in early childhood environments. When intentionally supporting children’s play, adults can ask reflective questions, provoke new thinking with additional materials, or collaboratively create new understandings.

    I fear that this intentional teaching requires a very high skill level and great program support, elements frequently lacking in early care and education program. I am concerned that when “results” are needed and resources/skills are scarce, teachers and programs often default to worksheets and direct instruction, leaving powerful play behind.

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  4. Marsha Ashley on

    Great article. Same problems early literacy faces. Enriched play is the way to go. Thanks.
    Marsha Ashley

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  6. I am a Developmental therapist/special educator working with toddlers and preschoolers. You are only partially correct. Yes, children are naturally and intuitively incorporating “math concepts” from their everyday play/interactions. However, to prematurely divorce or abstract the conceptual content (i.e. as systematic “intentional instruction”) from the concrete content (i.e., the context in which the affect reciprocal social-emotional interactions and pretend play are occurring) can indeed be detrimental and actually suppress the ability for toddlers and older to use their social emotional empathic thinking.

    Rather than focus on the fact that Abby (your example above) is engaged in the mathematical concepts (bringing to her attention by pedagogically focused on number in and of itself as a means and method to begin to direct leaning as concepts,or “tools of learning” in and of themselves, addition, subtraction, algebra), unwittingly removes her from the MUCH MORE critically Developmental-social emotional empathic focus that she talk about her trains with respect to symbolic conceptual thinking and content.

    For example:

    Where her trains are they going? What are they doing? Are they sure they want to go in ‘that direction?’ Why? What if the train crashes, what will happen? Now, contrary to popular belief, this is NOT focused on in the typical preschool or in the early intervention settings for children. Granted, all this involves time, distance, calculation but it is MUCH MORE IMPORTANT with respect to simple to complex symbolic and higher critical and social empathic thinking skills (nuanced simple to complex executive functioning) that she is able to think through and through and project symbolic based scenarios rather than have the rarefied insightful “good-doer teacher” who shall point out and intentionally teach the “math concepts” she is indulging in.

    Somewhat analogous, it is like a child “knowing” his/her shapes colors, letters and numbers, Entirely meaningless if s/he is not, more importantly, engaged (and is moreover, proactively encouraged to be engaged) in the pragmatics of higher critical reflective and social-emotional empathic based thinking, that is thinking with and through scenario based thinking. .

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