I’ve been reading to make sense of how play relates to mathematics. The reading trail led, as it so often does in education, all the way back to John Dewey — this time to his chapter ‘Play and Work in the Curriculum’ in the 1916 book Democracy and Education. Reading a text more than a century old can be heavy going: unfamiliar sentence constructions, words that have fallen out of use, and words whose meanings have shifted over time. I won’t claim reading this chapter was easy, but it was certainly worthwhile.
Dewey’s central claim is that play and work are “by no means so antithetical to one another as is often assumed”. That assumption is as familiar now as it was in 1916: typically work is regarded as serious business, and play as its reward. Dewey saw the pair differently: for him, what they share is purpose. Both involve “ends consciously entertained” with the means chosen and adjusted to achieve them. What differs is the kind of end each holds, and so the way the means are used. An end can be more of the doing itself, or a result that stands beyond the doing.
The difference in the ends is the first of two distinctions Dewey draws. He terms it psychological: Does the emphasis fall on the doing, or on the result? This is the line that separates play from work. The second distinction, which he calls economic, is, despite its name, a question of value. It does not separate play from work at all. Rather, it runs through each of them and asks: Is it worth doing for its own sake, or for what it leads to?
Dewey is explicit that the two questions must be kept apart, saying “it is important not to confuse the psychological distinction between play and work with the economic distinction”. Thinking of these as a pair of axes, one for each question, helped me make sense of what Dewey was saying. I’ve drawn them below as a 2×2 grid, and used Dewey’s writing to place an occupant in each cell.
| Valued for its own sake | Valued only for what it leads to | |
|---|---|---|
| Emphasis on the doing | Play | Idle excitement |
| Emphasis on the result | Work | Drudgery |
Play occupies the top-left cell. The emphasis falls on the doing: the aim of play is more of itself, not a finished object. Dewey illustrates this with a child “playing boat” who can “change the material that serves as a boat almost at will”, because the point is to keep the play going. The child values the activity of playing.
Work occupies the bottom-left cell. Fix the end, and play becomes work: the emphasis falls on the result, and an idea of an outcome is always held in mind. The same child, now making a toy boat “must hold on to a single end and direct a considerable number of acts by that one idea”, because the point is to finish the boat. The child values the outcome of a boat.
Play and work both sit in the left column because each is valued for its own sake. The player seeks nothing beyond the act of playing. A worker seeks an outcome. Work, for Dewey, is “equally free and intrinsically motivated”. This usage may jar with our modern-day perceptions of work, but that’s because Dewey cut work loose from its usual synonym, drudgery.
Drudgery occupies the bottom-right cell and is a hollowed-out form of work: keep the fixed end, strip the intrinsic value, and this is what remains. Dewey notes that it “is not intrinsically satisfying; it is a mere means for avoiding some penalty or for gaining some reward at its conclusion”. The joylessness is not caused by having a fixed end; it is caused by the value sitting outside the activity. Drudgery only looks like work, but it is means to some other kind of end, perhaps a paycheck.
Finally, idle excitement occupies the top-right cell: strip play the same way, and this is what remains. The emphasis falls on the doing: it is “recourse to idle amusement, to anything which passes time with immediate agreeableness”, stimulation sought in the moment with no result carried beyond it. The doing is not valued for itself: what is wanted is “stimulation by any kind of means”; the activity is interchangeable, which is why “anything” will serve. What drudgery is to work, idle excitement is to play. Dewey thought the stripping had a source: “false economic conditions which tend to make play into idle excitement for the well to do, and work into uncongenial labor for the poor”. Put another way, the right-hand column is not a different kind of activity; it is what happens to the left-hand column under the wrong conditions. That’s a question worth returning to.
Having drawn the grid, let’s think about how it might connect to mathematics. Consider adding the odd numbers and observing the running totals: 1 = 1^2, 1 + 3 = 4 = 2^2, 1 + 3 + 5 = 9 = 3^2, 1 + 3 + 5 + 7 = 16 = 4^2. In other words, the running totals are the perfect squares.
Tapped into a calculator, the odd numbers may simply be a way to pass the time, the squares sliding by unnoticed: idle excitement. Or a worksheet might ask students to find the sum of the first forty odd numbers. The result might be observed, but only because someone else required us to do it: a form of drudgery.
To someone who wants to know why the squares keep appearing, there is a desired outcome: wanting to know why the sum of the first n odd numbers is n squared. The proof might come algebraically, by pairing the numbers from the ends and counting the pairs, or geometrically, by drawing the totals as squares of dots and seeing each square grow into the next by an L-shaped border that is always the next odd number. The why is its own reward.
Or imagine a child on a mat, surrounded by coloured blocks. She starts arranging them in L-shaped borders, noticing that they form squares. She’s not trying to prove anything, just absorbed in continuing the pattern in front of her. She is playing.
The same mathematics can occupy all four cells. I’m reminded of John Mason’s observation that “there are no rich tasks, only tasks used richly”. It isn’t necessarily the teacher who chooses the richness; it’s the emphasis of the activity and the value placed on it by the person doing the task. Dewey made a similar observation a century earlier: “It is not enough just to introduce plays and games… Everything depends upon the way in which they are employed”.
The grid is just a snapshot, though. Dewey’s own account is more dynamic: “When fairly remote results of a definite character are foreseen and enlist persistent effort for their accomplishment, play passes into work”. That is, people move between cells; another idea for us to pick up later.
Dewey’s observations on play resonate with more modern writers. Stuart Brown, in his book Play, characterises players as people who “want to keep going and find ways to make it keep going”. For David Butler, play is “trying out ideas inspired by your curiosity, and being absorbed in the experience”. And Francis Su, listing the qualities of play, includes having “no long-term stake in the outcome”. Like Dewey, they are describing how a person approaches an activity, not the activity itself.
I opened by calling this chapter worthwhile, and it was. (There was a lot more I took out of it too.) Dewey gave me new questions about how people move between cells in the grid, and about the conditions that strip value from work and play. But he also sharpened my thinking: Where are we putting the emphasis in mathematics? And what value do we place on it? Both in school and outside of it. The school half might be the post to explore next.
References
Brown, S. (2009) Play: How It Shapes the Brain, Opens the Imagination, and Invigorates the Soul. New York: Avery.
Butler, D. (2017) ‘Playful and joyful maths’, in Seah, R., Horne, M., Ocean, J. and Orellana, C. (eds) 2017 Mathematical Association of Victoria Annual Conference Proceedings. Melbourne: The Mathematical Association of Victoria, pp. 156-157.
Dewey, J. (1916) Democracy and Education: An Introduction to the Philosophy of Education. New York: Macmillan. Chapter 15, ‘Play and Work in the Curriculum’.
Mason, J. (2020) ‘Effective questioning and responding in the mathematics classroom’, in Ineson, G. and Povey, H. (eds) Debates in Mathematics Education, 2nd edn. Abingdon: Routledge, pp. 131–142.
Su, F. (2020) Mathematics for Human Flourishing. New Haven: Yale University Press.
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