Plato, on the use of play in education

I’m currently working on a paper investigating a specific stage of education of the philosopher kings in the Republic.¹ The question that I’m trying to answer is whether that stage of education (from the ages of 20-30) is a mathematical education, whether it has turned to dialectic, or whether it somehow straddles the two stages.

The relevant passage reads:

And after that, that is to say, from the age of twenty, those who are chosen will also receive more honors than the others. Moreover, the subjects they learned in no particular order as children they must now bring together to form a unified vision of their kinship both with one another and with the nature of what which is. (537bc)

Many, probably most notably Myles Burnyeat,² argue that this ten year stage is a mathematical education. What the students do is spend ten years engaged in very high-powered mathematics. It isn’t until the end of this ten year stage when the students begin to engage in dialectic and philosophy. Let’s call this the mathematics interpretation.

I disagree with this reading of the passage. Instead I think that at this point the youths have learned the mathematics prescribed as the propaideutic education – they get those subjects in childhood – and in this stage they’re being asked to begin to engage in philosophy, by treating the mathematical subjects they learned in a more ‘meta’ way. They’ve learned the mathematics involved in harmonics, geometry, calculation, and astronomy and they’re being asked to think about the truths learned and how those truths relate to one another. They’re being asked to tie the various subjects together to form a coherent whole. And this should not be characterized as mathematics but is, instead, philosophy. It is engaging in dialectic (or something akin to dialectic). Let’s call this the dialectics interpretation.

One passage that is very relevant to this issue is at 536d-537a. There, Socrates says

Calculation, geometry, and all the preliminary education required for dialectic must be offered (προβάλλειν) to future rulers in childhood, and not in the shape of compulsory learning either (οὐκ ὡς ἐπάναγκες μαθεῖν)…don’t use force (βίᾳ) to train the children in these subjects; use play (παίζοντας) instead. That way you’ll also see better what each of them is naturally fitted for.

Proponents of the mathematics interpretation argue that if the mathematical education is introduced in childhood by play then it simply doesn’t seem likely that an extensive mathematical education will result from this approach. Moreover, we don’t see Socrates say anything to indicate that there will be any other method of teaching mathematics in childhood. If we are to locate just where such an extensive education may occur, then, it seems reasonable to locate it in the 10 year stage, where Socrates makes mention of the propaideutic studies in a much more rigorous way.

And I think that this line of reasoning is quite compelling. If the students engage in a rigorous mathematical education in their childhood, we should expect to see Socrates say so. Instead he says that they are taught the mathematical subjects through play. If I want to defend my dialectics interpretation, then, I need to have something to say about Socrates’ claim that the mathematics will be introduced to the children by play and show how that claim is compatible with my claim that the youths master mathematics before they’re 20.

One of the big problems with this issue in general is that Plato doesn’t say much more than the above quoted passage about the role of play in the Republic. So it would be fruitful to look to places outside of the Republic where he does say more. And I think one of the best resources for this is the Laws³ .

Consider the following passage:

What I assert is that every man who is going to be good at any pursuit must practice that special pursuit from infancy (ἐκ παίδων), by using all the implements of his pursuit both in his play and in his work (παίζοντά τε καὶ σποθδάζοντα). For example, the man who is to make a good builder must play at building toy houses (οἰκοδομημάτων παίζειν χρή), and to make a good farmer he must play at tilling land; and those who are rearing them must provide each child with toy tools modeled on real ones. Besides this, they ought to have elementary instruction in all the necessary subjects (τῶν μαθημάτων ὅσα ἀναγκαῖα προμεμαθηκέναι προμανθάνειν),—the carpenter, for instance, being taught in play the use of rule and measure, the soldier taught riding or some similar accomplishment. So, by means of their games, we should endeavor to turn the tastes and desires of the children in the direction of that object which forms their ultimate goal. First and foremost, education, we say, consists in that right nurture which most strongly draws the soul of the child when at play to a love for that pursuit of which, when he becomes a man, he must possess a perfect mastery. (κεφάλαιον δὴ παιδείας λέγομεν τὴν ὀρθὴν τροφήν, ἣ τοῦ παίζοντος τὴν ψυχὴν εἰς ἔρωτα μάλιστα ἄξει τούτου ὃ δεήσει γενόμενον ἄνδρ᾽ αὐτὸν τέλειον εἶναι τῆς τοῦ πράγματος ἀρετῆς:) Laws 643bd

This characterization seems quite similar to that in the Republic. Just as there, the youths in the Laws are going to be sorted according to that “special pursuit” that they are best suited for. An education in that skill or craft doesn’t begin when he is an adult but, rather, they are educated from childhood on. In childhood the children play at those skills that they will develop and practice in adulthood. Thus a child destined to be a carpenter will build toy houses in his childhood. Why? The Athenian says that doing so leads the child to love those subjects that he will have mastered as an adult.

What we don’t get is any indication of just when and how long the play will last. A five year old may use blocks to build houses, or collect spare stones and boards to build forts. But would a fifteen year old do the same thing, or should we expect him to have moved on to a more advanced practice of his craft? Common sense tells us that as the youths get older their studies and play should become more advanced. But this above passage doesn’t, alas, give us much to work with.

Another passage from the Laws gives us a bit more content about just what this education may involve. This passage – beginning at 819a – has narrowed the focus to the education of rulers. It is these rulers alone who should study the various mathemata closely, because it is only they that have the capacity to do so and because such studies will be important for ruling. To avoid “complete and absolute ignorance” of these mathemata, the Athenian suggests the following:

One ought to declare, then, that the freeborn children should learn as much of these subjects as the innumerable crowd of children in Egypt learn along with their letters. First, as regards counting, lessons have been invented for the merest infants to learn, by way of play and fun (παιδιᾶς τε καὶ ἡδονῆς),—modes of dividing up apples and chaplets, so that the same totals are adjusted to larger and smaller groups, and modes of sorting out boxers and wrestlers, in byes and pairs, taking them alternately or consecutively, in their natural order. Moreover, by way of play (παίζοντες), the teachers mix together bowls made of gold, bronze, silver and the like, and others distribute them, as I said, by groups of a single kind, adapting the rules of elementary arithmetic to play (εἰς παιδιὰν ἐναρμόττοντες τὰς τῶν ἀναγκαίων ἀριθμῶν χρήσεις); and thus they are of service to the pupils for their future tasks of drilling, leading and marching armies, or of household management, and they render them both more helpful in every way to themselves and more alert. The next step of the teachers is to clear away, by lessons in weights and measures, a certain kind of ignorance, both absurd and disgraceful, which is naturally inherent in all men touching lines, surfaces and solids. (Laws 819ad)

From here the Athenian has a brief discussion with Clinias about line, surface, and solid and questions about their commensurability. The Athenian claims that many have very mistaken beliefs about such matters. Moreover, there are worries not simply about whether what things are commensurable with other things (are lines commensurable with other lines? with solids?), but there are also:

Problems concerning the essential nature of the commensurable and the incommensurable (τἀ τῶν μετρητῶν τε καὶ ἀμέτρων πρὸς ἄλληλα ᾗτινι φύσει γέγονεν). For students who are not to be absolutely worthless it is necessary to examine these and to distinguish the two kinds, and, by proposing such problems one to another, to compete in a game that is worthy of them (φιλονικεῖν ἐν ταῖς τούτων ἀξίαισι σχολαῖς),—for this is a much more refined pastime than draughts for old men. [820d]…I assert, then, Clinias, that these subjects must be learnt by the young (τοὺς νέους); for they are, in truth, neither harmful nor hard, and when learnt by way of play (μετὰ δὲ παιδιᾶς ἅμα μανθανόμενα) they will do no damage at all to our State, but will do it good. (Laws 820cd)

In these two passages I think we get much more detail about just what the content of these childhood educations will be like.

In childhood, we discover first that there are teachers of the material. Just because subjects are introduced through play does not mean that there is not a set structure of education. The teachers choose their subjects carefully. At the youngest age, they provide ways to divide up apples and chaplets, they create lessons by which the children can sort boxers and wrestlers into a proper bracket system,⁴ and they mix together various types of metals and let the students sort it out. This seems very similar to games that today’s parents play with their very young children. We have toys that have children match shapes⁵, that teach them basic counting, basic colors, and so forth. In these tasks, the very young children learn the basic skills of mathematics – their shapes, how to count, how to do basic addition and subtraction, recognizing certain things as belonging to the same natural kind.

What is important in the above passages is that it does not stop with the characterization of the very basic education. “The next step,” according to the Athenian, is going to involve much more complicated sorts of reasoning. The teachers will be tasked with clearing up a certain sort of ignorance that is natural to men about commensurability. From basic reasoning about shapes and numbers, then, the students will be asked to think about what things are commensurable with other things. And this reasoning is going to get difficult quite quickly.⁶ Indeed, the students will be asked not only to reflect on whether certain things are commensurable with other things but they will be asked to reason about the “essential nature of the commensurable and the incommensurable”. They must be able to examine them, distinguish them from one another, and will compete with one another by proposing various problems.⁷

These latter studies are a long way from learning shapes and basic arithmetic. And yet the Athenian ends the passage above saying that “these subjects must be learned by the young…and…by way of play.”

What can we get from this discussion of play and education in the Laws? Several things, I think.

First, it looks like one of the primary reasons why play is encouraged is to cement in the young a love for the thing he is best suited to. If he associates his task with fun, then he is more likely to pursue it throughout the course of his life. Play is meant to shape the character and tastes of the young, then.

Second, the Athenian makes reference to teachers of the young and much of what he says is best understood if we think that there is a curriculum to their education. We need not think that just because subjects are introduced in play, that they are not part of a structured and rigorous curriculum.

Third, play can be simple or it can be complicated. Simply because subjects are introduced and taught through play does not mean that the play involves basic mathematics. When they get older, the students compete against each other about the nature of the commensurable and the incommensurable. The Athenian likens this to the draughts that old men play (saying it is far more refined than draughts). And yet this seen as a sort of game. It’s still play. It’s just far more complicated play.⁸

We can apply these lessons, I think, to the account of mathematical education in the Republic. One of the most important reasons why the childhood mathematical education is through play in the Republic is that it should not be compulsory. Children should not be forced to learn math because (1) forced learning never works and (2) it’s important to see what the children are naturally suited for, and this is most easily done by offering the subjects through play.⁹ The subjects, then, are taught through play. Which is to say they’re taught the subjects in a way that is fun and engaging. The students who find it fun and engaging pursue it voluntarily and those who are most successful show themselves to be best suited to the theoretical studies required for knowledge of the Form of the Good.

None of what was said above precludes there being a rigorous curriculum and teachers who guide the students in their studies. Indeed, given how structured and controlled the moral education is as described in books two and three, it would be shocking if the mathematical education is thereby completely unstructured or the students are left on their own to pursue the studies as they wish. It’s more reasonable, I think, to suppose that there are teachers and a curriculum for the mathematical studies.¹⁰

In support of this, consider the account of the ascent in the cave. The philosopher does not make it out of the cave on his own but is, rather, released from his bonds, compelled to turn around, dragged (by someone (τις) ) up the path of the cave.¹¹ There seems to be someone acting a guide for philosopher, helping him turn his soul around and progress out of the cave. It is not something the philosopher could do on his own.¹²

I think, then, that the passage at 536-7 is not problematic to my dialectics interpretation whereby the youths of Kallipolis will have extensive training in mathematics by the time they’re 20. Even though the subjects are introduced through play, we need not think that the play is thereby simple or unstructured. Instead what the reference to play emphasizes is that the education must not be coercive, it must be something that the youths voluntarily turn towards and study, something they find intrinsically compelling and engaging. And those who are most keen and most suited for this study will be in the best sort of environment for such study and will have truly skilled teachers who are best able to guide and structure their education. In such an environment, those with the best natures will be able to go far. They will be able to become masters of mathematics while still young.

And those who have been most successful at these studies will be chosen and, at the age of twenty, they will be asked to take that knowledge and think about it in a fundamentally different way, in a philosophical way, so as to develop accounts best suited for dialectic. But that? That’s a topic best suited for another entry.

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1. Writing about the Republic has apparently become the way that I’m procrastinating on my dissertation. There are worst ways to procrastinate, I suppose.↩
2. In “Plato on Why Mathematics is Good for the Soul”↩
3. Translations from the Laws are taken from Perseus, which uses a translation by R.G. Bury↩
4. this is, I take it, what he means when he says, “sorting out boxers and wrestlers, in byes and pairs”↩
5. put the triangle shaped block through the triangle shaped hole↩
6. “Again, as regards the relation of line and surface to solid, or of surface and line to each other, do not all we Greeks imagine that these are somehow commensurable with one another…but if they cannot be thus measured by any way or means, while, as I said, all we greeks imagine that they can, are we not right in being ashamed for them all…” (Laws 820ab).↩
7. I’m thinking here about debates and dialectic. Though I’m sure that’s reading a bit more into the passage than is warranted.↩
8. Consider the logic puzzles that today’s students are given. This is still in some sense a game; it’s still play. But it is also far more complex sort of play than simple matching games.↩
9. Those students who are eager to keep going, who are genuinely curious and puzzled are to be selected for further training.↩
10. And a reasonable question to ask is why we don’t see much mention of the structure of the curriculum. And, to be honest, I don’t know. Of course we get an extended discussion of just what topics are to be prescribed as preparatory to dialectic.↩
11. I’m going to overlook all of the language of force and pain in these passages; I have absolutely no idea what sense to make of it.↩
12. So how do we get the first philosopher? Well, consider Socrates. We might say that he had divine help to get where he did. And this seems to be the conclusion that Socrates comes to at 492a: “I think that the philosophic nature as we defined it will inevitably grow to possess every virtue if it happens to receive appropriate instruction, but if it is sown, planted, and grown in an inappropriate environment, it will develop in quite the opposite way, unless some god happens to come to its rescue.”↩

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