The study of Euclid’s Elements serves as an excellent example of the contemplative learning process. Proposition 5, an early proposition in the text, marks a turning point for most students, where they must not only identify a chain of equalities (something akin to a hypothetical syllogism), but do so in transposition.
Whereas students needed only identify equality by imposing or fitting one triangle upon another in Proposition 4 (superposing ∆BCA upon ∆EDF, cf. Fig.1); Proposition 5 demands an attention to multiple equalities and permutations, some which cannot be imposed without mental transposition (e.g., ∆FBC and ∆GCB, cf. Fig.2). Students must hold a complex array of sameness and otherness at once in their minds and complete several chains of reasoning which extends beyond the mere mechanics of measurement.
While the details of this proposition are not crucial here, what I would like to discuss is nature of the learning experience itself, as typified in Euclid. Students read propositions, sometimes multiple times, until they have a general sense of what is being said. They may create their own drawings of a proof or attempt to construct an image in their imagination. They do this in order that they might see what is going on. The goal is not simply to memorize a formula or conclusion, but to grasp how the demonstration works.
During such studies, a student may suddenly glimpse the middle term or pivot of a proposition, and experience that much talked about ‘aha!’ moment. But remarkably, this moment does not mark the completion or even the culmination of such work. The ‘aha! often occurs relatively early in the learning process. Euclid has merely ‘set the hook,’ but has not yet reeled the fish in, so to speak. An initial moment of insight gives students a sense that there truly is something out there, something beyond memorization which is to be grasped and understood. Students who have experienced such insight have yet the hard work of clarification, in which they must struggle to forge a clear connection between the text and the ‘aha!’.* The big difference is that such a student now labors under the power of a kind of faith and hope.
Such a student now must remain faithful to this insight, returning again and again to one’s confusions and difficulties until the proposition’s pattern is ‘fully formed’ within him. If one stop before this, if one mistakes the ‘aha!’ moment for the end and fulfillment of the learning process, one will be left with little but the sense of once having seen something. Ask such a student about what they have seen or how the proposition works, and he will have little to say. Tragically, he may even come to think that nothing true can be said about such things, or worse that such insights are always illusory. But let him toil over the problem until he grasps the whole, until he can, in his own words, comfortably walk through the demonstration, and he will now have an articulation (a logos) which joined together with insight makes manifest the meaning of the proof.* He will then know that initial insights can indeed be clarified and given utterance. Having worked in darkness, a darkness guided by faith, he will see the fruit of such labor, and long for greater fruit. Perhaps he will even long to encourage others on the educative journey.
*The idea of joining the ‘aha!’ to an account (a logos), particularly as depicted in Euclid and the Gerard Manley Hopkins poem, To R.B., which will be discussed in a later installment of this series, was presented to me by Joseph Keating, a good friend and fellow graduate student at St. John’s College in Annapolis Maryland.