Socrates and the Good Square Lesson
One of my favourite scenes from a Platonic dialogue is the scene where Socrates guides Meno’s slave boy through a geometric problem (82b-85b). Socrates (presumably drawing a diagram on the ground) asks the boy if he the shape he has drawn is a square. He then questions the boy about the properties of the square (equal sides, equal dividing lines along the diagonal) and then asks about the area. The boy seems to be following Socrates’ logic and to understand his questions. One could argue that the questions are leading, however (82c). The boy does give the correct answer about the area: if the square is two ft by two ft, the area is four (82d). So far, so good. Socrates then asks the boy about the area of a square twice the size. The boy initially answers, ‘eight’. At this point Socrates pauses to ask Meno whether he is teaching the boy anything or whether he is recollecting, i.e., drawing on knowledge inside himself (82e). Socrates probes the boy on his answer, and he confidently responds that eight is correct and the length of each side must be doubled, although Socrates has reminded the boy that both sides of the square must be equal (83a). Socrates draws the shape and the boy realises that this in fact produces a square four times the size. Again, one could argue that Socrates is leading the boy: ‘Four times that space, is it not?’ However, when Socrates asks the boy whether four times is equal to double, the boy begins to realise his error. So, if one doubles the lengths of the sides, one does not get a shape with an area that is double the size. At 83c, Socrates gives a brief recap of where they have got to, going over the area of the original square. Socrates asks the boy whether the line-length for an eight ft area square is longer than the original, but smaller than the larger one. The boy replies that he thinks so. Having agreed this, he then suggests a length of three ft for each side. The boy then answers correctly that the new shape will have an area of nine. Still too big for the eight that they are looking for. So, where do we go from here, Socrates asks? The boy hits a bit of a wall at this point and seems unsure. Socrates turns to Meno again, pointing out how the boy had been recollecting rather than simply having the problem spelled out for him. Certainly, logic has guided him through a process of elimination (84a). Socrates says the boy is now better off, since he has discovered his error and why he was wrong (84b). At 84d, Socrates returns to the problem. He adds another three squares of four ft in area until he achieves something looking like this:
Four equal squares, all four ft squared in area, all with lines of two ft. Socrates then draws diagonals in all of the mini-squares.
The boy does not grasp what Socrates is getting initially. Socrates points out that each diagonal has cut each square in half. Gradually, the boy realises that the diagonals create a square of eight ft in area. Simple application of Pythagoras’ theorem proves this correct. The diagonals each work out as 2.83, the square-root of eight.
Now, although Socrates is prompting the boy a little more than he claims, he is not actually giving him the answer. Socrates does successfully tease encourage the boy into thinking logically through the problem and reflecting on the steps he takes and where he makes mistakes. Yesterday I began my EducationInfluence blog on Tumblr called, ‘Ancient Musings, Modern Lessons’.(1) I critiqued what I consider to be the artificial and unhelpful division ‘teaching’ versus ‘learning’, on the grounds that both are too narrowly defined, thereby creating a dichotomy which does not exist. Socrates seems initially in this passage to advocate such a division saying that he is not teaching the boy. However, what he is saying is that he is not telling the boy what to think or what the answer is, the kind of teaching that Meno is familiar with. The debate about what virtue is in the dialogue ends up being a debate about what knowledge is and whether it can be taught. It is hard to prise Meno away from the notion that if something is knowledge, it must be teachable. A few sections before the maths problem, Socrates introduces the theory of recollection, quoting Pindar on the rebirth of souls at the behest of Persephone (81b). The idea is that the soul has loved many times, seen things both in the upper and lower worlds and that with each new life it is recalling the knowledge it has already observed, but forgotten. How sincerely Socrates is advocating a doctrine of reincarnation, I am not sure. I have often wondered whether he is simply trying to spur Meno on to greater independent thinking, or whether he is trying to convey to Meno the notion that some knowledge is innate, such as our sense of right and wrong, and is not, therefore, teachable in the factual transmission sense.
Whatever Socrates means, there is certainly a lesson for us here. A major part of learning comes from our own efforts and how these aid the development of our independence of thought and application. A major part of teaching, therefore, is to facilitate and aid this process. There is no conflict in my opinion between a teacher transmitting key topics of knowledge and balancing this with enabling pupil effort and development.
Once again, it is Socrates we must thank for his approach and his application.