With MOOCs fast becoming teaching trend-du-jour in western universities, it is easy to imagine that all disciplines and all ways of thinking are equally amenable to information technology. This is simply not true, and mathematical thinking in particular requires hand-written drawing and symbolic manipulation. Nobody ever acquired skill in a mathematical discipline without doing exercises and problems him or herself, writing on paper or a board with his or her own hands. The physical manipulation by the hand holding the pen or pencil is necessary to gain facility in the mental manipulation of the mathematical concepts and their relationships.
Keith Devlin recounts his recent experience teaching a MOOC course on mathematics, and the deleterious use by students of the word-processing package latex for doing assignments:
We have, it seems, become so accustomed to working on a keyboard, and generating nicely laid out pages, we are rapidly losing, if indeed we have not already lost, the habit—and love—of scribbling with paper and pencil. Our presentation technologies encourage form over substance. But if (free-form) scribbling goes away, then I think mathematics goes with it. You simply cannot do original mathematics at a keyboard. The cognitive load is too great.
Why is this? A key reason is that current mathematics-producing software is clunky, cumbersome, finicky, and not WYSIWYG (What You See Is What You Get). The most widely used such software is Latex (and its relatives), which is a mark-up and command language; when compiled, these commands generate mathematical symbols. Using Latex does not involve direct manipulation of the symbols, but only their indirect manipulation. One has first to imagine (or indeed, draw by hand!) the desired symbols or mathematical notation for which one then creates using the appropriate generative Latex commands. Only when these commands are compiled can the user see the effects they intended to produce. Facility with pen-and-paper, by contrast, enables direct manipulation of symbols, with (eventually), the pen-in-hand being experienced as an extension of the user’s physical body and mind, and not as something other. Expert musicians, archers, surgeons, jewellers, and craftsmen often have the same experience with their particular instruments, feeling them to be extensions of their own body and not external tools.
Experienced writers too can feel this way about their use of a keyboard, but language processing software is generally WYSIWYG (or close enough not to matter). Mathematics-making software is a long way from allowing the user to feel that they are directly manipulating the symbols in their head, as a pen-in-hand mathematician feels. Without direct manipulation, hand and mind are not doing the same thing at the same time, and thus – a fortiori – keyboard-in-hand is certainly not simultaneously manipulating concept-in-mind, and nor is keyboard-in-hand simultaneously expressing or evoking concept-in-mind.
I am sure that a major source of the problem here is that too many people – and especially most of the chattering classes – mistakenly believe the only form of thinking is verbal manipulation. Even worse, some philosophers believe that one can only think by means of words. Related posts on drawing-as-a-form-of-thinking here, and on music-as-a-form-of-thinking here.