This is excellent! Not only is it a great concept and a well-written article ("reify" is my new favourite word), but it completely succeeds in its goal of teaching a non-trivial concept. I'm very excited to see where this goes.
I've followed the work of some others in this space (like Bret Victor) for quite a while, and inspired by what I've seen and read (and also by frustration in college courses and elsewhere) I've started prototyping a new way to do derivations. It's effectively only a mock-up in its current stage but as it fits in the "new media for thinking" category I thought I'd share it anyway:
I guess you are working on purely symbolic thinking? I think the approach works best when it’s just not equations being manipulated, but visualized instantiations of specific (and concrete) examples.
Yes, but I feel there is room for more than one approach. For something as specific as derivations--which tend to have a linear, step-by-step approach I think this makes sense. Similar to how different programming languages excel at different tasks I imagine different thinking media will also excel at different cognitive tasks.
There is also the terseness of mathematical notation to compete with--I can't think of a better way to replace the sum of something over the range 1 to 1000 than the current sigma notation, or even a for loop or some other construct like "sum(f(x), 1...1000)". I don't know if there is a good way to make something like a large summation visual but I am definitely open to any ideas. I wish that every mathematical concept or object could be visualised effectively but again I doubt this is the case with the hope of being proven wrong.
If your interest is future pedagogic utility in academia (funny etymology for pedagogue: 'leader of children') that's one thing. If your interest is in what can be called today the cognitive dynamics of creativity as experienced by highly creative individuals then who better than Albert Einstein to report on his experiences. In mathematician Jacques Hadamard's 'The Psychology of Invention in the Mathematical Field' (Dover (1954),ISBN 0486201074, pp. 142-3) Einstein reports his creative cognitive processes in response to a questionnaire titled 'An inquiry into the working methods of mathematicians' as follows:
A) The words or language as they are written or spoken do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be "voluntarily" reproduced and combined...
B) The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associated play is sufficiently established and can be reproduced at will.
Or what might be called today visual/tactile qualial imagery manipulated in cognitive workspace (Baars). So yeah it would seem, 'a visual medium (can) be used to do serious mathematical exploration', at least as a lead-in to transcription from conceptual imagery to reporting.
This is an incredibly interesting comment! I think you've hit the nail on the head that this is better suited for pedagogy than for exploration. The extracts from the book are especially insightful and really something I had not considered before so thank you for that--I may just have to read the whole book (and it's by the same Hadamard of the Hadamard gate! It seems that anyone even slightly related to quantum computing has an interest in exploring and enhancing creative thought--that's 3 just in this thread).
I hope I didn't come across as suggesting that a visual medium couldn't be used to do serious mathematical exploration--I definitely think it can (and Strilanc's comment proves that it can). To clarify, I was more suggesting that it could be the case that not every mathematical concept is conducive to being explained or taught effectively using a visual medium, or even more loosely that in some cases a symbolic approach could be more effective than a visual one. However, I agree that for most concepts visual trumps symbolic and I currently have no concrete examples of the opposite.
I’ve heard even the most accomplished mathematicians build concrete visualizations in their head, they work with non-symbolic models when solving problems. I think that these mental models is what we should try to capture in the computer. So I guess it is a matter of what mental model is being used to make the derivation, and this would probably involve why the derivation is being made.
Anyways, I don’t want to discourage you, your work is definitely interesting. It’s just something you might want to keep in mind.
Yeah that definitely sounds right--at some point the brain will need to do the understanding as the paper won't. So I also agree that capturing more facets and the richness of thoughts is a much better goal than simply enhancing symbol-shunting. I suppose my idea could best fit in as either a transitionary model, or more likely as more of a replacement for the current method of teaching and learning derivations. In a sense just decreasing how bad the current worst case of teaching is rather than trying to increase the best case. Focusing on didactic rather than exploratory although I realise that both are possible with more expressive systems.
No, don't worry you have not done any discouraging. I really appreciate all your thoughts on the matter and will definitely think more about what (or if) this should be. I currently don't have much time to work on it but definitely have time for some good mulling.
I've followed the work of some others in this space (like Bret Victor) for quite a while, and inspired by what I've seen and read (and also by frustration in college courses and elsewhere) I've started prototyping a new way to do derivations. It's effectively only a mock-up in its current stage but as it fits in the "new media for thinking" category I thought I'd share it anyway:
https://oisinmoran.com/projects/derivation.html
If you want to work on or talk about this open source project I'd love to hear from you! (Email is in bio)