This does happen sometimes, by accident. I wrote Quirk  as an educational tool for learning about quantum circuits. But then I started finding it useful as a tool for optimizing circuits to use fewer gates, just by dragging things around and seeing what happens (there's still enough low-hanging fruit for that to be viable). The paper "Halving the cost of quantum addition"  only exists because of a "that's strange..." moment while I was messing around with a decomposition of the Toffoli gate in Quirk.
(I bet Michael Nielsen, the author of the linked article, gets a particular kick out of this example. He's also the co-author of the de-facto standard textbook for quantum computing.)
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:
If you want to work on or talk about this open source project I'd love to hear from you! (Email is in bio)
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.
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.
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.
Anyways, I don’t want to discourage you, your work is definitely interesting. It’s just something you might want to keep in mind.
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.
1. tr. Convertir algo abstracto en una cosa concreta.
2. tr. Reducir a la condición de cosa a una persona.
The semantic distinction between abstract and concrete idea is still virtual, though. The ripe apple you think about is still a virtual image. Or rather original, because even scientifically in e.g. biology, specific specimen are used as arch example to define clades and such. We care very much about relational structure, after all. An abstraction thus is something loosely or not at all connected to the currently favoured theory of the overall structure of life.
It's nice to see more stuff coming out of this.
Pretty amazing stuff, even today.
I really believe the matter is just human. We just need to take time to speak to each others peacefully. Instead we delegate to hurried teachers in potentially shamefull classes, or technology to do exercises..
"I'm less and less into speech.
I really believe the matter is just human. We just need to take time to grunt and mime to each other peacefully. Instead we delegate to sitting lazily and screeching loud sounds and noises"
Just as the spoken and written word opened up the world to us in new ways in the past, I feel that these new ideas may also lead us to new understandings.
I don't dismiss tech, it's just that it never address the issues to me. So it's not the solution, it's a tail chase.
We're humans, we like human relationships, sharing beauty/knowledge is part of that, that's why hurried teachers and classrooms are subpar, they turn it into a domination scheme (harsh words but I stand by them). On the other hand someone who shares the beauty he feels in a subject, his passion, will drive a pupil in the deepest of ways.
It will also remove some imaginary need for "school" to understand the world.
Lastly it will make people sense that sharing and exchanging is key. I'd like to see more of that.
When it comes to more modern technologies, like computers, I think we need to remember that, in historical terms, we're only just starting to make use of them. I would bet that in 100 years from now, there'll be much more beneficial applications of them in education than now.