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We're near a visual turning point. Conventional math notation coevolved with prevailing display technologies, from cold type to early serial computer interacti

We're near a visual turning point. Conventional math notation coevolved with prevailing display technologies, from cold type to early serial computer interactions. It took a generation for people to learn to chat on a phone, and a generation for the shift from typewriters to word processors to change how people compose text. It will take a similar generation drawing on a digital tablet to change how people compose mathematical thought. However, teaching through a pandemic, this evolution has been radically accelerated. Apps like Notability fluidly replace blackboards for teaching on Zoom. They however are primitive drawing editors. One outgrows them like a beginner windsurfing board. Apps pioneered by digital artists such as architects support much more advanced algorithmic drawing. Concepts is my current favorite; an example of math notes in Concepts is here: https://www.math.columbia.edu/~bayer/S21/Combinatorics/Week7/Feb23.pdf https://www.math.columbia.edu/~bayer/S21/Combinatorics/Week7... Still, these are closed editors. Unlike Emacs, they can't be used to modify themselves. One longs here for Lisp-universe live drawings, a break from the misconception that a static final visually appealing canvas is the objective. Rather, how much thought can we externalize and reify as live doodles? What notion of space will replace artboards and layers, for supporting drawings as live computation, completely accessible as if one had a time-traveling functional programming debugger? When software supports this, crosses that threshold like a general purpose programming language where users can create arbitrary modifications and extensions in this visual medium, then we'll see more papers like the present.

p1esk 6y ago
I don't see how the ability to automate drawings of the figures in your Concepts example makes either a student's or teacher's life easier. If anything it just overwhelms a reader with excessive visual information. Typing on a smartphone, using a word processor, or drawing on a digital tablet hasn't produce better communicators, typists or artists. People today can accomplish tasks faster, but not necessarily better. If you and I were having this conversation 100 years ago, I'd be typing these words on a mechanical typewriter, on a piece of paper which would reach you in a couple of weeks. If anything, the quality of my writing would probably have been better in that scenario. Same goes for tablets: there are lots of "learn to draw" apps on my iPad - but do they make it easier to learn how to draw? I'm not sure. The things I need to learn don't change significantly whether I draw on a tablet or on paper. The apps can certainly help me draw something, or even draw something for me, just like a human teacher would. And that's great, but it does not lead to any significant changes in how I learn or how I think. If you want to change how people think, you should focus on that, on the thinking part, not on the tools. When you succeed in that, you end up with something like this: http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ To me, the visualizations there is an ornament that makes the ideas a little more entertaining, a lubricant that makes the material a little easier to follow. However, if you remove all those diagrams and animations you don't really lose much - everything in that essay is described in words well enough to get the idea across. In fact, if you try to read it carefully while deliberately ignoring any visual information, you might actually get more out of it! That is not to say visualization are useless. Sometimes a picture is indeed worth a thousand words. Sometimes it does make learning dramatically easier. This has been true for thousands of years, and the recent technology you mentioned does not really change much in this regard.
- mncharity 6y ago
  > Typing on a smartphone, using a word processor, or drawing on a digital tablet hasn't produce better communicators, typists or artists. People today can accomplish tasks faster, but not necessarily better. If you and I were having this conversation 100 years ago, I'd be typing these words on a mechanical typewriter [...] "If you and I were having this conversation 1000 years ago, I'd be a scribe in a royal court writing these words with a quill. If anything, the quality of my writing would probably have been better in that scenario. [...] but it does not lead to any significant changes in how I learn or how I think." Though I think it fun that someone in 1500's Spain might be rumored to have made a deal with Devil, based on being able to read not only silently, but, gasp, without moving their lips! How to put this. I suggest you are vastly underestimating the importance of environment to accomplishment. The technology version of a wealthy kid's "growing up in poverty wouldn't have made any difference - I'm just naturally good at things". > but do they make it easier to learn how to draw? iPad, drawing apps, and an ecosystem of artist youtube videos, versus... let's say, a snail-mail booklet from scam "Learn to Draw!" advertisements in the back classifieds of 1960's Popular Mechanics? Um, yes. > However, if you remove all those diagrams and animations you don't really lose much - everything in that essay is described in words well enough to get the idea across. One bit of progress in teaching over recent years, has been the slow realization that "get[ting] the idea across" isn't the hard part. But rather pruning misconceptions and firming understanding. Students and their teachers remain steeped in misconceptions. A main lesson of physics education research has been that if you think you're teaching successfully, then you're also not testing successfully. worrydream's work is an attempt at being less badly unsuccessful.
- mncharity 6y ago
  > does not lead to any significant changes Revisiting this the next day, perhaps it might be usefully thought of as like a customer acquisition funnel? One with lots of friction/leaks. Say a bad landing page, broken install instructions, no examples, poor documentation, no community, and unreadable source. Given that, still, some visitors will survive the gauntlet, to become minor and major customers. And customers will spend far more time working the product, than working around these not-critically-significant-for-customers frictions. And no amount of fixing, will make most visitors into minor customers, nor most minor into major. And yet. You'll have more visitors, and more visitors will become minor customers, and more minor customers will become major ones, if these frictions are improved. Perhaps similarly, for tool quality and people's skill?
draugadrotten 6y ago
> It took a generation for people to learn to chat on a phone What does this even mean? If it means that only children born by parents who had a phone can chat using them, it is a false claim, seeing that even the Queen has a phone and uses it to chat. https://www.cheatsheet.com/entertainment/queen-elizabeth-ii-smartphone.html/ https://www.cheatsheet.com/entertainment/queen-elizabeth-ii-...
- hsgdh3487 6y ago
  Contextually, I interpret "generation" as a period of time, something like 20-40 years.
mathgenius 6y ago
> We're near a visual turning point. When Newton wrote his Principia, he had to turn all of his algebraic calculations into geometric manipulations, so that people would actually believe his results. Now we are faced with the reverse: graphical calculi manipulations are having to be rewritten into algebraic notation in order for papers to be published [1]. [1] https://twitter.com/coecke/status/1362800194895642631 https://twitter.com/coecke/status/1362800194895642631
Kaida_Smile 6y ago
Regarding modifiable, fluid, math notation editor - I heard of this post through a friend and would like to bring your attention to the Graspable Math Canvas with embedded Desmos graphing features: https://graspablemath.com/canvas?load=_607afeebfb63a5be https://graspablemath.com/canvas?load=_607afeebfb63a5be (Sorry if my formatting is weird, I've never used this platform before).
taliesinb 6y ago
Notability (and also concepts) has turned into a big part of my process of "live thinking". A lot of what you said has resonated deeply with me. Is there any way I can get in touch to discuss further?

p1esk 6y ago

I don't see how the ability to automate drawings of the figures in your Concepts example makes either a student's or teacher's life easier. If anything it just overwhelms a reader with excessive visual information. Typing on a smartphone, using a word processor, or drawing on a digital tablet hasn't produce better communicators, typists or artists. People today can accomplish tasks faster, but not necessarily better. If you and I were having this conversation 100 years ago, I'd be typing these words on a mechanical typewriter, on a piece of paper which would reach you in a couple of weeks. If anything, the quality of my writing would probably have been better in that scenario. Same goes for tablets: there are lots of "learn to draw" apps on my iPad - but do they make it easier to learn how to draw? I'm not sure. The things I need to learn don't change significantly whether I draw on a tablet or on paper. The apps can certainly help me draw something, or even draw something for me, just like a human teacher would. And that's great, but it does not lead to any significant changes in how I learn or how I think. If you want to change how people think, you should focus on that, on the thinking part, not on the tools. When you succeed in that, you end up with something like this: http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ To me, the visualizations there is an ornament that makes the ideas a little more entertaining, a lubricant that makes the material a little easier to follow. However, if you remove all those diagrams and animations you don't really lose much - everything in that essay is described in words well enough to get the idea across. In fact, if you try to read it carefully while deliberately ignoring any visual information, you might actually get more out of it! That is not to say visualization are useless. Sometimes a picture is indeed worth a thousand words. Sometimes it does make learning dramatically easier. This has been true for thousands of years, and the recent technology you mentioned does not really change much in this regard.

mncharity 6y ago
> Typing on a smartphone, using a word processor, or drawing on a digital tablet hasn't produce better communicators, typists or artists. People today can accomplish tasks faster, but not necessarily better. If you and I were having this conversation 100 years ago, I'd be typing these words on a mechanical typewriter [...] "If you and I were having this conversation 1000 years ago, I'd be a scribe in a royal court writing these words with a quill. If anything, the quality of my writing would probably have been better in that scenario. [...] but it does not lead to any significant changes in how I learn or how I think." Though I think it fun that someone in 1500's Spain might be rumored to have made a deal with Devil, based on being able to read not only silently, but, gasp, without moving their lips! How to put this. I suggest you are vastly underestimating the importance of environment to accomplishment. The technology version of a wealthy kid's "growing up in poverty wouldn't have made any difference - I'm just naturally good at things". > but do they make it easier to learn how to draw? iPad, drawing apps, and an ecosystem of artist youtube videos, versus... let's say, a snail-mail booklet from scam "Learn to Draw!" advertisements in the back classifieds of 1960's Popular Mechanics? Um, yes. > However, if you remove all those diagrams and animations you don't really lose much - everything in that essay is described in words well enough to get the idea across. One bit of progress in teaching over recent years, has been the slow realization that "get[ting] the idea across" isn't the hard part. But rather pruning misconceptions and firming understanding. Students and their teachers remain steeped in misconceptions. A main lesson of physics education research has been that if you think you're teaching successfully, then you're also not testing successfully. worrydream's work is an attempt at being less badly unsuccessful.
mncharity 6y ago
> does not lead to any significant changes Revisiting this the next day, perhaps it might be usefully thought of as like a customer acquisition funnel? One with lots of friction/leaks. Say a bad landing page, broken install instructions, no examples, poor documentation, no community, and unreadable source. Given that, still, some visitors will survive the gauntlet, to become minor and major customers. And customers will spend far more time working the product, than working around these not-critically-significant-for-customers frictions. And no amount of fixing, will make most visitors into minor customers, nor most minor into major. And yet. You'll have more visitors, and more visitors will become minor customers, and more minor customers will become major ones, if these frictions are improved. Perhaps similarly, for tool quality and people's skill?

draugadrotten 6y ago

> It took a generation for people to learn to chat on a phone What does this even mean? If it means that only children born by parents who had a phone can chat using them, it is a false claim, seeing that even the Queen has a phone and uses it to chat. https://www.cheatsheet.com/entertainment/queen-elizabeth-ii-smartphone.html/ https://www.cheatsheet.com/entertainment/queen-elizabeth-ii-...

hsgdh3487 6y ago
Contextually, I interpret "generation" as a period of time, something like 20-40 years.

mathgenius 6y ago

> We're near a visual turning point. When Newton wrote his Principia, he had to turn all of his algebraic calculations into geometric manipulations, so that people would actually believe his results. Now we are faced with the reverse: graphical calculi manipulations are having to be rewritten into algebraic notation in order for papers to be published [1]. [1] https://twitter.com/coecke/status/1362800194895642631 https://twitter.com/coecke/status/1362800194895642631

Kaida_Smile 6y ago

Regarding modifiable, fluid, math notation editor - I heard of this post through a friend and would like to bring your attention to the Graspable Math Canvas with embedded Desmos graphing features: https://graspablemath.com/canvas?load=_607afeebfb63a5be https://graspablemath.com/canvas?load=_607afeebfb63a5be (Sorry if my formatting is weird, I've never used this platform before).

taliesinb 6y ago

Notability (and also concepts) has turned into a big part of my process of "live thinking". A lot of what you said has resonated deeply with me. Is there any way I can get in touch to discuss further?