If I had to guess why "math reform" is misinterpreted as "math education reform", I would speculate that school is the only contact that most people have had with math. Like school-physics or school-chemistry, math is seen as a subject that is taught, not a tool that is used. People don't actually use math-beyond-arithmetic in their lives, just like they don't use the inverse-square law or the periodic table.

Teach the current mathematical notation and methods any way you want -- they will still be unusable. They are unusable in the same way that any bad user interface is unusable -- they don't show users what they need to see, they don't match how users want to think, they don't show users what actions they can take.

They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly. And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface.

Anything that remains abstract (in the sense of not concrete) is hard to think about... I think that mathematicians are those who succeed in figuring out how to think concretely about things that are abstract, so that they aren't abstract anymore. And I believe that mathematical thinking encompasses the skill of learning to think of an abstract thing concretely, often using multiple representations – this is part of how to think about more things as "things". So rather than avoiding abstraction, I think it's important to absorb it, and concretize the abstract... One way to concretize something abstract might be to show an instance of it alongside something that is already concrete.

The mathematical modeling tools we employ at once extend and limit our ability to conceive the world. Limitations of mathematics are evident in the fact that the analytic geometry that provides the foundation for classical mechanics is insufficient for General Relativity. This should alert one to the possibility of other conceptual limits in the mathematics used by physicists.

be inspired by the untapped potential of human capabilities. Don't just extrapolate yesterday's technology and then cram people into it.

You can look at an interface and see it as a clearly signposted user journey you should follow. Or you can see it as a collection of functions and affordances to repurpose. As raw material, rather than a guided path.

Let’s say you present 3 directions to a client: directions A, B, and C. These are our initial 3 branches. You have a client review, direction C is the winner, and so you iterate again. 3 more branches: C1, C2, and C3. Another review, another winner, another round of iterations: C2.1, C2.2, C2.3. Branch out, choose one, zoom in, branch out, repeat.

Sometimes, the design process requires us to zoom out. Let’s say you present those 3 creative directions, A, B, and C, but nothing lands. Back to the drawing board. You might keep pushing forward with branches D, E, F. Nothing lands. You’re forced to zoom out and realize that you’re not even on the right parent branch.

The industrial designers talked about contours that felt gratifying in the hand and actions that provided a fidget-like comfort such as flipping the lid of a Zippo lighter or the satisfying click of a pen.

In video games, the button you press to make a character jump is often a simple binary input (pressed or not), and yet the output combines a very finely-tuned choreography of interactions, animations, sounds, particles, and camera shake to create a rich composition of sensations. The same jump button can feel like a dainty hop or a powerful leap. “Game feel” (a.k.a. “juice”) is the “aesthetic sensation of control” (Steve Swink, Game Feel) you have when playing a game.

The difference comes down to choice—which is to say, Design (with a capital “D”). Game feel is what makes some games feel gratifying to play (a character gliding down a sand dune) and others feel frustrating (sticky jumping, sliding). These decisions become a signature part of a game’s aesthetic feel and gameplay.

The Browser Company has written that software can optimize for emotional needs rather than just functional needs. Jason Yuan has promoted the idea of “fidgetability” where, similar to a key fob or lighter, digital actions can be designed to feel satisfying. Rahul Vohra has talked about making interfaces that are first fun as a toy—enjoyable to use without any greater aim.

The 2D portion is a particle simulation that “feeds” the growing sphere made with Lottie. It’s inspired by the charging animation common in games before your character delivers a big blow. Every action needs a windup. A big action—in order to feel big—needs a big wind up.

This is the big moment—it has to feel gratifying. We again combine 2D and 3D elements. The sphere and checkmark pop in and a massive starburst fills the screen like an enemy hit in Hollow Knight.

Our digital products are trapped behind a hard pane of glass. We use the term “touch”, but we never really touch them. To truly Feel a digital experience and have an app reach through that glass, requires the Designer to employ many redundant techniques. Video games figured this out decades ago. What the screen takes away, you have to add back in: animation, sound, and haptics.

In other words, when Kylie Jenner posts a petition demanding that Meta “Make Instagram Instagram again”, the honest answer is that changing Instagram is the most Instagram-like behavior possible.

The first trend is the shift towards ever more immersive mediums. Facebook, for example, started with text but exploded with the addition of photos. Instagram started with photos and expanded into video. Gaming was the first to make this progression, and is well into the 3D era. The next step is full immersion — virtual reality — and while the format has yet to penetrate the mainstream this progression in mediums is perhaps the most obvious reason to be bullish about the possibility.

The second trend is the increase in artificial intelligence. I’m using the term colloquially to refer to the overall trend of computers getting smarter and more useful, even if those smarts are a function of simple algorithms, machine learning, or, perhaps someday, something approaching general intelligence.

The third trend is the change in interaction models from user-directed to computer-controlled. The first version of Facebook relied on users clicking on links to visit different profiles; the News Feed changed the interaction model to scrolling. Stories reduced that to tapping, and Reels/TikTok is about swiping. YouTube has gone further than anyone here: Autoplay simply plays the next video without any interaction required at all.