kisonecat

It looks like the text uses the more natural R R a few pages later.

Using the ssh keys to provide the usernames is clever.

This reminds me of [self-promotion] https://github.com/kisonecat/countdot.hs which "algebraically" describes some patterns of dots.

At least a 2D visualization is at https://word2vec.xyz/

For math, I have a lot of videos that use pens and little bits of paper. An example is https://youtu.be/7oWMaNg4Ezw which discusses the rearrangement of the terms in a conditionally convergent series.

There's definitely some connections between analysis and category theory though! There are "self-adjoint operators" in analysis and there are "adjoint functors" in category theory, and it is sort of fun to think about the analogy there.

TeX's source code is available in a "literate" format, so you can download tex.web, run weave tex.web, and then pdftex tex.tex. This will produce a large book with the TeX source code. A challenge is that the resulting TeX engine won't support LaTeX3, so you'll need to use "changefiles" to modify tex.web to add epsilon-TeX and other updates.

You can use the same code for server-side rendering: with dvi2html, you can run TeX on a .tex file to get .dvi and then produce a static html file. This renders not just the math but also the text using TeX's layout engine, which is often not what people want. Running it in the client means the TeX knows the width of the view.

Unfortunately, the current platform we're using isn't very accessible: the \answer{...} blanks aren't converted to speech and don't navigate correctly. The new backend parses some TeX anyway to an AST, and the idea is that we could use that to produce alt text for the math. There's certainly a lot to do.

I started by playing with emscripten which manages some of the initial complexity.

I implemented a TeX engine in WebAssembly so you really can run TeX in the browser. You can see a demo of this at https://tex.rossprogram.org/ and at https://github.com/kisonecat/web2js you can find a Pascal compiler that targets WebAssembly which can compile Knuth's TeX. Interesting primitives like \directjs are also implemented, so you can execute javascript from inside TeX. The rendering is handled with https://github.com/kisonecat/dvi2html for which I finally fixed some font problems.

To make it relatively fast, the TeX engine gets snapshotted and shipped to the browser with much of TeXlive already loaded. So even things like TikZ work reasonably well. There is of course a lot more to do! The plan is to convert ximera.osu.edu to this new backend by the fall.

Maple and Mathematica are not open, and it's important that not only the resources be open-source, but also the underlying technology stack, because we're trying to not only teach students mathematics but the most engaged students need to "look under the hood" and learn how the technology works.

So we do support SageMath for more complicated answer validation. This depends on the public SageMath cell servers, which provides cloud-hosted SageMath and enables things like \sage{derivative(x*x,x)}. It's also more performant to rely on JavaScript.

For developmental math, answer validation is somewhat harder than for calculus, because often you want to distinguish between say (x+1)^2 and x^2 + 2x + 1.

We run workshops on how to use Ximera. The next one is at UFlorida in October. If you're interested, there is some travel support available. Contact us at ximera@math.osu.edu

I don't think Jupyter offers a facility like $\frac{\answer{2}}{4} = 1/2$ for putting answer blanks in the middle of mathematical expressions. For sagemath, code like \sage{derivative(f,x)} also works, even inside an \answer.

An activity in Ximera has state, and this state is synchronized (via Fraser's differential synchronization) with all open clients, so the instructor can watch a student while they work in Ximera, and even edit the page alongside them.

The magic number is currently 100 samples; the perturbation size is "adaptive" in that the perturbation size is changed when the expressions appear to equal but they're also close to zero.

The main goal is to make sure that we pass the tests listed in https://github.com/kisonecat/math-expressions/blob/master/sp...

The biggest trick here is that TeX source like $\answer{2} + 1 = 3$ provides "answer blank + 1 = 3" on the web, but you can do more complicated things like $\sqrt{\answer{4}} = 2$.

The website uses Fraser's differential synchronization algorithm, so a student's work is saved as they do it -- and that happens in real time, so an instructor can "look over their shoulder" and watch a student working in real time. The server is actually a git server, so an author publishes content by performing a git push, and because the server knows the history of the content, the server provides the version of the content on which the student last did work.

I'm one of the developers of Ximera.

Expression equality is handled by https://github.com/kisonecat/math-expressions using a few heuristics, which is necessary because Richardson's theorem says this problem is undecidable. The main trick is to regard the expressions as complex-valued functions, find a point where the value is (nearly) equal, and then check for numerical equality in a neighborhood. (Doing this over the complex plane addresses the problem that many common expressions like sqrt(small number-x^2) have a restricted domain over the reals).

We rely on unit tests from Chris Sangwin (who wrote the textbook _Computer Aided Assessment of Mathematics_) to ensure that we're adjudicating expression equality mostly correctly.

(I'm Jim Fowler.) Thank you---I certainly appreciate it.

The Calculus One videos are available at http://youtube.com/kisonecat and the new Calculus Two: Sequences and Series videos will be posted soon.