Accessibility in digital mathematics learning resources - Christian Lawson-Perfect - IDPWD2020

[Andrew] The next speaker is Christian Lawson-Perfect, who's a member of the school's e-learning team who are responsible for Numbas, I think Christian's probably the main man as far as Numbas goes and have developed Coursebuilder and several other things that I'm hoping Christian will tell us a bit about that make it possible for lots of different people that wouldn't otherwise be able to to access mathematics and technical material that we produce.

So this has been a big effort and a huge program, and they've done really well and there've been awards and prizes, and it's a really good thing for us.

So I'll hand over to Christian.

[Christian] Thank you Andrew.

Yeah, so I'm going to talk about the stuff we do in the e-learning unit.

Turn that on...

right. Yeah, so as Andrew said, we won a collaborative award for teaching excellence from AdvanceHE and we got the VC's Education Excellence Award the year before that, so that's encouraging.

So first of all I've put this slide in.

I've assumed everyone present in Zoom can hear me.

We haven't been told, I don't think, of anybody that requested captioning.

The recording of this talk will be subtitled, so I'll make sure to do that.

Right, so the e-learning unit is - on the left, there's me, I'm taller than George, who is taller than Chris, who's in charge of all of us.

And we, as Andrew said, we make Numbas the assessment tool, Coursebuilder for making online versions of course notes which we've used very heavily this year.

And any other e-learning stuff that comes up.

We tend to be - people come and knock on our door when they've got a computer problem.

So we do quite a few things.

Now about me.

This is the day of people with disabilities. So here are mine.

It's quite a long list, I'm autistic, I'm colorblind, protanopia, I have recently pinned down that I have hypermobile Ehlers Danlos syndrome, and I think as anybody with a chronic illness knows you're never finished being diagnosed.

So just last week someone said maybe I've got dysautonomia so I'm going to do a fun tilt table test at some point and find out about that.

So for personal reasons, accessibility is important to me.

So, maths sometimes looks like this.

This is a picture of some handwritten maths.

I wrote it out myself this morning. I thought can I go and trawl through my box of course notes to find some stuff or just write out again?

I think the important things to notice about handwritten maths are that it's two-dimensional.

It's not like prose text where you have lines of characters, mathematical notation relies on the vertical position of things, the sizes of symbols...

Here I've got a derivation of the quadratic formula.

I've gone line by line, the notation uses some symbols like the line for fractions - is that a symbol or a thing, or is it like a graphical element?

And I've put on the end of each line an explanation of what I'm doing.

I've also drawn a diagram of a parabola on the right hand side, it's the kind of thing you might do while you're doing some working out, it's a little sketch to help your thinking.

So math looks like that, it's complicated.

And the challenge is to put this kind of thing online in a way that's accessible and useful.

Maths also looks like this sometimes.

Here's a clip from a group theory textbook I picked off my bookshelf.

It's typeset, there's a very long history, I think, maths has been typeset as long as anything has been typeset.

This one may in fact have been done pre computers.

So it's text and the important thing is there are all sorts of symbols in maths.

Things like Greek letters, divider lines, braces.

The "belongs to a set" relation.

There all sorts of things that are specific to maths and maths keeps coming up with new symbols and new forms of notation.

It's not really the case that you could implement all math notation forever.

There will always be new stuff.

And then also this has got a few diagrams at the bottom of this picture, some polygons labelled with letters and numbers and dotty lines.

And describing those just in words would be really difficult.

Maths is also spoken.

Something I'm interested in, but never taken the time to really go into in depth is the semantics in the way we speak maths.

So for example, if I say "x + y ...

squared", That's different to "x + y squared".

There's something to do with the cadence of the way you speak, to group things together, you can leave gaps, or you can say stuff more quickly.

So there's a lot of information in the way we speak maths as well, so if you want to computer to read maths out to you, it has to do that as well, or it has to do things like say "left bracket X + y right bracket squared".

Which is not how we're used to hearing maths. Maths is also drawn as I showed in my handwritten maths and you quite often do sketches.

You might do a commutative diagram laying out objects in a grid, roughly, with lines going between each other.

There are so many different ways of representing maths and it's not always the case that you can just sort of produce a long passage of text, which is equivalent.

I think I've only shown ways of math that are accessible to people who can both see and hear.

There is a Braille code for maths called the Nemeth code, which is sort of a different way of interpreting written Braille in order to represent all the different kinds of mathematical notation that I was talking about earlier.

It's not very widely used, but that is a thing that some people use.

So.

My job and the job of the e-learning unit is to make learning materials accessible.

We're legally required to make all our stuff accessible to anybody that might need to use it.

And I think that the right way to do that is through universal design.

And the idea with that is designing something so that it can be accessed, understood and used to the greatest extent possible by all people.

And that only really makes sense if you contrast it with the idea of making a separate, accessible version for somebody.

So if we had a student with a visual impairment come in and all our course notes were pieces of paper that we hand out, if we made a separate audiobook version just for that person, and didn't make that available to the other students, that's not universal design.

So the idea is to make something that just from the outset everybody can use.

And that pays pretty big dividends.

You make something that you might have a certain group of people or certain person in mind when you make something, but then everybody else might find it useful too.

It might be nice to sit back and listen to Andrew Duncan reading his course notes out.

Or having a text description of a diagram might make it easier to pick out certain bits of information.

So that's something that happens quite often, I think Universal Design is really an important principle.

OK, so I'm going to talk about a few of the things we do or a few of the problems we face and how we've tackled them.

So since time immemorial, when there have been handouts for our lecture courses, we'd either have photocopies of handwritten notes, or something that's been typeset, typically using LaTeX.

I think the last few people insisting on using Word have now retired.

So we will hand out something on paper, or even a PDF file.

None of that is particularly accessible for people who need things like a screen reader, or to be able to magnify text or change the spacing.

Even what you might think is quite simple stuff like having it in different colours are not easy to do if you've just got a PDF.

HTML, which is what web pages are written in is, I think, the most accessible format around at the moment.

You can add all sorts of semantic information that assistive software can use.

Assistive software is things like screen readers.

A person using a web browser can scale the page, they can apply their own styling as they wish.

The problem is, it's not very well set up for writing maths.

There's no standard mathematical notation that is easy to write built into web browsers.

And lecturers like writing TeX or LaTeX anyway, so we have all these course notes, we have a computer readable format for them in LaTeX.

So we, and by we I mean mainly George Stagg, wrote Coursebuilder which is a tool which takes LaTeX source.

It produces the PDF version, for students who want that, it also produces an HTML version which is as accessible as possible and that was produced in response to a student coming to the university who had some very specific requirements, which really prompted us to make this accessible stuff, but now some lectures are saying "I'd like all my course notes to be available this way, it's just better".

We make that available to all students so that people who haven't specifically asked for the ability to magnify the text or to increase the spacing between things, whatever, they can just do that.

I think that's really a real benefit of having this. So here's a example set of course notes.

I've picked it out of George's demo of Coursebuilder.

This is how things used to look, it's a PDF file, typeset using LaTeX.

We've got headers, some equations with mathematical notation and things are numbered.

OK.

Here's the equivalent in Coursebuilder.

This is now a web page with some links.

And it uses MathJax to render mathematical notation, and there's been some fantastic work done in MathJax to add some accessibility tools.

I won't turn it on because I've managed to crash my browser every time I've tried it this morning, there's something to do with the way I've set up my slides.

A tool for, if you're navigating through a page with just a keyboard, which typically many blind people do, there's not much point using a mouse if you can't see where the cursor is.

If you're navigating through the keyboard, you can step through an expression.

sort ofsemantically, they call it.

So if you've got a product, you can say go down the left hand side of this thing, or have a look inside this function, which makes it a lot easier to examine an expression just in a piece rather than having it all read out to you in one go.

Yeah, so there it is and there are some easy tools that can do stuff like change the colour scheme immediately, increase the font size, I mean, that's something that's built into browsers, but it's good to have that there as well.

So that's that.

I think that's a real achievement. Here's the same thing rendered using a slightly different layout, which we made for this student that that caused us to make Coursebuilder.

They wanted an off white background.

It's all about low strain on the eyes.

Lots of space between paragraphs.

And the text is a lot bigger as well.

So that's good, but what I've been pushing for in Coursebuilder is for this kind of stuff to just be in the standard theme that we had the choice of color schemes there, the ability to change the spacing between stuff I think would be good to have in the main thing as well.

So that you don't have to ask fro this separate thing, we just put our notes out and then you pick whichever settings work for you.

Right, so that's one half of it, we give maths out to the students.

We present them with stuff.

They have to write it as well, and working mathematicians have to write maths as well.

So how do we do that?

We - well, I - do a lot of handwritten work.

First of all, I'll get a piece of paper and write out some mathematical notation.

I might use Microsoft Word to write up maths.

We have a lot of students coming out of school who use Word's equation editor to write maths as an accessibility tool because it's graphical.

If they can't do handwritten work, that's identified as a way for people who can use maybe a single button clicker to write out maths.

Many working mathematicians use LaTeX, which I think everybody present knows.

It's got quite a forbidding syntax.

You don't really see what you've written until you compile it all through.

It's not the easiest to write.

For a student with limited mobility who had been using first of all Word and then some other tool they got DSA to pay for, the idea was, using a keyboard is a massive strain for this student, typing speed was very, very low, so the game is all about limiting the number of keypresses.

The tool that they had been using was quite good, but not quite up to university level maths, there's all sorts of notation, symbols and things that weren't easy to produce using this tool.

So I found a tool called Stackedit, which I'll show you in a second.

I made it use instead of LaTeX for mathematical notation, which it was set up to use normally, something called ASCIIMathML, the aim is to use a slightly simpler syntax than LaTeX.

So I'll show you that now, this is what it looks like.

You get over on the left hand side, you write some text, there we go, and over on the right hand side you get shown a rendering of it because what's on the left is Markdown.

So what can I do? Remember how to do bold.

There we go. I put a couple of asterisks around something to do bold.

To make it bold. So you could set up an accessible keyboard to have shortcuts for certain strings, like to turn stuff like bold on and off.

But the real point of this is about maths.

Here I've typed a dollar sign, it's put another one on the end which saves me one keypress.

Um, in the middle I can write 'x' and I get an 'x'.

Now in LaTeX it would be X and then a hat and then two to get 'x squared'.

This has XX produces X squared.

I'll just put that on another line.

And things like blackboard bold N, you just right two Ns.

Now what I want to put here.

So the idea is this simpler syntax is quicker to write for the student because it requires far fewer keypresses to produce pretty good looking mathematical notation.

Now this student's been using this for a few years now, for homeworks and exams.

I think that's been quite successful.

That was something we put together quite easily.

Not having to pay some company to make us a tool for this.

I think it's really good. Right, so my job, the main part of my job is to do with assessment.

I'm the main developer of Numbas, the system we use for online assessment in MSP and around the University.

I think some of the important points to do with using Numbas from an accessibility point of view are: can you use it with just a keyboard?

Paying attention to the contrast and colors used.

As someone who's very colorblind, it often catches me out with online tools that will say that the bit highlighted in red is important and I say which bit?

So, paying attention to that kind of thing.

Being able to use the interface just with the screen reader, so don't assume you can see it.

I'll show you some of what the interface is like next.

Another important point to mention is, as a developer of the tool, I can make the interface accessible, but then also you have to write the questions.

And it's really important to be aware of accessibility when writing a question.

Simple things like write full sentences, always name stuff that you're going to use later, like stuff that you wouldn't necessarily say is an accessibility issue, but quite often - we all remember maths textbooks when we were in school.

They have just a list of equations and sort of implicit was you have to solve this.

Or it's implicit that you have to simplify in a certain way.

People with learning disorders or neurodivergent people, you need to be very specific about what you're doing and what you're telling people to do.

Particularly, I think when it's marked automatically by computer.

So that's something to think about. So here's a a Numbas question.

What have I got?

Again, maths rendered using MathJax.

I think that's really the best thing at the moment for mathematical notation.

Now the important parts here were, I have to make sure that all the inputs have labels on them.

So I'm hovering over this thing and it says it's the answer for part A gap 0.

When I get some feedback, let's try and answer this.

I'm going to do this very quickly.

8, 1, -17 and I think 6...

there we go. Now, I haven't turned on my screen reader because it just kept crashing.

I think again to do with the way I've set up my slides today.

But the point is some new feedback popped up here but I'd just pressed this button.

So web browsers have a way of marking that as something that needs to be read out as soon as it changes.

So I press this button, the screen reader would immediately read out this feedback.

And that should be a green tick there, it was just before I started my talk.

Never do a live software demo.

Yeah so.

I think Numbas is, compared to other assessment systems around, pretty good as far as accessibility goes.

It's something I've spent a lot of time thinking about.

So now I think, here's a problem that we have yet to solve, or something that's even more complicated than just mathematical notation in text.

How do you make an accessible diagram?

So I had a picture of a parabola - a graph of a parabola and a couple of axes and a curve that looked parabola shaped back at the start of this talk.

How to make that accessible to someone who can't see it?

I could write some alt text, so that means alternative text that when you can't see the image they would have read out to them.

So I would say, I don't know what would I say?

I would say "Plot of a parabola".

That might not be enough information. I might say "two axes, one labeled X, one labeled F of X, curve starting at the top left, going down to a minimum at 0, minus 2, then going up to infinity again.

That might be too much information for the purpose of asking a question about the graph, so it's really tricky to come up with a good text description of a diagram.

We really don't - we're not in the habit of thinking about all the information that's in the picture.

I think it's stuff we all have to be taught when we're taught how to draw a graph.

It's something you have to be really mindful of when you're making things accessible.

Another way of presenting diagrams is f instead of it being a bitmap image, just like a grid of pixels, there's a format called SVG, which stands for scalable vector graphics.

You have a load of elements, like you might say, put a circle at this position or a line between these two points, colour them in this way, in that way, and you can attach screen-readable descriptions to individual elements in those diagrams, so a user could navigate their way through a diagram.

You might have a large group that says this is the set A and they go in and then read out the individual points.

So you can do a lot with that to make it easier to navigate within a picture.

I think within Chris's talk we had this about how do you get a global view of something and then a local view of individual bits?

So that is possible. It's a huge amount of work to set these things up.

There's a company called Desmos which makes a graphing calculator.

It's web based. They've been working on providing audio traces of graphs, which I don't think I've got time to show you.

So the idea there is, you might say y = X ^2 and it'll map tone to Y and X to time.

So there are some solutions.

It all gets much, much more complicated once you start thinking about interactivity.

Maybe you want to provide -- compass and straightedge constructions are something we're taught.

If you can't see the construction, what would the individual steps look like?

Suppose you had some controls for like move the compass around, draw a circle here, what would the computer be saying to you as you do all these things.

It's tricky.

So I...

A problem we also have in Numbas is that questions are randomly generated, so diagrams need to match whatever random set of numbers we come up with.

So I found an old scripting language whose aim was to produce diagrams a bit like TikZ, you might use inside LaTeX.

I reimplemented that and then added in features with a view to making accessible diagrams both easier to produce and to just happen by default.

So for example, it's quite hard in my implementation to specify a colour by saying how much red, green and blue it's got in it.

There's a palette of colours, one, two and three, which I know are accessible to pretty much anybody with colour vision, with any amount of color vision, so even me with protanopia deuteranopia or the other one, you can all distinguish these three colours.

So as long as you stick with that you're fine.

It makes it easy to add these screen readable descriptions to individual elements as well.

So here's a diagram I made with Eukleides, and I'm going to try to use my browser's accessibility inspector to show you what's going on here.

So the diagram, I'll describe it quickly in words first, it's a Venn diagram with three sets, A, B and C and there are 10 labelled numbers that are in all, some or none of the sets.

The idea was the question was like "say what's in a?" In my accessibility inspector I can see individual bits of the diagram.

At the top level it says "a set diagram".

If you go further in, you get some circles.

I've got a label A for this circle, it's telling me there's a transparent, color one filled circle.

I was making changes to this just before the talk and I didn't add on a proper description for that..

So we have circles.

Circle A, circle B, circle C, and then we have all these points and I've said number one is outside all of the sets at the top of the diagram, two in all sets, three - I've made an error.

That should be in just B.

So the point is you could go through this, read all these bits and then answer the question what's in the intersection of A and B?

Yeah, so that's half a solution.

The really, really hard problem is how do you assess a diagram that a student has produced?

So, for example, consider the prompt to sketch y = one over x squared minus two plus 4.

I would expect a student to do a drawing on a piece of paper maybe with a pair of axes and a curve.

I guess some kind of line to mark an asymptote.

What does this task mean for somebody with limited motor control, who it would either take them ages to draw this diagram or they can't hold a pen, they would have to get a computer to draw it for them.

Or someone with a visual impairment for whom a visual representation of the function doesn't really make much sense.

What's the equivalent task?

I've been talking to the people who make Isaac Physics, which is a online formative assessment aimed at A level largely, to do with physics.

They're looking at assessing sketches.

They want to provide a graphical interface for drawing a freehand sketch with the mouse and then marking it.

I think where I am at the moment is I think you want to ask the student for certain properties of the sketch.

So if the student manages to say the curve never goes below a certain point or the curve has a stationary point here, that's an asymptote and describe it.

I consider that the same amount of information as a freehand sketch.

So the hard part now is to implement that.

So that's where I am at the moment.

I think that's probably just about my time, so thanks.

[Andrew] Thank you Christian.

Put up your hand if you'd like to ask a question.

I think there's probably...

[Christian] Angela? I just wanted to ask about Coursebuilder, yeah?

So I could see that it had converted from the LaTeX, all the sans-serif text into like something like Ariel, but the maths remained with the serif.

Is that intentional and because I know dyslexics sometimes have a problem with serif text, so does that continue to cause a problem?

It's something I've often wondered about it, not just particularly related to what you've done.

[Christian] There's a real problem with that.

Well, OK, so first of all, the really straightforward reason is, MathJax only implements a couple of the TeX fonts, or a handful, and they're all with serifs.

So that's why it's not a sans-serif font.

You might not want a sans-serif font for mathematical notation anyway, because fonts convey a lot of information in mathematical notation.

I think at primary and secondary level textbooks quite often use a sans-serif font because they're not at that level of, you have to spot the difference between "is this more bold than that, or is this italic?" They don't get into that.

But for high level maths, the font really matters and I don't know how I feel about, should we be teaching students to get used to the conventions that they will encounter, reading published papers, or should we try and push for it to be more acceptable to use a sans-serif font, and declare up front this font means scalar, this font means vector, and so on.

Yeah so, the reason it's not there at the moment is no one's converted the font files, but in the future I don't know if it's something we'd really want anyway.

Certainly something you could do some research into.

[Angela] Thanks.

[Christian] David [David] That Venn diagram, were the colors chosen meant to be robust to the various colour blindness?

[Christian] Yeah, I forgot to say that, yeah, I added some colours into that late on just to show my set of accessible colours.

The important thing to bear in mind when using colour in a diagram is never to use colour solely to convey information.

I told the computer to make those three things three different colors.

The colors just on their own.

Actually, I'll screen share again so I can go back and talk about what I can see.

There it is.

So the the main part of A, and the main part of B and the main part of C all look -- I can distinguish them.

The intersections, I can't tell the difference between A and B, and C and B.

So I'm not using colour.

I'm not relying on color to convey information there, but I'm using it for anybody who could benefit from it.

It livens it up a bit.

I've made sure that even in grayscale, you can still read this diagram.

[David] Yeah I see.

OK yeah, great thank you. That's interesting thanks. [Christian] Andrew.

[Andrew] The things that you've done so far make all of my notes, if I was to use it, readable on a phone or a tablet or a PC.

I'm guessing in five years' time people will have thrown all those out of the window and there'll be some new gadget that comes along.

Havge you any idea what - I mean, do you think these things are going to be robust to the next generation?

[Christian] Well, I mean there are two ways of looking at it, aren't there?

Suppose you want it to still be accessible in the future.

So, when I was little we had stuff on the BBC Micro which is, now you can't use anymore, because no one's got a BBC Micro.

I think having stuff as web pages, that's probably the format that's most likely to still be readable in the future.

Yeah, maybe not all the JavaScript that MathJax has to do, but at least we've got the got the LaTeX there.

That might still be possible.

So I think there's a good chance that it will still be readable for a good long time.

The other part of that is, what if we come up with something really whizzy in a few years' time and have to remake it all again?

Which has happened with e-assessment, we had a previous system before we made Numbas and then we had to rewrite it all in Numbas format and then if something else comes along we have to rewrite all our stuff again.

You can't account for that.

Cedric Villani got involved a year or two ago, in some projects to present 'Holo Maths'.

So using this this Microsoft VR thing, you know they've got these holo VR goggles things that augment reality.

Looking at, once you can present stuff in three dimensions, can you do more than you would just do on a flat screen?

And I haven't heard anything good come out of that, but VR is getting more and more common, the technology is getting better, it may be that we're just habitually using VR for certain things in a few years' time, and will suddenly start asking, well, why haven't we got 3D, why can't we walk around graphs of this function?

Yeah.

[Andrew] So, I had another question actually also involving things to make more work for you guys, so if I put my notes into Coursebuilder at the moment, I haven't tried with anything more than very simple things, but I have a suspicion that because I use every possible fancy LaTeX package that I can possibly find, that that's not going to go very quickly.

So is that an aspiration to make Coursebuilder so that I can use all parts of LaTeX and then just slap my notes in to Coursebuilder or is that not desirable, perhap?

[Christian] George has done more actual handling of real Courebuilder stuff, so might say something in a bit, but before George starts talking - I can see he's unmuted himself - I think there are lots of LaTeX packages that are designed within a producing a PDF context.

All the stuff to do with layout, making it so that you can put something on top of something else which you don't want to repeat on a webpage.

So actually, you probably don't need as many of those things as you might think, to produce PDF stuff, but for other stuff where there's lots of nice macros that you want to use, yes.

We should aspire to have all that usable in Coursebuilder.

Coursebuilder builds on a thing called Plastex, which was developed by someone somewhere.

It tries to have both reimplementations of popular LaTeX packages, but then it can also read LaTeX itself and make a good effort to, because LaTeX is all about just expanding macros out, just to do that, which eventually boils down to something that it knows how to produce an HTML version of.

[George] Just jumping quickly under Christian, although he basically said everything I really wanted to say, which is that in a lot of cases when you take someone's LaTeX notes, you really don't want to exactly reproduce what they've done.

The reasoning being that lots and lots of LaTeX packages were all about to do with things about laying out things on a page, which was a fight with type.

You know it's not quite the same as a web page, and in particular there are things that people do that that are wholly inadequate, things like they write solutions and then put a white box over the top, or they write solutions in white text so when you print it off, it's blank, but if you have the PDF you can actually extract the solutions or even worse if you were using accessible web notes, you just change the colour of the text and there are the solutions, so what I don't really want people to do is to just think I'll just not even think about this, I'll throw my notes into Coursebuilder and it will deal with everything else.

It's not quite like that, it's a bit more interactive but saying that, as Christian says it's building on a system called Plastex, which is very different from other implementations of TeX to HTML that do exist where it's basically just a fancy find and replace for bits of mathematical notation.

That doesn't work in LaTeX, for technical reasons.

It's not a regular language, so you can't use regular expressions for example, so it really is a true reimplementation.

And because of that, it should be able to reimplement LaTeX packages.

And it does in some cases that we have added. It actually just straight runs packages from LaTeX.

The problem is that a lot of popular packages don't just run LaTeX code, they use what's called PDF specials or Postscript specials where it really is - the output of those packages really are very specifically defined to work in a PDF file, rather than some kind of abstract representation for documents.

It really is going into the minutiae of the PDF format.

And it's those kind of stuff that we can't reproduce.

So how we're handling at the moment is in the same way that Plastex handles it, which is to re implement those LaTeX packages.

So we hope to reimplement as many as possible, but there's always time against us and a lot of work.

As Christian says it's not necessary.

[Andrew] Good.

So maybe I should be starting by writing my notes in Coursebuilder without thinking about LateX at all, would that be right?

[George] Really, I don't think you have to go that far, I think really it's just that while you're writing your notes in LaTeX as you normally would, it's just trying to bear in mind accessibility, and I think that's true for anything, not just Coursebuilder but in anything.

When you're when you're working with any type of software package, try to bear it in mind and then later on whether accessibility matters or not, at least you thought about it.

[Andrew] Yep.

Do we have any more questions?

[Christian] I can ask and answer my own question.

I just remembered something I thought about putting in but I didn't have time to.

While we're thinking about the way that we write maths, there's a contingent that say let's all just use Mathematica syntax or Sage or whatever.

Let's use a computer readable representation of maths or even going back MathML which MathJax appeared to actually implement.

The idea was if we write stuff in a computer readable form, that's just how we should do maths from now on.

And I don't think - I think your question about in 10 years' time will we still be using this?

I don't think there's anything, any computer readable maths syntax that we'll still be using in 10 years' time, so we're not there yet.

But you never know.

It may be that we just give up on this huge lexicon of 2D mathematical notation that we've built up over the years and say we're just going to write linear strings of characters to represent every bit of maths.

I don't know if that will happen or we might come up with some better way of inputting math into the computer.

[Andrew] Handwriting maybe?

Which you do on a tablet all the time.

Good, that's really interesting.

Sounds like there's actually potential for a lot of overlap between those two talks.

Sonification of my notes is what I want next.

So, if there's no more questions, let's thank the speakers again, that was very good.

Thank you very much, both giving up your time and doing that.

That was great.

Thank you both. Thank you and thanks for coming along guys.

Nice to see everyone.

Bye bye.