Towards an improved UX for mathematical modelling: symphonies locked away from the world, painting fences & where is ‘technical concepts’ Hendrix?
By: Matt Garrod
Tags: Mathematics Education, Pedagogy, Visual Design, Interaction Design, Mathematics

[Cover image taken from: https://www.pexels.com/photo/blue-orange-and-black-abstract-painting-3418325/ ]

“The ultimate goal here is however not art itself. It is to curate this tiny world of obscure artists with the power to liberate the source code of reality & directly communicate the underlying truth without use of jargon or symbols w/ beauty the vehicle and reality its payload.” — Eric Weinstein (Tweet lifted from: https://twitter.com/EricRWeinstein/status/1200964277399478272 01/12/2019).

In my previous article I gave an overview of the different elements required for building a simulation model, integrating in with data and presenting the output in and an accessible manner. I referred to this as a “simulation intelligence pipeline”. I outlined areas where good design and communication might play roles at every stage in creating a simulation model and integrating it with data. In this article I focus on the first element: mathematical modelling.

In a 2019 medium article Stephanie Evergreen argues that the next big challenge for designers is data visualisation. She discusses the need for design in data driven enterprises and how those who traditionally work either in data or design would benefit from learning the basics of the other side’s craft. Data visualisation is a rapidly growing field with many people doing a lot of good work (some great examples can be found on Andy Kirk’s website. Therefore, I would argue that this battle may have already been won (or at least substantial progress has been made). I get the impression that many individuals with a background in design (or related disciplines) are flocking to get involved in representing data in various guises (I admit my viewpoint is biased as a data scientist who has attended data visualisation conferences). One area that has received less attention, as far as I can tell, is mathematical modelling. Various mathematical models underpin our understanding of climate change, the dynamics of populations in biology, the behaviour of fluids, economics, social behaviour and numerous other phenomena. Therefore, being able to communicate how these models work is clearly useful.

A common misconception about mathematical modelling is it proceeds purely through solving equations and arranging mathematical expressions. This is one method mathematicians use, however, there are numerous visual tools and other approaches used to gain intuition. For example, in the domain of dynamical systems visual representations such as phase portraits and bifurcation diagrams are used to examine the properties of systems which evolve with time. Complex systems and network science also tend to be relatively visual domains. These visual ways of representing mathematical models of systems are often useful and sometimes aesthetically pleasing. However, they are still mostly tools developed by specialists for specialists. As I noted in my previous article, failure in communicating the details of a mathematical model can lead to issues. These might include failure to understand the assumptions behind a model and its limitations.

In many enterprises the technical details underpinning mathematical models are not communicated to the end users. Why is this? The underlying answer is likely to be a complex mix of educational factors, economic incentives and research culture. However, I would argue that two of the key factors are:

Education. The way that mathematics is presented to students focuses on rules, techniques and formal notation. This (and other factors) encourages many people to stray away from mathematics from an early age due to it being “too abstract” and only understandable to people who put in the time to learn the symbolic language. Research dissemination. Research in technical subjects is predominantly presented in academic journals. The presentation in journals generally includes a large amount of technical jargon and heavy usage of formal notation. There are some brilliant counter examples (presented below), however, journals still seem to be the dominant medium used for research communication.

In both cases we are not fully utilising technologies and known methods for communicating ideas to full effect. The second point above is discussed in more detail in this article on “research debt” in the Distil journal. This more recent article on argues how we can use web based technologies and other methods to improve our level of communication in research, education, journalism and policy making. I would strongly recommend looking at some of the articles in the journal itself as they provide some great counterexamples to my points above.

In this article I have collected together some analogies which make clear where we fail at communicating mathematical content in education and research based settings. I believe they form the basis of some good “intuition pumps” (as philosopher Danial Dennet calls them) which can be used broadly in the context of designing processes to make technical subjects understandable. I summarise these analogies in the table below, however, I will go into more detail in the sections below.

Table 1: Three some analogies for poor exposition and teaching of technical concepts.

Symphonies locked away in sheet music

In Episode 14 of his podcast “The Portal” mathematician Eric Weinstein sits down with the artist London Tsai in order to discuss how poorly we do at visualising the most important concepts in mathematics and theoretical physics. I’d recommend listening to the whole clip, however, I’ve pulled out a few highlights. Eric likens the way mathematics is currently treated in society as if music were kept locked away as sheet music which is rarely performed:

“My belief is that we have the most beautiful symphonies in the world locked inside of our journals and our math libraries, lets say, as text, and you know you can ask well, if you have a symphony of Brahm’s: is it a symphony if it only residues in sheet music and is never performed? That’s a very disturbing question because the instructions for its performance are present and it may be that a tiny number of people can actually read a sheet of sheet music and say oh my god that’s gorgeous because they can hear it in their head but the rest of us actually need the thing to be performed and yet there is no orchestra or analogue of an orchestra that performs works of great mathematical of physical beauty.” [Transcribed from ~7 min 30s of https://www.youtube.com/watch?v=pVYCqK19-ww&t=237s ]

I love this analogy and think it can be more generally applied to mathematical modelling and technical concepts in general (rather than just maths and physics as they primarily discuss in the podcast). It also made me think: if “traditional” mathematicians or scientists play the role of arcane classical composers: who would the musicians who never learnt to read sheet music be? Some of the most acclaimed musicians in history (e.g. Jimi Henrix and Eddie Van Halen) are well known for their lack of ability to read sheet music. Granted, transcribing their music later on provides a great way for musicians to replay their classics. However, this demonstrates that knowledge of the notation was not necessary during the creation process. I cannot say exactly who these new iconoclasts of math communication will be. However, based on the above we might expect that they might not have learnt mathematics through traditional channels such as university education. Maybe we’ll know when these new individuals have arrived as we will see the existing establishment playing the cliche role of 20th century teenagers’ parents reacting with disgust at the next “non-classical” genre of music arriving on the scene.

One element of Eric’s analogy is the idea of a performance. What might this look like in the context of mathematical modelling? One domain where we are arguably seeing mathematical ideas played out is in the machine learning algorithms controlling many real world systems and even influencing people’s behaviour via social media. The huge impact of these algorithms on society has a knock on effect of encouraging more interest in maths and statistics. Machine learning is not “just statistics” (as academics in technical subjects won’t tire of snarkily remarking — see this article for a discussion), however, its widespread use does make the understanding of maths and algorithms relevant than it has perhaps been previously. The rise of ML has coincided with that of browser based visualisation technologies. Consequently, there are numerous great web based visual explanations of machine learning concepts (see: http://www.r2d3.us/visual-intro-to-machine-learning-part-1/, https://playground.tensorflow.org/ and https://distill.pub/2020/communicating-with-interactive-articles/ for some examples).

It is great that web based visualisation technologies are being used to disseminate technical ideas in this fashion. However, I think it is worth not to just privilege the currently “fashionable” areas of applied mathematics & statistics such as machine learning. Firstly, we arguably have a “research debt” in subjects such as mathematics and physics which extends back at least 100 years or so. Many of the deep ideas in mathematics and theoretical physics still have relatively little exposure. Secondly, as Peter Rowlett discusses in this article, there are many cases where abstract mathematical techniques have completely unexpected applications. Therefore, exposing niche abstract ideas that have little application is likely to have some payoff. I should say here that applicability of abstract mathematical ideas should not be our sole criteria for exposing them. Works in mathematics are cultural artefacts just like pieces of music and art. Therefore, there is room for taste (i.e. different people may prefer different genres) and appreciation of the content itself. The latter becomes easier given an effective way to “perform” pieces of mathematics.

Later on in the podcast, London goes on to say how he does not think of his art as mathematical art as he does not want his work to simply be considered part of mathematics outreach:

“That’s partially why I don’t think of my art as mathematical art because I don’t want it to be associated with this kind of like “oh bring math to the people” that kind of thing. It has to be somehow something… I mean look, the wonder of math we experience studying it, that is not… I don’t feel that when I go and look at these kinda displays of mathematical art.”

[Transcribed from ~57:22 of https://www.youtube.com/watch?v=pVYCqK19-ww&t=237s]

I agree with London’s perspective on this: seeing mathematics as something obscure which needs its own artists and outreach specialists can help people understand but it may not necessarily lead to people actually being ultimately more involved. Going back to the sheet music based analogy above: there is a huge difference between creating accessible educational tools for people to appreciate sheet music and actually performing the music.

This point leads Eric and London on to discuss how well known “mathematical artists’’, such as, M.C Escher may have some great work, but that this work only ends up illustrating a small subset of the ideas from the numerous of mathematical disciplines (e.g. tessellations and tilings on the plane). Broader communication of all of the different areas (especially those which get less exposure) is to be encouraged. As an aside, Quanta Magazine’s maps of mathematics is a nice well designed illustration of several of the different concepts in mathematics and how they fit together. Another recent article tells the story of some of the unsung female heroes in applied mathematics. Championing both less represented groups in mathematics and less well published disciplines gives us the combined bonus of encouraging more different people to get involved in more completely different activities.

London’s quote above demonstrates how existing mathematical art does not necessarily allow people to experience the same emotions as someone actually practising it. We cannot truly know what is going on inside someone’s head, however, effective communication techniques can get us much further than we currently are. The idea of connecting emotion to technical subjects is something discussed in the context of data art. For example, Kirell Benzi defines data art as:

“Data Art or data-driven art is an artistic practice that relies on the usage of a dataset to convey emotions to the audience.”

The idea of having an emotional connection to mathematics (or data) is so foreign to people that it may strike many as either an oxymoron or a pathology. Even as someone familiar with mathematics, I’m not quite sure what it means to express mathematics emotionally. This may be because I’m not used to talking about it in these terms. Perhaps those wishing to communicate the workings of mathematical modelling techniques must keep an eye on the techniques developing in disciplines such as data art (at least until technologies like Neuralink allow us to realy see what is going on inside someone’s head).

Work by artists such as London Tsai and journals such as Distil do a great job at exposing some of the most technical ideas in a fresh manner. There is certainly a cultural shift required in academic research to make these methods of exposition more standard. Arguably, this is work for the practitioners in technical disciplines. In the next sections below I focus on how changes in how we educate people about technical subjects might make things easier for those on the receiving end of research dissemination.

Painting the fence

Aside from the poor communication of technical research, another underlying cause of people’s lack of engagement with mathematical models (and keeping them as the reserve for the specialist) has its root cause in education. Suggesting how to revamp the education system is not the purpose of this article (and certainly not something I am qualified or informed enough about to discuss). However, the two analogies I introduce here provide a framing of the issues which could be useful for designers aiming at creating more accessible interfaces for mathematical models.

In his book “Love and Math” Edward Frenkel imagines how people would react to art if it were taught in the same way as mathematics:

“What if at school you had to take an “art class” in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: “Learning art at school was a waste of my time. If I even need to have my fence painted, I’ll just hire people to do this for me.” Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away”.

-Quoted from preface of Frenkel, Edward. Love and math: The heart of hidden reality. Basic Books, 2013.

This analogy has some themes common to the sheet music analogy above. However, it emphasises how the thing taught at school referred to as “maths” is not necessarily what practitioners of maths actually do. That is, there is only a small overlap between what research mathematicians think about all day and what people get taught in school. A cynic might claim that mathematics education is designed to only provide the most basic ability to keep the economy going whilst deliberately excluding any content which might risk children actually becoming inspired (there are of course some brilliant teachers who manage to successfully inspire their pupils, however, I get the impression that they have the odds stacked against them).

I may be exaggerating above, however, there is evidence that pupils’ perception of technical subjects can vary drastically from practitioners. Research presented in this study shows how students of mathematics and physics perceive words completely differently to researchers in those areas. The authors found that researchers generally have a positive perception of mathematics and physics related concepts, whilst students associate maths and physics with words which have a “negative emotional aura”. For example, they show how researchers identify physics with positive words, such as, “adventure” and “creative”. In contrast, for students, the negative perception of mathematics comes from negative association to specific tools such as “derivative”, “test” and “integral”. How can we be expected to communicate technical concepts to people when the words we use have completely different associations?

Given the way that research is usually presented (discussed above), it is no surprise that the majority of teachers do not regularly discuss research level mathematics or the details of what famous historical mathematicians worked on. Whereas, it is comparatively easy for them to exhibit or discuss recent work by artists. Exhibiting works of “great mathematics” to people will become significantly easier given the appropriate communication tools. I believe there is a role for practitioners in interaction design to develop the frameworks which will allow people to have a frictionless experience of mathematics rather than unnecessary frustration with jargon and formality.

Teaching people how to speak

It is not just the lack of inspiration provided in mathematics education which puts people off. The dry and formal way which the subject is traditionally taught also disaudes many from becoming involved. In his foreword to Alexander Bogomolny’s Probability Riddles Nassim Taleb compares learning mathematics to learning a language:

“How do you learn a language? There are two routes; the first is to memorize imperfect verbs, grammatical rules, future vs. past tenses, recite boring context-free sentences, and pass an exam. The second approach consists in going to a bar, struggling a little bit and, out of the need to blend-in and integrate with a fun group of people, then suddenly find yourself able to communicate. In other words, by playing, by being alive as a human being. I personally have never seen anyone learn to speak a language properly by the first route. Also, I have never seen anyone fail to do so by the second one.

It is a not well-known fact that mathematics can also be learned by playing –just watch the private correspondence, discussions and pranks of the members of the august Bourbaki circle. Some of us (and it includes this author) do not perform well on tasks via “cold” approaches, unable to muster the motivation to do boring things. But, somehow we upregulate when stimulated or when there is play (or money) involved. This may disturb many people married to cookie-cutter pedagogical methods that require things to be drab, boring, and bureaucratic for them to be effective –but that’s reality.”

He encourages a playful and pragmatic way for people to learn mathematics which differs from the standard teaching methodology. In my reading of this, being able to do mathematics might be less about ability and knowledge but more about encouraging people to develop the courage to get involved and try and speak the language.

What might playing with mathematics actually look like? Traditionally this might mean solving problem sets with pencil and paper. This can be an effective way to learn maths, however, going back to the sheet music introduced above, this might not be effective for people who struggle with the notational side of things. As an example, it’s possible for me to pick up the bass guitar behind me and have a quick strum along (without the requirement for sheet music), however, there is no analogous instrument to “interact” with mathematics. Software tools such as Mathematica and interactive environments like observable allow us to go further with code and playfully interact with mathematical ideas (though the current versions do tend to require some experience with mathematics or programming to be able to operator effectively). Furthermore, developing mathematical diagramming frameworks such as Penrose promise to make it easier for people to translate between formal notation and intuitive visualisations. As they put it in their paper:

“…the ability to translate abstract concepts into concrete illustrations is often limited to the select few individuals who have both a deep understanding of mathematics and an intimate knowledge of graphical tools.”

These tools and the communities formed around them are all fantastic. However, there is surely scope to go much further than this. We could imagine that combining ideas from interaction design, electronics, and mathematics might allow us to create technologies which will allow groups of individuals to discuss and “play” with mathematics in a social setting. Trained mathematicians can do this at ease in the presence of a chalkboard (which retains its nostalgic allure within academia), however, one might hope that we can go further and use technology to communicate mathematical concepts with no need for notation.

Taleb also makes the point that the natural way to learn mathematics is through more of a recreational approach rather than a classroom setting. The idea of learning through play rather than a formal setting should perhaps encourage people to stop taking mathematics so seriously. Granted, the world of formal institutions and rigour does have its own appeal and serves an important function for filtering out and refining the most important ideas. However, as I’ve tried to argue above, the current approach fails at providing a vehicle for widespread dissemination and frictionless consumption of mathematical content.

Summary

The common theme between the analogies presented above is that there is a huge gap in our ability to communicate technical concepts to a general audience. The difficulty that people have in communicating technical concepts generally means that we choose to not communicate at all. In the context of simulation models this hinders our ability to make all of the consumers of the output aware of both the scope and the limitations of the chosen modelling approach. In some cases this might lead people to blindly trust models which only work given a strict set of assumptions, while in others it may lead people to remain sceptical about well established results. Climate change is a good example of the latter, we have massive confidence about the existence of climate change, however, the complexity of the system hinders us from making a precise prediction about exactly where the system will end up. People tend to conflate scientists’ lack of ability to make an exact prediction with lack of any knowledge of the system at all.

In this article I have tried to portray mathematics as some kind of avant-garde art form which is worthy of more attention. Some may argue that this comes at the expense of emphasizing its place as a crisp, logical and rigorous way of thinking about problems. However, I believe that if we can do the former and make it easier to surface ideas without difficulty in communication, then the latter will naturally follow.