Diverse Modes of Human Thought in Mathematics

I still haven't figured out how my explorations in mathematics fit into my overall research. Why is reading math notation difficult for my brain? What is my personal relationship to mathematics? How has it become dysfunctional? And how can I fix it?

At first, I tried to fix the issue with better tools and memory system. I thought that my math notes are fragmented, because they are spread across different devices, so if I just OCR'ed tablet notes into a folder, it would make it easier to continue where I left off. Then, I realized that my problem wasn't a formatting problem at all, because mathematical thinking is operating beneath the level of language, symbolism and formalism. I realized that I was forcing myself to practice math in ways that don't work with my way of thinking.

This frustration prompted me to write this article. Here are some thoughts, quotes and links on math education that I collected over years.

Talia Ringer 🕊 on Twitter / X

Terry Tao and I spoke over coffee for like two hours yesterday, in part about diversity in how people think about math. We both agreed that people who hit these walls early mostly don't learn the way of thinking about math that works for them. It's an educational failure https://t.co/1JMrP7H3uy— Talia Ringer 🕊 (@TaliaRinger) July 18, 2023

The educational system for some reason gives people the impression that if they cannot think about math in the way they are taught to think about it, they are just bad at math. But often they just need a different way of thinking about it

I think it's extremely silly to imagine that mathematics exists on some hierarchy from "easy" to "hard." Mathematical thinking is many things, and many people who find earlier courses hard (arithmetic, calculus) excel at, say, abstract algebra

Terence Tao (@tao@mathstodon.xyz)

A small anecdote in relation to a recent coffee conversation I had with @TaliaRinger@types.pl (which they relate over at https://twitter.com/TaliaRinger/status/1681410191278080000 ): Yesterday I spoke with a children's book author who was interviewing me as part of a series she was writing on contemporary scientists. She freely admitted that she did not have great experiences with her math education at an under-resourced school and chose very early on to focus on writing instead. Nevertheless we had an excellent conversation about many mathematical topics that she was not previously familiar with, such as proof by contradiction, Cartesian coordinates, Mobius strips, or compressed sensing, all of which she found fascinating (and said she would read up on more of these topics herself after our interview). I posed to her the isoperimetric problem (using the classic story of Queen Dido from the Aeneid as the intro) and she correctly guessed the correct shape to maximize area enclosed by a loop (a circle), and instantly grasped the analogy between this problem and the familiar fact that inflated balloons are roughly spherical in shape. I am certain that had her path turned out differently, she could have attained far greater levels of mathematical education than she ended up receiving. This is not to say that all humans have an identical capability for understanding mathematics, but I do strongly believe that that capability is often far higher than is actually manifested through one's education and development. Sometimes the key thing that is missing is a suitable cognitive framework that a given person needs to align mathematical concepts to their own particular mental strengths.

Sometimes the key thing that is missing is a suitable cognitive framework that a given person needs to align mathematical concepts to their own particular mental strengths.

On proof and progress in mathematics

In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory...

People have very different ways of understanding particular pieces of mathematics. That's good! Mathematics lives and breathes from human diversity.

There are two counters to this trend, so that mathematics does not become entirely mired down in formalism. First, younger generations of mathematicians are continually discovering and rediscovering insights on their own, thus reinjecting diverse modes of human thought into mathematics

... we need to pay much more attention to communicating not just our deﬁnitions, theorems, and proofs, but also our ways of thinking. We need to appreciate the value of diﬀerent ways of thinking about the same mathematical structure.

This is a list of diﬀerent ways of thinking about or conceiving of the derivative, rather than a list of diﬀerent logical deﬁnitions. Unless great eﬀorts are made to maintain the tone and ﬂavor of the original human insights, the diﬀerences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit deﬁnitions.

vitalik.eth on Twitter / X

It's very easy to have full up-to-the-frontier knowledge of one part of a field, even to the point that you're making original contributions, while still being "first-year-undergrad" level in a directly adjacent area.— vitalik.eth (@VitalikButerin) April 7, 2020

The fundamental problem with even thinking about it in terms of "high school / undergrad / postgrad" is that it assumes that everyone takes the same path to learning a field. This may have been true 20 years ago. This is increasingly not true today.

... remember that your readers will have very uneven levels of understanding of any background material a topic requires

What's a mathematician to do?

I have to apologize because this is not the normal sort of question for this site, but there have been times in the past where MO was remarkably helpful and kind to undergrads with similar types of

[..] mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification

David Deutsch on Multiple Worlds and Our Place in Them (Ep. 124)

Plus, the defining human attribute visible from galaxies far, far away.

If I can answer in a single word, the way I would improve it is diversity. There should be diversity of funding criteria. There should be diversity of funding sources. There should be diversity of criteria for choosing research projects, and there should be diversity of criteria for choosing people for promotion and for being funded.

Any kind of standardization is the opposite of diversity. Just like I say you should have disobedience lessons in schools, so you should have unstandardizing objectives for science education and for how you run scientific research.

A Mathematician's Lament

It is far easier to be a passive conduit of some publisher's "materials" and to follow the shampoo-bottle instruction "lecture, test, repeat" than to think deeply and thoughtfully about the meaning of one's subject and how best to convey that meaning directly and honestly to one's students. We are encouraged to forego the difficult task of making decisions based on our individual wisdom and conscience, and to "get with the program." It is simply the path of least resistance

The most striking thing about this so-called mathematics curriculum is its rigidity. This is especially true in the later grades. From school to school, city to city, and state to state, the same exact things are being said and done in the same exact way and in the same exact order. Far from being disturbed and upset by this Orwellian state of affairs, most people have simply accepted this "standard model" math curriculum as being synonymous with math itself.

This is intimately connected to what I call the "ladder myth" - the idea that mathematics can be arranged as a sequence of "subjects" each being in some way more advanced, or "higher" than the previous. The effect is to make school mathematics into a race- some students are "ahead" of others, and parents worry that their child is "falling behind." And where exactly does this race lead? What is waiting at the finish line? It's a sad race to nowhere.

Real mathematics doesn't come in a can - there is no such thing as an Algebra II idea. Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of the subject, and a teacher's own path through the standard curriculum reinforces this myth and prevents him or her from seeing mathematics as an organic whole. As a result, we have a math curriculum with no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques, united only by the ease in which they can be reduced to step-by-step procedures.

I'm Paul Lockhart, author of A Mathematician's Lament, Measurement, Arithmetic, and The Mending of Broken Bones. Ask me anything!

401 votes, 128 comments. Thanks again everyone!

What do you think you mean by being good or bad at math? The question is your own personal relationship with Mathematical Reality. Do you want to go there and check out all the interesting beauty and pattern? Then just go. You do not need anyone or anything, just desire and curiosity. So what if you are not very good at it?

Bad design

Kill Math

The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols.

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.

Which is the premise of this project, of course -- people don't use math. But everyone seems to believe, if only math were taught better, they would use it! And my position (and the entire point of the project) is: No. 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.