Why is the state of mathematics education so abstract and uninspiring?
I came across this article by V.I. Arnold : https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html which is rather old. But some of the points mentioned in the article can be related to problems in the classroom today also.
People have an idea that being abstract and talking in abstract terms creates some sort of elitism. But it hampers understanding and excitement at the nascent stages. Abstraction is required to tackle complexity. But that is not the all and be all of the domain.
It can be taught like other natural sciences starting with real life examples and building up. It is much more clearly written in the article.
I would very much love to hear about books or courses that teach mathematics in the way mentioned in the article.
155 comments
[ 1.7 ms ] story [ 222 ms ] thread> But it hampers understanding and excitement at the nascent stages
Ultimately, self motivated learners dramatically outpace anyone else. A genuine interest will overwhelm any other factor, because the self motivated learner will not need to invest additional time finding excitement.
Artificially motivating learners is a difficult, time consuming, and often marginally effective task.
The only way I've seen consistently work is to have truly interested and fascinated public speakers.
Who then teach to the 5%, and perpetuate the cycle.
The most valuable learning you can get from a book is reading the table of contents, then the preface & foreword, then leaving it alone for 5 days.
In the interim, the table of contents seeps into your brain. You will be BRILLIANTLY prepared for the content.
We should not try to motivate students, we should try to not kill students enthusiasm. And for this the very early classes are essential.
If you kill a persons enthusiasm in a subject or in learning itself, you will afterwards toil years trying to motivate them.
I agree with the rest of your comment, but this is not true. I teach university maths. We are well aware that most of our students are not majoring in maths and will not go on to teach it. Most of them at my university will go on to be engineers, so we teach for that in their courses, light on proofs and derivations and focused on grounded practical knowledge and techniques.
> We are well aware that most of our students are not majoring in maths and will not go on to teach it. Most of them at my university will go on to be engineers, so we teach for that in their courses, light on proofs and derivations and focused on grounded practical knowledge and techniques.
One important part of why I have this opinion, mathematics is poorly taught to the lower 95th percentile, is historical.
Historically, pre 1950 & the GI Bill only the wealthiest students (who sat at the back of the class, partied, and generally focused on Latin not the hard sciences), and the most able students, went to university.
The reason I find that's incredibly relevant is this Hackernews post. Easily my favourite of all time.
https://news.ycombinator.com/item?id=32974959
In my mind, the consequence of all that is teaching the 5% is a well worn, highly polished process.
Hundreds of years of effort has been invested in the 5%.
For the 95th, barely a few decades. And even then, many academics I find take the elitist point of view.
In my experience, your department's point of view is very much the exception. A majority of academics I find follow one of two options:
-> The lazy option.
-> Train my future graduate students, bugger the rest.
IMO, the methodologies for training that lowest 95% are not at all developed.
Arguably for example, one theory I have is proof based mathematics should be introduced in Grade 7.
Proof based mathematics is the simplest, quickest, easiest way for an engineer to learn a new piece of mathematics.
A proof is a quicker, and more time efficient, teaching tool than 5 exercises. Is my theory.
And yet, the status quo is only pure mathematicians need to know proofs.
I'm reminded of Lockhart's Lament, https://www.maa.org/external_archive/devlin/LockhartsLament....:
> "Um, these high school classes you mentioned..."
> "You mean Paint-By-Numbers? We're seeing much higher enrollments lately, [...]"
> "And when do students get to paint freely, on a blank canvas?"
> "You sound like one of my professors! They were always going on about expressing yourself and your feelings and things like that—really way-out-there abstract stuff. I've got a degree in Painting myself, but I've never really worked much with blank canvasses. I just use the Paint-by-Numbers kits supplied by the school board."
OTOH, that was [mumble] decades ago, and I've not heard much good about trends in university teaching since then.
It's that "intermediate definitions" that causes too much indirection, just like you read a codebase with too much abstraction and redirection, you've lost.
It's like learning programming by first learning how transistors work, then microprocessors, then assembly language, then compilers, etc. While across the hall, the engineering department is learning Python directly.
But I think it is necessary, because rigor is why mathematics is useful. We can (and do) teach statistics without calculus, but the result is often bad stats and bad science.
Going the non-rigorous, fun way you get results faster, but you're foundation will be weak. That Python coding engineer - he will be limited in what he will be able to do. Fast, production level code more often is written in C, C++, Go etc. than Python.
Why do people in the west always think everything needs to be "fun" from the beginning? It can become fun at some point, but it doesn't need to be from the start.
Due to some reason (I'm not sure why) their work made a huge imprint on how math is taught around the world.
https://en.wikipedia.org/wiki/Richard_Feynman#Pedagogy
For advanced mathematics, I guess it has to be abstract. I mean this is the nature of the subject. Normally you are taught group theory with examples of groups like natural numbers or geometric symmetries, but it soon becomes groups derived from other groups etc. To make it non abstract would be to make gymnastics that doesn’t require flexibility or a fighter pilot that doesn’t need to handle high g force. I don’t think you can do it. Ink spilled on Monad tutorials a case in point. See also: programming!
Money is an incredibly boring topic unless you're making a lot of it.
I had an opportunity to see how "workplace" (remedial) math was taught in an Ontario school in the early 2000's. It was basic financial literacy, more along the line of keeping one out of financial troubles than handling large sums of money. Even though many of the students struggled with basic arithmetic, they tried to keep it concrete and the students seemed to remain sufficiently motivated to stick to it even though they faced huge hurdles.
Interesting article! It argues that when mathematics became dissociated from both physics and geometry, it became this horrible abstract algebraic kabbala that's at once disgusting and pointless.
The article talks a lot about France; I think in France the problem is a problem of selection / elitism: how to rate lots of students in a reproducible way in order to keep only a few. This process doesn't leave much room to rêverie or the joy of learning. It's a disease that modern proponents of "le mérite" don't understand. ("Mérite" is hard to translate; maybe "self worth"; it's a vast debate in France).
More generally, the problem of teaching is that it tries to put results in the heads of students. Some results took years or centuries to discover, and it would be impractical or impossible to tell the story of that discovery. But our minds are story machines; stories are the only thing we understand and crave for. A list of facts is the most boring thing imaginable and the hardest to remember. Yet it's not clear if there's any other way, esp. because there are so many facts to learn, and so little time.
Yet that doesn't excuse everything. It should always be possible to connect theory to real world applications. Here's a little observation about geometry for example.
My kids are currently learning about the Pythagorean theorem, where in a square triangle, the length of the "hypotenuse" is the square root of the sum of the square of each opposite side. But what does "hypotenuse" mean?
In math books (as well as on every website that I could find, see https://www.google.com/search?tbm=isch&q=pythagorean+theorem) the square triangle is always shown with the square angle at the bottom, usually to the left, and the long side (the hypotenuse) is a slope going from north-west to south-east on the page.
It turns out, hypo-tenuse means "that which supports". Obviously if the square triangle sits on its square angle, the hypotenuse doesn't support anything. But if the hypotenuse is instead horizontal and the square angle at the top, then we begin to see the front of a temple.
The Pythagorean theorem helps solve a myriad of problems, but one of them is extremely practical: how to build a temple with a square top at the front. That's more interesting than a²+b²=c².
Spoken like a true engineer :D
Discusting? Maybe.
Pointless? Gröbner basis!
You have to think up a new ontological idea (eg, imaginary numbers to “complete” the solution space of polynomials) before you can apply that to modeling things, eg signal analysis in electronics.
Category theory to some degree has this problem: the ideas are useful, but only once you’re at the level of “I know many areas of math and have noticed proofs have similarities… how do I discuss that?” …but it turns out that abstract problem is useful for computer science, eg design patterns.
What about number theory? Or theory of computation? I can see them happily exists while not related to physics/geometry hmm...
(Read the entire article. It's hilarious, and surprising that Nature decided to publish it.)
"Euclid is the worst. It is shocking that young people should be addling their brains over mere logical subtleties, .... I hold the view that it is essentially an experimental science, like any other, and should be taught observationally, descriptively and experimentally." - Oliver Heaviside, "The teaching of mathematics", Nature 62, pp 548-549, 1900. [1]
[1] https://ia600708.us.archive.org/view_archive.php?archive=/22...
As for Heaviside he was self taught, an engineer in spirit rather than a mathematician. In a more modern mathematical perspective, not derived by intuition, Maxwell equations are just one very simple equation in Clifford algebric spaces. This formulation will not prevail since the engineering (intutitive + sleigh of hands) approach has dominated instant gratification higher educational level curricula.
I am not saying that what I am doing is different, I am from an engineer background also.
But having had that mindset, I eventually met mathematical minds and was humbled by their vastly different approach. This approach is not for everybody though, the yellow books are difficult to parse. Having said that I find that the mathematical mathematics approach is applicable in all of engineering whereas the engineering mathematics approach is not applicable in all of math.
But teaching every kid as if the best thing school can do for them is to guide them to the door of calculus through which they will pass unto untold collegiate adventures has been a disaster and should stop.
You will still have your math majors. The school system should make an effort to conduct those kids towards that end but that should cease to be the emphasis. The school of wizarding has a way of finding its own and we should be focused on more people getting more use out of the math they do learn.
I wouldn’t mind if math education followed science: split up the course between lecture and lab!
Just listen last Klaus Schwab speech at G20/Bali https://youtu.be/DQjXODh0TOg where he describe the near future he want:
- IoT, poetically named Industrie 4.0, to allow OEMs control from remote, witch means de facto owning hardware formally sold to someone who legally/theoretically own it now;
- Sharing Economy witch means re-sell continuously the very same object, as a service/leasing etc making peoples compete to access such scarce and needed resources;
- corporatocracy, prosaically named Stakeholder Capitalism or Corporate Governance, but practically meaning: Democracy must end, substituted by a dictatorship of those who own and know against all others.
Followed by it's interview at APEC 2022 https://youtu.be/NWHDgXhMkgs where he state that China "it's a role model for many countries" and "we should be very careful in imposing systems. But the 'Chinese model' is certainly a very attractive model for quite a number of countries".
You need Gustave Le Bon bipedal bovines to realize such vision. Pushing the old « you were not made to live like brute beasts, but to pursue virtue and knowledge » would be a disaster for such model... People must know just the very little they need to do some tasks, NOT the whole picture.
They are answering why education is "so abstract and uninspiring", talking about the educational system specifically. Not peoples abilities. They are actually implying that people could understand if given the chance.
Bernays says this in his book Propaganda;
"The conscious and intelligent manipulation of the organized habits and opinions of the masses is an important element in democratic society. Those who manipulate this unseen mechanism of society constitute an invisible government which is the true ruling power of our country. We are governed, our minds are molded, our tastes formed, our ideas suggested, largely by men we have never heard of."
The same idea has persisted in different forms into the current "Modern Times".
How would you know if you're yet to learn the stuff?
Some math can be taught that way, not all.
most likely the debate reflects two distinct modes our brain is handling mathematical notions, and different people being more adept in one or the other
anybody who went through V.I.Arnold's mathematical methods of classical mechanics has no doubt which way his brain works :-)
The Bourbake school was in a response to the disaster of the Italian school of algebraic topology which put a lot of focus on rigor and formal proofs vs the hand waving Italians.
They where good in reforging existing proofs but when it came to imagining new ideas it where the visual and experimental mathematicians who kept pushing the needle forward.
[1] https://www.youtube.com/watch?v=drZEPzb3JY0
It's that type of introductory book written for the expert that already knows the theory.
The abstract nature of maths is what I find appealing about it. I prefer explanations that explicitly treat the abstract nature of what is being taught to ones that try to bootstrap it from examples. I like baby Rudin. I find pure maths easier and more enjoyable than applied.
There are different kinds of people in the world.
Mathematics usually always incorporates application. Rates of change is an example where you are applying the concepts of calculus along with geometry to solve real world problems.
As the article describes, you have to /feel/ certain concepts before you can apply the abstract terms to them. Getting students “playing”, seeing many many examples, especially from one another, and building their own examples is key. We build upon the open Illustrative Mathematics[1] base for scope and sequence, but dramatically improve the pedagogy IMO.
One good lesson to see what it’s like is the Line of Best Fit:
https://teacher.desmos.com/activitybuilder/custom/56fab6bc1a...
It’s hard to do well and not every topic is fully amenable to it! But we’re working hard on it (and PS, hiring engineers… email me!)
[1] https://illustrativemathematics.org/
https://mathigon.org/polypad#numbers
That is not an excuse to avoid good methods of learning.
https://www.maa.org/external_archive/devlin/LockhartsLament....
If we taught music like math, you'd spend years learning sheet music and theory before being handed a musical instrument. Note, I love the triangle area example and how it is extended to a cone's volume.
I used to teach high school math (turns out you can earn more as a software engineer), and I respected the curriculum we adopted, CPM. It wasn't a silver bullet, but it focused on conceptual understanding by investigating relationships and patterns and formalizing those into "normal" math.
By way of example, linear graphing is explored through tables, plotting, and uncovering y=mx+b over a couple projects. When I was taught the same, it was explicitly instructed starting with "m represents slope and b the y-intercept, now match graphs to their equations" - building mental models that connected concepts was left up to the pupil.
One teacher colleague who fostered lots of kids became convinced of the CPM model after working with algebra tiles with one of her fostered/adopted daughters who really, really struggled with symbolic and abstract representations. The tiles made the abstract real for the kid and allowed them to complete high school math.
Many concepts I learned as pure arithmetic are taught as visual and/or spacial concepts. Often there are multiple strategies for reaching the same answer.
This is a kind of math I first encountered while working as a contractor and later expanded upon when studying computer science. It is ubiquitous in the trades. I think parents don't like it so much because it draws upon a flexibility with math they were basically discouraged from exercising.
If you have a figure like this, you write this, then this, then "according to Thales' theorem", then that result.
No, you didn't write "according to Thales' theorem", the result is wrong. Write it again.
No, you can't write it in this equivalent way, you have to write it as taught. Write it again.
I went a bit further in math afterward so I could at least grasp at the essentials of calculus, but that one moment took a lot of the wind out of my sails.
At the faculties of two other universities where I worked, statistics was taught at a lower level, with as few formulas as possible, and a great number of examples, often with (subsets of) real data, directly related to the topics they were studying. Most students did not get it, many failing their exams multiple times. Then again, they were psych students. There was only a handful with some of interest in statistics, and they usually went on to study cognitive/experimental psychology.
There's no method that suits all, and not every student should be expected to reach the same level. It's ok if students don't master anything beyond the most basic of maths. It's a pity, but considering that many leave secondary education barely literate, it's not high on my list of priorities.
If people wanted to learn, they'd clearly go back after the course finished, and fill in the gaps. Much like you'd have a dentist fill in the cavities they found. You don't want a painful gap in your stats understanding, right?
But that isn't done. Unfortunately, the point of stats class is to pass stats class, not learn stats. The students are only responding to incentives, and society (in general) does not care about long-term understanding. You scrape by, say you "passed" -- like being fit 30 years ago -- and that's enough for the HR screen. Nobody ever follows-up.
- At the Open University in the UK you can get degree in "Mathematics and its Learning": https://www.open.ac.uk/courses/maths/degrees/bsc-mathematics... There is a reasearch group at the OU behind this.
- Department of Mathematics Education at Loughborough University: https://www.lboro.ac.uk/departments/maths-education/
- Mathematics Rebooted by Lara Alcock: https://global.oup.com/academic/product/mathematics-rebooted... (She's at Loughbourgh, linked above. She has also written some books for University level mathematics.)
- Frank Farris' gorgeous book "Creating Symmetry" (https://press.princeton.edu/books/hardcover/9780691161730/cr...), which has this brilliant passage:
This belief that my motivation deserves mention moves me to call this a postmodern mathematics book. By contrast, modern mathematics books were written in the twentieth century by intentionally voiceless authors for an intended audience of "the hypothetical anybody", which made the books feel cold and inaccessible, at least to me. Postmodern books are situated in time and place, taking into account the identities of both reader and author. Here I am, writing to reach you: please join me.
“The future is already here. It's just not evenly distributed yet”
Most people just want arithmetic, what's the tip on this bar tab? What do I put in the capital gains box on this tax return?
Universities, businessess and most who will go into other STEM paths want mathematics.
And mathematics is incredibly abstract and the reward are plenty of it's your thing.
I haven't read any recent research on this (cause I'm not much interested) but there is potent correlation between IQ and interest/ability in mathematics (and physics). This is telling (or concerning depending on your viewpoint) as IQ is a normal distribution meaning actually statistically some people are unlikely to grasp or be interested in the maths we care about.
TL;DR You wont understand why quadratics are so important until you encounter integral calculus. You wont appreciate complex numbers until you encounter complex analysis (well maybe some physics).
The same way you wouldn't appreciate cement until you encountered a brick wall.
It's doubly damaging in countries where the government works by majority voting. I saw that when comparing family members (one of whom dropped out after middle school) watching the news. They just don't have a good enough understanding of basic math and logic to have a useful opinion, so they take the mental junk food some politician throws at them.
All of that to say, teaching as advanced math as possible to as many people as possible is in everybody's interest. It's not about enjoyment or being able to write proofs, it's about developing children's and teens' reasoning skills.
Many simply aren't interested and seem to resent abstract mathematical education. And simultaneously desire better accountancy skills. This would be a way to achieve that
Maybe as a one-hour-a-week high school course. The "economics" taught there were so basic and incomplete as to be useless.