I used to be a professor. I did find I got more engagement from students when course material is grounded in reality, but not the same way a lot of posters are suggesting.

Using math to solve real world problems is all well and good, but it has the same problem as teaching math in the abstract. The theorems and techniques are all handed down from on high, like some divine miracle given to humanity by the gods of logic and reason.

This works fine for the tippy top of the class who just take the material and run with it (I was once one such), but it loses everyone else. People need context. I tried to, as much as time permitted, give the historical context for the math I was teaching. To show that mere men developed it, that they were trying to solve a specific problem, that they built everything up on what came before.

I got great results. I wish I had more time to do it, but, well, when the department standardized end of course test requires that they be able to do n kinds of derivative rules and solve m kinds of integral, I only had so much time to talk about how Isaac Newton invented the cat flap.

I really wish mathematics was taught like a history class. Mathematics, the way it’s taught, rarely ever explains the Why of a concept. For example, I understand that calculus is useful. You’ve shown me a whole bunch of uses for it. But why did we come up with calculus in the first place!

Learning about the motivations of the historical actors involved in making discoveries, and learning about the steps that led to those discoveries gave me a far more complete understanding of mathematics.

It also removed this massive whiff of arbitrariness that has always existed with mathematics for me.

Me too. I think Physics taught as history class would be extremely fun. Say you have to follow the steps of Kepler and do a few observations by yourself.

In college, I took a History of Science class about Copernicus and Kepler. It totally could have been adapted into an engaging physics course.

> I tried to, as much as time permitted, give the historical context for the math I was teaching.

This is great.

Funny how teachers generally aim for their students to 'understand' instead of merely 'replicating' (from memory). Yet math is often taught as a complex, fully formed system (especially in university).

Teaching math along the same path / outline it was historically developed teaches the 'why' alongside the 'how'.

Exactly. The why is missing. The how is inadequate in a vacuum. Bright students make the connection on their own, but the others, even highly motivated ones, get left behind.

To make the problem worse, quite a lot of my former colleagues don't know the why themselves. They've always thrived on abstraction, and are unable to understand or help when motivated students come to them seeking the why.

Bit of a chicken and egg problem, really. The professors who made it did so without the why, and therefore don't understand the need for it.

I'm as guilty as anyone. I couldn't tell you much about where my PhD research came from and why. At least my calculus students got some context.

Agreed.

I occasionally teach a strength of materials class, for mechanical engineering students, as a contract lecturer (my main job is research).

I love starting the fatigue part of the class with some historical context such as the Versailles/Meudon train accident (https://en.m.wikipedia.org/wiki/Versailles_rail_accident). I think it really helps to motivate the content and how the "laws", which can be sort of dry and boring out of context, came to be discovered by scientists and engineers who wanted to prevent these kinds of accidents from happening again.

I enjoyed your example. When I was a student, these types of stories made the math more relatable. I remember these apocryphal stories; I doubt their authenticity but found them amusing.

- Pythagoras was part of what we would call a religious cult. They were trying to discover the secret of the universe when they stumbled on PI and the Pythagorean theorem.

- The Sumerians counted in sixes. From them, we have sixty seconds in a minute, sixty minutes in an hour. Don't know why a day only has 24 hours. Maybe the royal mathematician goofed up and got sacrificed to Tiamat.

- Today, we count in tens. But you see artifacts of counting by twenty in our language. Lincoln's Gettysburg address begins with Four score and seven years ago our fathers brought forth, upon this continent, a new nation. Four score means four twenties or eighty.

- Trigonometry was created by enterprising Greek merchants who had fleets of ships. One problem they had was determining how far a ship was from port. They solved the problem by creating a virtual right triangle. On shore, they marked a "shore line" which they would place two observers. The two observers always stood on the line and at a known distance from one another by means of a fixed rope. The first observer would walk up and down the line. The second observer would adjust his position on the line to keep the same distance from the first. The first observer would walk up and down the line until he thought he was "perpendicular" to the ship which was at sea. The second observer would then write down the estimated angle between the shoreline and the ship. By knowing this angle (theta), the fixed distance between observers (adjacent distance), and the right/90 degree angle... they could calculate the distance to the ship (opposite distance). In other words, the distance to the ship is equal to the tangent of theta multiplied by the adjacent distance.

Two of your anecdotes are wrong, as far as I can tell. Please give sources.

Could it be due to the fact that, college level mathematics are meant to prepare extraordinary students to graduate into research. I imagine research often involves not-so sexy parts of mathematics.

I hear a lot of similar sentiments in school level mathematics in the country i live in. The problem comes in the trade-off. Do you cater to the topmost/ most promising students, who will end up contributing a lot in science, or do you cater to the ones who need most help, who will end up getting some skills on this, to make a living ?

Would love if you could name some books that teach like that, since we can't get you as a teacher.

I thought William Dunham's Journey through Genius: The Great Theorems of Mathematics did a great job of presenting the very real people and world behind several of results from history, from 400 BC (Hippocrates' Quadrature of the Lune) to recent-ish times (Cantor's transfinites).

There's discussions of feuds between mathematicians, how they used to keep results secret and literally duel over ideas (e.g. Cardano's chapter), the historical context and lives of the people who brought theorems into being (like Newton tasting his own chemicals as part of his alchemical research!), etc.

I’m a fan of Mathematics for the Nonmathematician [1]

[1] https://a.co/d/grHB5he

It doesn't go into specific stories but it does explain the process: How to Solve It by George Pólya.

Story telling is very powerful, including for educational purposes.

It could clearly be used more often. Doing real-world projects is excellent for engagement but also costly and not always practical.

teaching the history of science along with science is a much more effective way of teaching for the average student but alas schools are more concerned with operating in an assembly line fashion in which they try to stuff as much knowledge into a person as possible (which the student will undoubtedly forget soon after after they study enough to pass exams)

It can be perilous if you don't do it right - where new discoveries and methods are taught as if they are inevitable because you only look at the history when it is related to modern understanding.

Time for you to release a set of video lectures! [crosses fingers…]

Absolutely love this approach!

I've always loved math, but one class I didn't expect to like as much as I did was The History of Mathematics. The mathematical content was usually very basic of course, but putting all the things I was learning in my other classes in a historical narrative was eye opening and magical.

It also gave me a way better understanding of why we do math the way we do today. Of course mathematical truths are the same regardless of notation and method, but even simple things like the prevalency of the "=" sign are choices we made along the way in history. Today algebra is almost synonymous with math for many people, but it didn't even exist as a branch of mathematics until al-Khwarizmi he just wrote his equations out in words! But we made a concise notation over time and we're all much better for it.

And some parts are just funny. Like math "duels" over cubic equations in 1500s Italy

Mathematics: From the Birth of Numbers is an illustrated, well written history of mathematics, helped me see how all the different concepts came about. https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullber...

Because mathematics, like most tertiary disciplines, has always been taught in a teaching style suitable for the 5% of the audience who aim to go on to be mathematics professionals (who then teach mathematics).

> 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.

> Because mathematics, like most tertiary disciplines, has always been taught in a teaching style suitable for the 5% of the audience who aim to go on to be mathematics professionals (who then teach mathematics).

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.

Thank for your response. That's actually hugely insightful.

> 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.

You teach abstract algebra and real analysis that’s light on proofs?

No, engineering students don't take those courses. We teach as appropriate to the students in each course, unlike what urthor claimed.

> Because mathematics, like most tertiary disciplines, has always been taught in a teaching style suitable for the 5% of the audience who aim to go on to be mathematics professionals (who then teach mathematics).

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."

I think you are conflating enthusiasm with motivation.

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.

Enthusiasm can be contagious.

Yes, my brother, then 17 explained to me, then 10, the principles of calculus because he was so excited by it. He did it in pictures (little slices etc) and I got the basic idea, though I had no use for it. I just caught his enthusiasm.

You'd be surprised just how significant that prompting was to your future learning.

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.

Thanks for the suggestion. I’ll try this next book I read :)

If you're reading a math or other computational or textbook, I'd also suggest to read through the exercises in the back of the chapter before diving in. It primes you with keywords from the questions and helps you to think of how you can use what you've just read.

Yes, but so poor is my memory that when I eventually sat my exams I had to do most of it by those first principles instead of the known formulas!

I can see how that sucks during the exam, but knowing how to do it from first principles will stick with you for much longer. When I told my students that I went into mathematics because my memory is so bad, they never believed me. But it is the truth (at least somewhat).

Honestly a pet peeve of mine is how everyone in math tends to pretend that it is easy, at least once they cleared their own personal hurdles. This is, I guess, a side product of the fact that the world of math harbours almost exclusively people people that "got it" early.

If you hear or read discussions between undergraduate math people, they tend to act as if they discovered calculus, the basics of linear algebra, and analysis from the ground up all by themselves!

Same thing with professors constantly scoffing at all the "boring" and "trivial" parts.

What changed my life back then was a math professor for a mandatory course I had to take. He always emphasized that this stuff is hard, even for him, even the "boring" undergraduate basics. Hard, but doable, and trainable. After years of "this is easy, why don't you get it?" math education I was stunned by these displays, it's not that everyone just "gets it", aside from certain geniuses, it's that some get a headstart.

This left such an impression on me that I switched to math major and got my bachelors degree in math (I was very bad at school math). And that one professor is still a huge motivator whenever I try to overcome something I struggle with.

Fully agree.

There are places in math where you can find some cool examples. You go on YouTube and find videos of 3Blue1Brown, or Veritasium, or Mathologer, and you fall in love with them.

But unfortunately, not 100% of math can be taught this way. Not even 10% or 1%.

And math is not a spectator sport. You need to get your hands dirty.

Still, can it be taught a bit better? Khan Academy seems to me as close as you can currently get to teaching better math.

Methods I would use to teach math from scratch to day would be:

- get in a boat at night and do celestial navigation to get to a camp site (geometry)

- prove the earth is round using the moon. (geometry, proofs)

- count cards in blackjack and poker (combinatorics, probability, game theory)

- decode an ancient manuscript by constructing a grammar from frequency analysis (probability, statistics)

- decode a real american civil war cryptogram. (Vigenere, number theory)

- make a radio out of found stuff and transmit a message in various encodings by, finally with binary (from ohms law to information theory)

- reproduce Turing's Bomba from first principles in code to crack Enigma (number theory, computation)

- bet on a stock market return (brownian motion, randomness, shannon portfolio / information theory)

- structure an an election strategy for your candidate across multiple polls (conditional and independent probability, use dice and weightings for each poll result)

- work on a motorcycle / pocket-bike engine and optimize it win a drag race against someone elses engine configuration (differential calculus, sprocket sizes, power curves)

- synthesize a drum sound using an oscilloscope then sequence a loop and drone composition from oscillators and an ADSR filter (integrals, fourier analysis, feedback, deterministic chaos, complexity classes)

- configure or code a basic neural network to identify a signal or encoding. (complexity classes, godel's incompleteness)

- determine whether a piece of music is related to another (graphs, cosine similarity, pythagorian distance, homomorphisms)

That's off the top of my head, but each of these are 1-day to 2-week projects that give you a working competence in the area, imo. You could run a month long kids/teens camp on them.

These are all good projects but you are underestimating the difficulty of them by a lot. Even something conceptually simple like card counting has a lot of details that are difficult to get right. Just programming the rules of blackjack is a fiddly problem in itself (no standard ruleset and hard to explain the edge cases).

In my opinion these would be great things to introduce at the start of a course for motivation, but they either need a lot of handholding/scaffolding or a lot more time per project. There is a tradeoff between how much handholding you do and how much people learn. It's very easy for people to fool themselves into thinking they understand something just because they can follow along while someone else does all the thinking for them.

I'd say this view overestimates both the depth and abstraction at which people actually understand the things they do in the course of normal living. These exercises won't turn anyone into a trivia master or gatekeeper, but it will give them working knowledge.

The point is that each of these things requires actually doing them and they are fun enough to practice, so nobody merely follows along. Sure, a class of 30-40 kids won't be able to do it, but that's the worst possible way to teach anyone anything while only guaranteeing they are all evenly disadvantaged.

A blackjack tournament or poker tournament over a few days is enough to get card counting principles down, and even if you are crap at it, you understand that probability is a thing, it has distributions, it's not always right, and you can bet against other peoples ability to interpret them.

> - bet on a stock market return (brownian motion, randomness, shannon portfolio / information theory)

You can observe that many many day traders with real money on the line don't see these as important. Students who aren't even in it for real money need a lot of guidance to avoid chasing the greatest (fake) return by pursuing high variance strategies. Or maybe you discuss that's what they're doing. Could be interesting.

All of these are much more interesting if you do them after you understand the mathematics behind them.

To me the greatest feeling of understanding always came if I saw a complex system and could see how and why all the parts worked. I had a grear course about computer tomography and the one thing which really struck with me was that fractional detivatives have saved lives.

Teachers need to ask every student in the class one basic question right from the offset:

- Are you more interested in things or are you more interested in people?

Once you’ve separated the students into these two camps you can tailor the content to match the interest much better. You’re probably going to have to tunnel down some more into their interests and get creative to really engage them but in my opinion this is the most bang for your buck question that teachers should be using to tailor their mathematics education to the individual student.

Forcing a group of kids to not only rote learn formulas but also to rote learn formulas to solve problems that have no relevance to any of their interests in real life is the reason mathematics is so hated by so many people. Creating problems about “how many sweets can Gary buy if…” is such a poor attempt at making it interesting to kids it’s actually insulting. Educators need to start putting some proper effort in and stop treating kids like they’re robots.

This is a great idea!

You should write a book about this, and include some sample calculations in the appendix

Love this list, thank you. My simple contribution is building a parabolic solar cooker.

My company (Amplify) is trying to change this. We make the Desmos Math curriculum ( https://teacher.desmos.com ), which teaches math with digitally-enabled live, interactive, social exploration of mathematical concepts.

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/

Amplify also owns Mathigon, which is amazing.

https://mathigon.org/polypad#numbers

This looks cool. But do I have to be a student at a school to use it? Can I pay a few bucks a month to do it in my free time as an adult? Thanks!

Thanks for the interest! It’s designed for use in schools, but let me see if there’s something reasonable we can do. I’ll reply back here.

I love your scientific calculator and graphing calculators!

Thanks! Those are made by a close partner, the Desmos Public Benefit Corp, which split off from the curriculum to focus on keeping the calculators free and widely available. But much of their DNA is in the curriculum :)

Related article, Lockhart's Lament.

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.

I am looking for ways to "supplement" (or compensate ) for my sons math schooling - CPM looks interesting but if it is just solo study is it any more effective than the usual textbook and exercises ?

CPM benefits from a group setting where a couple (or more) of students can strive together, but even without a group setting there are lots of treasures.

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.

A quote by the founding father of modern electrical engineering, and the formulator of the current version of Maxwell's equations, Oliver Heaviside:

(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...

Real World intuition is a habit one needs to abstain from to be able to be creative in the field of higher level mathematics. I have a bad feeling that also modern physics who has been pretty much stale since the early twentieth century is not due to lack of intuition but due to lack of mathematical models.

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.

My grandfather counted in threes for his entire life because working in the stock yards of Omaha the cows came down the schute in threes. The application of such math fed him until he died at the age of 85. I'm sure he never heard of an algebraic space, Clifford's or anyome else's.

What's your point?

Teach people the kind of math that feeds them. The kind they can hack their way out of a box with. Teach them math which like a good screwdriver can be a lever and a hammer too when necessary.

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.

Leaving behind real world intuition doesn’t mean that math can’t be experimental in nature. The mathematical proof is itself experiential, is it not?

I wouldn’t mind if math education followed science: split up the course between lecture and lab!

New intuition and abstraction are what allows us to make progress where we previously were stuck.

Nice article from Oliver Heaviside. Essentially, it comes down to the realisation that experimenting with mathematics holds as much virtue as reason with logic.

There is a massive opportunity cost to teaching kids the amount and level of math that we do. The vast majority of people aren't going to use more than basic algebra and geometry. The common answer is that, even though they have zero practical use for higher math, they will develop reasoning skills. Teaching logic in HS would give reasoning skills that are directly practical to everyone.

Other things that we could teach kids if we made higher math elective: basic statistics, practical economics, entrepreneurship, emotional literacy and other soft skills. And, by far, the most important thing we could be teaching kids is a deep understanding of all the possible ways they can make money in this world. That's just a quick list off the top of my head.

This is coming from the perspective of an auto-didactic HS dropout who ended up achieving success in a tech career. Math was a big part of why I dropped out and not because it was hard. I got to polynomial equations in algebra and thought: I could do this, but it is repetitive and pointless, which was exactly how the rest of HS was to me at that point.

> The vast majority of people aren't going to use more than basic algebra and geometry

Most high school students take nothing past algebra and geometry, so what specifically are you complaining about?

As far as I can tell, most adults in the US working in a non-technical field do not understand very basic mathematical concepts like fractions, percentages, or negative numbers, and can’t really follow a line of logical reasoning. It’s definitely an indictment of mathematics education, but I can’t support the conclusion that students should spend less time learning mathematics.

> repetitive and pointless, which was exactly how the rest of HS was to me at that point.

So this wasn’t a complaint specific to math after all?

The point is what you mentioned. Teaching algebra when there are people struggling with fractions. Of course this gets us into the weeds of socioeconomic stratification, school funding, tracking for gifted kids or no tracking, etc.

I have come to appreciate the mathematical approach of definition first then examples after. I used to prefer examples first, but examples can cloud the concept and conflate their importance. Definitions (what you call abstractions) get at the heart of the matter, and staring at it for a while is actually the fast route. Baby Rudin, for instance, is entirely unapologetic and thus is one of my favorite math books. The book is mathematical in the way it treats mathematics.

There are many people interested in improving the teaching of mathematics. Here are a few links:

- 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”

One funny thing is, Math requires too much definition. I always laugh at the fact that, in order to define a theorem, i must learn a bunch of intermediate definitions, and it said: Yes, it's a must. So i think, math theorems lack ability to "shorten" a theorem by bypassing all intermediate definitions (too many).

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.

Math is frequently taught by skipping the underlying definitions...in other classes. Those other classes take the fun and useful results and leave the actual math class with the difficult, tedious process of learning mathematics rigorously.

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.

IMHO skipping underlying definitions makes it much harder to understand math, because you have to grok new concepts as a whole, while the definitions build these concepts from elements you already know.

I don’t get why maths teachers tend not to take the time to explain the definition properly. Even the symbols aren’t explained, they just assume students know the symbols.

They also assume the students know that there is no logic behind the choice of symbols but rather that they are often arbitrary non binding conventions that can be changed at any point in time.

I have a kid going through the much criticized "common core" math as we speak and it appears to me that this is one of the problems it tries to address.

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.

Old school math teachers are basically drill seargants, teaching you to use a hammer for one nail type of problem, right swing technique, and never to question the other ideas concerning the hammer.

I remember one particular turning point for me, I think about in 9th grade, where I asked my math teacher to go into more detail about how the quadratic equation was discovered and what the purpose for each part was. She seemed to have no idea why anyone would want (let alone need) to know that and doubled down on the advice to "just memorize it".

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.

Middle school flashbacks.

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 was very satisfied with being taught highly abstract mathematics.

I think that the abstraction is both helpful and neccesarry. It allows you gain knowledge which can easily transfer between different area and problems. To me rigour and abstraction are the cornerstones of understanding mathematics.

I am very glad about my mathematical education, because it allows me to reduce even complex problems to special cases of things I already know. Abstractness is not boring or uninspiring. What it gives you is a wider view, again and again you will run into applications see, that you already understand so much.

I agree with this. For example, teaching Lie groups / algebras from the historical perspective of the problem Lie was trying to solve would be a tedious exercise in multi variable calculus and PDEs. On the other hand, once you have a sense for the abstract and how it’s used, going back to read Lie’s original work (there’s a translation my Merker) gives interesting historical perspective, context, and new understanding. But I wouldn’t want to learn it from that perspective.

Something like group/ring theory needs to be abstract. In fact, many practical applications of those things are themselves in applied math and physics.

Something like differential equations or the calculus of variations is fun for examples but then you’re just teaching a physics class ;)

Indeed, even the most practical applied mathematician needs to have put in hours learning the abstract things.

Yea, Arnold’s ODE book is great, but it’s basically a physics textbook.

You equate abstractness with uninspiringness and lack of excitement but that's just how you feel about it, not an absolute.

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.

The way that mammals learn is by play. Seems like the more the abstract something is, the harder it is to play with it, since it is by definition non-concrete. In order to play with these concepts, you have to do that in your head, which is non-intuitive (or maybe even impossible) for a large part of the population.

If you want some excellent(the best I've ever seen) resources to learn about mathematics in general, https://www.youtube.com/c/3blue1brown arguably has the best content for learning about maths. His videos singlehandedly made me much more interested in maths than any textbook or teacher.

He not only relates topics he teaches to real life examples of using them, but his main goal is to provide an intuition for a lot of the theory that goes behind the maths we learn.

I understand what you are saying about abstraction being used to create elitism, and I think that you need to have a concrete understanding of an idea to understand an abstraction/generalisation of it. One problem with a lot of maths we tend to learn is that it's often taught in the reverse order of how it's discovered. Typically, mathematicians intuitively "know" that something is true(I think the term for it is a moral proof, but I could be wrong here), and then work their way up to a rigorous proof. In my experience, the maths I was taught was often starting with the rigorous proof and then the intuition behind it.

This style of teaching is good for passing an exam but it does leave gaps in groking stuff you learn. One question I always had was "how in the world did the mathematicians that discovered this come across this?".

Good luck on your learning journey :)

Anecdote: Back in the mid-80's, I talked to a Professor of Mathematics Education, from a university with a large Mathematics Education program. Her quick take - everyone in the field agreed that how we (Americans) teach mathematics stinks. Beyond that, there was nothing resembling agreement.

At my alma mater, statistics was taught almost using definitions, and a few bare bone examples about coins and multiple supposedly identical machines producing something. It was boring, and I remember the difficulty overcoming the use of linear algebra, but most of us got it. Then again, we were CS and EE students.

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.

The larger problem is that these subjects aren't studied for their own sake. They are obstacles on the way to a degree.

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.

Because math is a social field as much as it is an intellectual one.

What gets rewarded in mathematics is what other mathematicians respect. What other mathematicians respect is math that is hard, rigorous, and abstract. So that's the type of math they teach because the people who do well in modern mathematics are the people who excel at that style of math.

For what it's worth there are fields of mathematics that are taught well and that are very concrete. They also tend to be lower prestige. I'm particularly fond of numerical analysis. Nick Trefethen at Oxford put out a good series of lectures from his course that is understandable to anyone who remembers the basics of calculus and linear algebra: https://podcasts.ox.ac.uk/series/scientific-computing-dphil-....

The "what other mathematicians respect" aspect can be rather hegemonical. This thesis I came across recently is such a breath of fresh air in that space and I wish more people did this - http://www.theliberatedmathematician.com/thesis/

We are talking about tertiary level, right? K-12 they try to make it less abstract, but kids not into it find it boring (some might be future mathematicians and find it boring too!). Money is a good way to make it relevant. Which of these is the best deal? For example.

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 a good way to make it relevant. Which of these is the best deal? For example.

Money is an incredibly boring topic unless you're making a lot of it.

I don't know about that.

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.

Money is the conduit though. Would you buy this or that. It is the this or that which is interesting, or at least shows it is useful.

Richard Feynman had similar thoughts on how math should be taught:

https://en.wikipedia.org/wiki/Richard_Feynman#Pedagogy

I always thought that mathematics would be a much more interesting subject if it was taught along with the history at the early stages, like other sciences are.

It’s abstract for the very reason that mathematics can be applied everywhere. You can’t get to lhopital without understanding limits.

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.

I taught (ahem) data structures and algorithms from a book I really liked (T. Standish, Data Structure Techniques, 1980) which emphasized the mathematical aspects of the subject. (It's still available on Amazon for the bargain price of $6.40.) My students hated it. I told them that it can't be read like a novel; that every line has to be studied and understood, individually, in order. I failed as a teacher, either because they didn't follow my advice, or it was the wrong advice. My suspicion was that Standish's approach required too much effort; that the reward was not instantaneous.

I was taught high school "new math" from an English (O-level?) text that started with sets and was not divided into algebra, trig, geometry, etc. but provided an integrated approach that covered topics as they naturally arose. My teacher (Eric Turner, now deceased, so I was unable to let him know how much he affected my life) probably had much to do with it. He also taught physics, so perhaps he really understood both the abstract and the practical.

I have two only complaints about my mathematics education: the knowledge of how to construct and deconstruct proofs, and no statistics (too much calculus).

My kids, 17 and 15, don't grok mathematics the way I think they should. Tis a pity since underneath it all is mathematics.

By the way, there are some really excellent mathematics video channels on YouTube: 3 Blue 1 Brown comes to mind.

About ten years ago we rejoined with my classmates for some days (30 years since graduating highschool) and talked a lot about our teachers, about our experience with math/physics/literature etc. It seriously amazed me how different our experience was. We were in the very same class with the very same teachers. Some of us loved how math was taught and some of us hated. Some loved math since grade 7, some "finally started to get" it in grade 10 and some still don't get it why it all was even needed. Some hated the math, but started to love it because of physics. Some loved the geometry and algebra, but still don't get for what this calculus is even needed. And so on with every single subject. And most of these people had very strong feelings how things should be taught in schools – based THEIR personal experience of course!

I have now some experience teaching math for 11-14 years old kids and this experience strongly resonated in me. The most important thing I learned about teaching is that we don't learn about what we read, see, listen etc. We learn about what we think. A teacher or not, we have very limited influence about other peoples' thoughts. The idea that there are some magical methods all or even most of kids will understand all the math is naive at best.

Most of kids in the class don't get "why?" regardless what you do. But that's OK. You of course will talk about "why?", but it's important to give them "how?" skills for the times they will decide to think about math and they will find out "why?".

Great communication is a skill. Being good at mathematics is a skill. The fraction of people who have both skills is tiny. If you have to pick one, then it is better to spend time being taught by a great mathematician over a great communicator.

Then add to that the effort required to learn the subtleties of maths and compare that to the time it takes to get exposure in the real world, and it quickly becomes clear why mathematics education is unsatisfying.

1. Learning mathematics is a skill. If you teach/learn this skill, the effect of other barriers will be reduced.

2. Learning to communicate is a skill. Instead of saying "the fraction of people who have both skills" you could say "the fraction of people willing to learn both skills"... then it becomes more clear that if someone is teaching poorly it is because they are not putting in the time to acquire the skill of communicating/teaching.

I don't think people are talking about problems with great communicators in this thread. The complaints are usually about below average communication.

Anyway, I am grateful to have been taught by both great communicators and great mathematicians. The great communicators have a huge impact, don't underestimate it. The issue is that you might not even understand the impact of what the great mathematician is telling you, unless it is communicated in a clear way. All I remember from one geometry class is the pictures... and they make the fundamental idea crystal clear.

> it is better to spend time being taught by a great mathematician over a great communicator.

At university levels, maybe. It's certainly not overwhelming better, so you want at least a basic capacity of communication.

But for most teaching, a merely good understanding of math, with heavy knowledge of communication is clearly better. For a start, young students do not learn any advanced math.

> I came across this article by V.I. Arnold

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.

This ignores that imaginary numbers (useful for both electronics and Pythagorean triples) were originally thought to be that kind of “abstract nonsense”.

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.

>> 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.

What about number theory? Or theory of computation? I can see them happily exists while not related to physics/geometry hmm...

> this horrible abstract algebraic kabbala that's at once disgusting and pointless.

Spoken like a true engineer :D

Discusting? Maybe.

Pointless? Gröbner basis!

More than abstraction, it is the vagueness and ambiguity and the general feeling of handwaviness in the way mathematics is taught which perhaps is the bigger problem.

Take a concept of vector it is unclear from the way it is taught if it has an endpoint, that is if it is anchored to a point in space.

The problem is, Arithmetic is taught like math, instead of being taught like it is a language, a rich, beautiful language, with nuance and poetry.

I love the abstractness of Mathematics so I dispute the thesis. And I'm (wrongly) a Physicist.

Abstraction do not "hamper understanding" but the rigor necessary to define the abstract objects of Mathematics is ruthless on your misunderstanding, that's why people do not like it: they cannot hand-wave it.

I've always hated the "example-first" classes: they create the illusion of understanding and then you have dozen of seemingly disconnected facts (examples) that are just variations of one simple abstract structure that has not been defined because... I don't know.

It strikes me that many people have pet theories to the effect of: “if only the idiots teaching did it _this_ way, the problem of pedagogy would be solved”.

In other fields, non-experts recognize that the business is subtle, involves trade offs, and if you haven’t ever done the thing maybe you should be careful about presuming to know the best way to do it.

I think it’s because everyone has been a bored student in a class at some point so feels entitled to an opinion, although few have actually taught a class or truly weighed all the considerations in doing so.

the division between the more geometrical or "physical" perception of mathematics and the "pure" or abstract and more algebraic runs very deep. I remember reading somewhere about a 19th(?) century mathematician bragging there was no figure in their book.

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 :-)

There is similar anecdote in the documentary The man who saved geometry[1] about the life of the famous geometer Coxeder.

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

Mathematics is a language. Like any language it is difficult to master. For the vast majority of people - conversational minimum is enough. For some - ability to write an essay is necessary (these are the engineers). But some learn to really master it like a poet masters a language. And create beautiful poetry with it. Such talent is rare, both in actual poetry and in math.

Sometimes "X is a language" is used to naturalize poor pedagogy or when knowledge is made to be hard to learn. Math is a tool. Even if it's a language. Even a language itself is a tool. You wouldn't learn to read klingon if you don't like Star Trek and you will never need it.

That is not an excuse to avoid good methods of learning.

First of all, some context about this article. Vladimir Arnold was a great mathematician, but his opinions were often extreme, and expressed in a way that drew many antagonisms. In other words: he was respected for his technical works, but his philosophical positions were often decried. See for instance https://hsm.stackexchange.com/questions/12322/some-reference...

What Arnold criticizes is "Maths modernes", a way of teaching mathematics from theory down to practice. It was introduced in France by the Bourbaki group, and it was a great failure! Though "Maths modernes" have been removed from schools, I believe its influence is still strong at higher levels: I remember an elite student (from ENS, which is the top post-graduate for this in France) at an oral fro the national exam that recruited teachers. He clearly had no practical experience on the subject, but he could get through thanks to a higher-level theory. The jury was impressed, though that was inappropriate since the theory he used was not in the program of the exam.

As Arnold expresses in this article, anyone interested in mathematics should build a personal understanding of the concepts. And the stress on modelling is also right. But saying that everything should be rooted in the physical world is more dubious. That same argument was used, centuries ago, to reject √-1 and complex numbers. By the way, the fact that theory sometimes goes far from the initial modelling of reality is not restricted to mathematics, see Feynman famous quote "Nobody understands quantum mechanics."

Now your question: "books that teach mathematics" is too broad a category. For instance, I liked the (French) books on geometry (differential or classical) by Michel Berger, and I think they were prioritizing the practice and introducing concepts very gradually. But I had years of practice and solid bases in mathematics before wandering in these texts, so what was simple and evocative to the past me could be abstruse and theoretical to the present me.

My own experience is that introducing mathematical concepts through physical representations is useful but not sufficient. To build a personal representation, I often needed practice. After many exercises and questions, even if some of it was repetitive and calculatory, my inner understanding grew and I had a mental image of "how it works".

I once took a course on "Experimental mathematics" and it was one of the best classes ive taken.

Traditionally mathematicians have focused on getting good at proving theorems, which is important but boring. The idea behind the class was that finding interesting conjectures is actually much more important to advanced math research. The class relied heavily on computer-aided techniques to run simulations and search for interesting patterns.

In my final project i played conways game of life on different sized universes with different boundary conditions and surveyed the kinds of stable cyclical patterns that could emerge. People who were math majors were doing more rigorous stuff i couldn't explain.

In uni I never met a single prof in math/compsci that was passionate about teaching the beauty of the field. Neither had they had training in didactics or something else. They are too removed from the student to even understand their problems.

I took a fair number of uni math courses, and experienced a wide range - with many clearly passionate about (& very good at) teaching.

OTOH, that was [mumble] decades ago, and I've not heard much good about trends in university teaching since then.

Sure, YMMV, I had very little passionate courses I'm afraid :(

Many universities set up the rewards to greatly discourage teaching efforts, so even a person with an inclination that way is pushed in the other direction. (I'm at a SLAC and the great majority here are passionate about teaching.)

The title makes it quite obvious that the poster isn't enjoying deeply abstract concepts. To me that's what math is about at its core and is also what was appealing about it. There's nothing wrong with applying math or visualizing concepts etc but if you don't find joy in abstractness then math maybe just isn't for you. Also to bring "the state" into play as if this was a more recent trend math sadly got stuck in is BS obviously. And then "elitism" just because the poster doesn't grok it ... sorry, but you studied the wrong subject - it's as simple as that.

to simplify what's wrong with mathematics education. I have been trying to go back and re-take so to speak, everything from basic algebra onwards. It struck me that my mental math chops were appalling. I've stumbled on a recommendation, here i think, of "Secrets of mental math" by Benjamin and Shermer...and goodness, as the book states several times - "why is this not taught in schools!"

i mean, folks constantly say "when will i need this"...well, in my opinion, all the time...strikes me as odd that even folks I know who are decent at maths, cant do double, triple digit multiplications and the like in their heads. Beyond the math olympiad trickster outcomes you might get from this sort of study...having an intuition for COUNTING just seems so foundational to make sure you're...you know, not getting screwed over...in many areas.

I also picked up an older popular mechanics "The Art of Mechanical Drawing" - by Willard, and within are some incredible drawings, of screws, nuts, bolts, sacred geometric patterns, celtic style crosses and so on. The ability to draw such things with set squares, compasses and so on is deemed an "obsolete skill"...but once again like counting...developing such a knack for this kind of drawing would be beyond handy for someone trying to diagram an idea they have, or even to explain things to others, or purely as a mental exercise of being able to manipulate shapes.

Thirdly, take a look at the field of STEM - more popular than ever. But where are the incredibly detailed Mecanno, and Lego technics sets? Stores are filled with the shittiest looking banal "robots"

I dont want to come across as tin-foil-hatted - however it just really seems that there is an active force at play to basically limit the strength of the average reasoning mind. I'm about 1/3 of the way through the above mental maths book. And once getting into it - i noticed a distinct increase and improvement in general focus in completely unrelated tasks. Once again, doubtful this is just some sort of coincidence.

Maths - should be taught, as all subjects - in a transdisciplinary fashion. Engineering, Biology, Physics, Electronics, Computer Science - share so much common ground. Yet - we're still keeping them apart. We are still forcing people to choose 'one'... Sad.

I agree that the instruction of calculus in abstract is more difficult for many students to understand vs. learning it as applied to a practical application such as physics.

If someone had told me that driving my car down the street was a differential equation:

- instantaneous position in some coordinate system

- velocity on the speedometer

- acceleration on the gas pedal

The whole class could have sucked much less.

I think much of mathematics teaching could effectively be gamified. Kids will spend a lot of mental energy learning, playing, and getting good at games. Games which have arbitrary rules and interactions. Games show that level of interest doesn't have to be very 1:1 with connection to reality. This is the core of math, e.g. define an axiomatic set of rules, define production rules, then set various objectives and find ways to achieve them.

Well, there are structural problems and personnel problems. Most big, intractable problems come down to a combination of these two -- and the personnel problems are from structural causes.

The biggest issue is that we have no systematic tracking in the U.S. That means students are not given the opportunity to go to a vocational school, everyone is routed to a "college prep" high school. That causes people who would rather work on more practical things to listen to more abstract things, but watered down to the point of being uninteresting. Math is too boring for the smart kids, and irrelevant for everyone else.

The second biggest issue is personnel. Our math teachers aren't very good at math, and tend to be at the bottom of the barrel in terms of SAT scores. In other countries, the brightest college graduates go into teaching, or at least those in the upper quartile. In the U.S., it's those in the bottom quartile. I used to tutor Math education majors in Arizona - of all the majors I tutored, the math education majors were the worst, in terms of their knowledge and enthusiasm for the subject. At that time they needed only a single class beyond the standard freshman math curriculum. It was called "Advanced Calculus", but was basically just an introduction to real analysis. Basically you just go back over single variable calculus and are told the completeness axiom and Cauchy limits, and then you prove the major theorems of single variable calculus. That's it. The Math majors generally took this class in their freshman year (since they would AP test out of calculus, which was the only pre-requisite). The Math Ed majors would typically take it in in their Junior year, because almost none of them took AP calculus and many had to take remedial math classes. And it was a massive struggle for them. I would hear so many math ed majors complain that they hated math that one time I asked one "Why don't you try another major, if you hate math?" and the response was "Oh, I like teaching, but I hate math." That was really a moment of epiphany for me since I looked back on all the terrible math teachers I had, and saw the other side of the coin -- they hated the subject. They were having as much of a bad time as I was. That comes through, and influences the pedagogy.

In terms of books, I recommend anything by George F. Simmons, who adds historical notes and examples to his books. For example, "Differential Equations with Applications and Historical Notes".

Do you mean abstract, or rote?

I think maths is necessarily abstract, in fact that's where the beauty lies IMHO.

Learning maths in terms of calculating your taxes, or 'how long does it take to fill the bathtub, if both taps are open' type problems would be extremely uninspiring.

OTOH, I see a lot of teaching is just learning the 'abstract' mechanics of maths, without any understanding of why, and how it all fits together, etc.

The two seem to share some properties.

Arnol'd is just one opinion at the end of day - and I'm not even sure if he was as distinguished as a teacher as he was a researcher.

His (elementary) books on differential equations and on classical mechanics are the pinnacle of mathematical exposition. I cannot fathom the idea that the person who wrote these was not an excellent teacher.

Nah, I think they suck (and I'm not the only). A lot of handwaving, proofs are usually just sketches. It's bad.

It's that type of introductory book written for the expert that already knows the theory.

This article is about French mathematics education, which is proof-heavy and intuition-light. Elementary math education in the US eschews proofs almost entirely, which is a different problem.

The problem is not abstraction, which is necessary to develop new concepts like fractional exponents and i, but motivating the abstraction.

This can be traced back to Bourbaki: https://en.wikipedia.org/wiki/Nicolas_Bourbaki

Due to some reason (I'm not sure why) their work made a huge imprint on how math is taught around the world.

i'm now reminded of https://csclub.uwaterloo.ca/~pbarfuss/18093757-Fuckin-Concre...

I think the problem is that some people seem to just innately like mathematics, they like learning and applying formulas, they don’t need any background context or practical application to find it interesting - and they don’t realise how dry and boring this is for other people.

I find that people with some advanced math education are just better at reasoning and more successful in general. Teaching some people arithmetic and others the "rest" would be a huge shame.

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.

I whole heartedly agree. But people have finite teen years, and finite attention.

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

What I'd have loved to be taught is basic financial skills, as you said. But you need to cram that into an already crammed schedule that overworks children for diminishing returns (I found school miserable) and the math taught in my (first-world, very rich) country is already too basic IMO.

Maybe as a one-hour-a-week high school course. The "economics" taught there were so basic and incomplete as to be useless.

> 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.

How would you know if you're yet to learn the stuff?

Some math can be taught that way, not all.

Allow me to extend in "why is the state of education so abstract and uninspiring?" witch have a simple answer: because people without culture are easier to master. The target of all education reforms in the west (and I suspect anywhere) was and still is the creation of big flocks of ancient Greek's « useful idiots » to be employed as slaves, without even being aware of their condition.

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.

You have a point though it takes reading Herbert Spencer/"Social Darwinism", Gustave Le Bon, Edward Bernays, Walter Lippman etc. to even think on these lines. People forget how these people and their theories gave rise to and defended institutional Racism/Colonialism at the crucial "Dawn of Modern Industrial Society" starting in the 19th century. They put forth the notion that "the general populace are idiots that need to be managed" by a select few.

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".

So, the reason why some people cannot understand abstract math is... a Conspiracy by Western Elites? Let me wear my tinfoil hat.

> "why is the state of education so abstract and uninspiring?" witch have a simple answer

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.

I was wondering on this recently and without going into specifics it appears to me as if some aspects of science and learning are purposely anti human or at least designed to be as inefficent as possible.

Teaching abstraction is the most efficient way of learning, since your skills are widely applicable.

Says who?

Linear algebra for example applies to many domains. If you do not understand it abstractly but e.g. as deformations in 3D space you can not apply that knowledge in other domains.

What an idiotic way to describe an entire profession.

Surely mathematicians only use abstraction to feel superior to others. It can not be that abstraction is the most useful tool in all of mathematics, because it allows one idea to be used in a wide varity of areas.

>Best example: how do you compute the moment of inertia (or the kinetic emergy, it does not matter) of a rotating object? That is the reason for integration in several variables. Not exactly measure theory or Fubini’s Theorem. And measure theory only exists because arrogant mathematicians couldn't take it that physicists were calculating moments of inertia. There simply is no other use for it! Stupid insecure mathematicians.

Are you seriously claiming that measure theory came about because .... mathematicians were insecure about people doing multiple integrals?

case in point! https://twitter.com/Rainmaker1973/status/1596478077378252803

The fact of the matter is that math is difficult. Some don't find it so, but most do.

But even if one is pretty sharp, one single math class that is taught by a great teacher will not overcome 9 years of crappy math teachers. By the time you get to a great 11th grade teacher, most people have already decided that they suck in math and just try to get through it. The lowest possible "C" grade of 2.0 is good enough. No amount of "real world" examples will overcome this. In my classes, teachers tried to do this, but the little "trickery" to attempt to get it "relatable" was pretty transparent, and had nothing to do with me, because in my mind, I sucked in math no matter how teachers tried to present it in some kind of interesting way. It made zero difference, I had teachers that tried this. No go.

However, at one point in my life, somehow an academic trigonometry book came into my possession. This was when I was 35-years-old. I decided that there were people in my high school trig class that did well in it, and they were not THAT much smarter than me. So I decided that the problem was me. I then thought about it for a little while and decided that I never learned trig because I really didn't study the first 4 or 5 chapters, where they really teach the basics, and that I would not move on to the next sentence unless I completely understood the one I was on. It was slow slogging but that is what I did. And then, at the 5th chapter, bang, flash of light on the road to Damascus moment, and I understood all of trigonometry, all of it, in one single second. It was pretty wild, actually, I've never had that kind of revalation before or since, not like that, anyways. I thumbed through the rest of the book, stopping here and there, and confirmed that, yes, in fact I understood all of trigonometry in the book after only reading the first 5 chapters. It was simple. I didn't even need to read the rest of the book, because why? I already knew it deep down understanding.

I wish I'd have figured this out in high school about not moving on until you learned a single sentence before moving on to the next sentence, but I didn't, I just kinda skimmed the book as I went along. :(

I didn't find that having a great teacher or comparing it to practical examples helped me at all. What helped me is first, determination to understand, without that there is nothing. And second, to completely and totally understand each and every sentence and not move on until it is understood, even if it takes 5 hours of work to understand one sentence. And to do this, I went online and read other trig articles/tutorials because sometimes someone else explains it differently - by the way, this is also important - if you have a great teacher it doesn't matter because they only present it in one single way. Sometimes you have to read 5 different articles on it in order to get a 360 view of the issue to understand it from all sides, if that makes sense.

But the bottom line is that I didn't have a live teacher, at all. If one is truly motivated, one can learn it all directly from a well-written book, with using some alternate texts to help understand certain aspects.

I think the purpose of education is not to learn "things" or "skills" for jobs, as much as it is to help one become an autodidact - self learning. Not needing a teacher to help.

I think approach is true with all things. You have to understand the basics to understand the entire topic. As such, if you are in school, you should really pay attention and work extremely hard on the first 5 or 6 chapters and everything else for the year will be fairly simple. I don't know if this is for every subject, but it is for most. And if it is for most, that is good because then you will have the time to spend extra time for a class that is just fucking difficult every week.

You can learn anything on your own, especially now with the internet. I'm still blown away by people who said they never learned about finances in high school and it should be taught. That might be true but there is a plethora of online videos that teach personal finance. It's everywhere.

There might be a XKCD for this

https://m.xkcd.com/435/