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David Bessis was drawn to mathematics for the same reason that many people are driven away: He didn’t understand how it worked. Unlike other creative processes, like making music, which can be heard, or painting pictures, which can be seen, math is for the most part an internal process, hidden from view. “It sounded a bit magical. I was intrigued,” he said.
His curiosity eventually led him to pursue a doctoral degree in math at Paris Diderot University in the late 1990s. He spent the next decade studying geometric group theory before leaving research mathematics and founding a machine learning startup in 2010.
Through it all, he never stopped questioning what it actually means to do math. Bessis wasn’t content to simply solve problems. He wanted to further interrogate — and help other people understand — how mathematicians think about and practice their craft.
In 2022, he published his answer — a book titled Mathematica: A Secret World of Intuition and Curiosity, which he hopes will “explain what’s going on inside the brain of someone who’s doing math,” he said. But more than that, he added, “this is a book about the inner experience of humans.” It was translated from the original French into English earlier this year.
In Mathematica, Bessis makes the provocative claim that whether you realize it or not, you’re constantly doing math — and that you’re capable of expanding your mathematical abilities far beyond what you think possible. Eminent mathematicians like Bill Thurston and Alexander Grothendieck didn’t owe their mathematical prowess to intrinsic genius, Bessis argues. Rather, they became such powerful mathematicians because they were willing to constantly question and refine their intuitions. They developed new ideas and then used logic and language to test and improve them.
According to Bessis, however, the way math is taught in school emphasizes the logic-based part of this process, when the more important element is intuition. Math should be thought of as a dialogue between the two: between reason and instinct, between language and abstraction. It’s also a physical practice of sorts, like yoga or martial arts — something that can be improved through training. It requires tapping into a childlike state and embracing one’s imagination, including the mistakes that come with it.
“The mathematician’s message is for everyone: Look at what you can do if you don’t give up on your intuition,” Bessis said.
Everyone, Bessis says, has some experience with this process, meaning that everyone has practice thinking like a mathematician. Moreover, everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique.
[...] At its core, Bessis says, mathematics is a game of back-and-forth between intuition and logic, instinct and reason.
[...] It would be dishonest to deny that there are people who are incredibly good at math. There are 5-year-olds who are already genius mathematicians. You can see it: It looks like they’re communicating with aliens from outer space.
But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.
Math is a journey. It’s about plasticity. I am not saying that math is easy. Math is hard. But life, whatever you do, is extremely hard.
[...] Whenever you spot a disconnect between what your gut is telling you and what is supposed to be rational, it’s an important opportunity to understand something new. And then you can start this game of back-and-forth. Can you articulate your gut instinct and place it within a rational discussion? If there’s still a disconnect, can you visualize why? As you play that game, your imagination will gradually reconfigure. And in the end, if you’re persistent, your instinct and your reason will align, and you will be smarter. This is mathematical thinking.
Children do this all the time. That’s why they learn so fast. They have to. Otherwise, I mean, nothing makes sense. I think this is also why babies are super happy — because they have epiphanies all day long. It’s wonderful.
“When you do math, you’re exposed to the human thought process in a way that is really pure,” Bessis said. “It’s not just about understanding things, but about understanding things in a very childish, deep, naïve, super clear, obvious way.”
For adults, this way of thinking can be very slow. But if you don’t give up, what you can do with your intuition is way beyond your wildest expectations. And this is universal. My book is a life lesson for all creative people, not just those who want to learn mathematical concepts. The mathematician’s message is for everyone: Look at what you can do if you don’t give up on your intuition.
(Score: 5, Interesting) by aliks on Tuesday November 26, @06:30AM
Like everything in life, maths is a mixture of innate ability and learning by practice.
When you look at a school maths class you will have some kids that "just get it" on the first time of seeing a new concept, and others that have to work hard to drill it in.
And sadly there will be those that don't get it, don't do the hard work, and then hate the class for making them feel foolish.
I think Bessis is talking about that latter group, when he says everyone can do maths.
As for being a maths genius, nope, you can't learn it. You have to be born with fantastic pattern matching skills far out at the end of the spectrum of human abilities. So far out that you are close to seeing patterns that are not there and get accused of being close to madness.
To err is human, to comment divine
(Score: 5, Insightful) by r_a_trip on Tuesday November 26, @10:40AM (23 children)
Maybe I am the unluckiest guy in the world and I had the rare experience of almost all my math teachers being impatient, arrogant pricks. Or math is for people who get it and people who don't latch on immediately are being left by the wayside quickly and unceremoniously.
Intuitively I felt math was a language that makes sense once you learn how to use it. Trouble is, no one ever really took the time to properly take me through the process. Yes, maybe I am annoyingly slow. Yes, maybe more gifted people latch on sooo much quicker. For me, even when seriously asking for help, it always has been a situation where a teacher is juggling around with equations at breakneck pace and then getting annoyed that I already lost the plot 7 steps ago. Most heard complaint, "How can you not get this?!" Well, maybe you are the math equivalent of a native Spanish speaker who doesn't slow down and I only know the sentence ended because your lips stopped moving!
@Aliks No, math class didn't make me feel foolish. I know I am not a fool. I hated math class, because it was this impenetrable wall and it was mandatory. I worked as hard as I could (given how demotivated it made me), while barely understanding what I was doing and what the applicability of it was, praying to the Gods of Fortune that I scrape by with a pass.
(Score: 2, Insightful) by shrewdsheep on Tuesday November 26, @10:55AM (16 children)
In my experience enthusiasm for math is seldom inspired by teachers. I myself took my motivation from popular mathematics authors, the likes of Martin Gardner and Ian Stewart. Nowadays there is a lot of good stuff on Youtube (e.g. Mathologer). If you solve the riddles, do the exercises yourself, you are off to some good experiences.
Although I haven't tried myself, I guess, you could ask a LLM for a hint if you are really stuck nowadays. I myself only solved a fraction of the riddles when I tried as most were too difficult for me. With a few hints my experience would have been better. Still, just spending some time then checking the solution teaches a lot.
(Score: 4, Interesting) by r_a_trip on Tuesday November 26, @11:27AM (15 children)
My biggest problem with math teachers in the past has been their stubborn refusal to give a clear answer to the question, "What is the practical applicability of this? What can we do with this in the real world?" Most answers have been either describing what it does within the given assignment, e.g. integration, which is to determine the surface area under a function in an interval or "This is part of the curriculum and you need it to get a passing grade." For me the how in itself has never been satisfying. I am motivated by the why.
(Score: 2) by c0lo on Tuesday November 26, @11:52AM (6 children)
Did you get you practical examples what integration is good for, or did you give up (even until today)?
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 5, Interesting) by PiMuNu on Tuesday November 26, @12:46PM (2 children)
I was asked recently "why calculus" - I had to think about it, and the answer is incredibly deep I think.
Science gives us the state of a system - temperature, density, field strength, coronavirus infection density or whatever - and the rate of change of those variables with respect to each other.
Calculus let's us convert from the state of a system at one time to the state of the system at some later time. By integrating the rate of change we get to the new state. So it is a (the?) fundamental tool of science.
I never quite realised why it is such a fundamental tool until I tried to answer the question.
(Score: 3, Interesting) by c0lo on Tuesday November 26, @03:57PM
If I loan you $1 with an annual interest of x, and I choose to apply the compound interest every 6mo, the amount you owe me after an year will be larger than if I simply compute it once for the entire year.
On the same path, it will be even better if the compound interest is applied 4 times (every quarter) than only 2.
Question is: if I'm applying the compound interest in ever increasing number of increments, will the owed amount go infinite?
That question brought the Aha! to me when it came to understanding limits.
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 3, Interesting) by aafcac on Tuesday November 26, @09:05PM
I recently installed new thermostats to replace the less efficient ones that the electric company was wanted out. One of the biggest differences is that the newer one has a PID controller which allows for it to better correct for the bias that comes from not being located in the center of the room and better decide when to cut the power and how much to use to get the temperature to the correct value.
I hadn't really thought about it, but I was initially confused because the reading was noticeably, and laughably, off when I first installed it, but after a bit, the actual temperature felt consistent with what the display was showing. Essentially it knew with nearly 100% precision how much heat was being generated from the power going to the heaters, it just didn't know how much was being lost or where it was with respect to the heat sources. From what I can tell, it knows that the rate of heating is proportional to the energy going past it minus the heat being lost to the outside and some bit because it's not located in the middle of the space. And, the actual temperature that it's measuring. And, provides a more accurate estimate of what the room temperature closer to the center probably is. It's imperfect, but a pretty cool use for calculus in real life.
People obsess over AI, but this sort of controller has been popping up more and more in recent years as it helps to eliminate the need to over or undershoot your target when it came see what the rate of change and the actual temperature are doing. My espresso machine has one built in for deciding how much heat to apply.
(Score: 3, Informative) by r_a_trip on Tuesday November 26, @12:53PM (1 child)
I have slogged through my exams (decades ago), gotten a job where these higher concepts aren't necessary (bean counter) and said my goodbyes to math beyond arithmetic. I also don't feel the call to pursue it on my own. I will leave it to the people who like that stuff, my money just as easily buys the items from the fruits of their labor as anyone else's.
(Score: 3, Touché) by c0lo on Tuesday November 26, @02:48PM
Given the plot of inflation over time, how much $1 depreciated over the last 4 years?
https://www.youtube.com/watch?v=aoFiw2jMy-0 https://soylentnews.org/~MichaelDavidCrawford
(Score: 2) by VLM on Tuesday November 26, @01:27PM
The percentage who need to know how to design, build, use, know about, use without knowing, are pretty wide. Think of PID controllers. Everyone uses them, almost no one designs the hardware and firmware for them.
Another example showing wild changes in recent decades is "in the old days" if you wanted to know the volume of some part (to figure out how much it weighs, assuming you know the density of the material) then you integrate it BY HAND but now a days you ask the CAD program and there's like one guy on the entire planet (or maybe a very small team) who needs to know how the CAD program integrator works.
To some extent the entire education system suffers from this; theoretically all business students supposedly got training on being a CEO even if almost none of them will ever be a CEO.
(Score: 4, Informative) by Anonymous Coward on Tuesday November 26, @12:45PM (3 children)
Try the book Calculus Made Easy: https://calculusmadeeasy.org/ [calculusmadeeasy.org]
It's old enough to be freely available in the public domain, it's good, and it doesn't mess around with a bunch of nonsense like teachers in schools. Right up front he gives you good reasons *why* you want to learn how to do any operation, and presents things much more clearly and concisely than any of my teachers did.
(Score: 3, Interesting) by Unixnut on Tuesday November 26, @02:20PM (1 child)
Thanks for this, while I seem to posses natural understanding of logic and mathematics a string of awful teachers pretty much ruined me in maths. Programming and computers I was self taught which is probably why I am now much stronger in that area, but I always felt that without a grounding in maths that something was missing.
I personally went and downloaded the PDF [gutenberg.org], just reading the first two chapters has both enlightened and entertained me, including clarifying some things that teachers thought "too obvious" to explain to me.
Also I like the way it is written, I don't know why but a lot of modern books either are very "dry" (in the sense of nothing but the facts) making it hard to keep your concentration and other books that approach the subject informally, like a classmate is explaining to me rather than a teacher. So far when reading this book I feel like I am an apprentice in the presence of a master who is teaching and engaging me in the subject. I can't vouch for the entire book but its definitely a good start.
(Score: 1) by shrewdsheep on Wednesday November 27, @10:46AM
The reason for dryness (in a good sense) is that once you master the language of mathematics, explanations, examples or anecdotes become a distraction, basically dryness turns around, surrounding text become dry and only the math is vivid.
That, of course, has no bearing on introductory texts which should always give motivations, examples and practical applications (to remain fair, also check exercises!). Like, if you learn a new language an introductory book should be written in your native language. That being said, dryness in the eye of the beholder. There are different matches between textbooks and learners, which is why we need several/many. You seem to be a difficult match.
(Score: 0) by Anonymous Coward on Tuesday November 26, @08:35PM
That book literally saved my grade during my first semester of college calculus, the night before the final. I was desperate enough to seek out something like "Calculus for Dummies", which might not have even existed at the time. I saw Calculus Made Easy in the card catalog, went down in the stacks, found it (yes I'm that old), read it one of the little desks in the dank stacks, illuminated by a cheap florescent lamp and achieved enlightenment. It was the original British English version but that was just mildly annoying to this American. As a side effect I learned that what we call "grades" in regards to your year in school, the Brits call "forms".
(Score: 3, Insightful) by VLM on Tuesday November 26, @01:17PM
They don't want to admit in public it's mostly a filter.
You must be able to learn this hard thing this well to get a job solving other unrelated hard problems.
This is endemic in engineering where there is at least some excuse for lower to mid level math skills.
The medical field has an interesting, different solution of forcing med students to take organic chemistry as their filter class. So I took ochem with a weird mix of people who simply didn't care beyond getting the grade to move on with life along with people who wanted to make it their lives work. Meanwhile I'm sitting there in lab for four hours in ochem lab one afternoon thinking to myself how I'd rather be working on my programming homework seeing as I've been doing computer stuff all my life up to that point and I enjoy it ... may as well CS so I did.
(Score: 2) by JoeMerchant on Tuesday November 26, @03:56PM
> their stubborn refusal to give a clear answer to the question, "What is the practical applicability of this? What can we do with this in the real world?"
Oh, I got to the point in math where they would stand up and tell you "there's no clear application for this, it's just the basis for higher mathematical concepts for which there is also no clear application..."
I already wasn't getting laid, didn't need to work toward guaranteeing that in perpetuity.
🌻🌻 [google.com]
(Score: 3, Interesting) by Anonymous Coward on Tuesday November 26, @04:09PM
Math is actually a very, very precise language.
It's practical use is to break down complex things into simpler things. For example, the wind. How to describe it? With a single number - speed. Nope, you also need direction. So 2 numbers, a vector. Nope. It also has gradients, so 3 vectors. And so on. Mathematics is a language - much like regular language - that allows more and more precise depiction of reality.
My theory is that language is a tool that evolves over time that carries the human experience. We have stopped evolving but language gets more and more precise. It shapes our thinking because it is such an amazing tool. It gives us the ability to talk about past events, future events, relations between things with very high precision.
(Score: 2) by sjames on Wednesday November 27, @07:19AM
For me, it was understanding in physics how we get from v=at to d=½at².
(Score: 3, Interesting) by JoeMerchant on Tuesday November 26, @03:28PM (3 children)
First, I feel the need to share my prejudicial observation that Bessis' enthusiasm for the passionate intuition side of things is quintessentially French, in my experience. Not all French, but those with this particular flair do seem to be mostly French.
Second, I had a statistics professor - Turkish but lived, studied and worked in Paris for 30 years - who was quite the impatient, arrogant prick, not so much expecting people to "get it and latch on immediately" but rather the opposite - quite possibly because he himself didn't "get it" easily, he was capricious and arbitrary in his enforcement of "following the forms" - should a stats 101 quiz question "show the work" and do both a multiplication and an addition in a single line, that's full points withheld. In a post-mortem office visit to attempt to understand where my C came from, since a straight average of the marks he showed on my papers would be an F, and a straight consideration of the correctness of my answers in all quizzes, tests, and the final exam would be an A+, he confided in me: "you remind me of my son, he hates me too."
Third, I closed out the offerings of my University's mathematics department during undergraduate, they didn't have any further courses for me when I continued on in graduate school, but the Physics department did... so I took the first of those courses which taught Green's functions, closed path integrals around singularities on Gaussian surfaces, and one other thing which was apparently quite easy for me, and equally useless in the real world. Triaging the situation, it became apparent that with the final exam being "answer 5 of 9 provided questions" and since I never really mastered Green's functions, I would instead study the other two topics - what are the odds that 5 of the 9 questions would requite the use of Green's functions? Well... 4 of them started off "Using Green's functions...", but as I flipped to the 5th page it asked "solve the following differential equation" (implying use of Green's functions, but not explicitly requiring it...) having economized my study time by skipping Green's altogether, I embraced the technicality - drew the answer to the diffeq from memory of the previous exams which the professor had provided us to study, recognized that the current question was a simple transform of a previous question and so... demonstrated that the answer "by inspection" was correct, quickly solved the other 4 non Green's function problems, and handed in my exam in record time for the course - apparently Green's functions take a long time to work out.
Point of the third story: why? Why study Green's functions? Nearly every possible application of them has been worked through, cataloged in easy reference books, and while I'm sure it's an intriguing technique with potential application in higher levels of abstraction - in actual life applications? Not so much. My "intuitive passion" for abstract mathematical thoughts ends at some point once or twice removed from any foreseeable applications in actual life.
If I were to have the luxury of a decade of academic pursuit, I would much rather spend it on development of a dead-simple to operate cryptocurrency/blockchain based decentralized / federated social media platform... distilling the essence of "what makes these things tick in the real world" into something that at least has the potential to be used by hundreds, if not hundreds of millions, of people... Even if it never gets used by anybody, I suspect that getting it working and published would in some way influence the platforms that _do_ get used by people. As opposed to solving a few of Hilbert's remaining unsolved Problems [wikipedia.org] which would make my name immortal in mathematics circles, but otherwise...?
🌻🌻 [google.com]
(Score: 2) by corey on Tuesday November 26, @09:57PM (2 children)
Sorry but this all sounds like typical wishy-washy language from an academic. None of it connects with me. I tried to understand what he's on about but it sounds like lovely wordcrafting with a lot of cliche thrown in. You might guess I sound like an engineer, because I am!
At uni, I struggled through engineering maths. I mean, I do have an attraction to it and want to see this light they talk about. Maths is a very nice language to talk. But I remember classes (and exams) where for example, I was required to start with some simple circuit/electromagnetic equations and end up with one of Maxwell's/Gauss's/Faraday's equations (derive/prove them). The steps were (for example) to first divide both sides by something, then integrate over a closed curve around something, then assume some constant is zero because of something, then some group of terms on one side get replaced by one other term, some other steps, then rearrange, and there's the answer. I clearly remember thinking it's like walking around in a thick fog, on a dark night, trying to find something. Absolutely no reason why you should take any of the steps and you just have to rote memorise it. I recall getting the steps in the wrong order and ending up going down pages of derivation and not getting the answer. Or missing a step. There's no gut instincts, no "pure" / "childish" or obvious ways about it. It felt like pure maths for the sake of maths, no "we integrate over a closed curve to find the flux entering and exiting, we need to do this step because the next step depends on this." This is why I think this dude sounds like he's on crack with the flowery language but it's not the first time I've heard maths people talk like this. But I've been an engineer now for over 15 years and use a lot of electromagnetics and circuit equations in my work, I never need to derive anything, it's just there in the book. I'm glad I managed to get through it because I am a decent engineer and love the work.
Maybe one day when I'm retired, I'll have time (without the pressure of getting a 50 to pass the subject/unit) to sit and just work through the maths.
(Score: 2) by JoeMerchant on Tuesday November 26, @10:54PM
>typical wishy-washy language from an academic.
It's "passion." It's romantic aggrandizement of "intuition." It's wishful thinking that you're going to have a leap of inspiration that will take you to new pinnacles never before ascended by formal mathematicians... Yeah, it's mostly a bunch of bunk, until it's not and you really do come up with something novel and valuable (to someone) and if that ever happens then you have a romantic story to tell about it. Certainly more fun to toil at as a career for decades that way than just believing it's all an endless grind looking for a needle that may not be in any of the millions of haystacks you are scheduled to inspect first.
>I clearly remember thinking it's like walking around in a thick fog, on a dark night, trying to find something.
I had a filters professor who wrote his own textbooks and liked to concoct "intuitive" scenarios for discussion in the lectures. There were about 15 students in the class, and only two of us who could keep up with his maths as he walked through his demonstrations. I would follow what he was saying and put approximate placeholders on my mental scratch pad: area under this curve is 2, we're squaring that and then dividing it into this thing that seems to be the product of about 10, and 2... so... I would say "5". Meanwhile, there was an exceptional maths student from China who would furiously scribble out everything on paper - he'd still be scratching away when I would proclaim "5" then a few seconds later he'd stop scratching, mentally tally up all the integrals and other evidence he had worked out on paper and nod "yeah, 5." Of course, this was all a setup by the professor - any real world situation would have some horrendously irrational result like 3.5837. The other 13 students in the room just stared at us, vacantly, as if they were sitting in a fog on a dark night hearing something in the distance...
>Absolutely no reason why you should take any of the steps and you just have to rote memorise it.
That's what Fortran punch cards were for, in 1977.
>use a lot of electromagnetics and circuit equations in my work, I never need to derive anything, it's just there in the book.
When I took the first "path integrals on a Gaussian surface" maths course, I had a very good study partner. She would take detailed notes of the lecture - the lecture explained a lot that just wasn't in the text - and I would simply sit and listen and mentally follow what was being said in the lecture. Some hours after class, we would get together with the text and the assigned problem set, and I would explain her notes to her. She spent so much mental effort in writing down the lecture material that she couldn't really follow the concepts in class - but later with the luxury of time and practical application in the problems she'd get it fine. Without the step of explaining her notes to her, I would forget what I briefly knew during and immediately after the lecture, but again: working the problem sets together would cement the knowledge in my head, at least long enough to get through the term end exam. I went on to one later maths course that used those concepts again, then never had use for them in "real life." I suppose there was that MRI safety of implantable neurostimulators project that could, potentially, have used that kind of math to calculate the antenna efficiency of the lead wires, but you're talking about arbitrary shaped antennas coiled up under skin, muscle and fat, over multiple bones, in an MRI bore with the excitation pulses delivering RF energy that you ultimately want to know: how hot are the electrode tips of those lead wires going to get... Simulation of all that, in 2005, would have been Quixotically ambitious - particularly on our "as soon as possible" schedule so, we built simulated people out of acrylic tubs full of gelled saline and scanned them in actual MRIs at various facilities around the country, from Los Angeles to Charleston, paying at least $500 per hour for scanner time... As I understand things, in the ensuing nearly 20 years they have started moving away from entirely experimental heating measurements to more of a maths/simulation based protocol. But, still, even if you can work out the math yourself, the real battle is to present it in such a way that the regulatory reviewers will believe what you have done is adequate, regardless of how correct it may or may not be.
The "deepest" math I dove into after college came at my first job, there was a derivation of how to do a least-squares error fit of a 5th order polynomial with constrained endpoints to a measured data set - we did indeed get the least squares error curves, and they did have the bumps like we expected them to, and when the data was clean the fits were nice and accurate, but when the data was noisy the curves were worthless - even if they did look nice, they were terrible at "teasing out" the underlying physiological information swamped in the noise. Then there was a volume integral of the centrifugal acceleration of some arbitrary shapes rotating around a fixed axis - looking to balance nested spinning arms of mass in 3D so when they counter-rotated you only got sinusoidal forces along a single (chosen) axis... I actually tried to look up my college physics prof after I solved that one, just to share, but it had been nearly 10 years and he had retired and vanished.
🌻🌻 [google.com]
(Score: 2) by ChrisMaple on Thursday November 28, @06:49AM
For me, writing the steps for getting to some mathematical goal involves a lot of trial and error, lots of dead ends. I suspect that most mathematicians have the same experience. The seemingly purposeless process that the prof puts on the blackboard is something he had to memorize or copy from his notes; the first guy to derive it may have spent hours or years figuring it out.
I don't feel bad that I couldn't derive it on my own or follow the live demonstration. I just plug through it when I get home and concentrate on the difficult parts.
(Score: 2) by aliks on Tuesday November 26, @06:12PM (1 child)
I do understand your frustration, and yes I had at least one bad teacher who would be enough to put anyone off. In fact I also had maths teachers who only cared for the "naturals" and damn the rest.
There is no way to guarantee all maths teachers are top quality, but I think it would help everyone of the maths curriculum moved away from the Victorian era and into the 21st century with much more probability and statistics, vectors etc and less greek geometry, less number theory, and even less trigonometry.
To err is human, to comment divine
(Score: 2) by JoeMerchant on Tuesday November 26, @09:43PM
You remind me of my 8th grade art teacher. Her sole purpose in life seemed to be to crush the love of art in children, dissuade them from the notion that they had any talent or ability and belittle their every technical mistake as a sign that they "just don't have it." I spent the next 3 years ignoring art, but did take a class again in 12th grade with a teacher quite the opposite - and yes, much of what I did was dreadfully bad, but I improved - significantly - through the course of the year, unlike 8th grade when all our work seemed to get worse and worse as the year went on.
🌻🌻 [google.com]
(Score: 4, Interesting) by aliks on Tuesday November 26, @07:14PM
That is, the ability to strip away unecessary detail from a stack of examples to reveal the common underlying reality.
Its a great skill to have, and well worth learning from an early age. Thinking abstractly is valuable not just for about problems with numbers, shapes or functions, but all kinds of real life activity like computing, engineering, planning, architecture and design, and poetry (yes I really believe true!)
The real challenge for maths teaching is to somehow make it easier for students of all types to acquire this skill. Yes you need teachers with real teaching skills on top of their own maths qualifications, but also a good way to motivate the need to drill the seemingly impractical maths problems.
Maybe we should ask piano teachers how they motivate pupils to practice chords all day long?
To err is human, to comment divine