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Why is the state of mathematics education so abstract and uninspiring?

124 点作者 newsoul超过 2 年前
I came across this article by V.I. Arnold : https:&#x2F;&#x2F;www.uni-muenster.de&#x2F;Physik.TP&#x2F;~munsteg&#x2F;arnold.html which is rather old. But some of the points mentioned in the article can be related to problems in the classroom today also.<p>People have an idea that being abstract and talking in abstract terms creates some sort of elitism. But it hampers understanding and excitement at the nascent stages. Abstraction is required to tackle complexity. But that is not the all and be all of the domain.<p>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.<p>I would very much love to hear about books or courses that teach mathematics in the way mentioned in the article.

55 条评论

VyseofArcadia超过 2 年前
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.<p>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.<p>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.<p>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.
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urthor超过 2 年前
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).<p>&gt; But it hampers understanding and excitement at the nascent stages<p>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.<p>Artificially motivating learners is a difficult, time consuming, and often marginally effective task.<p>The only way I&#x27;ve seen consistently work is to have truly interested and fascinated public speakers.<p>Who then teach to the 5%, and perpetuate the cycle.
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Herbstluft超过 2 年前
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 &quot;got it&quot; early.<p>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!<p>Same thing with professors constantly scoffing at all the &quot;boring&quot; and &quot;trivial&quot; parts.<p>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 <i>hard</i>, even for him, even the &quot;boring&quot; undergraduate basics. Hard, but doable, and trainable. After years of &quot;this is easy, why don&#x27;t you get it?&quot; math education I was stunned by these displays, it&#x27;s not that everyone just &quot;gets it&quot;, aside from certain geniuses, it&#x27;s that some get a headstart.<p>This left such an impression on me that I switched to math major and got my bachelors degree in math (I was <i>very</i> bad at school math). And that one professor is <i>still</i> a huge motivator whenever I try to overcome something I struggle with.
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motohagiography超过 2 年前
Methods I would use to teach math from scratch to day would be:<p>- get in a boat at night and do celestial navigation to get to a camp site (geometry)<p>- prove the earth is round using the moon. (geometry, proofs)<p>- count cards in blackjack and poker (combinatorics, probability, game theory)<p>- decode an ancient manuscript by constructing a grammar from frequency analysis (probability, statistics)<p>- decode a real american civil war cryptogram. (Vigenere, number theory)<p>- make a radio out of found stuff and transmit a message in various encodings by, finally with binary (from ohms law to information theory)<p>- reproduce Turing&#x27;s <i>Bomba</i> from first principles in code to crack Enigma (number theory, computation)<p>- bet on a stock market return (brownian motion, randomness, shannon portfolio &#x2F; information theory)<p>- structure an an election strategy for your candidate across multiple polls (conditional and independent probability, use dice and weightings for each poll result)<p>- work on a motorcycle &#x2F; pocket-bike engine and optimize it win a drag race against someone elses engine configuration (differential calculus, sprocket sizes, power curves)<p>- 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)<p>- configure or code a basic neural network to identify a signal or encoding. (complexity classes, godel&#x27;s incompleteness)<p>- determine whether a piece of music is related to another (graphs, cosine similarity, pythagorian distance, homomorphisms)<p>That&#x27;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&#x2F;teens camp on them.
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aaronharnly超过 2 年前
My company (Amplify) is trying to change this. We make the Desmos Math curriculum ( <a href="https:&#x2F;&#x2F;teacher.desmos.com" rel="nofollow">https:&#x2F;&#x2F;teacher.desmos.com</a> ), which teaches math with digitally-enabled live, interactive, social exploration of mathematical concepts.<p>As the article describes, you have to &#x2F;feel&#x2F; 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.<p>One good lesson to see what it’s like is the Line of Best Fit:<p><a href="https:&#x2F;&#x2F;teacher.desmos.com&#x2F;activitybuilder&#x2F;custom&#x2F;56fab6bc1ab86b1f0600369d" rel="nofollow">https:&#x2F;&#x2F;teacher.desmos.com&#x2F;activitybuilder&#x2F;custom&#x2F;56fab6bc1a...</a><p>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!)<p>[1] <a href="https:&#x2F;&#x2F;illustrativemathematics.org&#x2F;" rel="nofollow">https:&#x2F;&#x2F;illustrativemathematics.org&#x2F;</a>
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sethammons超过 2 年前
Related article, Lockhart&#x27;s Lament.<p><a href="https:&#x2F;&#x2F;www.maa.org&#x2F;external_archive&#x2F;devlin&#x2F;LockhartsLament.pdf" rel="nofollow">https:&#x2F;&#x2F;www.maa.org&#x2F;external_archive&#x2F;devlin&#x2F;LockhartsLament....</a><p>If we taught music like math, you&#x27;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&#x27;s volume.<p>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&#x27;t a silver bullet, but it focused on conceptual understanding by investigating relationships and patterns and formalizing those into &quot;normal&quot; math.<p>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 &quot;m represents slope and b the y-intercept, now match graphs to their equations&quot; - building mental models that connected concepts was left up to the pupil.
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sn41超过 2 年前
A quote by the founding father of modern electrical engineering, and the formulator of the current version of Maxwell&#x27;s equations, Oliver Heaviside:<p>(Read the entire article. It&#x27;s hilarious, and surprising that Nature decided to publish it.)<p>&quot;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.&quot; - Oliver Heaviside, &quot;The teaching of mathematics&quot;, Nature 62, pp 548-549, 1900. [1]<p>[1] <a href="https:&#x2F;&#x2F;ia600708.us.archive.org&#x2F;view_archive.php?archive=&#x2F;22&#x2F;items&#x2F;crossref-pre-1909-scholarly-works&#x2F;10.1038%252F061606a0.zip&amp;file=10.1038%252F062548c0.pdf" rel="nofollow">https:&#x2F;&#x2F;ia600708.us.archive.org&#x2F;view_archive.php?archive=&#x2F;22...</a>
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wintermutestwin超过 2 年前
There is a massive opportunity cost to teaching kids the amount and level of math that we do. The vast majority of people aren&#x27;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.<p>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&#x27;s just a quick list off the top of my head.<p>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.
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avindroth超过 2 年前
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.
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noelwelsh超过 2 年前
There are many people interested in improving the teaching of mathematics. Here are a few links:<p>- At the Open University in the UK you can get degree in &quot;Mathematics and its Learning&quot;: <a href="https:&#x2F;&#x2F;www.open.ac.uk&#x2F;courses&#x2F;maths&#x2F;degrees&#x2F;bsc-mathematics-and-its-learning-q46" rel="nofollow">https:&#x2F;&#x2F;www.open.ac.uk&#x2F;courses&#x2F;maths&#x2F;degrees&#x2F;bsc-mathematics...</a> There is a reasearch group at the OU behind this.<p>- Department of Mathematics Education at Loughborough University: <a href="https:&#x2F;&#x2F;www.lboro.ac.uk&#x2F;departments&#x2F;maths-education&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.lboro.ac.uk&#x2F;departments&#x2F;maths-education&#x2F;</a><p>- Mathematics Rebooted by Lara Alcock: <a href="https:&#x2F;&#x2F;global.oup.com&#x2F;academic&#x2F;product&#x2F;mathematics-rebooted-9780198803799?cc=gb&amp;lang=en" rel="nofollow">https:&#x2F;&#x2F;global.oup.com&#x2F;academic&#x2F;product&#x2F;mathematics-rebooted...</a> (She&#x27;s at Loughbourgh, linked above. She has also written some books for University level mathematics.)<p>- Frank Farris&#x27; gorgeous book &quot;Creating Symmetry&quot; (<a href="https:&#x2F;&#x2F;press.princeton.edu&#x2F;books&#x2F;hardcover&#x2F;9780691161730&#x2F;creating-symmetry" rel="nofollow">https:&#x2F;&#x2F;press.princeton.edu&#x2F;books&#x2F;hardcover&#x2F;9780691161730&#x2F;cr...</a>), which has this brilliant passage:<p>This belief that my motivation deserves mention moves me to call this a <i>postmodern</i> mathematics book. By contrast, <i>modern</i> mathematics books were written in the twentieth century by intentionally voiceless authors for an intended audience of &quot;the hypothetical anybody&quot;, 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.<p>“The future is already here. It&#x27;s just not evenly distributed yet”
revskill超过 2 年前
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&#x27;s a must. So i think, math theorems lack ability to &quot;shorten&quot; a theorem by bypassing all intermediate definitions (too many).<p>It&#x27;s that &quot;intermediate definitions&quot; that causes too much indirection, just like you read a codebase with too much abstraction and redirection, you&#x27;ve lost.
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DubiousPusher超过 2 年前
I have a kid going through the much criticized &quot;common core&quot; math as we speak and it appears to me that this is one of the problems it tries to address.<p>Many concepts I learned as pure arithmetic are taught as visual and&#x2F;or spacial concepts. Often there are multiple strategies for reaching the same answer.<p>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&#x27;t like it so much because it draws upon a flexibility with math they were basically discouraged from exercising.
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constantcrying超过 2 年前
I was very satisfied with being taught highly abstract mathematics.<p>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.<p>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.
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omnicognate超过 2 年前
You equate abstractness with uninspiringness and lack of excitement but that&#x27;s just how you feel about it, not an absolute.<p>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 <i>like</i> baby Rudin. I find pure maths <i>easier</i> and more enjoyable than applied.<p>There are different kinds of people in the world.
hvasilev超过 2 年前
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.
wadd1e超过 2 年前
If you want some excellent(the best I&#x27;ve ever seen) resources to learn about mathematics in general, <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;c&#x2F;3blue1brown" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;c&#x2F;3blue1brown</a> arguably has the best content for learning about maths. His videos singlehandedly made me much more interested in maths than any textbook or teacher.<p>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.<p>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&#x2F;generalisation of it. One problem with a lot of maths we tend to learn is that it&#x27;s often taught in the reverse order of how it&#x27;s discovered. Typically, mathematicians intuitively &quot;know&quot; 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.<p>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 &quot;how in the world did the mathematicians that discovered this come across this?&quot;.<p>Good luck on your learning journey :)
bell-cot超过 2 年前
Anecdote: Back in the mid-80&#x27;s, I talked to a Professor of Mathematics Education, from a university with a large Mathematics Education program. Her quick take - <i>everyone</i> in the field agreed that how we (Americans) teach mathematics stinks. Beyond that, there was nothing resembling agreement.
tgv超过 2 年前
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.<p>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&#x2F;experimental psychology.<p>There&#x27;s no method that suits all, and not every student should be expected to reach the same level. It&#x27;s ok if students don&#x27;t master anything beyond the most basic of maths. It&#x27;s a pity, but considering that many leave secondary education barely literate, it&#x27;s not high on my list of priorities.
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jeffreyrogers超过 2 年前
Because math is a social field as much as it is an intellectual one.<p>What gets rewarded in mathematics is what other mathematicians respect. What other mathematicians respect is math that is hard, rigorous, and abstract. So that&#x27;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.<p>For what it&#x27;s worth there are fields of mathematics that are taught well and that are very concrete. They also tend to be lower prestige. I&#x27;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: <a href="https:&#x2F;&#x2F;podcasts.ox.ac.uk&#x2F;series&#x2F;scientific-computing-dphil-students" rel="nofollow">https:&#x2F;&#x2F;podcasts.ox.ac.uk&#x2F;series&#x2F;scientific-computing-dphil-...</a>.
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quickthrower2超过 2 年前
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.<p>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!
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runlevel1超过 2 年前
Richard Feynman had similar thoughts on how math should be taught:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Richard_Feynman#Pedagogy" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Richard_Feynman#Pedagogy</a>
synu超过 2 年前
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.
slashtom超过 2 年前
It’s abstract for the very reason that mathematics can be applied everywhere. You can’t get to lhopital without understanding limits.<p>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.
vgivanovic超过 2 年前
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&#x27;s still available on Amazon for the bargain price of $6.40.) My students hated it. I told them that it can&#x27;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&#x27;t follow my advice, or it was the wrong advice. My suspicion was that Standish&#x27;s approach required too much effort; that the reward was not instantaneous.<p>I was taught high school &quot;new math&quot; 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.<p>I have two only complaints about my mathematics education: the knowledge of how to construct and deconstruct proofs, and no statistics (too much calculus).<p>My kids, 17 and 15, don&#x27;t grok mathematics the way I think they should. Tis a pity since underneath it all is mathematics.<p>By the way, there are some really excellent mathematics video channels on YouTube: 3 Blue 1 Brown comes to mind.
obscurette超过 2 年前
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&#x2F;physics&#x2F;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 &quot;finally started to get&quot; it in grade 10 and some still don&#x27;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&#x27;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!<p>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&#x27;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&#x27; thoughts. The idea that there are some magical methods all or even most of kids will understand all the math is naive at best.<p>Most of kids in the class don&#x27;t get &quot;why?&quot; regardless what you do. But that&#x27;s OK. You of course will talk about &quot;why?&quot;, but it&#x27;s important to give them &quot;how?&quot; skills for the times they will decide to think about math and they will find out &quot;why?&quot;.
roenxi超过 2 年前
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.<p>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.
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bambax超过 2 年前
&gt; <i>I came across this article by V.I. Arnold</i><p>Interesting article! It argues that when mathematics became dissociated from both physics and geometry, it became this horrible abstract algebraic kabbala that&#x27;s at once disgusting and pointless.<p>The article talks a lot about France; I think in France the problem is a problem of selection &#x2F; elitism: how to rate lots of students in a reproducible way in order to keep only a few. This process doesn&#x27;t leave much room to rêverie or the joy of learning. It&#x27;s a disease that modern proponents of &quot;le mérite&quot; don&#x27;t understand. (&quot;Mérite&quot; is hard to translate; maybe &quot;self worth&quot;; it&#x27;s a vast debate in France).<p>More generally, the problem of teaching is that it tries to put <i>results</i> 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&#x27;s not clear if there&#x27;s any other way, esp. because there are so many facts to learn, and so little time.<p>Yet that doesn&#x27;t excuse everything. It should always be possible to connect theory to real world applications. Here&#x27;s a little observation about geometry for example.<p>My kids are currently learning about the Pythagorean theorem, where in a square triangle, the length of the &quot;hypotenuse&quot; is the square root of the sum of the square of each opposite side. But what does &quot;hypotenuse&quot; mean?<p>In math books (as well as on every website that I could find, see <a href="https:&#x2F;&#x2F;www.google.com&#x2F;search?tbm=isch&amp;q=pythagorean+theorem" rel="nofollow">https:&#x2F;&#x2F;www.google.com&#x2F;search?tbm=isch&amp;q=pythagorean+theorem</a>) 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.<p>It turns out, hypo-tenuse means &quot;that which supports&quot;. Obviously if the square triangle sits on its square angle, the hypotenuse doesn&#x27;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.<p>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&#x27;s more interesting than a²+b²=c².
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billfruit超过 2 年前
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.<p>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.
skymast超过 2 年前
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.
annyeonghada超过 2 年前
I love the abstractness of Mathematics so I dispute the thesis. And I&#x27;m (wrongly) a Physicist.<p>Abstraction do not &quot;hamper understanding&quot; but the rigor necessary to define the abstract objects of Mathematics is ruthless on your misunderstanding, that&#x27;s why people do not like it: they cannot hand-wave it.<p>I&#x27;ve always hated the &quot;example-first&quot; 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&#x27;t know.
superposeur超过 2 年前
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”.<p>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.<p>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.
college_physics超过 2 年前
the division between the more geometrical or &quot;physical&quot; perception of mathematics and the &quot;pure&quot; 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.<p>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<p>anybody who went through V.I.Arnold&#x27;s mathematical methods of classical mechanics has no doubt which way <i>his</i> brain works :-)
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graymatters超过 2 年前
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.
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idoubtit超过 2 年前
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 <a href="https:&#x2F;&#x2F;hsm.stackexchange.com&#x2F;questions&#x2F;12322&#x2F;some-references-for-vladimir-arnolds-thesis-mathematics-is-a-part-of-physics" rel="nofollow">https:&#x2F;&#x2F;hsm.stackexchange.com&#x2F;questions&#x2F;12322&#x2F;some-reference...</a><p>What Arnold criticizes is &quot;Maths modernes&quot;, 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 &quot;Maths modernes&quot; 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.<p>As Arnold expresses in this article, anyone interested in mathematics should build a personal understanding of the concepts. And the stress on <i>modelling</i> 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 &quot;Nobody understands quantum mechanics.&quot;<p>Now your question: &quot;books that teach mathematics&quot; 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.<p>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 &quot;how it works&quot;.
yummypaint超过 2 年前
I once took a course on &quot;Experimental mathematics&quot; and it was one of the best classes ive taken.<p>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.<p>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&#x27;t explain.
RamblingCTO超过 2 年前
In uni I never met a single prof in math&#x2F;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.
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2-718-281-828超过 2 年前
The title makes it quite obvious that the poster isn&#x27;t enjoying deeply abstract concepts. To me that&#x27;s what math is about at its core and is also what was appealing about it. There&#x27;s nothing wrong with applying math or visualizing concepts etc but if you don&#x27;t find joy in abstractness then math maybe just isn&#x27;t for you. Also to bring &quot;the state&quot; into play as if this was a more recent trend math sadly got stuck in is BS obviously. And then &quot;elitism&quot; just because the poster doesn&#x27;t grok it ... sorry, but you studied the wrong subject - it&#x27;s as simple as that.
an_aparallel超过 2 年前
to simplify what&#x27;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&#x27;ve stumbled on a recommendation, here i think, of &quot;Secrets of mental math&quot; by Benjamin and Shermer...and goodness, as the book states several times - &quot;why is this not taught in schools!&quot;<p>i mean, folks constantly say &quot;when will i need this&quot;...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&#x27;re...you know, not getting screwed over...in many areas.<p>I also picked up an older popular mechanics &quot;The Art of Mechanical Drawing&quot; - 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 &quot;obsolete skill&quot;...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.<p>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 &quot;robots&quot;<p>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&#x27;m about 1&#x2F;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.<p>Maths - should be taught, as all subjects - in a transdisciplinary fashion. Engineering, Biology, Physics, Electronics, Computer Science - share so much common ground. Yet - we&#x27;re still keeping them apart. We are still forcing people to choose &#x27;one&#x27;... Sad.
mullenat超过 2 年前
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.
smitty1e超过 2 年前
If someone had told me that driving my car down the street was a differential equation:<p>- instantaneous position in some coordinate system<p>- velocity on the speedometer<p>- acceleration on the gas pedal<p>The whole class could have sucked much less.
karmakaze超过 2 年前
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&#x27;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.
rsj_hn超过 2 年前
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.<p>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 &quot;college prep&quot; 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.<p>The second biggest issue is personnel. Our math teachers aren&#x27;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&#x27;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 &quot;Advanced Calculus&quot;, 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&#x27;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 &quot;Why don&#x27;t you try another major, if you hate math?&quot; and the response was &quot;Oh, I like teaching, but I hate math.&quot; 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.<p>In terms of books, I recommend anything by George F. Simmons, who adds historical notes and examples to his books. For example, &quot;Differential Equations with Applications and Historical Notes&quot;.
jacknews超过 2 年前
Do you mean abstract, or rote?<p>I think maths is necessarily abstract, in fact that&#x27;s where the beauty lies IMHO.<p>Learning maths in terms of calculating your taxes, or &#x27;how long does it take to fill the bathtub, if both taps are open&#x27; type problems would be extremely uninspiring.<p>OTOH, I see a lot of teaching is just learning the &#x27;abstract&#x27; mechanics of maths, without any understanding of why, and how it all fits together, etc.<p>The two seem to share some properties.
data_maan超过 2 年前
Arnol&#x27;d is just one opinion at the end of day - and I&#x27;m not even sure if he was as distinguished as a teacher as he was a researcher.
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lern_too_spel超过 2 年前
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.<p>The problem is not abstraction, which is necessary to develop new concepts like fractional exponents and i, but motivating the abstraction.
dottedmag超过 2 年前
This can be traced back to Bourbaki: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Nicolas_Bourbaki" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Nicolas_Bourbaki</a><p>Due to some reason (I&#x27;m not sure why) their work made a huge imprint on how math is taught around the world.
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Tycho超过 2 年前
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.
psychphysic超过 2 年前
Hot take, mathematics should be split into arithmetic and mathematics.<p>Most people just want arithmetic, what&#x27;s the tip on this bar tab? What do I put in the capital gains box on this tax return?<p>Universities, businessess and most who will go into other STEM paths want mathematics.<p>And mathematics is incredibly abstract and the reward are plenty of it&#x27;s your thing.<p>I haven&#x27;t read any recent research on this (cause I&#x27;m not much interested) but there is potent correlation between IQ and interest&#x2F;ability in mathematics (and physics). This is telling (or concerning depending on your viewpoint) as IQ is a normal distribution meaning actually statistically some people are unlikely to grasp or be interested in the maths we care about.<p>TL;DR You wont understand why quadratics are so important until you encounter integral calculus. You wont appreciate complex numbers until you encounter complex analysis (well maybe some physics).<p>The same way you wouldn&#x27;t appreciate cement until you encountered a brick wall.
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data_maan超过 2 年前
&gt; 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.<p>How would you know if you&#x27;re yet to learn the stuff?<p><i>Some</i> math can be taught that way, not all.
kkfx超过 2 年前
Allow me to extend in &quot;why is the state of education so abstract and uninspiring?&quot; 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&#x27;s « useful idiots » to be employed as slaves, without even being aware of their condition.<p>Just listen last Klaus Schwab speech at G20&#x2F;Bali <a href="https:&#x2F;&#x2F;youtu.be&#x2F;DQjXODh0TOg" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;DQjXODh0TOg</a> where he describe the near future he want:<p>- IoT, poetically named Industrie 4.0, to allow OEMs control from remote, witch means de facto owning hardware formally sold to someone who legally&#x2F;theoretically own it now;<p>- Sharing Economy witch means re-sell continuously the very same object, as a service&#x2F;leasing etc making peoples compete to access such scarce and needed resources;<p>- 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.<p>Followed by it&#x27;s interview at APEC 2022 <a href="https:&#x2F;&#x2F;youtu.be&#x2F;NWHDgXhMkgs" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;NWHDgXhMkgs</a> where he state that China &quot;it&#x27;s a role model for many countries&quot; and &quot;we should be very careful in imposing systems. But the &#x27;Chinese model&#x27; is certainly a very attractive model for quite a number of countries&quot;.<p>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.
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Guthur超过 2 年前
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.
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pfortuny超过 2 年前
Insecurity.<p>There are many maths professors who do not really enjoy them. This creates insecurity which the formalism alleviates (as an insecure person myself, I know the strength of formal rules).<p>They are afraid to do maths the true way: take a real example and “be led by it” to the formalism, not the other way round.<p>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.
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andrewfromx超过 2 年前
case in point! <a href="https:&#x2F;&#x2F;twitter.com&#x2F;Rainmaker1973&#x2F;status&#x2F;1596478077378252803" rel="nofollow">https:&#x2F;&#x2F;twitter.com&#x2F;Rainmaker1973&#x2F;status&#x2F;1596478077378252803</a>
FrontierPsych超过 2 年前
The fact of the matter is that math is difficult. Some don&#x27;t find it so, but most do.<p>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 &quot;C&quot; grade of 2.0 is good enough. No amount of &quot;real world&quot; examples will overcome this. In my classes, teachers tried to do this, but the little &quot;trickery&quot; to attempt to get it &quot;relatable&quot; 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.<p>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&#x27;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&#x27;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&#x27;t even need to read the rest of the book, because why? I already knew it deep down understanding.<p>I wish I&#x27;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&#x27;t, I just kinda skimmed the book as I went along. :(<p>I didn&#x27;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&#x2F;tutorials because sometimes someone else explains it differently - by the way, this is also important - if you have a great teacher it doesn&#x27;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.<p>But the bottom line is that I didn&#x27;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.<p>I think the purpose of education is not to learn &quot;things&quot; or &quot;skills&quot; for jobs, as much as it is to help one become an autodidact - self learning. Not needing a teacher to help.<p>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&#x27;t know if this is for <i>every</i> 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.<p>You can learn anything on your own, especially now with the internet. I&#x27;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&#x27;s everywhere.
ck2超过 2 年前
There might be a XKCD for this<p><a href="https:&#x2F;&#x2F;m.xkcd.com&#x2F;435&#x2F;" rel="nofollow">https:&#x2F;&#x2F;m.xkcd.com&#x2F;435&#x2F;</a>