It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than concept focused route. As soon as you start memorizing operations and treating math more like a narrow grind only approached through arbitrary problems solved primarily through computation or use of rote application of poorly understood technique, you lose the ability to understand math as anything other than a human attempting to be a computer.
The author expresses his misintuition of variables - there are plenty of simple examples and thought exercises which can inform the intuition. Later he expresses getting mixed up dealing with fractions - this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions. Modern math education is spectacularly guilty of going too far too fast.
Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary.
> ...still ended up going down an execution rather than concept focused route.
This is something that made me really disappointed when I realized it in school. It was always a step by step process taught to me, and I loved math. I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived. I was blown away. We had just learned and it were told to memorize it the previous year and I had just accepted it.
After that, math changed for me. Everything had a reason that was connected to everything else. If you understood where it came from, you could do even more than if you were just given a formula to follow. How beautiful.
I realized that I had 5 years (assuming we only cared to do so after algebra) of match education that could have been better taught if done conceptually. I don't blame teachers. You need to get everyone to pass your class for state requirements, so you try to streamline it to get this year done, and not prepare for the future. Also, most math teachers I had did it as a job, not because they loved it, and there's nothing wrong with that. When I had that calculus class during my senior year though, I realized that that man loved math. Hard but fair. I learned so much and that teacher taught me how to develop a work ethic.
I wish meta-learning and a more concept based approach could be applied to high school courses (at least looking back). I understand why they aren't, but man would I love to see how it would play out.
I think those of us who are good at math greatly underestimate the amount of rote needed to reach competence-- first, how much we actually needed, and second, how much more of it most people need.
A good math education in arithmetic, algebra, and calculus is a combination of concepts and drill. This was a concept that I rejected back in school. Now I'm teaching-- not mathematics, but I have tutored and helped students catch up who have had problems.
There's a fair number of students struggling in pre-calc who have all the concepts just fine, from the bottom to the top. But when they're dealing with lots of terms and keeping a higher level goal in mind, their performance on a few simpler things, like fractions and quotient properties, falls apart. Maybe they missed a week or two in 6th grade when this was really solidified and practiced.
The terrible thing that tends to happen, once you stumble in math: the amount of concept content you have falls. The focus moves even more to rote, but focused on the "more difficult" stuff--- leaving whatever core deficit there is intact. There's solid reasons for this, but the outcomes are not great.
I also have a kid who is going into AP Calculus pretty young. I'm kinda nervous that he just has not had enough reps of practice, even though he scores very high on placement tests.
I think there are smarter drills to do. In another comment I lauded the way I was taught math, and how it built intuition, but it certainly had lots of drills too. I think what worked was that we were taught more than one way to do each thing, usually one more theory-heavy and another more technique-heavy, and we had lots of drills focused just on the primitive operations. There was lots of focus on mental math, which is just super handy for understanding more complicated problems since you don't even have to think about the simpler steps. While they did go over how fractions were just division in disguise (with pictures, which I found very helpful), knowing how to find a least common denominator is much faster than improvising every time, so even if I hated it I'm glad we did lots of drilling with those too.
A lot of what you describe has become the standard way to teach elementary math-- from the "new math" onwards to Common Core pushing aspects of looking at problems the same way
Our school does the much-lauded Singapore Math in elementary, which definitely tries to build intuition and looks at many approaches, and supplements with drills.
And I teach a competitive math class which definitely is all about finding different ways around problems and comparing and contrasting.
I'm thrilled to hear it's popular! In my hometown it was killed by doubters, but perhaps with expanding evidence they'll reconsider. I think perhaps it was lumped in with disastrous testing efforts, but the math at least was pretty great.
When I was learning math in public school, my family couldn't make heads or tails of my homework. Even in Kindergarden we learned that '2 = 1 + 1' was the same as '1 + 1 = 2', and we got started with '1 + ? = 3' even in first grade. A focus was put on mental math, teaching techniques to break two digit numbers up, before we did pencil-and-paper work. Whenever possible there would be more than one technique, and the class would discuss which one we liked the most. Once we all got to high school, our teachers kept remarking on how much quicker our year picked up the new subjects than previous years (we were one of the first crops of kids taught in this way).
My little cousin goes to the same elementary school as I did. When I looked over his homework, I saw none of what I was taught. A glance at his textbook showed they had switched back to the old method of teaching by rote. Apparently school board members who had run in part based on their skepticism of the new techniques had voted in a change in curriculum. One funeral at a time indeed.
Seeing what people complaining about "common core" mathematics are complaining about, and it is understandable that they are complaining.
At some point the idea came about that for things like addition, subtraction or multiplication, we should teach multiple different ways of viewing the same concept (like addition) in the hopes that even if students don't really understand the classic approach one of the other approaches makes sense. So perhaps the classic explanation of borrowing in subtraction does not make sense to some students, but one of a few other equivalent ways of handling it does make sense.
But then it got turned into a system where all students need to learn all these different different methods, and apply them in both homework and on tests. Which totally defeats the point. The who idea was that some students may find some methods useful and intuitive, and others find different methods useful and intuitive, and as long as every student finds some method can can work with we are better off than only teaching the classic method of the concept.
When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer.
I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”.
Which was such a thoroughly disappointing response and I hated trigonometry since it was mostly about memorisation rather than applying logic from understanding.
About 10 years later I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle.
It made me so happy to finally understand and see the actual logic as opposed to punching numbers into a calculator, but I was also sad and a bit upset the unit circle was not taught in high school.
>I asked the teacher where the numbers which were spat out by the calculator came from
This is a bit of simple knowledge which is sadly unbeknownst to even most math majors and educators. Often, students are taught in calculus that cosines (and hence other trig functions) are computed with Taylor series, which is not really correct. In fact they use CORDIC, a highly optimized algorithm.
But CORDIC is based on repeated use of the sum formula, and a simple version of this can be taught without calculus. Just notice that:
cos(2x) = 2 cos(x)^2 - 1
Then for small enough x, you have (can be shown by drawing, but kind of annoying):
cos(x) ≈ 1 - x^2 / 2
Or rigorously: 1 - x^2 < cos(x)^2 < 1 / (x^2 + 1)
So divide x by 2 until you get a small number (x < 0.1 is usually good enough), use the quadratic approximation, and repeatedly apply the doubling formula. CORDIC uses a similar iteration based on the sum formula. There is no need to wait for Taylor series, limits, convergence tests, etc. You can even bound the error term!
I'm not entirely sure this would work (i.e. improve understanding), though — I have not taught high schoolers before. And there is a decent bit of work required.
I can’t remember the details now, but my recollection was that the unit circle enlightened me that they were ratios. It was even possible to reason about “round” angles (like 90° and 45°), in your head without the need of a calculator.
I never tried a random angle like 13° in my head, but I figured it was kind of on a scale between the “round” numbers which I knew, which would give a good sense of the actual number, even if not accurate enough.
The point is, the unit circle explained the figures to me instead of them being just some magic numbers.
Likewise, there was an actual explanation of concepts like cos and sin in that they are the names for relationships between angles/sides (again, I can’t remember now, but it made sense), rather than them just being tools you’re told to use when.
Wasn’t there at least right triangle math for sin = adjacent/hypotenuse and cos = opposite/hypotenuse? This sounds like serious educational malpractice.¹
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1. Although when I think about educational malpractice, I remember a program I was teaching in where the director asked me about why the numerator and denominator were called what they were and I explained the denominator identified what kind of fraction (and made the analogy to denominations of currency) and the numerator was how many (the “number”) of that kind of fraction. She then, that same day, gave a talk to the assembled teachers saying that she’d asked me and I’d said that’s just what they were called. I nearly stood up and screamed in anger. If I hadn’t desperately needed the money, I would have quit on the spot.
My recollection is that we were taught the hypotenuse rule earlier in the year, might have even been just before trigonometry, but I have no recollection of being informed of any "connection" between them during school.
We were simply taught which equation to use for particular scenarios and how to use calculators with particular equations.
When you bought it up, I do recall the connection being used by Open University to explain things, which I also found a bit thrilling to learn.
Although only anecdotal evidence, I have asked a few South African colleagues over the years about the unit circle and it seems like none of them were taught it during school, so it seems like it's just not in the curriculums over here, which I feel is a tragedy.
There’s a nice diagram I created an EPS of (which I can’t find now) that shows the unit circle derivations for tan/cot and sec/csc.
Here’s how to create this. Draw a unit circle centered on the origin. Draw a ray from the origin where θ will be the angle measured from the positive x axis. You know sin and cos already: you draw a perpendicular from the x axis to where the ray intersects the circle. sinθ is the distance from the origin to where your perpendicular hits the x axis and cosθ is the distance from the x axis to where the perpendicular hits the circle.
Now, draw another line perpendicular from the x axis at (1,0) to your ray (tangent to the circle). Let’s call the point where this line hits your ray T. The distance from T to (1,0) is tanθ and the distance from the origin (0,0) to T is secθ. (As an added bonus, you can easily see that 1+tan²θ = sec²θ). We can do the same process, drawing a tangent line from (0,1) instead of (1,0) to get cotθ and cscθ. You can do some simple math with similar triangles to get familiar formulae like tanθ=sinθ/cosθ etc.
Explaining signs of the functions outside the first quadrant is left as an exercise to the reader.
> but I was also sad and a bit upset the unit circle was not taught in high school
I'm not a mathematician, but that sounds like poor teaching to me. Even if it's not part of the programme, learning the unit circle takes literally a few minutes and is an invaluable tool afterward. I don't know if it's possible to develop an intuition for trigonometry without learning the unit circle.
The problem is, these tools always feel like they should be "the way" it's taught. I feel complex numbers should be taught as a special case of geometric algebra, though I suspect that would make things much more difficult. My understanding is that lots of university level maths _is_ taught like this: start from the most general case.
> I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle. It made me so happy to finally understand
It also has a downside... The British Empire depended on robust trigonometrical education. In those days, you couldn't efficiently navigate the world's oceans without it and they needed a steady supply of ship's masters for the thousands of ships that made the empire work. Earth is the unit circle.
> The British Empire depended on robust trigonometrical education [...] and they needed a steady supply of ship's masters
The British Empire was already collapsing rapidly when the Open University was founded in 1969. I doubt that ensuring a supply of ship masters was a motivation in choosing to include trigonometry in the syllabus.
Sure, if you want to believe that Open University invented a brand new pedagogy rather than continuing a traditional style of teaching trigonometry that had been honed over the course of 150 years of naval power projection.
> I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived.
This is pretty easy to do for a polynomial term, but much harder for x raised to a non-integer power[1], or sin(x). There is no one way to "derive the formula for a derivative"; different functions have to be analyzed differently.
[1] Wikipedia suggests that the simplest way to derive this formula is to begin by establishing that exp(x) is its own derivative. But that's not an approach I'd be likely to take with students new to calculus.
I'm no math pro, so my terminology is probably way off. I just remember a lot of h(x) in there and showing why that makes sense in the context of what a derivative is.
I kind of had a borderline religious experience in college calculus too. We had a handful of problems back to back: how long has a murder victim been dead? How hot does a fast food chain have to make their coffee so it burns their customers taste buds so they can't taste it? How long does it take the moon to orbit the earth?
And we solved all these problems the same way. There was finally a grand order and purpose to all of it that was everywhere all the time.
I left school at 15, never to return to education, in part because of how maths was taught.
There was never an explanation of the _why_ of things. It was very frustrating, and the teaching was very poor.
I ended up a programmer, in part, because I figured if teachers weren't teaching, i'd teach myself. This has been a good strategy for me, however there's serious holes in my pure maths knowledge.
One of the thoughts I've been developing over the past 10 years or so is that you can't teach someone the solution to a problem they don't have. This encompasses the "why" question but even goes beyond it, because even the answer to "why" is often just another level of "why" and/or "who cares", quite reasonably. To learn something, you need a problem, you need to grapple with the problem for a bit, and then you can be presented with a solution. Then, the answer to why is clear: Because it will solve this problem.
Education would still have to artificially give students problems; waiting for students to naturally have a problem for which taking the derivative of a tan function and then leaping in to discuss that just doesn't scale in all sorts of ways. But if we gave them problems first, and let them chew on them for a bit, I think it would work so much better.
But that's an anathema to the current system. It would require not moving students through in cohorts because you need to give students enough time to chew on things and there's no way that will be standardized. And of course it requires admitting that Very Smart People Are Totally Wrong About Education, and I might as well ask for a pony while I'm at it. The Curriculum Must Not Be Changed. The Curriculum Must Not Be Questioned.
I wish someone would pay me for a few years to try to develop a computer-based math curriculum based on this concept. I'm still waiting for education to move to the phase where computers are used as something other than "The Old Curriculum, But On A Computer!" I thought we'd be farther along on that path by now. I seem to have underestimated the inertia of the Holy Curriculum, Hallowed be its Name and Hallowed be The Heroes Who Practice It in its infinite glory, yea verily.
There is plenty of pure mathematics that is beautiful and worth learning on its own without any practical application. A great deal of joy I get from mathematics is the delight in seeing a novel structure I hadn’t before where the proofs fall effortlessly out of the definitions.
Most of pure math does have practical applications - to pure maths! My favorite professors never launched into a subject without a motivating example, even if that motivation was often "Look at x, y, and z. Aren't they awfully similar?". My first exposure to Abstract Algebra started with a little number theory, moved on to rings, then to ideals, and only then to groups. Many people I've talked to about it are surprised we took axioms away rather than adding them, but the way we learned motivated each step. Indeed, groups themselves were introduced with permutations.
Similarly, I found measure theory was best introduced by showing how handy cardinality was for finite sets. A "practical" application would have been probability, so perhaps this wasn't exactly application focused, but we certainly didn't start from the definition and work our way out.
I feel the same joy when seeing a derivation of a novel algorithm where the effective procedure falls effortlessly out of the definitions. A good example is Dijkstra's derivation of Smoothsort[1]. It's worth noting that he was educated as a professional mathematician, not as a computing scientist.
To be honest, even in pure math this approach ought to be taken. I love math too. But it's not really a very good pedagogical approach even in pure math to start out with a full week of unmotivated definitions.
I have no issue with the problems being posed being very abstract at a suitable level for the student. By the time you hit college, I have no issues with a professor introducing group theory with "Hey, look at this aspect of graph theory, and this aspect of topology, and this aspect of algebra... what commonalities do you think we could abstract from them?" But that's a way better introduction even at that level than "Let's spend 90 minutes giving unmotivated definitions and hoping you pick up the pieces later."
In a conventional school setting I expect the problems to be more concrete, by their nature. I can give another example myself: Taylor polynomials. In my opinion, they're one of the more important things to learn at that level. You can give the students a simple problem: "Having learned sin, cos, and tan, and by this point memorized some of the common values, please develop a procedure for taking an arbitrary sin/cos/tan of an angle." Give them some time to chew on it. They may even come up with some modestly clever things, maybe cover some more special cases or something. But then you can go into how we only "really" know how to add, subtract, multiply, and divide, and here's a tool that allows you to take a wide variety of functions that up to this point only existed in calculus and as magic buttons on your calculator, and turns them into problems we can do with real pencils on real paper using real human brains that do not come with a "sin" button. (And then, heh, be grateful you live in the 21st century and you don't actually have to.)
That's now how I learned them. I learned them as just "Here's some Taylor polynomials. Do these homework problems." And I did. I learned them, and could do the math. It wasn't until years later in my computer hardware class that I realized this is what was motivating them. (Not the literal hardware, because of course Taylor polynomials greatly predate that, but the need to be able to calculate these things prior to computers.) And I'm not saying "oh, that's what they are"; math very often has the characteristic that something is discovered for reason X but then has both mathematical and practical applications well beyond it. My point here is that my understanding of Taylor polynomials is now much richer than what I got in the class I learned them in... but there was no reason for that insight to be delayed and almost coincidentally obtained. It could easily have been conveyed via a different teaching method.
I think you have the right of it: it's hard to teach something like maths to someone who isn't curious or interested. And it is definitely difficult to hook someone's attention.
When my children were still babies and quite young I was reading Zvonkin's book, Math from Three to Seven. And when they reached that age I started playing games with them myself to try and introduce these ideas to them. Like Zvonkin I found that one of my kids was more keen than the other... but the only way to keep them hooked was to avoid the "M" word: maths.
What I think helped was to remind ourselves that we were playing games. Any time I went into an area that required calculation: determining some value -- they would catch on to that and shut down. However if we stuck to exploration and fitting things together and exploring games together I could keep them interested for an hour some days.
And as an adult that's what has kept me interested: Martin Gardners' articles in Scientific American and books; John Conway's playfulness (ONAG, the bloody game of life, etc) -- the stuff that wasn't simply rote calculation which I find many attempts at practical applications seem to focus on.
I can appreciate definitions and proofs now because I've learned the language well enough to piece things together. However it was the fun, the absurd, and the playfulness of the completely impractical that kept me going. Games, thought experiments, what-ifs. That sort of stuff.
Agreed with your overall point, and your specific example. Myself and one of my good friends I met in my physics classes in college both felt the importance of Taylor series had been massively undersold in our calculus courses, because it just kept coming up in our various physics courses. I learned it just as a thing that existed, but we kept relying on them when deriving things in courses like thermal dynamics or mechanics. We would joke that calculus professors should stop the class and just emphasize, "This is really important!" But of course, that wouldn't make the material land any better, for the reasons you've explained.
Re: computer based math curriculum, this is what Jason Roberts is doing with Math Academy here: https://www.mathacademy.us It's mainly targeted to kids, but has adult users as well. It is by far the best self-paced math program I've come across with the widest breadth (from the basics up to graduate level).
He's been working on it for years and talks about it a lot on his podcast: https://techzinglive.com
I’d subscribe to your math app. Khan Academy Kids has greatly accelerated my kids’ language and reading skills to the point where they were reading chapter books to us before Kindergarten.
Brilliant and KA seem to be the leaders in self-directed learning, but I’m still waiting for the device described in Neil Stephenson’s Diamond Age.
It's hard to explain to young kids what "out of scope for this class" means. And that the number of people who need these skills vastly the number of people who need to understand the derivation.
Lattice multiplication will probably take algebra to explain. Long division definitely will require algebra. In my school, there's a gap of 5 years between teaching the two. A lot more people in the world need arithmetic than they need algebra (easily over a factor of 10). We can't put off teaching arithmetic till they learn algebra.
A lot of people don't realize that this problem goes all the way to undergrad and grad education in engineering or science. Laplace transforms are very useful, but they require complex analysis to begin to understand. If you blindly apply the integration that is normally taught, the Fourier transforms of several simple functions have integrals that simply, clearly do not converge. Yet we're taught tricks to indirectly calculate them. How is that possible? How do we get a result from something that clearly diverges?
And don't even get me started on the Dirac Delta function.
Recently I picked up an introductory analysis book - it starts from Peano axioms and builds up natural numbers, sets, integers, rationals, and then finally reals. It requires a fair amount of mathematical maturity to explain simple concepts, like how multiplying a positive with a negative could result in a negative, or how multiplying two positive numbers can result in an even smaller number (something that I did get upset about in my school days).
While yes, it is convenient to cherry pick examples where it was taught poorly without intuition, the reality is that if you want to prepare someone to go into, say, engineering, there is a lot of math one needs to cover, and teachers just can't afford to spend time explaining things that are way out of scope.
I don't believe (nor do I think you were claiming) that this means math cannot be taught more intuitively. An even broader example to your point is that Calculus itself was invented without foundation, at least without foundation modern mathematicians would find satisfactory; intuition patched some holes around 'infinity.' But Calculus could none the less be developed by blackboxing its inner workings and justifying its existence by just how spectacularly useful and predictive it was.
Likewise with arithmetic and algebra itself: ancient civilizations who factored numbers or solved equations did not require Peano's work to justify inventing some math. Calculus and all the math before it would indeed rest on foundations absent any Set Theory. The wider perspective is that discovery starts in the middle of a concept, and works out towards its implications and inwards towards its axioms.
The justification taught to those learning math (or anything) need not start from its axioms, but it should start from its history in context to how humans found use for the concept - that is justifying enough and likely more satisfying than learning axioms developed after the fact when the question is 'why?'
Much of the math part was taught in math classes. Engineering classes worked with the math classes to ensure the students were ready for the math there.
Chemistry, physics, thermo, dynamics, electronics, fluid mechanics, electronics, etc., were all math classes, in addition to a solid slate of required math classes.
If one didn't care for math, Caltech was a very very wrong place to attend :-) You either got good at it, or you left. I definitely felt that 4 years of that rewired my brain.
One of my friends in HS was against calculators, and also very stubborn, and got in trouble for asking how to do sin, cos, tan functions without a calculator.
I highly recommend you try rekindling your relation with math one day. It has lots of knowledge buzz moments, when you see things fit together. Feel free to email me if you have any questions, or you can post them on this forum for readers of the book https://gitter.im/noBSmath/community
I think an introductory course would ideally hop back and forth between the explicit execution of specific examples and the understanding of abstract concepts. As you say the concepts are very important - for example, the author never mentioned the associative, distributive, or commutative rules which apply to all cases of addition and multiplication (but not necessarily to subtraction and division). Those are universally applicable, and indeed understanding them is important for moving on to higher maths (group theory etc.)
However, without doing a fair amount of ditch-digging, abstract general theories of ditches might be hard to grasp. For a programming example, consider data structures - lists, trees, graphs, and so on. The abstract concept is clearly what allows larger constructs to be made, i.e. entire programs, but if you've never actually implemented a linked list in any particular programming language, and try to, the result is almost certainly going to be a buggy disaster.
The optimal math education might be one where for every hour the instructor spends on the conceptual approach, the student then spends about three hours on the execution approach as applied to specific examples of those concepts (ideally with a teaching assistant around to help if the student gets stuck).
When I remarked yesterday that engineering curriculums that emphasized memorization of engineering formulas rather than understanding how to derive them were inferior, I was dismissed as arrogant and egotistical :-)
I work with K12 schools in the U.S. and one of our really sad stories is how a calculus students failed to solve a "A t-shirt that costs $15 is 20% off today. How much does it cost after the 20% discount (don't include tax)?" <-- not the exact wording, but you get it.
They responded, "I don't remember the formula for a sale". Only knowing formula's is terrible, it makes knowledge super fragile. So, I for one support your idea of emphasizing understanding over formula memorization!
Eh, the closer you can get to entering a formula without the mental effort of backtracking and lookahead the better. It's like fraction buttons; obviously you can just think ahead and use parenthesis with division, but a smart fraction button will save a lot of time
You'll note that English is written left to right. In the string "20%" there is a "2", a "0", and a "%" arranged from left to right. To type that string into a calculator, one could press "2", "0", and "%" in that order, or "0", ".", "2", in that order. To know to lead with "0." rather than "20", you have to look ahead.
For the others you'd have to do the mental work to append a 1 regardless, might as well stick with decimals
Was that related or unrelated to the comment? I can manage arrogant and egotistical without even remarking. #engineering.
But more seriously (although that is normally how engineering formulas work) if an engineer is actually using the formula for anything they tend to pick up the intuition quickly. The memorisation in formal teaching is more limbering up the mental muscles so that it is easier to learn when & if the time comes.
Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed.
Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and those who had memorized them were more likely to solve the problem than someone like me who happened to forget the theorem existed in some book and is trying to rederive everything from scratch.
Of course, those who mostly or only memorize perform the worst.
Basically, most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform. So some level of "take it on faith and memorize a few items" are needed. There are other examples of this.
Also, having studied under many top class physicists, I can tell you that a significant number of them cannot derive much of the mathematics they use, despite being wizards in applying the techniques.
1. none of the exams at Caltech required memorization. They were open book and open note. Memorizing simply wouldn't have helped. For example, one physics exam question was: "Assume magnetic monopoles exist. Derive how Maxwell's Equations would then look." If you didn't understand the ME derivation, you'd be completely lost. The same for FFTs, where the exam question was derive the hyberbolic transforms.
2. one winds up inadvertently memorizing things used often, like I knew all the trig identities from excessive use. I never attempted to memorize any of them. Just like I know a lot of the hex opcodes for the x86 :-)
3. we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.
4. I've forgotten an awful lot in the 40 years since. But I took an online MIT course and was pleasantly surprised that it was still there, it just needed a bit of oiling.
> most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform
Which is why some engineering schools are better than others. I applied to
the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.
I'm sorry you were left behind because of your insistence on learning it thoroughly. You were doing it right. You just were in a school that didn't value doing it right.
BTW, I did not mean deriving a formula from scratch every time you used it. That would be silly. Just that at some point you did, and thereby understood where it came from, hence understood its limitations, and knew how to adapt it to a situation not in the book.
Much of your experience was Caltech specific, and will simply not translate to other schools. Caltech is famous for this - more so than schools like MIT, etc.
1. I learned the hard way, as did others, that one should still memorize with open book exams (or at the least make a 1-2 page cheat sheet). Why? Because there was a time limit and most professors would not alot enough time for people to even look up everything they needed in the textbook. Sure, if they increased the time by 50-100%, you'd do just fine with memorizing.
I'm not even hypothesizing there. After one open book exam I went and asked everyone who got a good score - most had incorporated some level of memorizing. And clearly most who did not memorize at all got a poor score (which is less surprising than it should be, because most students will do poorly regardless ;-) Still, I was the clear exemplar of one who improved from "below average" to "one of the top students" within the duration of one semester when I finally embraced that some level of memorization would be needed.
2. True, and I can relate to trig identities - most of which I remember 25 years later - and even after over a decade of not needing them. However, when I got to more advanced topics in math, the frequency with which I would need to use them dropped significantly, and the approach of "Just do lots of problem sets and you'll passively memorize" failed me in grad school.
> we were not expected to use the Laplace transform until its derivation was demonstrated. This applied to all the formulas used. FFTs too.
Did they teach you measure theory before those transforms? Did they teach you measure theory before probability? Did they teach you the theory of distributions before the Dirac-Delta function?
> Which is why some engineering schools are better than others. I applied to the USN&WR list of "top 10 engineering schools in the US". Caltech was #2.
I went to a top 5 engineering school. Can assure you Caltech's approach is not the norm.
I am sadly well aware of that. Caltech was known as unique at the time, I wonder what it is like 40 years later.
1. It was rarely necessary to look anything up. What was on the exam was reliably in the notes or in the assigned textbook. I don't recall ever memorizing things, and managed an A- average. My innate abilities were completely average there, and we all knew who the really smart ones were, like Hal Finney. What I did do to prepare, however, was ensure I attended every lecture and took comprehensive notes, make sure I could solve every homework problem, and every midterm problem (in trolling for the final). I did not look at prior year's stuff.
2. I didn't attend grad school, so can't comment there. But I did match wits with Masters engineers at Boeing, and would wind up fixing their work, too, though far more rarely. That group eventually offered me a position, though they had a Masters as a requirement.
Measure theory wasn't taught, at least in the undergrad courses I took. Neither was the theory of distributions.
> Can assure you Caltech's approach is not the norm.
There's nothing wrong with practicing formal manipulations: these are just "the rules of the game" and are 100% rigorous. Now, if you want to know "why these rules and not others?", they just happen to follow from basic properties like associativity, commutativity, distributivity etc. There's not much else to it.
The author's "misintuition" of variables is not critically important; variables can literally just be arbitrary symbols, and substitution understood separately. But the other "rules" are far more relevant.
The one thing that gave me deep insights into algebra was discovering Peano's axioms.[0] I had been taught the associative, transitive, and distributive laws in 6th grade at school, but discovering the axioms really opened my eyes in high school; especially under what circumstances they could and couldn't be applied. The last penny to drop was to realise that where you have numbers (e.g. x) you could also substitute formulas, provided you adhered to the rules of when the axioms could be applied. Obviously for similar reasons this also applies to matrices (e.g. when can matrices be inverted (additive and multiplicative), and hence why matrix addition is commutative, but multiplication is not).
Don't get me started on subtractive anti-commutativity vs divisive anti-commutativity.
Another thing that bothers me is when helping the my own kids with maths I came to realise that their teachers don't know the Peano axioms, so how the heck can my kids even stand a chance at properly learning algebra.
> “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”
Many other comments here don't seem to have spent much time reflecting on why his niece said this.
Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, consider that learning math might increase your capacity for memorization as a byproduct. This skill carries over to so many others, but I think its importance gets glossed over as a result of it being simple to look things up using the internet.
Another thing to think about: if you have something memorized (e.g., an identity, a theorem, a formula...), then you can think about it when you're walking around. When you have many of them memorized, you'll be able to think about how they relate to each other. The more you have memorized, the fewer stumbling blocks there will be to trip over when you're mulling over a problem or trying to figure something out.
There's a lot of apocrypha out there about how if you're learning math, you shouldn't try to memorize things. You should instead try to "pick it up". This elides two important details: 1) there's selection bias at work---many people who are extremely good at math have minds like steel traps, remembering many things after seeing them only a couple times, 2) the expectation is that you spend a significant amount of time drilling and working, to give yourself an opportunity to pick it up.
Now, why did his niece say this? When you're learning the basics like this, becoming "comfortable with formal manipulation" is really a matter of running a large number of experiments. You try this and that manipulation and get comfortable with them over time. You apply a formula in error and observe the results. You puzzle over what happened. Over time, you discover the logical framework which holds everything together.
Asking "why" at this point (when you're learning very basic math!) is like putting the cart before the horse. You would receive an answer that you would be in no position to understand. Hence, "be like a child": play around, make some mistakes, try to understand what's going on. As you grow, you will become more sophisticated in your approach and better able to shift the balance so that you can attempt to simultaneously understand more of what you're learning.
Maybe "asking 'why'" isn't the right way to express it. Perhaps what the OP was really after here was a concrete illustration of the formula, rather than a rigorous derivation of the formula from first principles.
There are many concrete geometrical illustrations of fundamental concepts in algebra, but most people aren't even aware that they exist.
The niece's comment, oddly, both points to this and glosses over it. You don't teach a child multiplication by only forcing them to memorize times tables. You also show them stacks of coins, etc. But we do teach children algebra by only forcing them to memorize symbolic procedures. Meanwhile Book 2 of Euclid is free online, or very cheap from Dover.
I can distinctly remember in my second or third year of engineering undergrad doing some stuff with the Radon Transform. At the time I had nothing to map it too. The maths to me was completely abstract. I just sat there, thought hard, and worked through problem after problem until it became intuitive. Now I have difficulty understanding what I found hard, because it all fits together with loads of other things into a coherent mass of concepts.
It was interesting to me because it was one of a few times as an adult where I realised the framework was completely missing and I would just have to start from scratch, which meant lots of rote effort and thinking hard. There was no shortcut.
> ...still ended up going down an execution rather than concept focused route.
Everyone will say, of course, that "concepts matter". But the reality is that there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables.
One can't move on to new concepts in math until the previous dependent concepts have been mastered. Mastery means practice, practice means drills, and all of that is boring but necessary. The best we can do is to make the mathematics more meaningful by tying it to problem solving. At some point many students will experience an "ah ha" realization and be further stimulated, while for some others math will always be meaningless drudgery. For most it will be something in-between.
> there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables.
Sure, but imagine learning multiplication tables without having any idea what it means to "multiply"; literally just memorizing sequences of symbols, without ever looking at piles of coins or whatever.
Multiplication is so basic that this is hard to imagine, but I'm sure that, say, difference-of-squares rules feel like this to most beginning algebra students -- and for most people that probably never changes.
I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.
> I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.
Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretation is merely a manifestation of it that just happens to work in some domain.
As an example, the formula is equally valid for complex numbers, but I doubt you'd get there from Euclid's. No doubt some geometric interpretation can be found for that as well, but then I'd pick some other algebraic field where it's true and you'd have to search yet again for a geometric interpretation.
Fair enough. Maybe I should have just said that it's possible to give concrete illustrations of basic algebraic concepts, and that doing this would probably help some students learn algebra, and might help others retain it.
But for whatever reason this is generally skipped in middle/high-school algebra.
I'm torn. I think it's always good to show these - it certainly makes the subject more interesting!
At the same time, if one is to use algebra for future studies/work, one really needs to be able to manipulate those symbols in the abstract, without feeling a need for some deeper understanding. I can see teachers not wanting to deal with "But what does that really mean?" for every detail in algebra.
This goes to the question of why anything other than basic arithmetic is compulsory. The famous 10th-grader's whine "what are we going to use this for", which infuriated my own 10th-grade Algebra 2 teacher, and even made me roll my eyes at the time, is actually a fair question when algebra is taught as abstractly as it typically is.
I think you've explained exactly why here -- because the emphasis on abstract manipulation presupposes that this is useful for something that we need to get on to. But that's just false for almost all students. And yet they're required to take the class to get a diploma.
My vote would be to treat any math beyond basic arithmetic as a liberal art, and do a lot less of it in compulsory curricula, but spend a lot more time on deep understanding. This would benefit everyone. The current approach pretends that everyone in the class is going to be a certain kind of engineer or scientist some day.
I was disappointed by this. It’s just a caricature of only one side - It’s the easiest form of come. Almost things that are innovative will not be best on the first iteration. Why not offer (funny) ideas on how to improve it?
You could read the article I linked, if you want to know why new math is problematic. It's not that easy to improve things. And who said that comedy needs to be balanced?
If you know the history of education, the "new math" thing is a sad story of a mind-blowing stupidity that keeps hurting math students for decades, because it divided most people into two camps that keep promoting two different wrong ideas.
Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people.
If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is.
Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers.
This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera.
In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one.
An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc.
This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids.
And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all").
To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens.
So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked".
So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha".
And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks.
Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".
I don't really know much about the history of education, I only read the article I linked. From what I understand from that, it is not so much that constructivism has been debunked, but rather that in order to explain something properly, you need to understand it yourself deeply. Now, many teachers don't actually understand math on that level. So it will be difficult for them to teach it based on constructivism. That seems to me to be the main difficulty. Also, everybody understands things differently, so even if the teacher is capable, how do you scale that for an entire classroom?
> in order to explain something properly, you need to understand it yourself deeply. Now, many teachers don't actually understand math on that level. So it will be difficult for them to teach it based on constructivism.
Yes, this is a sad truth about teachers, many of them actually do not understand deeply the stuff they teach.
It seemed to me unlikely that e.g. a 10 years old kid could ask teacher a math question (related to what they are learning) that the math teacher couldn't answer. Like, any adult, especially one with a university education, should be able to answer any question from the first four grades of elementary school, right? Except... nope. I have friends that try to promote constructivist education at schools, and "what if the kids ask me something I will not be able to answer" is indeed a frequent objection made by actual teachers.
Like, okay, it is definitely possible to make an innocently sounding math question that is actually extremely hard to solve. My favorite example is: "can you cut a square into an odd number of triangles of equal area, but not necessarily the same shape?" (The answer is no, but the proof requires university-level math.) But in real life, this is extremely unlikely to happen; and if it happened to me, I would simply say "sorry, I do not know".
> everybody understands things differently
This is perfectly okay. You let the kids think about it individually first, then discuss their solutions with each other. If you get multiple explanations how things work, that's great -- having things described from multiple perspectives helps you understand it even better.
>Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything.
This is so true. I was exactly like the author. I could not understand any math beyond algebra. Trigonometry was way beyond my ability.
But one day, I found an old trig book lying around when I went to visit my parents - it was one of my sibling's books when she took it.
I picked it up and had a thought - all those people who DID understand trig and higher math were not that much smarter than me. I mean, yeah, the super smart ones were, but I'm just talking about the regular students. So I got a bug to show myself that if they could do it, then I could do it.
My approach that I would do is to read the first section of the first chapter until I understood it completely and not move on until I did. I would not even move on from a sentence, or indeed a word or term, until I understood exactly what was happening. I would look all over the internet, watch youtube videos on that one single concept, or that one single word, until I understood, and I was absolutely adament about not moving on until I understood.
Word by word, sentence by sentence, section by section, I rolled on. I wanted to completely completely completely understand the very very basics of trig. It was slow going, as you might imagine. Super slow.
Well........all of a sudden, about 4 or 5 chapters into the trig text, my mind exploded. It was blown apart by a blinding light. In one single instant, I swear, one instant, I knew all of trigonometry. I got it. All of trig. One single instant. It was probably one of the most singular moments in my life.
So I then started paging through the entire rest of the trig book, and I knew it all. I swear. I would just glance over the pages and know what everything was about.
Now, I was NOT an expert, by any stretch of the imagination, I'm not claiming I became a savant. No. I'm just saying I "got" it.
As you said, I has the exponential results. So exponential, it was instantaneous.
>If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary.
This is it exactly. You describe it perfectly.
I did not continue on, because I actually had other things in my life that I had to move on to. But it was great because I knew then that math was easy. No doubt I could do the exact same with any math class, no doubt in my mind whatsoever.
For whomever wants to do great in almost any area of study, study HARD on those first 3-5 chapters. They set the tone for the rest of the course. If you study hard on them, they remainder of the course is trivial, with some things here and there that you need help with, but for the most part it is simple.
Very well said! It is true that once you grok this technique, you are basically unstoppable. It does make for slow progress at first, but it's worth it.
Edit: I went to grad school for math, and I got there by pretty much doing what you described!
Ok you two have inspired me. I've been wanting to learn math (and relearn college math from decades ago) and I always get bogged down and quit. I'm gonna try it your way.
I recommend the Art of Problem Solving community as a great place to start. I have no affiliation with them, they're just awesome for math learners of all levels.
The problem with using the right techniques is that there are essentially three kinds of techniques:
a) fast at the beginning, then gradually slow down, e.g. rote memorization;
b) slow at the beginning, then gradually speed up, e.g. deep understanding;
c) slow at the beginning, then remain slow forever, e.g. doing something chaotic and stupid.
The problem is that sometimes you have incompetent people who use the third kind of method. Then, when things blow up, people start paying close attention to the speed, and reject any method that is slow at the beginning, because they suspect it would be the same story.
(In other words: premature optimization, technical debt.)
I went through a very similar process. Not moving on, not hand-waiving, not just executing steps before understanding them - it changed the way I learn math and learn and think in general for the better.
One analogy I've come up with to justify this method is that you should not measure your progress by distance covered when traversing terrain of varying elevation. Some days you will be walking up hill, and others downhill. To lament that you covered two miles today but ten yesterday without regard for the energy required to traverse each day's path is often the type of thinking that makes students give up on their math or technical abilities.
Often times when learning something technical, I will spend as much time on one sentence, one line of the proof, as the rest of it. Being honest with my own understanding makes it harder and makes these roadblocks more frequent - but the unquestioning mastery of the topic after the fact is well worth it.
I've had a couple moments like that in my education. I've described it as stumbling about in a dark room, and suddenly the lights came on. One was when I struggling to learn how a computer worked. It was just a collection of arbitrary facts to me, more and more, until suddenly -- shazaam! And the rest was obvious and trivial.
This is addressed in the article - just after the ‘(7/2) / 2 = 7’ misunderstanding:
[I called] Deane Yang, my friend who is a mathematics professor at NYU.
“The way you remember procedures is you remember why,” he said.
“Because?”
“Because people learn math as a collection of procedures,” he said. “When things get difficult, they’re lost, and math becomes religion class. The teacher says what’s right and wrong, and for all you know math came out of the sky, and some prophet told you how to do it, and it’s just blind belief then.
Interesting that he wants to know _why_ knowing why helps too!
I do wonder if this is covered more in-depth in his book.
> you lose the ability to understand math as anything other than a human attempting to be a computer
I get what you're saying and it's important, but I also disagree with how you're framing it. It's desirable to get to the point where you can automatically do purely mechanical symbol transformations as you work through a problem, thus saving your higher reasoning for the conceptual problem of searching for which transformations will lead to a solution.
Frankly, I find the continuing concern with "intuition" with respect to purely mechanical transformations to be rather medieval. It is slowly changing though as the influence of computing science osmoses back into more traditional mathematics. Using Mathematica or similar tools makes it abundantly clear how distinct the mechanical and conceptual challenges are.
I don't believe layers should not be built on top of to the point of rarely, or even never thinking about the inner workings again. I just believe that when building those layers, having an intimate relationship in your mind with how they work and relate to each other will have positive effects, even after you start executing steps without regard for their deeper concepts.
That's the case even for just the 'engineer' archetype, who uses math strictly as a tool. For any person doing theoretical work, discovery is often preceded by a fundamental understanding and tweaking of the most basic tools - so I believe the mindset is useful in both cases but borderline necessary for the second.
> this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions
Leaving aside that you concluded this from a one-sentence story about solving a single problem, it's funny that the idea that a "clear understanding of fractions" can make the skills easy to learn is something that virtually every trained teacher has believed for the last (I'm not sure how many) decades and yet nobody alive today is reporting that it happened that way for them.
It's possible that every future teacher getting certified to teach politely pretends to believe these things, but secretly wishes to spend their career torturing children, so they abandon the teaching of concepts as soon as they step into a classroom. Or maybe they go through the motions of teaching kids to understand the meaning behind fractions, but they intentionally do it wrong and laugh with each other about it in the teacher's lounge. That's one explanation.
Another explanation is that this opposition of "understanding" to "execution" and "concepts" to "skills" is an incomplete way of looking at things that has limited application to how people learn but isn't a panacea, and people still struggle with skills no matter how much you try to prepare them with concepts.
I think it does a vast disservice to anyone when we assume that their job performance would be radically improved by the first cliché we ever learned about their field.
If the idea is so magical, and we know that it has permeated the field of education for generations, where are the generations of students who remember math as being "trivial, intuitive, beautiful, clever?" Different educational philosophies have come and gone, emphasis has shifted, so where's the generation of parents saying math was much easier for them than for their kids? Where's the PTA meeting where all the parents say, "We think there's something wrong with the way math is taught here, because we all remember fractions being trivial and intuitive, but our kids are finding them hard?"
Maybe teachers aren't such idiots that they consistently overlook the value of the most cherished and shopworn truism their field has produced in its history?
I'll start my reply with this: I'm not a teacher, nor am I an understander of maths, but whenever I talk to teachers at any level, and especially mathematics teachers, they lament that they want to teach differently, but due to time constraints, or predetermined lesson plans, or any number of reasons, they simply can't. They have to teach in a way that gets students enough knowledge to pass the test, but only those that seem to have a natural aptitude end up really learning anything. The rest discard the information as soon as it's no longer necessary.
I don't think it's a disservice to teachers to say it's a better way of teaching, because they say it themselves. It's just a model that only really works one on one. It can't be done in a room full of 30 bored children, and especially in a room of 400 university sophomores who are ready to get out of class to go to a party.
You are saying some things that are guaranteed to generate skepticism. First, that the great power of this teaching method only manifests under circumstances that very few people get to experience. Second, that concepts make skills "trivial" and "intuitive" but testing skills prevents teachers from teaching concepts.
Both of those statements call into question the effectiveness of what you're promoting.
It is okay to be suspicious when it sounds like someone just made this up, but actually in educational research it is one of the few well known things.
Anyone who tried both teaching and tutoring knows that the difference is just incredibly large. A part of it is that 1:1 you can pay more attention individually; a similar argument can be used in favor of smaller classes. But the other part is that as a tutor, you are free to actually use your best judgment, while in school it is more of "yeah, I know that I should do X, but the rules say that I have to do Y instead".
The school is just insanely ineffective, for various reasons. You need to follow a predetermined schedule, whether it makes sense for the given classroom or not. Your students are expected to already have some knowledge from their previous grade, and if they don't (which happens quite often) you don't get any extra time to catch up. There are all kinds of disruptions, like students who never pay attention and interrupt you and their classmates during lessons, but you must proceed at the speed that was designed for a hypothetical classroom without disruptions. The school inspection randomly checks whether you follow the latest fad, usually based on some pseudoscience, like whether your lessons are okay for both visual and kinesthetic learners, or whether your math lessons are sufficiently decolonialized.
So the same teacher who fails to teach her class fractions at school, may be a successful tutor during the afternoon and explain the fractions properly.
I disagree. My favourite lecturer for general relativity said the important thing is fluency, which comes from repetition. It’s great to know everything, but you should be able to do the basics, such as calculus or algebra, very quickly and without thinking. You will need that brain power for the new stuff. The best way to gain mathematical fluency is by lots of practise.
You can understand something first, and practice for the speed later. Understanding does not prevent practice.
I know the result of 8×8 immediately, and I don't think I would ever forget it. But, hypothetically speaking, if I ever made a mistake here, I have the option to slow down and verify the result. If I wouldn't have the option, there is a risk that I would make a mistake and then keep making the same mistake forever, because once the lessons are over, there would be nothing to correct me.
I agree, but the fundamental problem is how most western math education is structured top-down to be a goals-driven curriculum. In other words, teaching to the test.
In the US, it’s a sprint towards meeting goals tied to government funding (“core curriculum “, etc.) And so many get left behind in that sprint, often feeling they’re ‘not smart enough’ or ‘not a math person’.
Yes. The clearest example of this for me was the fact that many things I was taught in high school were taught to me as an arbitrary concept first, without the accompanying visual relation.
Being taught a² + b² = c² with the triangle and accompanying squares drawn out much much later is the best example I can give.
I agree. He tried to memorize his way through Algebra. It is hard to understand his mindset because of the stream of consciousness writing. But he appears to be confused with definitions versus equality. E.g., By definition (a+b)^2 is (a+b)(a+b). It is not just "equal".
This is insightful and rings true for me as well. A good blend of theory and practical knowledge is important to get the most out of any education. Do you have any recommendations for learning math this way?
This person describes an experience similar to the one which inspired my mindset.
One other thing to note is that the order you might learn math, even the 'bundles' of math confined to the standard classes, is somewhat arbitrary. If you take on the freedom, you are responsible for the direction. That being said the linear path from arithmetic to Calculus or Linear Algebra, or Vector Calculus (to combine the two) will give you very strong fundamentals and chops that will apply broadly. It feels like once you get some understanding of Vector Calculus you can branch off wherever you want, though many advanced topics past that don't strictly require it, learning to derive concepts in that subject is excellent preparation.
All that said, there are plenty of interesting topics not involving Calculus or Linear Algebra. I really recommend what As_You_Wish has said as being honest with your own understanding and developing deep rigor, even across just one chapter or one sentence, will be your guide. Once you get deeply acquainted with one area, even if just a tiny area like "I finally understand trig functions," you understand better where you can jump off to next.
Unfortunately, I don't have an easy answer. My son recently graduated from my high school and I noticed that while his school used Pearson's website to drill him on the mechanics. My son struggled with word problems precisely because he did not have the fundamental concepts.
For example, I argued that division is more intuitive when we multiply using fractions.
- Whole numbers have an implicit denominator of one. E.g., 7 is equal to 7/1.
- It is more intuitive to think of division as multiplying by fractions. E.g., 7/2 is equal to (7/1) x (1/2). We multiply the numerators together. And multiply the denominators together.
- Where this is helpful is when we deal with complex arithmetic like (7/2)/2. We have 7 pizzas. Divide by 2. And divide by 2 again. If you break it into fractions, this is equal to (7/1) x (1/2) x (1/2). The numerator is 7x1x1. While the denominator is 1x2x2. Putting it together, it is (7x1x1) / (1x2x2) which is equal to 7/4.
This discussion usually leads to the question of what is multiplication (again)? I argued that multiplication can be thought of as a shortcut for addition or can be thought of as set theory. I have two bags of marbles and each bag contains 10 marbles each. How many marbles do I have? You can either write 10+10 (multiplication as a shortcut for addition) or you can write 2x10 (set theory). Taking it back to division. I have two bags of marbles and each bag contains 10 marbles and I have to share my marbles with my sibling (divide by two), how many marbles do I have?. You can write either (10+10)/2 or you can write 2 x 10 x (1/2).
I personally found that learning math was far easier as an adult. Like the author, I sucked in grade school. I learned multiple foreign languages. I did well in the sciences. History and geography were simple. Math? It was the one thing that escaped me. As an adult, after various misadventures, entering into college everything was different. My mind was more disciplined, my teacher was far better, and I was more humble. It was easy then.
Despite that — or, maybe, because — my father was a math academic, I did poorly in school with math beyond basic arithmetic. When I was about 25 years old, a friend gave me three algebra textbooks written by an ex-military pilot, John Saxon. In a marathon session lasting about two weeks, I went through all three books from end-to-end and really learned algebra. For whatever reasons, the Saxon books worked really well for me — better than any other learning I have ever gotten from a textbook. Yes, I was really motivated but I attribute a lot of my success learning algebra to those books. Learning that opened a lot of doors since then, mostly because my confidence in taking on matters technical was so much improved.
I’ve been tempted to pick up an adjunct section of algebra at a local community college and invert the usual style of teaching and start with word problems. A lot of people have an intuitive sense of how to figure out, say, how to scale a recipe but when it gets turned into symbolic math it becomes a challenge for them. I’d kind of like to take advantage of that and start from the word problem and then move to how to turn that into symbols so that instead of thinking y = 1.4x-2.8 we'd say think in terms of you pay $1.40 for each donut but the first two are free. If all the x² and x³ have particular meanings, you’re less like to think that x²+x³ = x⁵ but recognize that you can’t simplify the expression until you know what x is.
"Sometimes I dreamed that there were numbers falling from the sky into chasms I couldn’t see the bottom of."
Reminds me of the night terrors I had when I was younger. I remember often having dreams that involve the unknowable, unseeable or in some other way were impossible for my finite mind to comprehend. In the worst of those nightmares I was left shaken with fear — feeling as though I had peered from just over the precipice of sanity itself.
As a child I experienced sleep paralysis where I would hallucinate shapes of impossible dimensions, both simultaneously incredibly large and incredibly small, hanging over my bed and leaving me to question the nature of everything. Looking back on it I can't even begin to understand why I dreamed those dreams.
My friends dad in his late 60's wanted to learn math and used the No Bullshit math and physics from https://minireference.com/ and loved it. Their Linear Algebra is quite good too.
I recently released a new book with just high school materials, specifically targeted at adult learners. Check out the website here: https://nobsmath.com/
I have to sit on my hands and not help my son with his math homework. The way they have taught him to do math is great - he can solve just about anything in his head - but I can’t help him without introducing him to the old slow way I learned of doing everything.
I really liked the Common Core early math curriculum. My daughter was taught to do calculations the way I had to come to them myself later on: break the problem apart into easier pieces. A simple example: If you are adding 73 and 29 break it to 70+20 and 3+9. Quick calculations takes the burden off on harder problems later on. Also, she learned to round and estimate and they did lots of practical math applications -- especially in probability -- so she's learned to think of math in a "common sense" way, for lack of a better word. It's easier for her to look at an answer and say "wait, that doesn't make any sense considering the givens".
That link did mention the "Advanced Cognitive Training for Independent and Vital Elderly (ACTIVE)" study. Apparently it found that training can affect quality of life and reduced car accidents in driving seniors.
No, but once you have learnt some algebra you'll be in a position to learn other things - electrical physics for example, or audio engineering. These subjects, and many others, use the language of algebra, because algebraic explanations are very compact and easy to follow.
So that's how a packer thinks! I'm a mapper[1], and it's always been a mystery to me as to how packers got that way.
When I was an IT Admin, I worked with woman who VERY competent at her job, but always needed help using the computer, for even basic tasks, even with a list instructions. It took time before it dawned on me she was a packer, and my instructions weren't specific enough.
I started making exact instructions for her, accounting for any possible variance in each step. (As you would in a program) Once I did that, she was thrilled, and we got along very well after that.
An example of things that can throw off a packer... in Windows Explorer, drag a file from one window to another... depending on context, there are 3 possible outcomes (move, copy, shortcut creation)... that confuses the heck out of people, especially packers.
Probably not. Knowing more stuff doesn't make you "smarter". Nor would mastering kinds of math that many people find challenging prove you are more intelligent. Clearly many intelligent people are not great at math or don't know algebra.
But exercising your curiosity and practicing the art of learning will always increase your mental fitness and the ability to apply your intelligence better. It's always good to keep in shape - mentally too.
I don't think it matters much what the subject is. It doesn't have to be algebra, as long as it captures your interest. Algebra is very challenging because it requires thinking in abstract concepts, so it's a great subject. It will indirectly lead to deeper understanding in many other fields too, including computer science.
Intelligent people are often curious and desire to learn new things, so No I don't think learning algebra makes you smarter - but I think the desire to is a possible symptom of this kind of curiousity.
Scientists can't agree to what extent human intelligence is inherited vs. acquired, and so asking if doing some activity x will result in an increase in personal smartness y will result in a wide variety of answers. (sort of a differential equation isn't it? Change in intelligence over time as a function of doing algebraic manipulations every day).
There are some plausible benefits - it's likely to help with internal mental organization, for example. However, some thought should be given to how to set to work, in particular avoiding bad habits that eventually thwart future progress (this can happen in say, learning to play a musical instrument).
Learning algebra (or any other field) conceptually is like constructing a connected map of various islands, although from reading this article it isn't clear whether that was accomplished. There's no mention of concepts like associativity, distribution, commutation, how this applies to the order of operations when you start mixing up + * - /, and the rather strange but also fundamental notion of identity. For example:
This is a bit sad because understanding these concepts opens the doors to fun higher math. Matrix multiplication is generally non-commutative, and this property makes it useful for quantum mechanics calculations. Symmetry operations - rotations and reflections and so on - are not generally commutative, but are associative. This is all very important in things like protein crystallography. A solid grasp of these ideas also allows for the introduction of the concept of groups as a way of looking at algebra. Here's a great series on that whole business:
A good rule is to spend at least as much time understanding fundamental concepts and abstractions as on working out the results of specific explicit examples.
One of my retirement goals is to redo my math degree - turns out it's really easy to get hooked up with online tutors, and I figure it's totally worth paying $20-$40 an hour to work through a textbook.
If you have a math degree, you already know how to learn math. That hasn't changed. You're not going to try passing tests, so you don't really need a tutor. What you might want is a counselor to guide your "reading", as is done in places like Oxford. That can be worth the $40.
All you need is updated texts, a plan of action, and maybe Wikipedia for an alternate take on sticking points. (Just using Wikipedia doesn't work well.)
You could also look at the MIT or Harvard online curricula/syllabuses to get a feel for what the current subjects and texts are. Maybe take a couple of the free courses.
Also, scan Kahn Academy first - that site is good for identifying holes in your background.
I posit that learning anything (even fengshui or astrology), at any age, will make you smarter, as long as you bring a pair of critical eyes with you. Besides learning about the actual “thing” (which you can still fail at the end of the day), you can always introspect to figure out why/how you failed and what your limits are, and then you’ll maybe find closure in saying “I’ve tried. It didn’t work out. There was no 'wasted potential' - the potential was never there to begin with. And I’m a wiser person now.”
The title is there to get you to click, there's not much need to analyze it. The article has little to do with it.
This reminds me of On Cinema At the Cinema when Gregg Turkington, as a parody of the film critic persona, often bases his critical opinion of the films based on what he thinks of the title and runtime.
This quote reminded me of learning chess as an adult as well:
> In the paper Acquiring Skill at Mental Calculation in Adulthood, Neil Charness and Jamie Campbell say that middle-aged people perform as older ones do, but if they practise, they perform more as younger people do. If speed is valued more than accuracy, the decline in ability is obvious. If accuracy is valued more than speed, the decline is less obvious and maybe not even very pronounced.
In my opinion, nothing can make you smarter. Learning rationalist techniques may make you better at thinking through problems, learning memory techniques like Memory Palace may improve your memory, using learning techniques like Spaced Repetition may help you learn and retain large amounts of information with the minimum necessary effort, but nothing can make you smarter.
I'm pretty sure this is wrong. The brain adapts to whatever activity you do and, as far as I know, is way more plastic than researchers thought decades ago. If you're doing dumb things all the time you're making yourself dumber over time and when you're performing difficult tasks repeatedly and daily your brain adapts to them to be able to solve them easier. I realized that a long time ago in a completely different area, in martial arts. When you start it's extremely hard to follow complex movements shown by other people. Beginners struggle with this a lot. Over the years it's getting easier and easier and you can often grasp complicated movements after seeing them once. I suppose it's the same in dancing and ballet.
The brain adapts to sports, painting, writing poems, or solving complex math puzzles. It just happens that some of these activities are considered smart and also (partly) measured with traditional IQ tests. You can even train to become better at IQ tests if you desire to acquire a completely useless skill set.
I think you have to define what "smart" means. I mean if you dropped me into some of my undergrad courses today, especially my CS courses, I would dominate like Wilt Chamberlain and people would think I am a genius. But what does it even mean to be smart anyway? I think in school a lot of the people we deem to be smart simply have more prior experience in the subject or just work very hard.
Adderall doesn't make you smarter; it just makes you more alert. It helps you focus longer on whatever intellectual task you're working on, be it problem solving or studying. (Not /s)
So being able to work longer and focus harder than what would normally be physically possible is not the definition of “smarter” even if it’s temporary?
Math is so much easier when your using it to model something that is real. I remember in Physics 1 in college when the professor was going over how a baseball hits a bat and how it is modeled with all of its nth order derivatives. Everything clicked! Not to say theoretical math isn't fun though!
Probably yes, since past research indicates that learning delays cognitive decay.
Even more, there is research pointing that continuous learning will prolong your lifespan. Best indication is a study on how university professors tend to live longer.
Despite the title, I don't think the article is about algebra "making you smarter", but more whether you can learn it successfully and if by pursuing such an endeavor you can hamper age-induced cognitive decline.
My litmus test of someone being intelligent or not--among other such tests--is what they say about Shakespeare. This might anger the Brits, but Shakespeare is garbage--and the guy says: "I read .... most of Shakespeare."
One famous guy lambasting Shakespeare (Tolstoy) was only marginally smarter, because according to his "Confessions", he read it many times over (incl. in English) and STILL could not find any artistic value in it. I could not have put up with that--for me, it was the first few pages that I knew this was a waste of time. You don't need to drink the whole festering junk of a decayed meal to know it is not edible.
He also claims he wants to test the limits of his intelligence by tackling Mathematics once again--implying wrongly, as it were, that math education is fine and it is our brains that are incapable of learning. Not so fast: how am I less intelligent if the bozo teaching me does not know how to teach maths?
Citing Carl Jung does not help either, though he namesdrops to mitigate the shame of not being able to do maths...Jung was not as bright as is commonly believed as well. He once claimed that UFO's were all imaginary-- a sweeping generalization by someone claiming to be scientist-- whereas I for sure know that UFO's are real for I have seen one and to prove that I wasn't seeing things, I had a camcorder ready which recorded the space-ship going vertically up very slowly. I have lost the video though but it certainly happened 20 years ago and I still remember it.
>Jung was not as bright as is commonly believed as well. He once claimed that UFO's were all imaginary-- a sweeping generalization by someone claiming to be scientist-- whereas I for sure know that UFO's are real for I have seen one and to prove that I wasn't seeing things, I had a camcorder ready which recorded the space-ship going vertically up very slowly. I have lost the video though but it certainly happened 20 years ago and I still remember it.
Your evidence that Jung was not as smart as believed is that you recorded a UFO 20 years ago (but can't prove it) and he said UFOs don't exist? Forgive me, but that's the opposite of convincing.
"but can't prove it"--- I don't need to prove it to anybody, as long as I have a clear proof of it myself. I am with Descartes when I say my senses did not deceive me, but Jung claims that he can speak for EVERYBODY and EVERYBODY is imagining UFO's...
Writing is always primarily of benefit to myself--and hence of any benefit to anyone else....would you cook a meal you wouldn't eat yourself but wish others did? Writing that is geared towards explicitly benefiting others (at the cost of not benefiting our own selves) is shallow, useless and fake.
My point is that your posts read like you have a chip on your shoulder and are working that out in your writing, and that comes through so strongly that it overwhelms any other message you may be trying to communicate. The original post, for instance, isn't off-putting because it's contrarian, but because it reads like the contrarianism is mainly in service of presenting yourself as cleverer than three well-regarded historical figures, plus the author of the article—which, even if that's true, who cares? That's what I meant by the post seeming like it was more for you than for anyone else.
You seemed surprised enough at the reaction to your original post that I thought you might like to know what about it probably "touched quite a few nerves", is all.
It certainly only appears like I am "self-promoting" but I was not aware of that until you pointed it out....Which, even so, I don't care whether someone cares or not about it.
There are a number of "well-regarded" figures in history--Columbus is one, the Americans even celebrate a day in his name--who should be condemned in fact...
My comment is of service to the intelligent reader and its main purport is: it's ok to find fault with the seemingly intelligent "well-regarded historical figures" if they do not match your objectively-finetuned assessment of what it means to be intelligent.
The author expresses his misintuition of variables - there are plenty of simple examples and thought exercises which can inform the intuition. Later he expresses getting mixed up dealing with fractions - this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions. Modern math education is spectacularly guilty of going too far too fast.
Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary.