Learning algebra in my 60s
theguardian.com
theguardian.com
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.
>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.
Edit: I went to grad school for math, and I got there by pretty much doing what you described!
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.
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.)
Thanks for your version. I enjoyed reading it!
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.
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.
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.
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.
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.
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.
https://www.americanheritage.com/whatever-happened-new-math-...
I agree with you, especially if you are in your 60s and have a math professor as your teacher, one would think a new math approach works best!
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".
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.
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.
I think it was long division and lattice multiplication in elementary school
Doing math by drawing numbers in predefined shapes so that it magically worked out was the most ludicrous thing I'd ever seen
"Because that's how it works." Wasn't really a satisfactory answer lol.
There are shortcut formulas, and before calculators a lot of calculations were done using drafting equipment.
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.
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.
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?'
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.
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.
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.
[1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD07xx/EWD796...
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.
He's been working on it for years and talks about it a lot on his podcast: https://techzinglive.com
Edit: The link to use the beta software is here with more details on how the system works: https://www.mathacademy.us/beta-test-information
Contextless Algebra is not intuitive.
Here is a PDF preview to get an idea of the contents: https://minireference.com/static/excerpts/noBSmath_v5_previe... and a standalone printable concept map: https://minireference.com/static/conceptmaps/math_concepts.p... There is A LOT to learn, but I assure you that as an adult you can understand the material much more easily, especially with our background in computing now.
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
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.
The other issue is in my other comment: https://news.ycombinator.com/item?id=32388595
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.
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.
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.
So I found out later :-(
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.
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!
.2 * 59.22
2.99 * 1.092
5.00 / 1.092
For the others you'd have to do the mental work to append a 1 regardless, might as well stick with decimals
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 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.
Both of those statements call into question the effectiveness of what you're promoting.
See Wikipedia: https://en.wikipedia.org/wiki/Bloom%27s_2_sigma_problem
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.
When did I say anything about schools? My sadness at the opportunity lost came from 'avoiding conventional education.'
Being taught a² + b² = c² with the triangle and accompanying squares drawn out much much later is the best example I can give.
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 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'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.
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.
What exactly is the downside and for whom?
> 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.
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 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.
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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.
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.
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.
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.
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 do think most programs implementing it end up taking out a little too much rote and algorithms.
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.
COMMON CORE DOES NOT SPECIFY PEDAGOGICAL METHODS!
It is, essentially. a list of topics that should be mastered at each grade level.
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.
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 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.
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).
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.
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.
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.
But for whatever reason this is generally skipped in middle/high-school algebra.
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.
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.
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’.
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).
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.
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.
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.
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.
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.
and standalone concept maps that show the prerequisite graph (very useful when learning from any book): https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_...
I recently released a new book with just high school materials, specifically targeted at adult learners. Check out the website here: https://nobsmath.com/
We can't wait for your stats book to be released.
He also enjoyed reproducing geometric proofs of certain equations.
What's funny is he hated math as a student but for some reason or other got turned onto Euclid in old age.
He then tried to get my dad to read Euclid but that didn't take.
Then I came across the book as teen and took an interest so Grandpa and I bonded over that.
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.
https://wiki.c2.com/?MappersVsPackersBut 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.
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:
https://byjus.com/maths/commutative-property/
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:
https://www.socratica.com/lesson/groups-motivation-for-defin...
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.
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.”
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.
> 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.
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.
https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
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.
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.
EDIT: I see I have touched quite a few nerves.
It reads like a post you made for yourself, not for the benefit of any reader.
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.
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.
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.