I have to say, I actually loved this article. Especially “Understanding doesn’t build fluency; instead, fluency builds understanding.”
I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.
FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.
That talk is about something superficially different—ego in math—but on reflection, I think the desire to look smart actually really does set one up for success in math in the particular way that the OP article describes.
When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.
> When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.
What should one do instead, in order to avoid merely “looking”/“sounding” smart?
>>> The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind.
To be honest, this sounds like orienting one's self in the 'space of mathematics'. Is it not possible that, just like one can navigate by landmarks (proofs) or by the space itself (deep understanding), that there are in fact two roads to intuition in mathematics, of which ones is practice and fluency, and the other is deep insight and understanding?
To me, it seems that there are two general categories of things referred to as "math": A: the one used in this article: What people generally refer to as math. What's used by engineers, (most) scientists, etc. B: The one used by math majors and mathematicians. This type is abstract, contains things domains that end in "theory".
My question: Do you think an approach like in the article is possible to learn Math B? I have tried several times, unsuccessfully. I'm proficient in most domains of Math A. (Differential equations, linear algebra etc, symbol manipulation, geometry, and how tho apply them to practical problems).
Math B seems, in contrast, beyond me. There is a programming analogy: Math B is like Haskell, or pure functional programming, which also is as ungraspable to me. I am wondering if maybe this is partially genetic, partially something you have to learn at an early age. Or maybe it takes a formal learning path.
I think there is another one, Math C that involves day-to-day mental arithmetic which I am terrible at despite being good at Math A and holding engineering degrees. There might also be another element of Math C which is a feel for numbers and lets you know if an estimate or the value staring at you on the calculator screen makes sense or if it is obviously wrong.
I tie my poor mental arithmetic skills partly to never properly learning multiplication tables, at least not all of them and perhaps something lacking in my brain which also means I have a terrible sense of direction.
Yet, when it comes to symbol manipulation where the numbers don't matter until the very end, then I am good at that.
> I tie my poor mental arithmetic skills partly to never properly learning multiplication tables
I thought this too.
When you're young the multiplication table seems like a daunting thing to memorize, but after graduating university, it doesn't seem so bad.
So I went back and memorized my times tables using Anki. It was pretty easy, but ultimately changed very little and I easily forget them if I stop practicing.
I've come to realize that not mastering the times tables were a symptom, not a cause, of my learning difficulties.
I did learn my multiplication tables correctly, but I was always horrible at mental arithmetic—my elementary school periodically had formal exams for mental arithmetic, and I consistently failed them. It was the only type of task where I ever consistently scored less than 90%,
I am also terrible at directions, my parents used to be worried about what would happen to me when I lived on my own—thankfully, Google maps became a thing and nobody notices I am bad at directions anymore.
Like you, my learning difficulties are strictly compartmentalized to these two specific domains. I hold a PhD in engineering, and I even have a minor in math, lol.
It takes significant effort to get in the groove of a particular 'day-to-day' math. My job has regularly recurring parts where I spend a lot of time doing one or more of 3 basic skills: setting up integrals and differential equations, converting lots of numbers to binary or hex, and trig-based approximations.
When I spend too much time away from one, it really shows. The other day, I tried to do the trig approximations, and it was like getting up after sitting too long with legs crossed. The fluency just wasn't there.
In high school, I got a D in first semester calculus, and declared myself "done" with math. Up until that point, I had used a calculator as a crutch, but calculus required symbolic manipulation that could not be faked. My dad's influence was stronger than my mom's -- she was fearless, but he frequently spoke of how "bad at math" he was. And that was an easy out. I was just taking after my dad, "bad at math!"
Around that time, I went from noodling around with programming, to taking it seriously. I learned a bunch of programming languages, and landed a web development job straight out of school. I wasn't just done with math, I was done with school, too!
After a few years of that, I got bored with web dev, and decided I'd rather try my hand at engineering of some sort. I enrolled in community college, and quickly discovered that all of the engineering courses had... math prerequisites. So I bit the bullet, and for the first time, applied myself. Turns out that I wasn't intrinsically bad at math; I just hadn't been sufficiently motivated! I was paying my own way, so I ended up taking a job in the tutoring center. As I transferred to university, I found myself taking more and more of these math "prerequisites" and not following through on the engineering courses. I matriculated as a math major, and today I've got a PhD in math.
In my mid-20s, I didn't even believe that I could be Math A person. But I got good at that stuff, for the sake of engineering! And then I went straight through to Math B (and, almost amusingly, forgot most of those Math A skills -- watch out, unused skills get rusty!)
I actually credit my programming experience for the intermediate transition from my "bad at math" late teens to my "willing to try Math A" mid-20s. Programming taught me to think rigorously, and abstractly. So I must push back on the notion that this is intrinsic to a person, and must be learned at an early age: I wasn't doing Math B until after 25 when my brain was supposedly fully mature. And while I did have the benefit of a formal education, I would assert with some confidence that the relevant detail there was that I was in a cohort of students who were working together, beholden to homework deadlines and exams -- because math is hard and it's really easy to get demoralized without that external reinforcement.
> So I bit the bullet, and for the first time, applied myself
This is what I would tell people, but just use a weight lifting analogy. If you're out of shape, of course you will struggle to do any sort of exercises. But if you keep working at it in a disciplined way, while being kind to yourself and praising your progress, eventually you can get good at it.
Calculus is a tiny bit of new material plus a shit ton of rote algebra. Even undergrad prob and stat was 80% algebraic manipulation.
I have also found that programming is the gateway drug to Math B. Thanks to Functional Programming and Type Theory I eventually found may way into Abstract Algebra, Topology, and Category Theory... Wish I had time to go back and study these with a mentor, though!
Once you're fully ensconced in the major, it pivots into type B. And it turns out that I hate type B but slogged through it with medium-good grades.
looking back on it now, I've come to like type B and wish I could go retake those classes with my current perspective.
I think my original distaste was largely due to what felt like a bait-n-switch: start out majoring in something you like and are good at, but then pull the rug out and switch to something completely different
The fact that intro math classes don't do proofs (Type B) is because of the same pressure from people who only want to do Type A.
Due to internal changes in my uni, for the first time, my freshmen year, the math department taught proper proof-based Calculus 101 (from Apostle of all books) to all majors. Then the engineers and biologists complained so much, they had to cut out a lot of proofs from Calculus 102. There were even more complaints, so by second year, there were hardly any proofs in the core math courses. In a few years, the calculus courses had become devoid of proofs.
Some unis have separate intro courses for math majors, but it's very difficult to offer them in the current economic climate.
The real problem is that "type B", despite being much more important of an activity to learn (for mathematics or any other kind of technical problem solving) is almost entirely ignored in primary/secondary education.
While at it: pure functional programming is very easy to grasp. You should just think about programming as of not tinkering with the state, not altering things, but as of producing outputs from inputs.
Say, analog electronics mostly works in the pure functional domain. An amplifier does not try to change the input signal. Instead, it produces a more powerful output signal, following the shape of the input signal. A tone generator in a musical instrument does not try to make a key on the keyboard vibrate. Instead it produces a sound signal according to the key pressed (which note and what velocity).
The simplest way to try practical pure functional programming is to connect a few Unix processes via pipes:
cat somefile.py | egrep '^def \w+' | wc -l
The above is a pure function compositon, as a map-reduce pipeline, in point-free style. (Yay, buzzwords.) It counts top-level functions in a Python file.
But how to achieve something like updating with that? By looping the output back to the input, and switching o the "next version" once it's computed. Conway's game of Life looks like an ultimate "update in place" thing. But it's purely functional, too: the new state of the map is completely computed based on the previous state if the map. Then the new map is seen as "the current map". Similarly, frames in a drawn animation do not change, but they are shown at the same place one after another, giving the impression of motion and change of "the same" picture.
In general, our Universe may be seen as a purely functional computation: its next state is a function of its past states, and the past is immutable.
I like your conviction Re "functional programming is very easy to grasp".
Many won't but I agree in the purest (sorry) sense.
There is no scattered changing state. I think we all learned input-function-output as a construct in maths class?
Spreadsheets (sans-VBA) is arguably the most prolific programming language and simplest, being used by people who do not recognise they are programming. Felienne Hermans gave a good talk on this subject in GOTO 2016.
Spreadsheets have numerous shortfalls though, and "real" functional programming languages make it difficult to not feel intimidated: in my experience, but this is getting much better.
[1] is a game of life in calculang, functional language I'm developing, where for all it's verbosity at least I hope the rules and development over generation (g) can be reasoned with (sans-state!).
Not very practical but can show calculang computation/workings as it progresses and as parameters change - things that are easy for FP and otherwise intractable, and which further help with reasoning.
But, a big challenge is to be approachable (not intimidating), and I'm trying to make that better. I think it helps enormously to be focused on numbers as calculang is, and not general programming.
The secret is that you can convert most type B math into type A by looking at steps in a proof as rules in a term rewriting system where the terms are mathematical expressions.
I've not found a book that makes this point completely explicitly, but most of those which cover sequent calculus get you half way there.
The rest of type B math is intuition which lets you guess at new conjectures and how to get you from the assumptions that you've made and the conjecture that you want to prove efficiently.
A mathematics degree will have some kind of 'transition to higher mathematics' course that you take your freshman or sophomore year. You meticulously work with sets, definitions/theorem/proofs in a simple setting, and especially adding 'structure' to sets with axioms.
In regards to the article, the course of this type that i took had frequent quizzes that required nothing but reproducing precise definitions or proofs we had learned. Of course the ideal would be for the student to be able to reproduce these from understanding. But in practice i was doing a lot of brute force memorization of definitions - i just hadn't internalized the language of mathematical logic well enough to reconstruct a concept's definition yet. however, it got my foot in the door and having those definitions in my head made the next courses easier, so if i retook that transition course a few years later on, i would not have needed to do so much memorization. i got better at learning those kinds of basic definitions.
So my answer to your question is yes to some extent - the memorization aspect of learning described in the article is useful for learning the first step to Math B as well. Also if you want to make another learning attempt, be sure and go back and start at that freshman/sophomore level transition course i was describing!
Correct me if I’m wrong, but aren’t you simply referring to “applied mathematics” and “pure mathematics”, respectively? I skimmed through the replies to your comment and I don’t believe anyone mentioned these terms, although I did see one reference to “abstract mathematics” (a term used by you, as well).
I thought these were well-known terms and thus that the dichotomy you describe was itself well-known, but I thought I’d add this comment on the chance that you weren’t familiar with them.
I was just thinking about this the other day. Personally, I think that math falls into two categories, though I think I would distinguish them differently from you (If I'm understanding you correctly). Its kind of like the difference b/t the hammer maker and the carpenter, the producer and the consumer. For me, mathematics (the kind you research and which is abstract and theoretical) is largely in the hammer maker camp. We'll call this math X, these guys are creating and polishing tools (aka in analysis providing proofs and arguments for why the real numbers can be considered complete or that a derivative actually can be taken on a given class of functions).
Then there is Math "Y". This is all the guys who use those things the X guys are selling, the proverbial hammers they have produced. They assume the X guys did their work correctly and that when they use the products they've bought i.e. the rules, theorems and strategies developed by the X guys, to solve a particular equation or problem, the answer is correct. For example, they assume the limit of the sum of two polynomial functions on the reals is equivalent to the sum of the limits of those functions - they don't care about all the nitty gritty details and justifications - the X guys figured all that out for them. They Y guys are much more concerned with figuring out how to get the rocket into space or ensure the skyscraper is soundly built.
I would say from my experience, very little of mathematics education is in the X camp, I'm not saying this is a bad thing though, perhaps it is similar to the fact that most programmers are not compiler programmers or programming language creators :)
Abstract math (type B) is a very rigorous discipline that underpins the other kind used by engineers (type A). Type A is indeed learned by repetition along with understanding. It is very important to simply do the math to become better at it and understand what you can expect from your calculations.
Type B on the other hand far more about understanding. You will never understand the theory of a mathematical space and how to apply it, by simple repetition. That is a far more theoretical and creative endeavour. You need to learn it and apply it to understand it. I suppose you could call the process of applying it some kind of repetition, but in my opinion the insights comes from applying it to concepts you already know.
A formal learning path is a very good idea, because people with more knowledge know what order you can progress in, in such a way that you actually apply your knowledge in a natural way and build on previous learnings. And it is definitely a huge help that teachers can help you guide your learning when you are stuck.
Proofs in abstract algebra, for example, require the ability to quickly and correctly manipulate symbols on paper (using already discovered rules/lemmas/theorems).
The repetitive practice is in this manipulation of symbols. It takes a long time and deliberative practice to learn this skill. You just practice by doing symbol repetition in different contexts, instead of doing the same thing over and over again like multiplication tables, because your symbol manipulation abilities have to be general [1].
If you try to teach, you will quickly discover that there is a wide difference in this ability for math majors by their final years. And the students who have poor symbol manipulation abilities inevitably struggle at the higher level concept application, because they keep making mistakes in the symbol manipulations and having to redo it.
[1] Contrast the training of 100m sprinters (multiplication table), who only run 100m on a fixed track that they will eventually race on, and the training of cross country runners (symbol manipulation), who practice on a variety of routes, because their races are on different routes.
I studied both A and B. In college, I declared a double major in math and physics. Then I went to grad school in physics.
Granted, it was one brain (mine) studying both subjects, so it should not be shocking that I learned both in the same way. Of course I practiced lots of problems and derivations in my physics class, but I also practiced and memorized lots of proofs in my upper level (i.e., more theoretical) math classes.
And truth be told, maybe even in my liberal arts courses as well. Thanks to programming, I got really good at typing. Thanks to owning a personal computer (one of the first at my college) I started writing and re-writing a lot. Repetition and practice even got me through those courses.
It was simply mercenary at the time, not wanting to waste time during exams recalling the easy stuff gave me more time to think about the hard stuff. But I think it did help me in the long run. I still use a lot of that stuff today, at age 60, though it's certainly more computer-aided than it was back then.
If you feel comfortable with Math A but not Math B you might enjoy _Graphical Linear Algebra_[1], which is specifically that bridge!
As someone who is decent at Math B but mostly incompetent at Math A I suspect it comes down to the old analysis vs algebra opposition — being better at thinking about things visually/spatially vs linguistically. Both are trainable though.
I think the approach outlined here works well enough. When teaching us about adjunctions, our category theory lecturer used to have us recite the definition of a left adjoint at the start of every lesson, and he'd draw the diagram as we spoke. I can't say I can still recite it by heart but I do feel like I have a decent intuition for adjoint functors.
I think it's the key. Learning maths isn't something you can do on the side. It's countless hours of intensive active learning and problem solving. I don't see how it can be done outside an academic path. I remember my undergraduate program, we had 14 hours of class of maths alone a week, plus a 4-hours exam every other Saturday, and I was working several hours a day on top of this, most days of the week.
Maths can be fun, but who wants to do that kind of effort for the pure joy of learning?
Math as done by mathematicians 100% involves knowing the ins and outs of core concepts by heart. You can’t begin to derive new theorems about things you aren’t fluent with
> Math B seems, in contrast, beyond me. There is a programming analogy: Math B is like Haskell, or pure functional programming, which also is as ungraspable to me.
This is not just an analogy – Haskell (or pure functional programming in general) is a lot closer to math than other ways of programming. Specifically, it is derived from category theory (which fits to your description of math B being things ending in "theory" as well).
The answer is yes and it's the only way. You need to develop fluency to understand "Math B". It's only ungraspable right now because you haven't had enough of the right kind of practice. The right kind of practice is 1) well motivated for your curiosity 2) achievable.
Um... yes, it's even more important in math B to be able to have, at your fingertips, all the theorems related to Ideals, Rings, Groups, Categories, Topologies, etc. This is why I re-read my math textbooks from time to time. You always miss some theorems, and they're often key to higher-level understanding.
I commented on this on HN a couple months ago, but I had a similar conclusion regarding the value of memorization when I joined med school after studying computer science in undergrad and grad.
It took me a while to buy in to high-volume memorization as a learning technique (especially coming from CS, where memorizing facts is not a huge emphasis). After a while though, I started recognizing how the quick recall encouraged by the system enhanced my understanding of concepts vs replacing it (I wrote about this a couple years ago [0]).
That's my takeaway from The Shallows by Nicholas Carr. Knowing how to derive information or where to find information doesn't mean you know the information and knowing the information is necessary to form higher level associations in the mind.
I have sung the praises of memorization since I was a kid, and yet the attitude that, with the internet, memorization is no longer required because we have access to unending knowledge, seems oh so prevalent. One wonders what these advocates must think... do they claim to know Norwegian because they can use Google to translate at any particular instant, for example? One wonders what life might be like for people so confident in their non-existent abilities.
Huh. That's an interesting premise. I think I would split it though - knowing the basis well enough to derive results is probably fine for later deduction, knowing where to find the information is definitely useless for it.
"Oh, and check this out: I'm a bloody genius now! Estás usando este software de traducción de forma incorrecta. Por favor, consulta el manual. I don't even know what I just said, but I can find out!"
True, that is rote learning. It can be the basis of critical thinking. Critical thinking involves methods of inquiry...who, what, why, when and how. Rote thinking is limited to 'how'.
Wow. That author sure loves to talk about herself. I kept reading, but the whole article feels like an overdrawn introduction without payoff.
If you want to know how you can become better in math and rewire your brain to be math compatible I‘m afraid you will be none the wiser after reading this.
I skipped the purely biographical paragraphs. Some important parts:
- Memorization and rote practice are important for learning, not just the current Zeitgeist (2014) of “understanding” without the former. This becomes the foundation that allows you to focus on higher-level things like understanding and applying formulas.
- Experts develop “memory chunks” which allows for example chess masters to draw on thousands of different past games, openings, variations.
Memorization happens naturally through repeat exposure, and works better if this is exposure in some meaningful context rather than through cramming via flash cards or whatever. The best kind of practice is practice that you are motivated to do because is inherently interesting. The less "rote" you can make this the better.
For a primary school example, if you can solve basic arithmetic problems in service of a fun and challenging logic puzzle, that is more motivating than solving a page of arithmetic problems one after the other.
More generally, while mathematics certainly requires putting in time and actually doing the work of thinking a whole lot about a variety of hard things in the service of solving hard problems, very little of that is memorization per se.
> In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science.
The older teaching method also sucked.
In my opinion, the single most important thing primary school math education should be teaching is how to attack and solve nontrivial word problems. Unfortunately we did not have any of that before, and still do not have any now. Cf. https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/trav...
The thing is her definition of „understanding“ isn’t actually „understanding“ but rather surface level intuition.
Personally I only accept „understanding“ once I can explain it and reuse it in a different context. But I am not self centric enough to deny that there absolutely are plenty of people who love to memorize without having an abstract understanding. And they are doing just fine.
As an example from my life, my 8 year old learned how to calculate perimeter and area this year.
The cool thing about area of a rectangle is you can just turn it into a multiplication array. Which is something they learned to help them understand multiplication.
If she just memorized multiplication, she wouldn't actually understand the formula.
I never really know what people mean by "rewiring your brain". You just have to spend a lot of time studying it. The hard part as an adult is probably making the time, especially if your work is already mentally taxing.
Thoughts flow through my brain like electrons through wires, both at a speed I cannot truly comprehend. The paths of my thoughts, the way words connect together, the emotions they evoke, the feelings that are associated with - these are all malleable. I have been working to rewire myself away from pessimism and towards optimism for years now. It's not easy, and sometimes I fall back into old patterns. As years have passed, though, I've found the new pathways easier, the new roads getting more familiar. My thoughts have previously wanted to go one way, and I spent time yanking at them to go a different way. I spent enough time at it that I now on good days more naturally go they way I want to go.
It's not just sitting and repeating a single thing over and over again, it's working with your own natural inclinations so that you can recognize when you experience X and naturally reach for Y that perhaps Z is what your preference really is, upon reflection. If you can practice that enough then, in the moment, you can sometimes find yourself not following the old pathways but the new.
Rewiring seems like a pretty good analogy as when you rewire a house you work hard -- it is dirty, dusty work. Pulling wire is hard and thankless because when you're done you cover it all up and, if you're lucky, it works! Then at the end you're . . . back in a state where nobody but you knows any different what is happening behind the walls. Things work, and others might have no idea anything changed at all. But you put in the hard work and you know how things actually work on the inside now and it's exactly how you want it, not how it was before.
If you've got a better analogy, I'd love to hear it.
This was my experience with the article which could be consolidated down to a single sentence: "Manipulate and play with concepts you intend to internalize, rather than relying on rote memorization".
I would think that would be obvious to anyone that merely memorizing something like 'f=ma' would be meaningless without deliberate attempts at application (both theoretical and practical).
There was a kludgy attempt at tying the study of foreign languages to STEM, but it just amounted to, everything is ultimately a craft. You have to practice to perfect it.
Right?! I want to know "How they rewired their brain to be fluent in math" but so far all I've seen is a bunch of talking about how great they are. This article sucks.
feel bad that this article ended up on Hacker News (HN), since it seems additional context is needed.
Currently, the emphasis in training and education is on ensuring students understand the material, and rote memorization is viewed as a failure mode.
The author acknowledges this but introduces an argument that rote memorization is critical to achieving fluency.
I suspect that stating this position among typical education-focused circles will result in pushback.
To lend credibility, she adds her lived experience as a way to explain what she means—and to clarify that she isn’t saying everyone is wrong, just that we may be too harsh on memorization.
This is apparent to me because, frankly, if it weren't for the additional content she added, I wouldn’t have spent more than a few seconds before dismissing it.
The article even made me concerned about an internal project I am involved in, prompting me to verify that I hadn’t overlooked some issues.
If you want a TL;DR, the golf analogy matters: If you want to learn math, you need to understand it and then practice it so much that it becomes second nature.
I wish mathematics education would incorporate more history and philosophy. Personally, I was never great at math in school, less because of aptitude and more because I just found it boring and disconnected from the things I found interesting as a kid.
Years later, I’ve been slowly trying to “catch up” with my mathematical knowledge, and I find myself the most interested in topics that relate to the lives of mathematicians (and how events impacted their work) and to the philosophy of mathematics. I didn’t get any of this in school math classes, which focused purely on calculations and formulas.
I had the same experience with accounting, as well: boring in isolation but fascinating when connected to the history of double entry accounting in Italy, global trade from the 1500s onwards, and so on.
Similar in my case. I'm beginning to like math just now because after years of software engineering I'm seeing commonalities in math and software engineering. Math feels like "creating a software systems for numbers" where I also am the compiler at the same time.
From an intellectual perspective (different than my engineering perspective), I'd label mathematics as quantitative philosophy. I like philosophy too.
This applies to every field. Public education in general sucks. Cramming kids into a box 8 hours a day for 20 years forcing them to learn a bunch of topics in the most driest ways possible.
The lack of freedom to pursue wny individual interests aside, major subjects like History, mathematics, biology are all reduced down to a bunch of facts you can fit in a textbook and test in a multiple choice format.
Well, the goal here is to help people who don't like math learn to like it, or at least find it interesting enough to learn.
I think that people get slotted into the likes math or doesn't like math category quite early in life, often because they don't have an on-ramp to finding it interesting. They then spend the rest of their education avoiding it because "they're not a math person," when in many cases they probably just needed a bit more context/history/philosophy/something interesting to get them on the right track.
Yeah, the big lie of mathematics education is that everything currently known just appeared fully formed inside mathematicians heads because they are mathematicians gifted with mathematical thinking.
The reality is that every advance in mathematics evolved after decades of people crunching numbers (typically for some engineering application) with an earlier form of math until they worked out the advance. I think students would have more confidence in their own ability if they understood that the mathematical innovators were equally stumped for long periods of time.
I don't think that's true at all and would consider the history and deeper structures (philosophy) of something to be integral to that topic. In fact, I think the (very modern) rigid division of subjects into separate categories is part of the problem. Intellectuals from a few centuries ago would find it absurd that we consider the liberal arts and mathematics to be entirely opposite types of things.
I'd like to know what education reformers would say to the subtitle
> Sorry, education reformers, it’s still memorization and repetition we need.
It seems needlessly confrontational, and misses what the article is about. The article asserts that practice and repeated use of math is important. I don't think it's really suggesting that we should go back to how math was taught in the US 40 years ago.
But maybe I'm just out of touch with math education reform: In high school I was graded on how fast I could do matrix multiplication, and thought matrices were kinda stupid. Then I learned about linear algebra and coupled oscillators in college and thought they were awesome.
So I'd assumed the educational reform was about removing the busy work from math and focusing on what you'd actually use it for. Am I wrong?
I remember hearing an interview with a woman who was doing charter school-type work in the UK (IIRC), where most of her students were thought of as "underperforming".
She was successful because they emphasized drills, lots of drills, and more drills.
Teachers and students hate drills. Teachers, because they're tedious to grade, and students because they're boring. But they work. It's no different than doing the same Super Mario Bros. level again and again until you time your jumps just right.
I've often thought that gamification of drills would be a great way to get kids to learn their math facts or whatever, but there seems to be an allergy to doing this in the US education system. What the US education system seems to be addicted to is moving from one hype/fad to the next, as that's where the money trough seems to be.
I love learning high-level knowledge using conceptual overviews and I remember them very well too. I’m just not wired to remember the nitty-gritty of things and I found drills are the only way forward provided they are implemented intelligently like Anki.
My view on them changed as well. First I found them stupid and mind-numbing. Now my life is busy and chaotic and drills are one of the few easy, zen-like moments. Maybe students need harder lives to actually come to appreciate their repetitive and simple nature.
This actually is a fairly controversial stance in education, so there's reason to be combative about it. A lot of education tends to emphasize meeting students where they are to the point that it can completely subsume the irreducibility of complexity when confronting some knowledge conceptually; one _can_ be better served instead by attempting to memorize some bulky, impenetrable abstraction and instead make sense of it through its application. A lot of knowledge only becomes clearer when one forges ahead with a dim appreciation of what is being articulated but the confidence, willingness, and (most crucially) feedback mechanism for testing it out anyway.
I had a same experience with you but there was no algebra at art school. I learned why math was interesting from trying to figure out how a computer works. I know I have huge gaping blind spots but I can use math for what I need. I'm now trying to avoid math and only using the shapes of math.
On UK Teacher Twitter, there are two factions that disagree noisily but civilly: the 'trads' and the 'progs'. Both factions have educational psychology research to back up their claims: the progs lean mainly on research from big institutions, often international; the trads often do research in their own classrooms. The trads' pupils seem to do better in the UK's public examinations, but that might be an artefact of those exams.
It's also worth noting that half of all teachers leave the profession leave within five years. Forty years thus represents about eight generations of teachers being trained with their own biases, refining their ideas in classroom practice, and then training the next generation. On top of this are the cycles of trendiness in the various schools of thought in educational psychology, as well as the varying policy platforms of governments. Educational practice from four decades is less historical and more archaeological.
When I was teaching university level mathematics, there were many people in my classes who couldn't do fast matrix multiplication or even fast multiplication. The inevitable result was that they had to constantly drop from the higher level of abstraction that we were trying to learn, to the lower level abstraction of arithmetic, and failing to learn the higher level of abstraction.
On this forum, I will say, imagine your object-oriented programming language can build all sorts of abstractions but can't multiply numbers. So every time you have to multiply numbers in your algorithm, you have to instead write a few lines of assembly code that do the same thing. How much efficiency would you lose.
I didn't even know what a matrix was until I got to college, and I went to a supposedly well-ranked high school and was on the higher level math track. This was about 10 years ago. I think some of the education reform might've removed concepts altogether instead of actually improving the presentation? Though I suppose in order to present things well you might need to cut back on the total number of topics covered, I'm not sure I'd describe what I did learn as well-presented either.
You naturally memorize that which you are exposed to, but to say that that means we should discourage memorization in favor of purely exposure (which is the current status quo AFAICT), is completely misguided.
Yes, you will almost certainly memorize anything with enough exposure, but targeted memorization is also useful, if the former's not going fast enough.
Teach category theory, and if you can't, then don't bother teaching math at all.
Because you'll be teaching the awful wasteful rote math that everyone hates and can't use, instead of the nice universal stuff that lets you transfer intuition and see how ideas communicate between different knowledge domains far beyond what was traditionally seen as math.
The 20th century gave us real new ideas in math. But our primary math education is still 200 years out of date. Until that changes, math education will remain a deadening cargo cult that throws away far more human potential than it develops.
The rote stuff’s about all I’ve ever actually managed to find a use for, as an adult.
It’s useful daily. Pre-algebra is useful fairly often (even if I weren’t a programmer, plugging numbers into a formulas and basic graphing are very handy, quite often). Trig I think I managed to use once, but only because I didn’t know the right way (if you find yourself using trig on a minor home project you’re probably missing some trick or standard or something that lets you not do that—I suspect it was the case then)
That’s… about it. Stats, kinda, but mostly looking up the formula for the thing I want and plugging in the numbers, which barely counts.
Is this comment still taking into account all the ~8 yr old common core changes to math curricula around the country? Even that's 200 years out of date? What're we supposed to be doing with five year olds that's so much better?
When taking mathematics classes in university I always noticed an enormous gap between what I thought I understood compared to how confusing the problems were. I am glad the author mentions that phenomenon.
For (the few) students who actually understood the subject the problems are just busywork, for those who didn't it is the most important part of the learning process. There is exactly one way to understand mathematics, which is actually doing it. This can be many things, but actually solving problems is an important part. I believe that problems should be interesting, but repeated recall definitely is important as well.
Same goes for coding. You can go into a problem super well prepared and think you know what you need to do but often only through a lot of repetitive work of experiments and trying multiple times do you gain the knowledge to actually solve the problem smartly.
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I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.
FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.
That talk is about something superficially different—ego in math—but on reflection, I think the desire to look smart actually really does set one up for success in math in the particular way that the OP article describes.
When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.
What should one do instead, in order to avoid merely “looking”/“sounding” smart?
To be honest, this sounds like orienting one's self in the 'space of mathematics'. Is it not possible that, just like one can navigate by landmarks (proofs) or by the space itself (deep understanding), that there are in fact two roads to intuition in mathematics, of which ones is practice and fluency, and the other is deep insight and understanding?
My question: Do you think an approach like in the article is possible to learn Math B? I have tried several times, unsuccessfully. I'm proficient in most domains of Math A. (Differential equations, linear algebra etc, symbol manipulation, geometry, and how tho apply them to practical problems).
Math B seems, in contrast, beyond me. There is a programming analogy: Math B is like Haskell, or pure functional programming, which also is as ungraspable to me. I am wondering if maybe this is partially genetic, partially something you have to learn at an early age. Or maybe it takes a formal learning path.
I tie my poor mental arithmetic skills partly to never properly learning multiplication tables, at least not all of them and perhaps something lacking in my brain which also means I have a terrible sense of direction.
Yet, when it comes to symbol manipulation where the numbers don't matter until the very end, then I am good at that.
I thought this too.
When you're young the multiplication table seems like a daunting thing to memorize, but after graduating university, it doesn't seem so bad.
So I went back and memorized my times tables using Anki. It was pretty easy, but ultimately changed very little and I easily forget them if I stop practicing.
I've come to realize that not mastering the times tables were a symptom, not a cause, of my learning difficulties.
I did learn my multiplication tables correctly, but I was always horrible at mental arithmetic—my elementary school periodically had formal exams for mental arithmetic, and I consistently failed them. It was the only type of task where I ever consistently scored less than 90%,
I am also terrible at directions, my parents used to be worried about what would happen to me when I lived on my own—thankfully, Google maps became a thing and nobody notices I am bad at directions anymore.
Like you, my learning difficulties are strictly compartmentalized to these two specific domains. I hold a PhD in engineering, and I even have a minor in math, lol.
When I spend too much time away from one, it really shows. The other day, I tried to do the trig approximations, and it was like getting up after sitting too long with legs crossed. The fluency just wasn't there.
https://en.m.wikipedia.org/wiki/Fermi_problem
(Folks with neurodivergencies like autism/ocd/adhd etc are particularly more likely to have these.)
Around that time, I went from noodling around with programming, to taking it seriously. I learned a bunch of programming languages, and landed a web development job straight out of school. I wasn't just done with math, I was done with school, too!
After a few years of that, I got bored with web dev, and decided I'd rather try my hand at engineering of some sort. I enrolled in community college, and quickly discovered that all of the engineering courses had... math prerequisites. So I bit the bullet, and for the first time, applied myself. Turns out that I wasn't intrinsically bad at math; I just hadn't been sufficiently motivated! I was paying my own way, so I ended up taking a job in the tutoring center. As I transferred to university, I found myself taking more and more of these math "prerequisites" and not following through on the engineering courses. I matriculated as a math major, and today I've got a PhD in math.
In my mid-20s, I didn't even believe that I could be Math A person. But I got good at that stuff, for the sake of engineering! And then I went straight through to Math B (and, almost amusingly, forgot most of those Math A skills -- watch out, unused skills get rusty!)
I actually credit my programming experience for the intermediate transition from my "bad at math" late teens to my "willing to try Math A" mid-20s. Programming taught me to think rigorously, and abstractly. So I must push back on the notion that this is intrinsic to a person, and must be learned at an early age: I wasn't doing Math B until after 25 when my brain was supposedly fully mature. And while I did have the benefit of a formal education, I would assert with some confidence that the relevant detail there was that I was in a cohort of students who were working together, beholden to homework deadlines and exams -- because math is hard and it's really easy to get demoralized without that external reinforcement.
This is what I would tell people, but just use a weight lifting analogy. If you're out of shape, of course you will struggle to do any sort of exercises. But if you keep working at it in a disciplined way, while being kind to yourself and praising your progress, eventually you can get good at it.
Calculus is a tiny bit of new material plus a shit ton of rote algebra. Even undergrad prob and stat was 80% algebraic manipulation.
Once you're fully ensconced in the major, it pivots into type B. And it turns out that I hate type B but slogged through it with medium-good grades.
looking back on it now, I've come to like type B and wish I could go retake those classes with my current perspective.
I think my original distaste was largely due to what felt like a bait-n-switch: start out majoring in something you like and are good at, but then pull the rug out and switch to something completely different
Due to internal changes in my uni, for the first time, my freshmen year, the math department taught proper proof-based Calculus 101 (from Apostle of all books) to all majors. Then the engineers and biologists complained so much, they had to cut out a lot of proofs from Calculus 102. There were even more complaints, so by second year, there were hardly any proofs in the core math courses. In a few years, the calculus courses had become devoid of proofs.
Some unis have separate intro courses for math majors, but it's very difficult to offer them in the current economic climate.
Say, analog electronics mostly works in the pure functional domain. An amplifier does not try to change the input signal. Instead, it produces a more powerful output signal, following the shape of the input signal. A tone generator in a musical instrument does not try to make a key on the keyboard vibrate. Instead it produces a sound signal according to the key pressed (which note and what velocity).
The simplest way to try practical pure functional programming is to connect a few Unix processes via pipes:
The above is a pure function compositon, as a map-reduce pipeline, in point-free style. (Yay, buzzwords.) It counts top-level functions in a Python file.But how to achieve something like updating with that? By looping the output back to the input, and switching o the "next version" once it's computed. Conway's game of Life looks like an ultimate "update in place" thing. But it's purely functional, too: the new state of the map is completely computed based on the previous state if the map. Then the new map is seen as "the current map". Similarly, frames in a drawn animation do not change, but they are shown at the same place one after another, giving the impression of motion and change of "the same" picture.
In general, our Universe may be seen as a purely functional computation: its next state is a function of its past states, and the past is immutable.
Many won't but I agree in the purest (sorry) sense.
There is no scattered changing state. I think we all learned input-function-output as a construct in maths class?
Spreadsheets (sans-VBA) is arguably the most prolific programming language and simplest, being used by people who do not recognise they are programming. Felienne Hermans gave a good talk on this subject in GOTO 2016.
Spreadsheets have numerous shortfalls though, and "real" functional programming languages make it difficult to not feel intimidated: in my experience, but this is getting much better.
[1] is a game of life in calculang, functional language I'm developing, where for all it's verbosity at least I hope the rules and development over generation (g) can be reasoned with (sans-state!).
Not very practical but can show calculang computation/workings as it progresses and as parameters change - things that are easy for FP and otherwise intractable, and which further help with reasoning.
But, a big challenge is to be approachable (not intimidating), and I'm trying to make that better. I think it helps enormously to be focused on numbers as calculang is, and not general programming.
[1] https://6615bc99ad130f0008ecc588--calculang-editables.netlif...
I've not found a book that makes this point completely explicitly, but most of those which cover sequent calculus get you half way there.
The rest of type B math is intuition which lets you guess at new conjectures and how to get you from the assumptions that you've made and the conjecture that you want to prove efficiently.
In regards to the article, the course of this type that i took had frequent quizzes that required nothing but reproducing precise definitions or proofs we had learned. Of course the ideal would be for the student to be able to reproduce these from understanding. But in practice i was doing a lot of brute force memorization of definitions - i just hadn't internalized the language of mathematical logic well enough to reconstruct a concept's definition yet. however, it got my foot in the door and having those definitions in my head made the next courses easier, so if i retook that transition course a few years later on, i would not have needed to do so much memorization. i got better at learning those kinds of basic definitions.
So my answer to your question is yes to some extent - the memorization aspect of learning described in the article is useful for learning the first step to Math B as well. Also if you want to make another learning attempt, be sure and go back and start at that freshman/sophomore level transition course i was describing!
I thought these were well-known terms and thus that the dichotomy you describe was itself well-known, but I thought I’d add this comment on the chance that you weren’t familiar with them.
Then there is Math "Y". This is all the guys who use those things the X guys are selling, the proverbial hammers they have produced. They assume the X guys did their work correctly and that when they use the products they've bought i.e. the rules, theorems and strategies developed by the X guys, to solve a particular equation or problem, the answer is correct. For example, they assume the limit of the sum of two polynomial functions on the reals is equivalent to the sum of the limits of those functions - they don't care about all the nitty gritty details and justifications - the X guys figured all that out for them. They Y guys are much more concerned with figuring out how to get the rocket into space or ensure the skyscraper is soundly built.
I would say from my experience, very little of mathematics education is in the X camp, I'm not saying this is a bad thing though, perhaps it is similar to the fact that most programmers are not compiler programmers or programming language creators :)
Type B on the other hand far more about understanding. You will never understand the theory of a mathematical space and how to apply it, by simple repetition. That is a far more theoretical and creative endeavour. You need to learn it and apply it to understand it. I suppose you could call the process of applying it some kind of repetition, but in my opinion the insights comes from applying it to concepts you already know.
A formal learning path is a very good idea, because people with more knowledge know what order you can progress in, in such a way that you actually apply your knowledge in a natural way and build on previous learnings. And it is definitely a huge help that teachers can help you guide your learning when you are stuck.
The repetitive practice is in this manipulation of symbols. It takes a long time and deliberative practice to learn this skill. You just practice by doing symbol repetition in different contexts, instead of doing the same thing over and over again like multiplication tables, because your symbol manipulation abilities have to be general [1].
If you try to teach, you will quickly discover that there is a wide difference in this ability for math majors by their final years. And the students who have poor symbol manipulation abilities inevitably struggle at the higher level concept application, because they keep making mistakes in the symbol manipulations and having to redo it.
[1] Contrast the training of 100m sprinters (multiplication table), who only run 100m on a fixed track that they will eventually race on, and the training of cross country runners (symbol manipulation), who practice on a variety of routes, because their races are on different routes.
Granted, it was one brain (mine) studying both subjects, so it should not be shocking that I learned both in the same way. Of course I practiced lots of problems and derivations in my physics class, but I also practiced and memorized lots of proofs in my upper level (i.e., more theoretical) math classes.
And truth be told, maybe even in my liberal arts courses as well. Thanks to programming, I got really good at typing. Thanks to owning a personal computer (one of the first at my college) I started writing and re-writing a lot. Repetition and practice even got me through those courses.
It was simply mercenary at the time, not wanting to waste time during exams recalling the easy stuff gave me more time to think about the hard stuff. But I think it did help me in the long run. I still use a lot of that stuff today, at age 60, though it's certainly more computer-aided than it was back then.
As someone who is decent at Math B but mostly incompetent at Math A I suspect it comes down to the old analysis vs algebra opposition — being better at thinking about things visually/spatially vs linguistically. Both are trainable though.
I think the approach outlined here works well enough. When teaching us about adjunctions, our category theory lecturer used to have us recite the definition of a left adjoint at the start of every lesson, and he'd draw the diagram as we spoke. I can't say I can still recite it by heart but I do feel like I have a decent intuition for adjoint functors.
[1]: https://graphicallinearalgebra.net/
I think it's the key. Learning maths isn't something you can do on the side. It's countless hours of intensive active learning and problem solving. I don't see how it can be done outside an academic path. I remember my undergraduate program, we had 14 hours of class of maths alone a week, plus a 4-hours exam every other Saturday, and I was working several hours a day on top of this, most days of the week.
Maths can be fun, but who wants to do that kind of effort for the pure joy of learning?
This is not just an analogy – Haskell (or pure functional programming in general) is a lot closer to math than other ways of programming. Specifically, it is derived from category theory (which fits to your description of math B being things ending in "theory" as well).
Dead Comment
It took me a while to buy in to high-volume memorization as a learning technique (especially coming from CS, where memorizing facts is not a huge emphasis). After a while though, I started recognizing how the quick recall encouraged by the system enhanced my understanding of concepts vs replacing it (I wrote about this a couple years ago [0]).
[0] https://samrawal.substack.com/p/on-the-relationship-between-...
https://www.youtube.com/watch?v=F3TG1AzBJYo
If you want to know how you can become better in math and rewire your brain to be math compatible I‘m afraid you will be none the wiser after reading this.
- Memorization and rote practice are important for learning, not just the current Zeitgeist (2014) of “understanding” without the former. This becomes the foundation that allows you to focus on higher-level things like understanding and applying formulas.
- Experts develop “memory chunks” which allows for example chess masters to draw on thousands of different past games, openings, variations.
For a primary school example, if you can solve basic arithmetic problems in service of a fun and challenging logic puzzle, that is more motivating than solving a page of arithmetic problems one after the other.
More generally, while mathematics certainly requires putting in time and actually doing the work of thinking a whole lot about a variety of hard things in the service of solving hard problems, very little of that is memorization per se.
> In the United States, the emphasis on understanding sometimes seems to have replaced rather than complemented older teaching methods that scientists are—and have been—telling us work with the brain’s natural process to learn complex subjects like math and science.
The older teaching method also sucked.
In my opinion, the single most important thing primary school math education should be teaching is how to attack and solve nontrivial word problems. Unfortunately we did not have any of that before, and still do not have any now. Cf. https://cs-web.bu.edu/faculty/gacs/toomandre-com-backup/trav...
Personally I only accept „understanding“ once I can explain it and reuse it in a different context. But I am not self centric enough to deny that there absolutely are plenty of people who love to memorize without having an abstract understanding. And they are doing just fine.
The cool thing about area of a rectangle is you can just turn it into a multiplication array. Which is something they learned to help them understand multiplication.
If she just memorized multiplication, she wouldn't actually understand the formula.
Thoughts flow through my brain like electrons through wires, both at a speed I cannot truly comprehend. The paths of my thoughts, the way words connect together, the emotions they evoke, the feelings that are associated with - these are all malleable. I have been working to rewire myself away from pessimism and towards optimism for years now. It's not easy, and sometimes I fall back into old patterns. As years have passed, though, I've found the new pathways easier, the new roads getting more familiar. My thoughts have previously wanted to go one way, and I spent time yanking at them to go a different way. I spent enough time at it that I now on good days more naturally go they way I want to go.
It's not just sitting and repeating a single thing over and over again, it's working with your own natural inclinations so that you can recognize when you experience X and naturally reach for Y that perhaps Z is what your preference really is, upon reflection. If you can practice that enough then, in the moment, you can sometimes find yourself not following the old pathways but the new.
Rewiring seems like a pretty good analogy as when you rewire a house you work hard -- it is dirty, dusty work. Pulling wire is hard and thankless because when you're done you cover it all up and, if you're lucky, it works! Then at the end you're . . . back in a state where nobody but you knows any different what is happening behind the walls. Things work, and others might have no idea anything changed at all. But you put in the hard work and you know how things actually work on the inside now and it's exactly how you want it, not how it was before.
If you've got a better analogy, I'd love to hear it.
I would think that would be obvious to anyone that merely memorizing something like 'f=ma' would be meaningless without deliberate attempts at application (both theoretical and practical).
There was a kludgy attempt at tying the study of foreign languages to STEM, but it just amounted to, everything is ultimately a craft. You have to practice to perfect it.
The author, Barbara Oakley, has a free Coursera course that is pretty good:
https://www.coursera.org/learn/learning-how-to-learn
... then you just have to do it. And keep working on it, even though it feels awful
Currently, the emphasis in training and education is on ensuring students understand the material, and rote memorization is viewed as a failure mode.
The author acknowledges this but introduces an argument that rote memorization is critical to achieving fluency.
I suspect that stating this position among typical education-focused circles will result in pushback.
To lend credibility, she adds her lived experience as a way to explain what she means—and to clarify that she isn’t saying everyone is wrong, just that we may be too harsh on memorization.
This is apparent to me because, frankly, if it weren't for the additional content she added, I wouldn’t have spent more than a few seconds before dismissing it.
The article even made me concerned about an internal project I am involved in, prompting me to verify that I hadn’t overlooked some issues.
If you want a TL;DR, the golf analogy matters: If you want to learn math, you need to understand it and then practice it so much that it becomes second nature.
I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=33890921 - Dec 2022 (9 comments)
I Rewired My Brain to Become Fluent in Math (2014) - https://news.ycombinator.com/item?id=13674101 - Feb 2017 (46 comments)
The building blocks of understanding are memorization and repetition - https://news.ycombinator.com/item?id=12508776 - Sept 2016 (94 comments)
How I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=8402859 - Oct 2014 (144 comments)
How I Rewired My Brain to Become Fluent in Math - https://news.ycombinator.com/item?id=8400837 - Oct 2014 (6 comments)
Years later, I’ve been slowly trying to “catch up” with my mathematical knowledge, and I find myself the most interested in topics that relate to the lives of mathematicians (and how events impacted their work) and to the philosophy of mathematics. I didn’t get any of this in school math classes, which focused purely on calculations and formulas.
I had the same experience with accounting, as well: boring in isolation but fascinating when connected to the history of double entry accounting in Italy, global trade from the 1500s onwards, and so on.
From an intellectual perspective (different than my engineering perspective), I'd label mathematics as quantitative philosophy. I like philosophy too.
The lack of freedom to pursue wny individual interests aside, major subjects like History, mathematics, biology are all reduced down to a bunch of facts you can fit in a textbook and test in a multiple choice format.
I think that people get slotted into the likes math or doesn't like math category quite early in life, often because they don't have an on-ramp to finding it interesting. They then spend the rest of their education avoiding it because "they're not a math person," when in many cases they probably just needed a bit more context/history/philosophy/something interesting to get them on the right track.
The reality is that every advance in mathematics evolved after decades of people crunching numbers (typically for some engineering application) with an earlier form of math until they worked out the advance. I think students would have more confidence in their own ability if they understood that the mathematical innovators were equally stumped for long periods of time.
I mean it can be a great story hook to start off with a subject, but in the end isn't math about the calculations, formulas and proofs?
Dead Comment
> Sorry, education reformers, it’s still memorization and repetition we need.
It seems needlessly confrontational, and misses what the article is about. The article asserts that practice and repeated use of math is important. I don't think it's really suggesting that we should go back to how math was taught in the US 40 years ago.
But maybe I'm just out of touch with math education reform: In high school I was graded on how fast I could do matrix multiplication, and thought matrices were kinda stupid. Then I learned about linear algebra and coupled oscillators in college and thought they were awesome.
So I'd assumed the educational reform was about removing the busy work from math and focusing on what you'd actually use it for. Am I wrong?
She was successful because they emphasized drills, lots of drills, and more drills.
Teachers and students hate drills. Teachers, because they're tedious to grade, and students because they're boring. But they work. It's no different than doing the same Super Mario Bros. level again and again until you time your jumps just right.
I've often thought that gamification of drills would be a great way to get kids to learn their math facts or whatever, but there seems to be an allergy to doing this in the US education system. What the US education system seems to be addicted to is moving from one hype/fad to the next, as that's where the money trough seems to be.
My view on them changed as well. First I found them stupid and mind-numbing. Now my life is busy and chaotic and drills are one of the few easy, zen-like moments. Maybe students need harder lives to actually come to appreciate their repetitive and simple nature.
While maybe not ideal, it’s something I like as it puts problems into the game as a required mechanic for success.
[0]: https://www.prodigygame.com/main-en/
It's also worth noting that half of all teachers leave the profession leave within five years. Forty years thus represents about eight generations of teachers being trained with their own biases, refining their ideas in classroom practice, and then training the next generation. On top of this are the cycles of trendiness in the various schools of thought in educational psychology, as well as the varying policy platforms of governments. Educational practice from four decades is less historical and more archaeological.
On this forum, I will say, imagine your object-oriented programming language can build all sorts of abstractions but can't multiply numbers. So every time you have to multiply numbers in your algorithm, you have to instead write a few lines of assembly code that do the same thing. How much efficiency would you lose.
Just practice multiplying numbers.
Yes, you will almost certainly memorize anything with enough exposure, but targeted memorization is also useful, if the former's not going fast enough.
Because you'll be teaching the awful wasteful rote math that everyone hates and can't use, instead of the nice universal stuff that lets you transfer intuition and see how ideas communicate between different knowledge domains far beyond what was traditionally seen as math.
The 20th century gave us real new ideas in math. But our primary math education is still 200 years out of date. Until that changes, math education will remain a deadening cargo cult that throws away far more human potential than it develops.
It’s useful daily. Pre-algebra is useful fairly often (even if I weren’t a programmer, plugging numbers into a formulas and basic graphing are very handy, quite often). Trig I think I managed to use once, but only because I didn’t know the right way (if you find yourself using trig on a minor home project you’re probably missing some trick or standard or something that lets you not do that—I suspect it was the case then)
That’s… about it. Stats, kinda, but mostly looking up the formula for the thing I want and plugging in the numbers, which barely counts.
For (the few) students who actually understood the subject the problems are just busywork, for those who didn't it is the most important part of the learning process. There is exactly one way to understand mathematics, which is actually doing it. This can be many things, but actually solving problems is an important part. I believe that problems should be interesting, but repeated recall definitely is important as well.