Arithmetic is to mathematics as planing a piece of wood is to building a piece of furniture, or as typing is to programming. It's a critical, underlying skill, but it's not the whole thing.
The question of what mathematics might actually be is like the question of what pornography is. I can't define it, but I know it when I see it.
I see mathematics as the act of using symbols and rules for manipulating those symbols to generate statements that are true if the axioms are true. (The statements are generally more interesting if there is a correspondence with real life somewhere. But there doesn't need to be one. Abstract nonsense and all that.)
That's a fairly vague and and non-rigorous statement. Making it rigorous and specific could possibly involve mathematics, if desired.
So in this respect I see arithmetic is a bit like an instance (in the OO sense) of the act of mathematics, but mathematics is a class of thinking and, especially, expressing. In particular, the fact that most arithmetic has a fixed set of rules and you don't generally invent new rules consistent with some meta-rules - the algorithm being performed could be trivially done by a computer - suggests to me that it doesn't really require much mathematical thinking.
When I was younger (mid-teens, and aspiring programmer), adults used to ask me if mathematics was important for programming. I would reply that it is not, that it is very rare for complex arithmetic or calculus etc. to be useful in most programs. That statement was true for both how I and those I was talking to saw maths at the time. But now, looking back, I think programming does actually use some mathematical thinking - real mathematical thinking - albeit not requiring anything like the same level of rigour. You invent your symbols, and compose them to solve the problem, and try to ensure invariants are preserved, and convince yourself that all cases are handled and the result won't have holes in the proof - i.e. bugs.
In your description, mathematics is the pure study of theoretical computing machines. However, I think that the act behind mathematics is what people are interested in -- general or capricious analogy-making ability. That's what we want to develop in young people.
We want the thing or process that operates the math, including novel operations for novel situations, and I believe that analogy-making is that thing.
And this is may be why people have an emotional resistance to arithmetics in school. Because arithmetics is an instance of math, but what people are practicing is a particular instance of operation, rather than the skill of operation.
From my experience of professional mathematicians at university, arithmetic was usually their worst skill. They'd quite often struggle (for a short moment) to multiply, say, 12 and 16 together. The public perception of a mathematician is somebody who remembers Pi to a 1,000 digits. A real mathematician doesn't care less.
This. So much this. Arithmetic is bookkeeping, it's turning an elegant abstract process into an ugly solved instance through rote mechanical labor.
In most of my college math classes, I largely 'got' it, but what killed me, and many other students, was the arithmetic. We could take multiple integrals, but dammit, 2*3 /= 5. Thank God for partial credit.
A curious "definition" I once read (sorry, don't remember the original source) goes something like this. Mathematics is the smallest subject that satisfies the following:
1. Mathematics is the collection of topics for which there is a mathematician who considers the topic part of mathematics.
2. Mathematicians are those humans who devote their lives to expanding human knowledge of mathematics.
This is how I learned it - in hindsight, I feel this is easier to reason about, as it forces you to apply operations synchronously. (As opposed to, say, subtract 5 from both sides, then add 8 => which we can reduce to add 3. But this reduction might just be noise to your brain.)
If you have an intuitive understanding like "both sides represent a number, as long as I manipulate it the same way the equation stays true" then you wouldn't have problems with logic when dividing by zero
Yes, I understand the concept that you quoted. Theoretically, it's more sound.
However, I want to point out that the purpose of the 'magic mirror' method is to ease mental math. It frees your brain from having to spend time / space on the extra step of applying the same operation to both sides.
Putting this in the context of 'showing your work':
Normal: x / 3 = 4 => 3 * x / 3 = 4 * 3 => x = 12
Mirror: x / 3 = 4 => x = 4 * 3 = 12
It's a bit complex to describe - I feel how you do basic math such as this is hardwired into your brain at a very young age.
A good first step is to come up with some possible reasons for why we suck.
I know I felt that mathematics was completely devoid of context during my high school and undergraduate years.Now that I have problems to solve, I have a real purpose when I go back and relearn what I was taught years ago.
Perhaps we should not have mathematics classes at all. Instead, we should just expect to encounter mathematics every subject, and the mathematics is taught where appropriate.
Also, I'd use the natural explanation, which the evidence points to: schools are reasonably successful at inculcating conformity and ignorance. (http://www.youtube.com/watch?v=pFf6_0T2ZoI) If they didn't fulfill this social function, they'd be dismantled or fixed.
(Thinking otherwise is like thinking that invading Iraq was about WMDs and the US gov't was noble-but-fumbling. Contorting ourselves into logical pretzels to preserve an illusion.)
I wish people like Edward Frenkel well. (Author of "Love and Math: The Heart of Hidden Reality", who tries to undo the damage done by math education.) But they're fighting an educational system which is inherently opposed to supporting critical thought, which fires effective teachers who refuse to work in the correct ideological framework.
The point I was driving at is we still end up with people who don't understand basic algebraic manipulations. So the issue isn't "you never did math", the issue is we're failing to get people to understand it.
The converse side of the idea - which I'm presently experiencing - is to definitionally heavy and try and show the proofs for all sorts of things. From my personal experience, this is equally as bad - I have real difficulty following proofs, where applying the results of said proofs is straightforward - easier to memorize, and apply, and slowly from that work backwards to figure out what the ramifications are elsewhere. But that's a process which takes years - I have 1 semester so in a practical sense memorization is the key.
Yes, a lot of high school subjects focus on the results rather then the process. Community colleges are also guilty. It's not just contained to math and science but even the arts and philosophy.
It's a part of the prevailing attitude "Everyone should go to college." High schools focus on rote learning instead of critical thinking to improve their chances of admission to a good college. Then those same colleges frown on those mechanical methods. The worst part is we're training people to be spoonfed knowledge rather then seek it.
Ha, this problem is nowhere near as bad as it is in China, where your admissions are nearly (outside of connections) solely based on your college admission test results.
At least in the US, the most selective universities look at other aspects of your application.
That's not to say there aren't other problems in the States, such as negative discrimination / racial quotas.
I've been talking to a few chinese college students. I'd heard that the gaokao was everything in college admissions, and I'm prepared to believe it plays a strong role, but I was a little disoriented by the results of asking four students how they got into their particular college:
复旦 student #1: My gaokao score wasn't high enough for 北大, but it was high enough for 复旦, so I came here.
复旦 student #2: My high school recommended me to 复旦, they gave me an interview, and then they offered me a place in the school of social science. Because of the offer, I didn't need a very high gaokao score.
财大 student #1: I took 财大's own entrance exam and qualified for admission, so I didn't put in any effort on the gaokao and got a low score. That meant my gaokao score wouldn't get me in to any other universities, so I came to 财大.
财大 student #2: I took the gaokao and my score was high enough for 财大, but not for 复旦, so now I'm at 财大.
So half of everyone I've asked fits the mold, but that seems like a low figure to me. The sample size here is only four; have I stumbled upon a wildly unrepresentative group?
In the case of the second student, I feel there's some missing context there. My guess is it was an alumni recommendation, which would explain the circumstances.
For student #3 - I'm unaware of how popular the practice of having a test per university is (or how practical it is to take them as a prospective applicant). Maybe this is restricted to certain universities or majors, or works similarly to EA or ED in the US?
One thing I never understood about standardized testing is why do they create them in a predictable way that makes them easy to prepare for (and thus game)?
Why not create totally different tests each year with all kinds of difficult problems and then grade the whole thing on a curve? The only way to succeed is to actually understand what you're doing better than your peers.
"We are so cool here, that we exclude kids from our campus who achieved less than X score"
That PR message becomes a puzzle when X becomes a randomly distributed number from year to year. So is 1580 a great number this year while 1400 was a great number last year?
The greatest danger would be an "emperor has not clothes" moment when random test results randomly categorize random students into random schools. That might make the test irrelevant in the future, thus not taken. And the test providers income stream disappears.
The scores would be more like 99th percentile or 80th percentile etc. So the message would be "We are so cool, we only accept the top 1%" which works fine and is more meaningful than 1580 or w/e.
That would assume the cream rises to the top, rather than random testing smoothly distributing them into some kind of spread spectrum signal across the test results.
Or it creates a lotto effect. In some ways this is honest and respectable; if we accept that most people are where they are solely because of whom their parents are, providing a lotto to decide who gets the best deal is in some ways fair and honest. "I was lucky to be born to certain parents" rhymes well with "I was lucky to attend certain schools"
Because it's "standardized" testing, that's the whole point. If the nature of the test changes drastically yearly, how can you compare results from year to year?
It's "standardized" as in everyone gets the same test in a given year. Absolute results on something like a college admissions exam are pretty meaningless since getting into college is a zero-sum game for all but the worst schools. All that matters is doing better than X% of the population.
Seems to me like incentivizing deep learning over memorization outweighs losing the ability to compare absolute results from one year to the next.
Speaking of training, the article is a reasonably good long form analogy between music and math WRT training vs education.
If you want an education in math, you don't memorize multiplication tables, just use a calculator. If you want training in math, you don't talk about applications and word problems and proofs, you memorize certain symbolic manipulations.
If people learned programming like they were taught math, they would be forbidden from knowing the concept of a quicksort exists until they successfully memorized and recited a 3SAT proof, "just because". It would be interesting to ace an automata theory class before writing your first line of code. Probably not required, and probably not a good idea, but it would be interesting.
You can educate a kid about what a derivative is in grade school as soon as they can slap a ruler up against a graph, what maybe 2nd grade or so? But you can't train them what a derivative is until mid to late high school, at least post-geometry and post-algebra (and some aspects of 1st year calc require a post-trig background, but not all)
This is aside from the meta issue that math is usually weaponized into a tool of filtration, because anyone can master it given enough innate skill or sweaty effort, so it seems "fair" to use for filtration. If you eliminate its usefulness as a filtration system, that doesn't mean as a culture we're not going to filter, it just means we're going to torture undergrad aged kids by filtering them on a new criteria, perhaps how well they've memorized historical names and dates, or how well they've memorized geographic maps, or how well they've memorized the bones of the human body or the electron configurations of the periodic table.
We did as well, although it seemed to be kind of slipped in rather than purposely designed into the curriculum. What the school was judged on formally was mostly how well kids did at the standardized exams, both the state exams for regular classes, and the AP exams for advanced classes. Preparing to do well on those didn't really involve doing any science.
The main factor that I think led to some science happening anyway was that a "good" middle-class suburban school felt it had to own some fancy technology to come across as modern and well equipped. So we had computer labs decked out with Macs, and pretty decent chemistry equipment. But once you spent some money on some fancy stuff, you need to put it into the school day somehow.
The computers were mainly used as quasi-free-time, where you had some self-directed time to play on the Macs, as long as you were doing it in one of the "edutainment" applications. Some of those were designed by educators with a constructivist approach to free-form, experimentation-based education, which produces a kind of virtual-science environment (other applications, of course, were badly designed and not useful for learning anything). And then the chemistry lab had to be used for something too; that one was a bit more directed, generally going through some standard experiments.
I found both parts to be pretty disconnected from what we were tested on, which was maybe precisely why they were interesting and educational...
You really never did anything worth caring about in high school, academically, at least. Math was a series of formulas to memorize and various sequential combinations of said formulas to memorize. English and writing classes were focused on forced structure (Intro has 4 sentences, ends in thesis, each body paragraph's topic sentence must relate back to thesis, if discussing multiple stories - Intro-ABAB-Conc. structure is preferred, 3 pages long, must integrate quotes from narrative, etc.) using forced literary techniques (rhetorical devices, and general "proposals" that tell you how to write and what to write) as a (forced) lens to study ancient works of narrative that no student cares about. Then the essay is assessed on how well it fits each forced mechanism, instead of well, being assessed on whether it's good writing or not. History was a series of events and dates to memorize and recall. Natural sciences varied : HS physics was a lot like HS math, HS biology a lot like HS history, HS chemistry some mixture (zing!) of both, mostly the latter. You might notice a pattern (pattern-finding being a very valuable trait NOT taught in HS courses, because who needs patterns when you can just memorize each individual event in the textbook?) : a focus on memorization over analysis, a focus on pre-set structure over creativity.
At least Computer Science was all right. But that's only because I was lucky enough that my school's CS department was large enough to afford to be taught by some seriously smart people truly dedicated to both the study of computer science and the art of pedagogy, but small enough such that the principal let the CS dept operate as it wants without interfering with the College Board's awful way of treating every subject. Part of a course (AP Comp Sci A) absolutely required you to at least dip your toes into the College Board's bullshit, and skimming over the Barron's book/taking the actual test, it seemed like the College Board had planned a lot of tedious stuff like Java/Java's standard library details, manual loop evaluations, and that infuriating GridWorld bullshit (a complicated, but still incredibly awful simulation program; the test assesses your knowledge of GridWorld's actor types and which Bug goes which way rather than assessing ... computer science, which is honestly what I fucking signed up for). The stuff in my school's course that really intrigued me and got my mind jogging (working through sorting algorithms, data structures, and big O analysis on your own after you've been taught the absolute basics) was the stuff they cut out of the AP Computer Science AB program, which was an earlier program that was deemed too difficult, I guess. As if the College Board was intentionally avoiding stuff that required analysis or actual thought.
My chemistry class had experiments. We had to formulate a hypothesis and report how the results compared to it. Isn't that in keeping with the spirit of the scientific method? Granted I did not go to school in the US.
I did. But only because of doing the elective advanced mathematics in high school, which required me to learn proofs, work things out from first principles, and basically do everything I'd end up doing again in first and second year university (BAppSci in Mathematics). The thing is, it had an 80% failure rate, because the rest of what we'd learnt was exactly how the OP described. Such a shame.
I think a certain amount of blame has to go to the teachers though as well. My personal anecdote, I did extremely well in Calc AB in high school, aced the AP exam, aced first semester Calc III in college, then had a professor in linear algebra who, in hindsight many years later, was a terrible teacher. He zoomed through everything, didn't explain, and just presented rather than taught. I still remember his comment to help us understand - "if you're having trouble picturing 11 dimensions, picture a 3D picture, but in 11 dimensions." Thanks! My last math course, Partial Diffs I did well again. To a certain degree I feel "math is math" but how it's taught is different from prof to prof.
I had an excellent math teacher in High School who would intersperse questions in his lectures, turning around and waiting long enough for a few hands to go up and then choosing someone to answer.
I was never able to adjust to the recitation-style math lectures in University and ended up having to, as you say, teach myself by reading the book and posing questions to myself as I went along.
I was and still am angry that quality of education I was receiving in exchange for tens of thousands in tuition was considerably poorer than what I was used to receiving for free in public High School.
I feel your pain. But part of college is learning how to learn things on your own. It varies a bit by field, but in theory by the time you end your undergraduate work you are almost out of work that other people have done before, and now it's up to you to figure out new things.
Probably one reason (among many) I didn't bother trying for a PhD.
Learning on your own is certainly core to college - but lectures that just where the TA just takes an outline from the book and regurgitates it are a waste of time. I certainly never attended those classes. I imagine they are useful for auditory learners but I’m not one of those.
Even at the college level there are plenty of class formats where either teaching was done, or at the very least was done via discussion.
The "here’s are the course materials read to you by a very bored smart person who has better things to do", isn’t a class format - it’s a series of exams.
Halfway through Calc III it was interesting to watch the visual students, who normally would do very well, have a rough time, while other students, who just treated it as an abstract system had more overhead/trouble/were slower when learning initially, but it paid off when getting to un-visualizable systems.
In my experience, virtually all of Calc III is "visualizable" (even somewhat esoteric stuff like Lagrange multipliers[1]), because it's mostly about vectors (which have a natural geometric interpretation). My claim would be that to be really good at math, you need to be skilled at both visualization (and other intuitions) and abstract systems. They complement each other well.
My intro physics class was kind of like that. There was a departmental standard that all the profs were instructed to use, and it apparently just involved showing us equations and how to derive them (generally after we had already had to derive them on online coursework).
The tests were actually applying those equations, being able to pull apart a problem into what bits of information you had, what bits you needed, and being able to combine and substitute out formulas to get the missing bit you were asked for.
In hindsight, I can vaguely see how the instruction might have helped on the test (I'd have gotten used to substituting out and deriving new equations from the old), and I can see how I might have done better on the tests (write out what equations I remembered. Write out what data I was given. Write out what data I was missing. Start substituting things out until I found a way to calculate the thing they asked for), but at the time, the only thing that helped me pass was actual instruction and practice outside of class on solving problems.
Geoffrey Hinton uses this same "picture a 3D picture, but in 11 dimensions" quote in his Neural Networks Coursera class. Honestly I think it's the only way to imagine >3 dimensional objects.
> I still remember his comment to help us understand - "if you're having trouble picturing 11 dimensions, picture a 3D picture, but in 11 dimensions." Thanks!
To be fair to the prof, you cannot actually picture 11 dimensions. That looks like a joke to me (hey, I laughed).
I was decent at math pre-college - placed in top 5 in several state-level competitions, and enjoyed my fair share of more obscure branches that weren't traditionally taught in school (number theory, combinatorics).
I agree with the majority of the author's points, but I despise his quick judgement on freshman students complaining about calculus.
I also said the same 'ironically stupid thing' in my freshman year, but that's because I _dreaded_ doing calculus as it's traditionally taught. It's much harder to find elegance in calculus than it is in say, algebra or geometry. (Mostly because the 'grunt' work behind it is so much more tedious.) Those are similar to programming in the sense that coding has elegant, extensible solutions and quick, dirty hacks. With calculus, I always felt like I was a inadequate human version of Mathematica.
More simply put, I could always solve problems using shortcuts in high school both to save time and to give myself more of a mental challenge. In intro calculus classes, there is no such thing.
I'd say derivatives/integrals being applied to velocity/acceleration/etc, and using integrals to find volumes of revolution and such, are pretty elegant.
I agree with you here, but I meant more from a process rather than application perspective.
By analogy, it's like choosing to use options / maybe monad instead of null in a service that takes input. The end service's functionality is the same regardless! (maintainability might be a different matter though, hehe)
In South Africa, with its bottom of the world rankings in school mathematics, this problem is particularly acute. With the government-set matric (school-leaving) exams it is sufficient to work through past exam papers and memorise the answer patterns, to be guaranteed good marks. This isn't a recent phenomenon, but has been the case for many years, although it seems to have gotten worse in recent years. Although there is a more realistic matric exam used by most private schools (Independent Examination Board), they will inevitably have to lower their standards as well to remain competitive with the government-set exams.
I am ashamed to admit that even when I got to university, I preferred the handful of maths and physics lecturers who followed a similar approach - work through the homework, memorise the answers, and pass.
>I am ashamed... I preferred the handful of maths and physics lecturers who followed a similar approach
no shame in admitting that you've been thru a broken system. Tertiary education isn't about setting a bar, it's about learning and discovery - setting a bar should be left for vocational training institutions, where you get certified that you are capable of doing such and such. Universities _should_ ostensibly be about personal learning and inherent motivation. I would garner that you shouldn't even be rewarded with any sort of formal certification from a university. Those who need such a formal cert ought to take an exam from a vocational training and certification institution.
(Fair disclaimer, this analogy is borrowed from a much better writer, Paul Lockhart). And the words "Paul Lockhart" are hyperlinked to http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf which Typhon already pointed out several comments up-thread! Your reading comprehension needs a tune-up.
Yeah I thought the exact same thing. To me, this isn't particularly interesting (because it's a re-hash), but the author's lecture for HS students on graph theory, linked in the post, is.
I try to teach genuine mathematics to elementary school pupils in supplementary classes during the weekends. Because what I do in my classes is intentionally quite different from school lessons in mathematics in most elementary schools, I have to explain my approach to new clients. My FAQ "Problems versus Exercises"[1] is the first in a series of four FAQ documents about how genuine mathematics involves problem-solving, and sometimes doing something that looks frighteningly hard at first. I think this kind of approach to mathematics can be helpful to hackers and to their children.
I had an interesting transition from high school to college as well - I aced math classes beforehand, doing well in courses such as college Calculus III, Linear Alegbra, and Differential Equations, but ended up failing an intro proof class my first semester of college.
After failing that class, I still took enough from it to pass subsequent proof-based math classes with good grades. Ultimately I left a top PhD program in math after 4 years and did well for myself since that failure, but it's interesting to note that transition from easy & menial calculations to full-on hard logic that challenges you at the highest level mentally. Our education system does a poor job of preparing us for it.
Edit: For further context, I was a top math & science student in NY, having placed top 50 in competitions nationwide and similarly competitive state and region-wide
There is a huge disconnect between intuitive mathematics and the formalized one taught at universities since the middle of the 20th century due to Bourbaki's group. For many people this emptied mathematics and made it inaccessible to a large portion of population, making them 2nd class citizens of the future, group which would be otherwise capable of mastering it with a proper pedagogical style.
IMO this is a pedagogical insanity, flooding young kids with formalisms that took centuries to emerge without any explanation about their background and enforcing form over content, which is what cuts many super talented people and forces them to focus at different fields.
There are many problems with contemporary math that are conveniently avoided (binary logic for example - most of the population doesn't believe it has any connection to thinking due to weirdness of material implication and teacher's insistence that this is the right way to think, never mentioning that its distant father Aristotle was so discontent with it that he immediately developed a first proto-modal logic), etc. If some constructionists and intuitionists weren't going against the scientific current, we wouldn't have had computers for a long time.
I always really liked math in school, cause it was just logic and that appeals to my lazy side. I was good at it too.
The last 2 years of high school, however, I had picked the 8 hour math options (25% of total course time) and the fun was quickly beaten out of it by having to learn formal ways to write a proof. Saying the same thing in plain language was 'invalid'.
From that point on math felt more like learning a foreign language than about doing logic.
Math is a formalism of everyday logical reasoning. It's the difference between a formal language and English.
Learning rigorous mathematics is frustrating at first because the veracity of a statement can seem intuitively obvious but difficult to prove. However it is a key stepping stone to modern mathematics and will considerably sharpen your intuition after you've gone through the process.
That's because your brain implements a weakly truth-preserving plausibility logic, whereas real mathematics makes use of strongly truth-preserving formal logic.
Blaming Bourbaki for this seems to give them credit for far more than the influence they had. They did cause an entire generation of professional French mathematicians to waste effort on re-proving things that had already been proven.
But the trend toward (excessive?) formalism at the expense of intuitive understanding is much larger in scope than Bourbaki.
No-one in logic even pretends that the material conditional is in general an adequate representation of natural-language conditionals anymore. It's fine in the context of mathematics though.
This is timely for me as I've just started a course in Abstract Algebra after taking a class on proofs. There doesn't seem to be a great way to get through this course without full understanding of the concepts since it is so proof laden.
The idea struck me yesterday that there's no real reason one couldn't teach this in elementary school or Jr. High and maybe that would be amazing for students? These kind of courses shape the way you reason.
It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them". That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.
Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.
Mathematics has always seemed the same to me. I don't really use much of it day-to-day, but occasionally I'll come across a geometry problem or something when I'm building software; maybe I end up doodling triangles, and using basic trig and algebraic manipulation to understand more or solve my problem.
Much of our teaching processes focus on skills, rather than a more abstract notion of "education." There's been much said about why this is a bad thing; I'm rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don't teach math, but we do teach the skills that are required to "experience" math at a later date.
So maybe I've convinced myself of the validity of the title, if not the individual arguments.
You and I live in different worlds. It seems to me that the bulk of what I do writing software is related to math in some manner. I guess I have a lot of "number cruncher" sort of clients.
I guess most people build CRUD stuff without (knowingly) using functional principles.
Programming is inherent mathematical, but if you don't progam functional, the math "behind" your software gets more abstract ... or should I say obscure?
In the end your computer is a big function with input, calculation and output. But if you write your code with side effects you can't do a fine function based split-up of your code, you have to consider bigger chunks of code, consisting of many dependent functions as one "mathematical" function.
If the math is more complex and logical reasoning about it gets harder, people start to think about it as "not-math" but something different.
I take issue with this because this is only one way to look at programming, not the inherently correct way. Yes, functional programming is a useful abstraction, but at the hardware level your processor uses something closer to a procedural paradigm. So your computer is not just a big function, not literally. Or rather, the "functions" your computer uses don't correspond 1:1 with the functions you write in functional computer language abstractions.
Good question. Why aren't processors optimized for functional programming? I think one reason is because of the way hardware memory works. So to answer what else you need, one other thing you need is physical memory and a way to organize the data in it. Forgive me for what follows if I misunderstand some functional programming aspect, because I'm primarily a digital hardware person, and programming in general is not my forte. Also I apologize if I explain things you already knew.
In hardware, CPU instructions are read sequentially from memory. These instructions are pretty basic... add two numbers, load a data word from a certain memory address, jump to a new address if two numbers are equal, etc. Modern processors do have some pretty fancy instructions but what I said is still basically true. So those instructions are our primitives. The only way to make abstract procedures from those primitives, from the perspective of assembly programming, is to make a sequential series of these primitive instructions starting at a known address, and then branch to that address and read those instructions in order. When you abstract many times out, as is common in functional programming, there starts to be a lot to keep track of. When you evaluate a function that is very abstract, how it looks in hardware is a whole lot of branching and returning to and from different memory addresses. Not necessarily bad, but it's starting to look pretty different in code vs. in hardware. And if you branch to uncached ("unexpected") locations, you add latency when you have to fetch instructions from RAM. You also have to keep track of any data needed at a higher level of the function, which necessitates automated memory management, including garbage collection. These things can introduce a lot of overhead in the program, especially when you have things in it like deep recursion.
tl;dr: There's no such thing as stateless assembly programming.
> Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.
For the author's analogy, music is not being taught like what you describe. In his analogy it's being taught as several years of learning to read and transcribe music, without listening to or performing it.
Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching "real math" once the students reach 13 or 14 years old. And throughout this, it's rarely hinted at how this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don't realize its real potential because the solution to the "problem" is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) / 2' and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won't and will get to college thinking they're "good at math" and then fail horribly because they don't have the skill set for college mathematics, they don't realize what college mathematics entailed (so many jokes about my "modern algebra" textbooks, "We took that in 9th grade!").
The thing is, I can't think of another way to teach the curriculum.
Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You've counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you've done that, you abstract all of arithmetic with algebra.
And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have "learned" everything up to calculus, but they don't actually understand what's going on. They just know the formulae and how to plug-and-chug.
How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood why the formula was used just as much as how it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they're fucked.
In any case, though, I think that the current curriculum is as good as it's going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren't going to learn calculus unless you understand algebra, and you aren't going to learn algebra unless you understand arithmetic.
A few weeks ago there was an article here on HN that suggested kindergarten students might do better learning an intuitive form of calculus and algebra before arithmetic. Yes, math is built out of layered abstractions, but we can rotate the entire conceptual space to use a different foundation and still get a complete picture in the end.
I don't disagree with the order of teaching, in general, but with its pacing. Arithmetic shouldn't be given 6 (K-5) years. Algebra shouldn't be 3-4 years (6-8 or 6-9). We bore students with the same material slightly stepped up each year for years at a time until they get to high school. Then we try and hit the accelerator and make them jump from the most rudimentary concepts in arithmetic and algebra and get them through trigonemetry or calculus in 3 more years. The concepts of trig, calculus, probability and stats, linear algebra can all be taught earlier. Students are capable of this, but the curricula aren't designed around it.
I didn't even realize until college that Algebra II was linear algebra. The notations used in college linear algebra would've been difficult for me to grasp fully at the time, but they make solving those systems of equations so much easier. And learning the notation [in high school], getting to the courses in college they'd be far less intimidating. We have 13 years with students before college, plenty of time to introduce notation and higher order concepts slowly rather than dumping it on them freshman year of college.
Kids deal with abstraction every day. The very idea of "color" and "number" are abstractions of concrete experience.
Teaching kids basic group theory is very possible. You can play games with shapes in the plane to learn about dihedral groups (without ever using those words). Graph theory, as the author says, is another avenue.
The problem is that what students are practicing isn't math, any more than running after the ball when you miss a swing in tennis is practicing tennis. And you improve at what you practice.
> The thing is, I can't think of another way to teach the curriculum.
But there's already a rich literature of real-world results for better ways to teach mathematics. See for example Seymour Papert's "Mindstorms" as a starting point (and much has been done since it was written ~30 years ago).
All children already learn quite a lot of fairly deep mathematical intuitions. We just take them for granted because everybody learns them.
For example: conservation of volume, the concept of "integer", order independence of cardinality, projecting orientation onto other reference frames, the equivalence between ordinal and cardinal numbers.
Everybody learns these things because they're embedded in our environments, and we can learn them playfully as children. When we create environments that embed even richer concepts, children learn those concepts just as easily. This is the explicit design goal of LOGO, and the whole family of descendants it has inspired.
Teaching in this way requires a degree of freedom and play that normal schools generally don't tolerate, which is why these proven, powerful tools still haven't taken over the world.
Where does, say, proof by induction fit into that linear set of progressing abstraction? Surely you don't need to understand calculus to understand inductive reasoning?
It ends up being on its own. You get introduced to it during geometry, but I don't think that it really gets taught until college.
I took BC calculus as a junior, so we got left with about a month between the seniors' leaving and the end of the year. My teacher said, "Okay, we're gonna learn number theory." Oh fuck, that was hard. It was completely unlike anything that we'd done before, and it required the development of completely different skills. I wasn't bad at it, but I definitely wasn't good at it either.
I'm glad that I got a taste of it, as that thought process has helped me in countless situations, but you're definitely correct in that it doesn't fit with the rest of the traditional curriculum.
> It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them". That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.
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I'd agree with you, if what we taught was an understanding of the concepts and established mechanisms. However, it seems to me that, most of what I saw in schools was just symbol manipulation.
For example, people didn't actually seem to understand that to get the area of a circle you took the radius, multiplied it by the ratio of the diameter to the circumference and squared it. They understood that you took the radius, multiplied it by a magic number, and for unknown reasons squared that.
The mapping of the symbols onto reality was often missing. It wasn't problem solving beyond the level of having a lookup table in your head that said 'When calculating an area do this, then this, then this.'
All that said, there are things it makes sense to memorise after you understand them - low level components where the speed gained in doing so allows you to use them in higher level abstractions. My point isn't that it doesn't make sense to teach people tools. But that to just give them the tools without the understanding of how they function seems harmful to their ability to create and adapt their own tools down the line.
The mapping of the symbols onto reality was often missing.
Perhaps because it's not a part of math? Math is an art. It's totally unconcerned with things like "reality". If you're so concerned with reality, you've probably never done math.
Some do, others don't. There is an interesting dichotomy: much of mathematics does appear to be completely dissociated from reality, and yet in the other direction, reality appears to be entirely mathematics. I think to reach a unification, it is absolutely essential that we have people working in each direction. You aren't likely to develop a theory of (∞,n)-categories by working backward from reality, and yet it turns out to be useful to have done so once you start exploring e.g. topological quantum field theories.
If that's your definition of art, then I can only consider myself fortunate to have been doing something else. Though, it's hard to see why anyone would gain anything in studying it were that the case. They could just make totally random shit up and claim, with as good a justification as any other, that it was as worthwhile as the work of any renowned mathematician.
However, I don't agree that art is unconcerned with reality, nor that maths is. We learn to draw by looking at things in the world, we get our rules about anatomy and so on from there; we learn maths based - at least initially - on physical examples; we tell stories based upon common themes and situations. The basic rules of these things are drawn from their correspondence to reality; with what people have experience with; and form the meaningful grammar of the system. Art is always a language, and a language is always representative. Even when - as arts - we may bend a few rules and simplify certain aspects to lend emphasis or make it better suited to approaching a particular problem.
To hold otherwise seems to me to deprive them of a foundation to communicate any common meaning. It would be no better to learn maths, under that definition, than to spend your life insanely scraping crayons across a page. (Indeed, given that the symbols would also be arbitrary in their meaning, it would be hard to tell the difference.)
Is maths an art, is it a science?
It's a language of enquiry. To claim that it proceeds solely by a 1-1 correspondence to reality is to picture mathematicians off somewhere counting things and deriving pi from taking increasingly precise measurements of actual circles. (And, really, even physics proceeds with the help of a healthy dollop of imagination that forms the foundation of hypothesis.) To claim that it has no correspondence to reality is to reduce the whole exercise to nonsense.
I do not see how either position taken as an absolute is tenable.
Your circle area formula is incorrect - you square the radius first, then multiply by pi. It's quite easy to see why if you substitute tau/2 for pi - 1/2 tau r^2 is clearly the integral of tau * r, where tau radians is a complete circle. Mixing diameter and radius in the same formula and hiding the resulting factor of two in a constant is a pedagogical disaster.
> It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them".
Is this so, or is it true that we merely do not allow any other path?
At least in the United States, very few children attend anything other than public schools, and these public schools have a strict curriculum that introduces few concepts but in a rigid order, interspersed with months of monotonous busy-work that comprises little more than arithmetic and solving equations.
As a whole, society has never ran the experiment to see if it is a necessary step, so claiming that it is necessary is silly. It is perhaps true that if we did run the experiment, it would fail repeatedly, but only then could it be claimed to be necessary.
>> Much of our teaching processes focus on skills, rather than a more abstract notion of "education." There's been much said about why this is a bad thing; I'm rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don't teach math, but we do teach the skills that are required to "experience" math at a later date.
Skill based education is better than not learning anything. No one argues this, it's a strawman. The point is, learning a method to do something without any kind of explanation or example of why you ever would do it is woefully suboptimal. In an attempt to come up with the most absurd example of this - it can feel like learning to scuba dive in a world with no water deeper than 8 feet, especially given that most teachers I had actually could not give me examples of how to apply the "math" I was learning (and I was a very immature handful back then, very unshy about demanding an explanation of why I was wasting my time learning something that I couldn't see a use for). The author isn't arguing for eliminating all the "skills-based" education he references, he's arguing that we move it closer to how music is taught - teach all the skills, but then immediately apply them to something the students can relate to, at that age (not in 7 years when they are working and run the risk of being like me, and not recognizing the importance of these concepts until then). Math is everywhere. Most people learn basic computation fairly easily, since it can be taught in the context of going to the grocery store or splitting a check. Get beyond this, and all of a sudden, schools quit even attempting to anchor math education in reality (by this I mean a situation that could conceivably occur in reality, not simply taking a number problem and putting words to it).
The argument in favor of mandatory musical education would actually probably benefit from stealing a bit of this piece: music is one of the best forms of education in terms of immediate application of the concepts and methods you are taught, perhaps only behind physical education.
The method of teaching might be the problem, but I think it's the wrong thing to focus on. A teacher who loves math, who is inspired by it, will teach it "right" (I think) even if ultimately they need to teach all of the boring routine steps to solving some particular kind of problem. At least that is what worked for me. Watching a teacher work with problems in class who loved math and loved patterns made me aware there was a "there" there.
I'm a former math teacher, now a programmer. I think he leaves out a few considerations:
1) You need to be able to do basic calculations before you can do advanced proofs. I taught a lot of high school seniors, and I had a ton of students who were smart enough to handle abstract concepts, but couldn't follow along when I showed them cool proofs because they got caught up on the basic calculations (because they hadn't learned them well in middle/high school).
2) Good high school teachers DO do a lot of pattern recognition/abstract reasoning. That's the entire idea behind a discovery lesson and constructivist teaching - having students learn formulas by discovering patterns and reasoning about them.
3) Again, as he points out, American high schools do do proofs in Geometry. He thinks they're really pedantic, but there are good reasons why 2-column proofs are so tedious. For one, students seeing proofs for the first time freak out, so giving them structure helps. For another, if the students write out every single step, it's easier to identify who really knows his/her stuff and who's BSing.
1. This. And many of the smarter students also gravitate towards these things.
2. It's easier for many students to grasp things when they are not abstract.
3. Yes, it still rattles my brain that Algebra teachers force students to memorize the quadratic formula. It's ridiculous. The method of completing the square is straight forward, more applicable in other situations, and can even derive that verbose formula. It only fosters the, "memorize every possible form of the question that could be on the exam" type of learning.
The formula for quadratic equation solutions is not that hard to memorise. It is useful to remember since it makes it possible to get the solutions instantly, whether numerically or algebraically. It also exposes discriminant of the equation, another useful concept (to instantly determine the number real solutions). Of course, the derivation of these formulas should also be taught, but it is really inefficient to derive things from scratch every time.
It was unpleasent for me to memorise it. Could you elaborate on why it should be a goal to make students calculate solutions to a lot of quadratic equations?
It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?
Quadratic equations are common. Very common. In physics, geometry, differential equations and so on. It is also the next step beyond linear equations. It is nice to be able to solve these quickly and to be able to tell their properties just by looking at the coefficients. The sum of roots, the product of roots. The axis of symmetry of the parabola. Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.
>Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.
Yes, but this is actually false, so it's probably for the best.
Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant.
Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in.
* - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.
I remember reading (with horror) the methods for factoring cubics and quartics and thinking, "Well, I guess that's what they did before they had Newton's method and computers." Ewwwww.
Just because the solutions are not guaranteed to have a closed form in radicals doesn't mean that my statement is false. In the field of real numbers any polynomial with a degree higher than two is reducible.
>it is really inefficient to derive things from scratch every time.
Well yeah, it would be really inefficient to derive the formula and then plug numbers into it, but you don't have to do that, you can just teach people to complete the square instead. It's pretty much as fast as using the formula (since it's the same operations), but it builds on your existing equation manipulation skills and so you understand every single step. And if you ever need to actually know the equation for whatever reason, you can derive it easily.
Isn't it the case that 99% of the population just needs to know it exists, and, in the very, very, very rare case they ever need it again, they would simply look it up?
I'd rather spend our children's time letting them know that these formulas exist, making sure they can perform them as needed, and then have them move onto something new. No need to drill / memorize trivial (as opposed to fundamental and foundation) stuff that is 2 seconds a way with google.
The worst kind of math education was that endless dribble associated with memorizing useless patterns that could easily be looked up in a book. At one point I had (painfully, oh my god, so painfully) memorized about 20 different patterns so I could chunk up integrals into products of u(x)v(x)only to spit them back on exam day, and never again look back on any kind of calculus. That was not a pleasant day week in high school. Would have been much better spent learning Geometry, Prob/Stats, Discrete Math, Linear Algebra, or any other host of mathematically oriented topics.
Kids forget. If you just introduce a concept, spend a day or even a week on it, then move on to a different concept, most kids (if not almost all) will forget it by the beginning of the next school year.
Personally, it think the time for manipulation of symbols has passed. Math should be only about concepts and hardware or software should do he manipulation. But then you run into systemic issues like "how do I test all students". The factory-styled learning environment needs to die.
I think I've actually used the quadratic formula maybe twice outside of an academic math class in which the quiz question was designed to invoke the quadratic formula.
I'm a computer scientist, one of the more mathematical occupations.
I've never been in a life-or-death situation where I thought, "aha, thank God I memorized that formula in high school!". Also, I don't actually remember it anymore, other than "negative b plus or minus the square root of b squared minus four-a-c, all over two-a".
Well huh. I guess I do remember it, but apparently at a verbal-auditory level rather than visual-symbolic. As usual for me.
Still: I've never had occasion to use that recitation I just made.
1. I'm implemented the quadratic formula in software more than twice over the years, I think. In, you know, real programs I was paid money to write.
2. Have you ever been in a life-or-death situation where anything you learned in class in high school was useful to you? "Well, I would have died, but then I remembered that Julius Caesar's last three words were not actually 'Et tu, Brute!' according to Shakespeare..."
I remember struggling greatly in algebra because it was like a boatload of recipes to remember. There was never much discussion on what these actions meant or why you do them.
There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle."
Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.
Recipe's are a very helpful fallback where you might be struggling with understanding the origins of material in a rigorous manner. You can inspect a triangle all you like, but at the end of the day it's much easier to simply remember the formula.
This, I am finding, is the only way I'm managing to actually understand complex analysis - take the formulas for the results, and remember how to apply them. It's revealing to me that what looks complex gets very simple in that manner (and also that I still get tripped up by elements of basic integration).
If I couldn't do this, then I'd be lost - and in a test simplification remembering how to work through the definition isn't possible (and isn't required thank god) because that's a path which leads to me spending 6 hours figuring out and trying to picture something in a way which makes sense.
Totally agreed with #1. I think it's easy to forget how deeply you need to get calculation and symbol manipulation into your fingers so you don't get stuck later on.
It would be like trying to learn "software engineering" with a weak understanding of syntax and variable manipulation. You can do it, but you are building on top of a house of cards.
There was something on HN years ago about the problems students encounter when they are missing a bit of the preparatory work for the next steps in math. Wish I could find it.
I have taught enough lectures to high school students (presenting "advanced proofs") and talked to enough geometry teachers who abandoned the two-column geometry proof to know that 1) is not true and 3) is not worth it. The problem is that people build up proofs like they're something to freak out about, or that the proofs that are presented are inherently mechanical because that's what students are taught. You can't expect someone learning to write proofs to be perfect any more than you can expect a first-time drawer to color within the lines. It's okay and should be embraced as an opportunity to reflect and improve. A proof is not complete just because you "got to the answer." It's complete when it's simple, elegant, and easy to explain to others.
I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples:
The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).
"You have to show your work Chance" was the sentence that drove me to despise school. As a 6th grade visual spatial student in a "gifted" algebra class, I could see the answer as if I were reading english but struggled to show my work.
I read/write slow and I have an incredibly hard time memorizing anything so I rebelled. Even after I got my act together, got my GED, and went to college I suffered through the various levels of Calculus because it was the same tune all over again. Classes like Linear Algebra were a lot harder for me to "see" but it was still faster & easier for me to take the time to visualize it.
My girlfriend's brother has this same problem - he definitely knows the material, but he gets really frustrated because it's so tedious to write out something when you can just write the answer down and go onto the next problem.
When I went over it with him, I showed him several spots where he made careless mistakes - he added where he should have subtracted, he multiplied where he should have divided, he screwed up a decimal point, whatever. I told him, "It's easy to spot your errors now because these are easy problems. But when you get to harder math, it's going to be much harder to find out what you did wrong, and a teacher isn't going know whether you made a careless mistake or just don't know it at all. By showing your work, you show the teacher that you actually know it."
Nowadays he understands the reason why he needs to show his work but still hates it. I'm hoping that he'll be like this only while the math is easy.
By showing the work, he's busy proving himself instead of learning. This demonstrates that, intentionally or not, public schools have become primarily credentialing institutions and not teaching institutions. I offer a thought exercise...
If you could be given a magical amulet that let you teach students better, more quickly, and more permanently than ever before but at the expense of never being able to test them to see exactly what it was that they learned, or you could be given a magical apparatus that let you test them perfectly so that you knew exactly what it was that they learned and did not learn but gave no insights or help into how to teach them those things that they failed at, which would you choose?
Which would your local school administrator choose? Which would your children's teachers choose? Which would the legislator writing education policy choose?
Everything else is post hoc rationalization. Having decided what it is that we want public education to be, we need to have some sort of justification for it even if it doesn't make sense.
Do you know what people who don't show work do when they move on to more difficult problems? They start scribbling it out on paper, without any prompting. The more difficult problems are interesting enough that they want to get them right, and when they notice that basic mistakes are interfering they strive to avoid those.
Or, in some cases, they just don't bother. When you solve the Poincaire Conjecture (spelling? didn't want to cheat and look it up) no one gives a crap whether or not you "showed your work" because most of the other mathematicians can also "just see" the boring details, and are interested primarily in the truly insightful portion of the proof.
I suspect that we're actually selecting for accountants and not math geniuses when we harp on "showing your work". How many Perelmans did we discourage and how many math stooges were praised last year in public schools?
As I said, the problem is that when a kid is having trouble, it's very difficult to figure out where he's going wrong if he isn't showing his work. To take a simple example, let's try factoring a quadratic. The kid doesn't show any work and just writes down "x = 1 and -5." He's wrong. Well, how did he get there? Did he make a careless mistake when factoring it? Did he try the quadratic formula but mess up a term? Is he just guessing? I don't know because he didn't show his work.
Meanwhile, if he shows that he's factoring the polynomial and writes a 5 where he should have written a 3, I can immediately tell that he knows what's going on but made a careless mistake. Alternatively, if he writes down a bunch of gibberish, it means that he doesn't know what's going on and needs someone to go over the concepts again.
It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions, or do you want a compiler that is slower but gives you detailed warnings and error messages? I'd rather take the latter. Maybe once I'm perfectly sure that my code works, I'll do it with the former.
> As I said, the problem is that when a kid is having trouble, it's very difficult to figure out where he's going wrong
The correct (though inefficient) approach is to keep trying until you see that he starts understanding. However, this is impossible when there are 25 other students in the classroom. Each might require a different manner of teaching to "get it", or learn at different speeds. And so if you're trying to crank out graduates on an assembly line this just won't cut it.
So instead of figuring out a solution where each student can get the education he deserves as a human being, we instead seek to change the student so that he can be programmed with the education that is possible in an assembly line system. This also explains the dearth of highly competent, highly respected teachers... you don't staff your factory with gifted artisans who could carve the pieces. You want someone who will push the button and have the product stamped out in 0.75 seconds.
If you calibrate everything perfectly, some number of students will get a highly optimal (for them) education where everything was timed perfectly, using the easiest-to-understand lessons. For everyone else, for the slow and learning disabled, for the quick and talented... it will be an awful experience. And, whether you call it luck or circumstance, neither of those groups will be educated well enough to be able to express their criticism easily.
> It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions,
But a compiler isn't a person, and a person isn't a compiler. I don't want to treat people as if they were machines... I especially don't want to treat children like they are machines, it's almost certainly even more damaging the earlier that happens to them.
I'm a programmer too, I do this for a living. I know all too well how easy it is to think of human circumstances and other people as if they were machines to be debugged, and it feels awful. Imagine what the 7 year old kid feels like in school when he's a bug to be solved on the teacher's trouble ticket system. Especially when he's probably marked "low priority, fix when time allows".
You're no longer talking about a system where learning is considered the primary goal. It may not even be a goal at all.
> If you calibrate everything perfectly, some number of students will get a highly optimal (for them) education where everything was timed perfectly, using the easiest-to-understand lessons. For everyone else, for the slow and learning disabled, for the quick and talented... it will be an awful experience. And, whether you call it luck or circumstance, neither of those groups will be educated well enough to be able to express their criticism easily.
This is a really good point - the public education system isn't an artisanal workshop; it's a large, industrialized factory where "raw materials" are turned into "product." Every grade is another step in the factory process. And while I guess it might be optimal given the very limited resources that we devote to education, it's heartless and doesn't work very well from an absolute standpoint.
Personally, I didn't get a lot of my education from school. Sure, I was there ten hours a day, but I mostly learned from my father and the homework that I did. I would get assignments, and my father was the one who really taught me whenever I ran into problems. I would then go back to school and pass tests.
Unfortunately, my situation was atypical and very lucky; I was blessed with a loving father who was fascinated with a large variety of topics and loved teaching. Most kids don't get a resource like that and get stuck with school as being the only avenue for learning. How can you reach them? I think the only answer is more money, which will go toward more teachers. Cut down the class size to ten kids per class, and you'll get a much more individualized curriculum. As long as you have 25 kids in the classroom, you're going to end up with the factory approach.
Lots of kids see the answer directly for the simple problems, and so don't see the point of showing their work.
The point of showing the work is to learn the mechanics of solving the problem. If you do not learn the mechanics of solving for simple problems, you will not be able to apply the mechanics to more complex problems where one can no longer intuitively see the answer.
Yep, I understand the motive behind the requirement. There's a lot of good that can come out of it; students are forced to learn the "steps" or "mechanics," teachers can see where things went south and give partial credit or offer assistance, and finally it is a great way to circumvent cheating.
I totally get it and it makes total sense except for when it doesn't. The problem with our education system is that it's fundamentally flawed. It is designed to work best with the typical student being taught from a skill set chosen for optimal widget-making. People learn/think differently and yet we cater more and more to the mantra of pump-and-dump where children with the ability to retain the most frivolous information wins.
It took until my junior year as a comp-sci student for me to figure out what I personally needed to learn the material. I absolutely had to understand the big picture before I could ever attempt to solve the problems. If I didn't and I relied entirely on memorization then I was destined to fail.
In order for me to understand the big picture, I had to ask questions, sometimes a lot of them. Abstract questions would annoy some professors and certainly other students. Engineering classes, like most college classes, are full of people brought up in a system where the slide-show-after-slide-show of formulas, facts, or bullet points was all they needed. My questions were irrelevant and an interruption to their note taking.
Point 1 is only true for a proper subset of mathematics. This subset tends to be the only mathematics that are taught in high school (or even in undergrad unless you're a math major), which I think is a huge part of the problem. I can't recall the last time I did a proof of something in abstract algebra, category theory, or algebraic topology that actually involved a calculation of any kind, so clearly a facility with basic calculations is unnecessary for those proofs. Instead what is required is a facility with understanding rigorous definitions and abstractions, which is extremely valuable and important, and of which the average high school mathematics education provides essentially none.
The author seems to be building a false dichotomy. Math is the combination of two. To continue the use of the music analogy, if you can sing and write songs but do not know any theory and cannot express your ideas to anyone but yourself, is it truly music? As an amateur musician, I can say it is absolutely frustrating to work with people who cannot transcribe or manipulate their work in any meaningful way--even a simple tab would do.
Mathematics are the same way. Yes, you need to solve problems, but you also need to solve problems in ways that can build on your past knowledge and be shared with others.
(FYI, I'm a lifelong amateur musician, programmer, and data analyst. My formal education consisted of a double electrical engineering / mathematics major.)
By the way, there are schools where you actually do math. I was lucky to get into one of these on the second try, after 7 series of exams and interviews. The math lessons (apart from algebra and geometry, which we had to learn too, of course), were set up pretty simple: you were given a single sheet of paper with some axioms and definitions about the topic at hand and a list of lemmas and theorems that you had to prove. When you thought you could prove on of them, you called on of the teachers (there were about 5 per class), sat down with them, and tried to defend your proof. No homework, nothing else but this sheet.
I didn't pursue a career in mathematics, like a lot of my classmates, but these lessons gave me more anything else I did in all years spend on 'education'.
Here's the book on math teaching lessons, and although it's in Russian, you can get an overall idea of the level of math involved by skipping through the pages: http://www.mccme.ru/free-books/57/davidovich.pdf
It describes a 4-year program, from 8 to 11 grade of russian school, for kids from 13-14 to 16-17 years old respectively.
>There’s one kind of student I routinely encounter, usually in a freshman calculus course, that really boils my blood: the failing student who “has always been good at math.”
I understand the larger point about the difference between high-school math and high-level mathematics, but come on, don't be so pedantic! Everyone calls whatever it is you study in High-School & Elementary school - Math.
On a related matter, physicists and other scientists have done a pretty good job of communicating to the general public what it is they do. On the other hand, very few people actually know what professional Mathematicians actually do - something I realized when I struggled to explain it to my dad the other day.
Also, put it into the terms he'd like to put it into: His failing student has always been good at symbol manipulation, but now that he's gotten to this teacher's more advanced symbol manipulation, he's failing. Suddenly it sounds like the kid has a perfectly valid concern, quite possibly with this guy's teaching style, and the teacher's response is to make fun of the kid for using the word "math" in the fashion 99% of the American population uses it.
Your two points are really the same point. The reason that people don't understand what mathematicians do, is because we've been referring to calculation as mathematics our whole lives. If we did a better part separating the two out earlier, people would understand which part mathematicians work on.
I find it interesting that the author states that essentially mathematics is formalized 'pattern matching', yet rails against the insidious imposition of these very patterns in the pedagogy (in the form of rote exercises).
Isn't naive pattern recognition the basis of deeper dimensional understanding (ya know, the 'theory')? Isn't this how intelligence is built?
It seems pretty easy to make rag on the lack of 'true understanding', when you've spent 25+ years recognizing the patterns.
As a person from Eastern Europe (where hard sciences were very important subjects) studied in a special school for math talented children and got dragged screaming and unwilling to math competitions - I definitely did math in high school.
Also some of the crazy stuff we did for physics/chemistry required a lot of math.
In my experience, and opinion, going somewhere for learning purposes should be a humbling experience. College should be a place where a student realizes how little he knows.
I say this because in my experience, the first symptom of ignorance is a feeling you know a lot.
I'm depressed since childhood because since a very young age, I had read about great minds. How could I ever feel I'm "good at maths" after reading about Gauss, or Galois?
It made me feel like the lowest form of life.
That is akin to the way Military Generals feel towards Alexander the Great: You can be a great General, but you probably will never be Alexander the Great.
Maybe this should be done freshman year: Before even a single "maths" course is dispensed, a session on the achievements of Gauss and Galois, at age 17 or 19.
Maybe a brief discussion on who Lagrange was, and what he did in his teens.
This should take out any feeling of being "good at maths", and make students shut their mouth and open their ears.
The (seemingly intended to sound ridiculous) description of teaching music matches pretty well with what I remember from my occasional K-12 music classes. There was also an element of listening to music, but it wasn't particularly interesting. My first exposure to what I see as a serious study of music was in college.
But why single out math in particular? I look back on pretty much all of my K-12 education as fairly trite and superficial, in terms of "doing real work in the subject". My experience with, say, college-level history was much more intense (and seemingly more true to the field of history) than anything in high school. On the other hand, I'm not sure I would be able to do well at "real math" like calculus or graph theory or whatever you wish to deem "real math" if I was struggling with adding numbers and solving equations, and I attribute getting past such struggles to doing bountiful rote exercises...
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[ 3.1 ms ] story [ 276 ms ] threadIs arithmetic not maths?
The question of what mathematics might actually be is like the question of what pornography is. I can't define it, but I know it when I see it.
That's a fairly vague and and non-rigorous statement. Making it rigorous and specific could possibly involve mathematics, if desired.
So in this respect I see arithmetic is a bit like an instance (in the OO sense) of the act of mathematics, but mathematics is a class of thinking and, especially, expressing. In particular, the fact that most arithmetic has a fixed set of rules and you don't generally invent new rules consistent with some meta-rules - the algorithm being performed could be trivially done by a computer - suggests to me that it doesn't really require much mathematical thinking.
When I was younger (mid-teens, and aspiring programmer), adults used to ask me if mathematics was important for programming. I would reply that it is not, that it is very rare for complex arithmetic or calculus etc. to be useful in most programs. That statement was true for both how I and those I was talking to saw maths at the time. But now, looking back, I think programming does actually use some mathematical thinking - real mathematical thinking - albeit not requiring anything like the same level of rigour. You invent your symbols, and compose them to solve the problem, and try to ensure invariants are preserved, and convince yourself that all cases are handled and the result won't have holes in the proof - i.e. bugs.
We want the thing or process that operates the math, including novel operations for novel situations, and I believe that analogy-making is that thing.
And this is may be why people have an emotional resistance to arithmetics in school. Because arithmetics is an instance of math, but what people are practicing is a particular instance of operation, rather than the skill of operation.
In most of my college math classes, I largely 'got' it, but what killed me, and many other students, was the arithmetic. We could take multiple integrals, but dammit, 2*3 /= 5. Thank God for partial credit.
1. Mathematics is the collection of topics for which there is a mathematician who considers the topic part of mathematics.
2. Mathematicians are those humans who devote their lives to expanding human knowledge of mathematics.
3. Mathematics includes geometry and arithmetic.
x + 5 = 10 the equals sign is a magical mirror so when you take operations across it it changes them to the opposite of what they were
so adding five becomes subtracting five, multiplying by two becomes dividing by two, etc.
x = 10 - 5 by way of magical mirror
If you have an intuitive understanding like "both sides represent a number, as long as I manipulate it the same way the equation stays true" then you wouldn't have problems with logic when dividing by zero
However, I want to point out that the purpose of the 'magic mirror' method is to ease mental math. It frees your brain from having to spend time / space on the extra step of applying the same operation to both sides.
Putting this in the context of 'showing your work':
Normal: x / 3 = 4 => 3 * x / 3 = 4 * 3 => x = 12
Mirror: x / 3 = 4 => x = 4 * 3 = 12
It's a bit complex to describe - I feel how you do basic math such as this is hardwired into your brain at a very young age.
I know I felt that mathematics was completely devoid of context during my high school and undergraduate years.Now that I have problems to solve, I have a real purpose when I go back and relearn what I was taught years ago.
Perhaps we should not have mathematics classes at all. Instead, we should just expect to encounter mathematics every subject, and the mathematics is taught where appropriate.
(Thinking otherwise is like thinking that invading Iraq was about WMDs and the US gov't was noble-but-fumbling. Contorting ourselves into logical pretzels to preserve an illusion.)
I wish people like Edward Frenkel well. (Author of "Love and Math: The Heart of Hidden Reality", who tries to undo the damage done by math education.) But they're fighting an educational system which is inherently opposed to supporting critical thought, which fires effective teachers who refuse to work in the correct ideological framework.
The converse side of the idea - which I'm presently experiencing - is to definitionally heavy and try and show the proofs for all sorts of things. From my personal experience, this is equally as bad - I have real difficulty following proofs, where applying the results of said proofs is straightforward - easier to memorize, and apply, and slowly from that work backwards to figure out what the ramifications are elsewhere. But that's a process which takes years - I have 1 semester so in a practical sense memorization is the key.
It's a part of the prevailing attitude "Everyone should go to college." High schools focus on rote learning instead of critical thinking to improve their chances of admission to a good college. Then those same colleges frown on those mechanical methods. The worst part is we're training people to be spoonfed knowledge rather then seek it.
At least in the US, the most selective universities look at other aspects of your application.
That's not to say there aren't other problems in the States, such as negative discrimination / racial quotas.
复旦 student #1: My gaokao score wasn't high enough for 北大, but it was high enough for 复旦, so I came here.
复旦 student #2: My high school recommended me to 复旦, they gave me an interview, and then they offered me a place in the school of social science. Because of the offer, I didn't need a very high gaokao score.
财大 student #1: I took 财大's own entrance exam and qualified for admission, so I didn't put in any effort on the gaokao and got a low score. That meant my gaokao score wouldn't get me in to any other universities, so I came to 财大.
财大 student #2: I took the gaokao and my score was high enough for 财大, but not for 复旦, so now I'm at 财大.
So half of everyone I've asked fits the mold, but that seems like a low figure to me. The sample size here is only four; have I stumbled upon a wildly unrepresentative group?
For student #3 - I'm unaware of how popular the practice of having a test per university is (or how practical it is to take them as a prospective applicant). Maybe this is restricted to certain universities or majors, or works similarly to EA or ED in the US?
Why not create totally different tests each year with all kinds of difficult problems and then grade the whole thing on a curve? The only way to succeed is to actually understand what you're doing better than your peers.
That PR message becomes a puzzle when X becomes a randomly distributed number from year to year. So is 1580 a great number this year while 1400 was a great number last year?
The greatest danger would be an "emperor has not clothes" moment when random test results randomly categorize random students into random schools. That might make the test irrelevant in the future, thus not taken. And the test providers income stream disappears.
Or it creates a lotto effect. In some ways this is honest and respectable; if we accept that most people are where they are solely because of whom their parents are, providing a lotto to decide who gets the best deal is in some ways fair and honest. "I was lucky to be born to certain parents" rhymes well with "I was lucky to attend certain schools"
Seems to me like incentivizing deep learning over memorization outweighs losing the ability to compare absolute results from one year to the next.
If you want an education in math, you don't memorize multiplication tables, just use a calculator. If you want training in math, you don't talk about applications and word problems and proofs, you memorize certain symbolic manipulations.
If people learned programming like they were taught math, they would be forbidden from knowing the concept of a quicksort exists until they successfully memorized and recited a 3SAT proof, "just because". It would be interesting to ace an automata theory class before writing your first line of code. Probably not required, and probably not a good idea, but it would be interesting.
You can educate a kid about what a derivative is in grade school as soon as they can slap a ruler up against a graph, what maybe 2nd grade or so? But you can't train them what a derivative is until mid to late high school, at least post-geometry and post-algebra (and some aspects of 1st year calc require a post-trig background, but not all)
This is aside from the meta issue that math is usually weaponized into a tool of filtration, because anyone can master it given enough innate skill or sweaty effort, so it seems "fair" to use for filtration. If you eliminate its usefulness as a filtration system, that doesn't mean as a culture we're not going to filter, it just means we're going to torture undergrad aged kids by filtering them on a new criteria, perhaps how well they've memorized historical names and dates, or how well they've memorized geographic maps, or how well they've memorized the bones of the human body or the electron configurations of the periodic table.
The main factor that I think led to some science happening anyway was that a "good" middle-class suburban school felt it had to own some fancy technology to come across as modern and well equipped. So we had computer labs decked out with Macs, and pretty decent chemistry equipment. But once you spent some money on some fancy stuff, you need to put it into the school day somehow.
The computers were mainly used as quasi-free-time, where you had some self-directed time to play on the Macs, as long as you were doing it in one of the "edutainment" applications. Some of those were designed by educators with a constructivist approach to free-form, experimentation-based education, which produces a kind of virtual-science environment (other applications, of course, were badly designed and not useful for learning anything). And then the chemistry lab had to be used for something too; that one was a bit more directed, generally going through some standard experiments.
I found both parts to be pretty disconnected from what we were tested on, which was maybe precisely why they were interesting and educational...
At least Computer Science was all right. But that's only because I was lucky enough that my school's CS department was large enough to afford to be taught by some seriously smart people truly dedicated to both the study of computer science and the art of pedagogy, but small enough such that the principal let the CS dept operate as it wants without interfering with the College Board's awful way of treating every subject. Part of a course (AP Comp Sci A) absolutely required you to at least dip your toes into the College Board's bullshit, and skimming over the Barron's book/taking the actual test, it seemed like the College Board had planned a lot of tedious stuff like Java/Java's standard library details, manual loop evaluations, and that infuriating GridWorld bullshit (a complicated, but still incredibly awful simulation program; the test assesses your knowledge of GridWorld's actor types and which Bug goes which way rather than assessing ... computer science, which is honestly what I fucking signed up for). The stuff in my school's course that really intrigued me and got my mind jogging (working through sorting algorithms, data structures, and big O analysis on your own after you've been taught the absolute basics) was the stuff they cut out of the AP Computer Science AB program, which was an earlier program that was deemed too difficult, I guess. As if the College Board was intentionally avoiding stuff that required analysis or actual thought.
I was never able to adjust to the recitation-style math lectures in University and ended up having to, as you say, teach myself by reading the book and posing questions to myself as I went along.
I was and still am angry that quality of education I was receiving in exchange for tens of thousands in tuition was considerably poorer than what I was used to receiving for free in public High School.
Probably one reason (among many) I didn't bother trying for a PhD.
Even at the college level there are plenty of class formats where either teaching was done, or at the very least was done via discussion.
The "here’s are the course materials read to you by a very bored smart person who has better things to do", isn’t a class format - it’s a series of exams.
Halfway through Calc III it was interesting to watch the visual students, who normally would do very well, have a rough time, while other students, who just treated it as an abstract system had more overhead/trouble/were slower when learning initially, but it paid off when getting to un-visualizable systems.
[1] E.g. http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.ht...
The tests were actually applying those equations, being able to pull apart a problem into what bits of information you had, what bits you needed, and being able to combine and substitute out formulas to get the missing bit you were asked for.
In hindsight, I can vaguely see how the instruction might have helped on the test (I'd have gotten used to substituting out and deriving new equations from the old), and I can see how I might have done better on the tests (write out what equations I remembered. Write out what data I was given. Write out what data I was missing. Start substituting things out until I found a way to calculate the thing they asked for), but at the time, the only thing that helped me pass was actual instruction and practice outside of class on solving problems.
To be fair to the prof, you cannot actually picture 11 dimensions. That looks like a joke to me (hey, I laughed).
I agree with the majority of the author's points, but I despise his quick judgement on freshman students complaining about calculus.
I also said the same 'ironically stupid thing' in my freshman year, but that's because I _dreaded_ doing calculus as it's traditionally taught. It's much harder to find elegance in calculus than it is in say, algebra or geometry. (Mostly because the 'grunt' work behind it is so much more tedious.) Those are similar to programming in the sense that coding has elegant, extensible solutions and quick, dirty hacks. With calculus, I always felt like I was a inadequate human version of Mathematica.
More simply put, I could always solve problems using shortcuts in high school both to save time and to give myself more of a mental challenge. In intro calculus classes, there is no such thing.
By analogy, it's like choosing to use options / maybe monad instead of null in a service that takes input. The end service's functionality is the same regardless! (maintainability might be a different matter though, hehe)
I am ashamed to admit that even when I got to university, I preferred the handful of maths and physics lecturers who followed a similar approach - work through the homework, memorise the answers, and pass.
no shame in admitting that you've been thru a broken system. Tertiary education isn't about setting a bar, it's about learning and discovery - setting a bar should be left for vocational training institutions, where you get certified that you are capable of doing such and such. Universities _should_ ostensibly be about personal learning and inherent motivation. I would garner that you shouldn't even be rewarded with any sort of formal certification from a university. Those who need such a formal cert ought to take an exam from a vocational training and certification institution.
https://news.ycombinator.com/item?id=7221713
http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf
EDIT : Oops, I hadn't seen that he already linked to this text. That'll learn me to post too quickly. Oh well.
[1] http://www.epsiloncamp.org/ProblemsversusExercises.php
After failing that class, I still took enough from it to pass subsequent proof-based math classes with good grades. Ultimately I left a top PhD program in math after 4 years and did well for myself since that failure, but it's interesting to note that transition from easy & menial calculations to full-on hard logic that challenges you at the highest level mentally. Our education system does a poor job of preparing us for it.
Edit: For further context, I was a top math & science student in NY, having placed top 50 in competitions nationwide and similarly competitive state and region-wide
IMO this is a pedagogical insanity, flooding young kids with formalisms that took centuries to emerge without any explanation about their background and enforcing form over content, which is what cuts many super talented people and forces them to focus at different fields.
There are many problems with contemporary math that are conveniently avoided (binary logic for example - most of the population doesn't believe it has any connection to thinking due to weirdness of material implication and teacher's insistence that this is the right way to think, never mentioning that its distant father Aristotle was so discontent with it that he immediately developed a first proto-modal logic), etc. If some constructionists and intuitionists weren't going against the scientific current, we wouldn't have had computers for a long time.
The last 2 years of high school, however, I had picked the 8 hour math options (25% of total course time) and the fun was quickly beaten out of it by having to learn formal ways to write a proof. Saying the same thing in plain language was 'invalid'.
From that point on math felt more like learning a foreign language than about doing logic.
Learning rigorous mathematics is frustrating at first because the veracity of a statement can seem intuitively obvious but difficult to prove. However it is a key stepping stone to modern mathematics and will considerably sharpen your intuition after you've gone through the process.
But the trend toward (excessive?) formalism at the expense of intuitive understanding is much larger in scope than Bourbaki.
The idea struck me yesterday that there's no real reason one couldn't teach this in elementary school or Jr. High and maybe that would be amazing for students? These kind of courses shape the way you reason.
It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them". That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.
Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.
Mathematics has always seemed the same to me. I don't really use much of it day-to-day, but occasionally I'll come across a geometry problem or something when I'm building software; maybe I end up doodling triangles, and using basic trig and algebraic manipulation to understand more or solve my problem.
Much of our teaching processes focus on skills, rather than a more abstract notion of "education." There's been much said about why this is a bad thing; I'm rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don't teach math, but we do teach the skills that are required to "experience" math at a later date.
So maybe I've convinced myself of the validity of the title, if not the individual arguments.
You and I live in different worlds. It seems to me that the bulk of what I do writing software is related to math in some manner. I guess I have a lot of "number cruncher" sort of clients.
Programming is inherent mathematical, but if you don't progam functional, the math "behind" your software gets more abstract ... or should I say obscure?
In the end your computer is a big function with input, calculation and output. But if you write your code with side effects you can't do a fine function based split-up of your code, you have to consider bigger chunks of code, consisting of many dependent functions as one "mathematical" function.
If the math is more complex and logical reasoning about it gets harder, people start to think about it as "not-math" but something different.
You got input, output and calculations.
What else do you need?
In hardware, CPU instructions are read sequentially from memory. These instructions are pretty basic... add two numbers, load a data word from a certain memory address, jump to a new address if two numbers are equal, etc. Modern processors do have some pretty fancy instructions but what I said is still basically true. So those instructions are our primitives. The only way to make abstract procedures from those primitives, from the perspective of assembly programming, is to make a sequential series of these primitive instructions starting at a known address, and then branch to that address and read those instructions in order. When you abstract many times out, as is common in functional programming, there starts to be a lot to keep track of. When you evaluate a function that is very abstract, how it looks in hardware is a whole lot of branching and returning to and from different memory addresses. Not necessarily bad, but it's starting to look pretty different in code vs. in hardware. And if you branch to uncached ("unexpected") locations, you add latency when you have to fetch instructions from RAM. You also have to keep track of any data needed at a higher level of the function, which necessitates automated memory management, including garbage collection. These things can introduce a lot of overhead in the program, especially when you have things in it like deep recursion.
tl;dr: There's no such thing as stateless assembly programming.
For the author's analogy, music is not being taught like what you describe. In his analogy it's being taught as several years of learning to read and transcribe music, without listening to or performing it.
Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching "real math" once the students reach 13 or 14 years old. And throughout this, it's rarely hinted at how this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don't realize its real potential because the solution to the "problem" is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) / 2' and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won't and will get to college thinking they're "good at math" and then fail horribly because they don't have the skill set for college mathematics, they don't realize what college mathematics entailed (so many jokes about my "modern algebra" textbooks, "We took that in 9th grade!").
EDIT: Grammar.
Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You've counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you've done that, you abstract all of arithmetic with algebra.
And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have "learned" everything up to calculus, but they don't actually understand what's going on. They just know the formulae and how to plug-and-chug.
How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood why the formula was used just as much as how it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they're fucked.
In any case, though, I think that the current curriculum is as good as it's going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren't going to learn calculus unless you understand algebra, and you aren't going to learn algebra unless you understand arithmetic.
HN Discussion: https://news.ycombinator.com/item?id=7333998
I didn't even realize until college that Algebra II was linear algebra. The notations used in college linear algebra would've been difficult for me to grasp fully at the time, but they make solving those systems of equations so much easier. And learning the notation [in high school], getting to the courses in college they'd be far less intimidating. We have 13 years with students before college, plenty of time to introduce notation and higher order concepts slowly rather than dumping it on them freshman year of college.
EDIT: Clarification on time of something
Teaching kids basic group theory is very possible. You can play games with shapes in the plane to learn about dihedral groups (without ever using those words). Graph theory, as the author says, is another avenue.
The problem is that what students are practicing isn't math, any more than running after the ball when you miss a swing in tennis is practicing tennis. And you improve at what you practice.
But there's already a rich literature of real-world results for better ways to teach mathematics. See for example Seymour Papert's "Mindstorms" as a starting point (and much has been done since it was written ~30 years ago).
All children already learn quite a lot of fairly deep mathematical intuitions. We just take them for granted because everybody learns them.
For example: conservation of volume, the concept of "integer", order independence of cardinality, projecting orientation onto other reference frames, the equivalence between ordinal and cardinal numbers.
Everybody learns these things because they're embedded in our environments, and we can learn them playfully as children. When we create environments that embed even richer concepts, children learn those concepts just as easily. This is the explicit design goal of LOGO, and the whole family of descendants it has inspired.
Teaching in this way requires a degree of freedom and play that normal schools generally don't tolerate, which is why these proven, powerful tools still haven't taken over the world.
I took BC calculus as a junior, so we got left with about a month between the seniors' leaving and the end of the year. My teacher said, "Okay, we're gonna learn number theory." Oh fuck, that was hard. It was completely unlike anything that we'd done before, and it required the development of completely different skills. I wasn't bad at it, but I definitely wasn't good at it either.
I'm glad that I got a taste of it, as that thought process has helped me in countless situations, but you're definitely correct in that it doesn't fit with the rest of the traditional curriculum.
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I'd agree with you, if what we taught was an understanding of the concepts and established mechanisms. However, it seems to me that, most of what I saw in schools was just symbol manipulation.
For example, people didn't actually seem to understand that to get the area of a circle you took the radius, multiplied it by the ratio of the diameter to the circumference and squared it. They understood that you took the radius, multiplied it by a magic number, and for unknown reasons squared that.
The mapping of the symbols onto reality was often missing. It wasn't problem solving beyond the level of having a lookup table in your head that said 'When calculating an area do this, then this, then this.'
All that said, there are things it makes sense to memorise after you understand them - low level components where the speed gained in doing so allows you to use them in higher level abstractions. My point isn't that it doesn't make sense to teach people tools. But that to just give them the tools without the understanding of how they function seems harmful to their ability to create and adapt their own tools down the line.
However, I don't agree that art is unconcerned with reality, nor that maths is. We learn to draw by looking at things in the world, we get our rules about anatomy and so on from there; we learn maths based - at least initially - on physical examples; we tell stories based upon common themes and situations. The basic rules of these things are drawn from their correspondence to reality; with what people have experience with; and form the meaningful grammar of the system. Art is always a language, and a language is always representative. Even when - as arts - we may bend a few rules and simplify certain aspects to lend emphasis or make it better suited to approaching a particular problem.
To hold otherwise seems to me to deprive them of a foundation to communicate any common meaning. It would be no better to learn maths, under that definition, than to spend your life insanely scraping crayons across a page. (Indeed, given that the symbols would also be arbitrary in their meaning, it would be hard to tell the difference.)
Is maths an art, is it a science?
It's a language of enquiry. To claim that it proceeds solely by a 1-1 correspondence to reality is to picture mathematicians off somewhere counting things and deriving pi from taking increasingly precise measurements of actual circles. (And, really, even physics proceeds with the help of a healthy dollop of imagination that forms the foundation of hypothesis.) To claim that it has no correspondence to reality is to reduce the whole exercise to nonsense.
I do not see how either position taken as an absolute is tenable.
If you're believing this, you've probably never done physics.
Is this so, or is it true that we merely do not allow any other path?
At least in the United States, very few children attend anything other than public schools, and these public schools have a strict curriculum that introduces few concepts but in a rigid order, interspersed with months of monotonous busy-work that comprises little more than arithmetic and solving equations.
As a whole, society has never ran the experiment to see if it is a necessary step, so claiming that it is necessary is silly. It is perhaps true that if we did run the experiment, it would fail repeatedly, but only then could it be claimed to be necessary.
Skill based education is better than not learning anything. No one argues this, it's a strawman. The point is, learning a method to do something without any kind of explanation or example of why you ever would do it is woefully suboptimal. In an attempt to come up with the most absurd example of this - it can feel like learning to scuba dive in a world with no water deeper than 8 feet, especially given that most teachers I had actually could not give me examples of how to apply the "math" I was learning (and I was a very immature handful back then, very unshy about demanding an explanation of why I was wasting my time learning something that I couldn't see a use for). The author isn't arguing for eliminating all the "skills-based" education he references, he's arguing that we move it closer to how music is taught - teach all the skills, but then immediately apply them to something the students can relate to, at that age (not in 7 years when they are working and run the risk of being like me, and not recognizing the importance of these concepts until then). Math is everywhere. Most people learn basic computation fairly easily, since it can be taught in the context of going to the grocery store or splitting a check. Get beyond this, and all of a sudden, schools quit even attempting to anchor math education in reality (by this I mean a situation that could conceivably occur in reality, not simply taking a number problem and putting words to it).
The argument in favor of mandatory musical education would actually probably benefit from stealing a bit of this piece: music is one of the best forms of education in terms of immediate application of the concepts and methods you are taught, perhaps only behind physical education.
1) You need to be able to do basic calculations before you can do advanced proofs. I taught a lot of high school seniors, and I had a ton of students who were smart enough to handle abstract concepts, but couldn't follow along when I showed them cool proofs because they got caught up on the basic calculations (because they hadn't learned them well in middle/high school).
2) Good high school teachers DO do a lot of pattern recognition/abstract reasoning. That's the entire idea behind a discovery lesson and constructivist teaching - having students learn formulas by discovering patterns and reasoning about them.
3) Again, as he points out, American high schools do do proofs in Geometry. He thinks they're really pedantic, but there are good reasons why 2-column proofs are so tedious. For one, students seeing proofs for the first time freak out, so giving them structure helps. For another, if the students write out every single step, it's easier to identify who really knows his/her stuff and who's BSing.
It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?
Could someone have told us that in high school?
Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant.
Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in.
* - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.
http://en.wikipedia.org/wiki/Irreducible_polynomial#Real_and...
Well yeah, it would be really inefficient to derive the formula and then plug numbers into it, but you don't have to do that, you can just teach people to complete the square instead. It's pretty much as fast as using the formula (since it's the same operations), but it builds on your existing equation manipulation skills and so you understand every single step. And if you ever need to actually know the equation for whatever reason, you can derive it easily.
I'd rather spend our children's time letting them know that these formulas exist, making sure they can perform them as needed, and then have them move onto something new. No need to drill / memorize trivial (as opposed to fundamental and foundation) stuff that is 2 seconds a way with google.
The worst kind of math education was that endless dribble associated with memorizing useless patterns that could easily be looked up in a book. At one point I had (painfully, oh my god, so painfully) memorized about 20 different patterns so I could chunk up integrals into products of u(x)v(x)only to spit them back on exam day, and never again look back on any kind of calculus. That was not a pleasant day week in high school. Would have been much better spent learning Geometry, Prob/Stats, Discrete Math, Linear Algebra, or any other host of mathematically oriented topics.
Personally, it think the time for manipulation of symbols has passed. Math should be only about concepts and hardware or software should do he manipulation. But then you run into systemic issues like "how do I test all students". The factory-styled learning environment needs to die.
I'm a computer scientist, one of the more mathematical occupations.
I've never been in a life-or-death situation where I thought, "aha, thank God I memorized that formula in high school!". Also, I don't actually remember it anymore, other than "negative b plus or minus the square root of b squared minus four-a-c, all over two-a".
Well huh. I guess I do remember it, but apparently at a verbal-auditory level rather than visual-symbolic. As usual for me.
Still: I've never had occasion to use that recitation I just made.
2. Have you ever been in a life-or-death situation where anything you learned in class in high school was useful to you? "Well, I would have died, but then I remembered that Julius Caesar's last three words were not actually 'Et tu, Brute!' according to Shakespeare..."
There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle."
Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.
Recipe's are a very helpful fallback where you might be struggling with understanding the origins of material in a rigorous manner. You can inspect a triangle all you like, but at the end of the day it's much easier to simply remember the formula.
This, I am finding, is the only way I'm managing to actually understand complex analysis - take the formulas for the results, and remember how to apply them. It's revealing to me that what looks complex gets very simple in that manner (and also that I still get tripped up by elements of basic integration).
If I couldn't do this, then I'd be lost - and in a test simplification remembering how to work through the definition isn't possible (and isn't required thank god) because that's a path which leads to me spending 6 hours figuring out and trying to picture something in a way which makes sense.
I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples:
[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/
The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).
I read/write slow and I have an incredibly hard time memorizing anything so I rebelled. Even after I got my act together, got my GED, and went to college I suffered through the various levels of Calculus because it was the same tune all over again. Classes like Linear Algebra were a lot harder for me to "see" but it was still faster & easier for me to take the time to visualize it.
When I went over it with him, I showed him several spots where he made careless mistakes - he added where he should have subtracted, he multiplied where he should have divided, he screwed up a decimal point, whatever. I told him, "It's easy to spot your errors now because these are easy problems. But when you get to harder math, it's going to be much harder to find out what you did wrong, and a teacher isn't going know whether you made a careless mistake or just don't know it at all. By showing your work, you show the teacher that you actually know it."
Nowadays he understands the reason why he needs to show his work but still hates it. I'm hoping that he'll be like this only while the math is easy.
If you could be given a magical amulet that let you teach students better, more quickly, and more permanently than ever before but at the expense of never being able to test them to see exactly what it was that they learned, or you could be given a magical apparatus that let you test them perfectly so that you knew exactly what it was that they learned and did not learn but gave no insights or help into how to teach them those things that they failed at, which would you choose?
Which would your local school administrator choose? Which would your children's teachers choose? Which would the legislator writing education policy choose?
Everything else is post hoc rationalization. Having decided what it is that we want public education to be, we need to have some sort of justification for it even if it doesn't make sense.
Do you know what people who don't show work do when they move on to more difficult problems? They start scribbling it out on paper, without any prompting. The more difficult problems are interesting enough that they want to get them right, and when they notice that basic mistakes are interfering they strive to avoid those.
Or, in some cases, they just don't bother. When you solve the Poincaire Conjecture (spelling? didn't want to cheat and look it up) no one gives a crap whether or not you "showed your work" because most of the other mathematicians can also "just see" the boring details, and are interested primarily in the truly insightful portion of the proof.
I suspect that we're actually selecting for accountants and not math geniuses when we harp on "showing your work". How many Perelmans did we discourage and how many math stooges were praised last year in public schools?
Meanwhile, if he shows that he's factoring the polynomial and writes a 5 where he should have written a 3, I can immediately tell that he knows what's going on but made a careless mistake. Alternatively, if he writes down a bunch of gibberish, it means that he doesn't know what's going on and needs someone to go over the concepts again.
It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions, or do you want a compiler that is slower but gives you detailed warnings and error messages? I'd rather take the latter. Maybe once I'm perfectly sure that my code works, I'll do it with the former.
The correct (though inefficient) approach is to keep trying until you see that he starts understanding. However, this is impossible when there are 25 other students in the classroom. Each might require a different manner of teaching to "get it", or learn at different speeds. And so if you're trying to crank out graduates on an assembly line this just won't cut it.
So instead of figuring out a solution where each student can get the education he deserves as a human being, we instead seek to change the student so that he can be programmed with the education that is possible in an assembly line system. This also explains the dearth of highly competent, highly respected teachers... you don't staff your factory with gifted artisans who could carve the pieces. You want someone who will push the button and have the product stamped out in 0.75 seconds.
If you calibrate everything perfectly, some number of students will get a highly optimal (for them) education where everything was timed perfectly, using the easiest-to-understand lessons. For everyone else, for the slow and learning disabled, for the quick and talented... it will be an awful experience. And, whether you call it luck or circumstance, neither of those groups will be educated well enough to be able to express their criticism easily.
> It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions,
But a compiler isn't a person, and a person isn't a compiler. I don't want to treat people as if they were machines... I especially don't want to treat children like they are machines, it's almost certainly even more damaging the earlier that happens to them.
I'm a programmer too, I do this for a living. I know all too well how easy it is to think of human circumstances and other people as if they were machines to be debugged, and it feels awful. Imagine what the 7 year old kid feels like in school when he's a bug to be solved on the teacher's trouble ticket system. Especially when he's probably marked "low priority, fix when time allows".
You're no longer talking about a system where learning is considered the primary goal. It may not even be a goal at all.
This is a really good point - the public education system isn't an artisanal workshop; it's a large, industrialized factory where "raw materials" are turned into "product." Every grade is another step in the factory process. And while I guess it might be optimal given the very limited resources that we devote to education, it's heartless and doesn't work very well from an absolute standpoint.
Personally, I didn't get a lot of my education from school. Sure, I was there ten hours a day, but I mostly learned from my father and the homework that I did. I would get assignments, and my father was the one who really taught me whenever I ran into problems. I would then go back to school and pass tests.
Unfortunately, my situation was atypical and very lucky; I was blessed with a loving father who was fascinated with a large variety of topics and loved teaching. Most kids don't get a resource like that and get stuck with school as being the only avenue for learning. How can you reach them? I think the only answer is more money, which will go toward more teachers. Cut down the class size to ten kids per class, and you'll get a much more individualized curriculum. As long as you have 25 kids in the classroom, you're going to end up with the factory approach.
The point of showing the work is to learn the mechanics of solving the problem. If you do not learn the mechanics of solving for simple problems, you will not be able to apply the mechanics to more complex problems where one can no longer intuitively see the answer.
I totally get it and it makes total sense except for when it doesn't. The problem with our education system is that it's fundamentally flawed. It is designed to work best with the typical student being taught from a skill set chosen for optimal widget-making. People learn/think differently and yet we cater more and more to the mantra of pump-and-dump where children with the ability to retain the most frivolous information wins.
It took until my junior year as a comp-sci student for me to figure out what I personally needed to learn the material. I absolutely had to understand the big picture before I could ever attempt to solve the problems. If I didn't and I relied entirely on memorization then I was destined to fail.
In order for me to understand the big picture, I had to ask questions, sometimes a lot of them. Abstract questions would annoy some professors and certainly other students. Engineering classes, like most college classes, are full of people brought up in a system where the slide-show-after-slide-show of formulas, facts, or bullet points was all they needed. My questions were irrelevant and an interruption to their note taking.
Mathematics are the same way. Yes, you need to solve problems, but you also need to solve problems in ways that can build on your past knowledge and be shared with others.
(FYI, I'm a lifelong amateur musician, programmer, and data analyst. My formal education consisted of a double electrical engineering / mathematics major.)
I didn't pursue a career in mathematics, like a lot of my classmates, but these lessons gave me more anything else I did in all years spend on 'education'.
It describes a 4-year program, from 8 to 11 grade of russian school, for kids from 13-14 to 16-17 years old respectively.
I understand the larger point about the difference between high-school math and high-level mathematics, but come on, don't be so pedantic! Everyone calls whatever it is you study in High-School & Elementary school - Math.
On a related matter, physicists and other scientists have done a pretty good job of communicating to the general public what it is they do. On the other hand, very few people actually know what professional Mathematicians actually do - something I realized when I struggled to explain it to my dad the other day.
Isn't naive pattern recognition the basis of deeper dimensional understanding (ya know, the 'theory')? Isn't this how intelligence is built?
It seems pretty easy to make rag on the lack of 'true understanding', when you've spent 25+ years recognizing the patterns.
Also some of the crazy stuff we did for physics/chemistry required a lot of math.
I say this because in my experience, the first symptom of ignorance is a feeling you know a lot.
I'm depressed since childhood because since a very young age, I had read about great minds. How could I ever feel I'm "good at maths" after reading about Gauss, or Galois?
It made me feel like the lowest form of life.
That is akin to the way Military Generals feel towards Alexander the Great: You can be a great General, but you probably will never be Alexander the Great.
Maybe this should be done freshman year: Before even a single "maths" course is dispensed, a session on the achievements of Gauss and Galois, at age 17 or 19.
Maybe a brief discussion on who Lagrange was, and what he did in his teens.
This should take out any feeling of being "good at maths", and make students shut their mouth and open their ears.
But why single out math in particular? I look back on pretty much all of my K-12 education as fairly trite and superficial, in terms of "doing real work in the subject". My experience with, say, college-level history was much more intense (and seemingly more true to the field of history) than anything in high school. On the other hand, I'm not sure I would be able to do well at "real math" like calculus or graph theory or whatever you wish to deem "real math" if I was struggling with adding numbers and solving equations, and I attribute getting past such struggles to doing bountiful rote exercises...