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.

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.

3. Mathematics includes geometry and arithmetic.

Arithmetic:Mathematics::Programming:Computer Science.

It is interesting that the author had to portray a hypothetical scenario of learning music theory and never practising to get his point (which I more or less agree with) across.

Sometimes, it is necessary to create unrealistic scenarios and problems because real world problems may be too complicated for beginners. So you end up oversimplifying. In this process, sometimes the essence of the material is lost.

A solution is to pay equal attention to the problem sets and examples as the rest of the course content.

Read "Lockhart's Lament." It's written much better than this rip-off: http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf

His point is that you're not doing "math" in Algebra class because "math" (as practiced by mathematicians) is actually a highly creative process, and there's nothing creative about, e.g., word problems.

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.

Hmm. I'm not sure I agree.

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.

>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.

99% of kids who were "always good at math" will continue to be good at math in college. So the entire article is a rant against a straw man to make a case for his beliefs on how math should be taught. Not that I disagree that math education is stupid, but just saying this rant is has no foundation.

The 1% of kids who did well in high school and then fail in college because they are so attached to their rote memorization of techniques have a profoundly broken approach to problem solving that is bigger than the education they received. I've tutored many kids exactly like that and it is very hard to pry them free of that mentality. It is part of their personality. Also, those kids were never really good at math in high school either and were battling (using tutors for help frequently) uphill to get through their entire primary curriculum.

The much bigger and real tragedy of math education in the US is the very large percentage of kids who have been labeled as "not good at math". Those kids 99% of the time are actually plenty good at math but have fallen out of the system because of frustration and a poor fit for their learning style. Those kids don't end up in universities trying to take calc for science majors at all because they believe they aren't capable and that is a crime.

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.

[1] http://www.epsiloncamp.org/ProblemsversusExercises.php

I've already read that somewhere...

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.

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 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.

Past discussion with lots of comments, circa 12 February 2014:

https://news.ycombinator.com/item?id=7221713

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 was in Math high school and in Math class. We had 12 of mixed math per week: algebra, geometry, linear algebra, probability and what not. Too bad in college we couldn't pick classes like in USA, so we had to do some of them over again in 1st and 2nd semester. Our math teacher was a very old, smart and animated Jewish guy. He received money from Soros fund for being awesome, and also survived Nazi blockade of Leningrad. In his 80s he still went there EVERY year to walk across frozen lake in memory of that event. For 4 years he taught us he always asked after explaining the task: so, Joe, what do you think we should do here? And we can go on and on about how we should find a common denominator or perhaps build an additional triangle and... he just smile and say "we need to think". And then he will tell us step by step his thought process. By the end of the 12th grade we all knew answer to that question.

In my years of mentoring and tutoring, I've run into a number of students of programming (and, indeed, teachers) with the same problem.

Programming is about problem solving first, and teaching programming is about teaching someone to look at a problem analytically, and how to use an abstract flow of logic to solve a problem, and how to diagnose issues with your own logic when things go wrong. It's about critical thinking. It's about attention to detail.

It's about all of these things first, and about slapping keys second. But too many people see learning programming as learning the act of typing code, and focus too much on rote memorization of syntax, or teaching tools instead of thinking. Someone should come out of a course saying that they learned how to think in this new way as programmer, not that they learned a new programming language.

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'.

Other things you probably never did in High School: Science

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.)

Some students are taught this way:

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

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.

I'd use different terminology.

Arithmetic, two column geometry proofs, and even transposing a sheet of music to another key, are all math.

I think the author is talking about education, not really math. Jeremy wants kids learning math to think about, be interested in, and search for meaning in math. He's right that teaching rote mechanics doesn't lead to curiosity for most people. But there are no high school classes that reliably lead to curiosity.

> You never did math in high school

Yes I did. Your claim is BS.

Your claim is based on no knowledge of me and what math I learned in high school and, thus, is incompetent.

Your claim is an insult to me and the math I did learn in high school.

In the article, you imply that a student's claim "I was always good at math" is poorly founded. But for some students, that student claim is correct. Your implication is based on no knowledge of me and is incompetent. Moreover, for me you claim is flatly wrong -- In high school I always was good at math. E.g., my plane geometry teacher was severe in the extreme, likely the most competent in the city, and I commonly toasted her. Your implication that the student's claim is poorly founded is an insult to my abilities at math.

The claim in your title is guaranteed to be wrong for some thousands or tens of thousands or more good US high school math students present and past.

From research in applied math, I hold a Ph.D. degree from one of the world's best research universities and have published peer-reviewed original research in applied math and, thus, know what the heck I'm talking about.

You owe many thousands of good math students a profound apology.

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.

This is an important topic, and I'm glad that it gets a lot of attention on HN. I hope this spreads.

I've participated in a few of these discussions so far (apologies if you're tired of hearing me repeat myself), and I truly believe that almost all these issues are downstream of a single and very fundamental problem: math teachers are rarely drawn from top math students.

Here in the US, we love a plan, and we have a potentially harmful concept of a career ladder. A classroom teacher is on the bottom run, and it's considered career progress to set the curriculum for all teachers.

My take on it is this - if the US drew it's math teachers from the top 10% of math graduates, the "plan" wouldn't be nearly as important. Yes, it's a good idea to have a general standard for where students should be, it's a good idea to have some kind of training for math teachers, and it's a good idea to check in every now and then. But think about the relative importance of "a great curriculum" vs "teachers drawn form the top 10% of math graduates".

We could change up the curriculum to reflect the problems Mr Kun has identified, but without an armada of top math teachers, it would make absolutely no difference. If we drew our math teachers from the top ranks of math students and allowed considerably autonomy (along with general guidelines), I suspect many of the improvements Mr Kun talks about (along with so many other complaints about math instruction) would happen on their own.

So... how do we get very strong math majors who are inclined to teach into the profession, and how do we keep them there? To me, this is the upstream bug fix that will be referenced (perhaps in one line) when all these other bugs are closed.

I started Electrical Engineering when I was 35 so I got to observe the high school to university math transition from a semi-unique position. Yes, as the author states, there were a lot of students with good high school math marks that didn't really know any of the sort of math taught in university. But that math turned out to be continuous functions (linear mostly), limits, and a whole lot of memorized integrals and derivatives of those continuous functions. This was in an era where people were carrying around lap tops.

The one token "numerical methods" class was all about solving ... wait for it ... continuous functions.

The problem is that to fix this weird situation we have to start teaching iterative methods of solving problems at all levels of the educational system. I have the distinct impression that people at those various levels tend to assign blame to those at other levels (this article could be an example of that).

So does it really matter that high school students don't learn the wrong math? There is a much bigger question here.

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...

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.

> 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.

---

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.

> 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!").

EDIT: Grammar.

You build software w/o using Mathematics?

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.

> 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.

  > 99% of kids who were "always good at math"
  > will continue to be good at math in college.

As someone who works as a part-time teaching fellow at a UK University, this does not match my experience. I find that many who do well at school do not, in truth, understand what they're doing. Some do, but many have become adept at magically divining the right process to follow, and then following it without mistakes.

So your claim that this is "a rant against a straw man" does not match my experience. As a result, I'd be interested to know what leads you to make the claim I quote. Do you have figures, or personal experience? And are you talking about math as in Analysis, Number Theory, Logic, Topology, and similar, or are you talking about "Math Methods" such as are required in subjects such as Physics and Engineering?

For reference, this:

  > The much bigger and real tragedy of math education
  > in the US is the very large percentage of kids who
  > have been labeled as "not good at math". Those kids
  > 99% of the time are actually plenty good at math but
  > have fallen out of the system because of frustration
  > and a poor fit for their learning style.

That is absolutely true., and is why I give some 150 talks every year to school age kids, trying to encourage them to engage (continue, return, or start) with math.

A friend of mine who does math tutoring for high school seniors and college freshman told me that many of her students who did well on their AP Calculus exams did not actually understand what a derivative/integral was. They could procedurally find the derivative/integral for a given function, but they didn't know what it meant.

Admittedly, this is anecdotal, but it does seem to support the argument/rant in the article.

I feel that I, personally, was very lucky in my high school mathematical education in that my teachers exposed us to the concepts/meanings of all of the operations we were doing long before they exposed us to the procedural trivia of finding integrals/derivatives and the like. It is unfortunate that not all schools are like this.

I disagree. This attitude...and teaching style often extend well into college.

I literally had a professor who read from the book for lecture and then did his tests out of the book. Granted, it's probably one of the best physics text books ever written (David Griffiths) but it's very, very difficult to learn this way. Students resort to memorization when they have no other options. Nearly every student could solve problems within a range of 'like' problems (change a variable or two, a power, etc.) but had no actual ability to solve truly unique equations in the subject or even formulate their own questions.

This was a 3rd year physics course at a top-50 school.

FWIW, I'd probably put myself in the same category the OP writes about.

I hit a wall in calculus 2 in University, because previously I had been able to rely on simply showing up and not putting in much effort to still do "pretty good". That was the first time I did poorly in any math/science course, and was a pretty big wake-up call. As a result, I ended up having to study more to fill in the gaps.

I was naive, came from a poor background with crappy public schools, and didn't think of math as particularly important to focus on until I saw how useful it was in physics. I don't think it's an all too uncommon story.

There's not really a conflict here.

"Math" up through high school in the US, and through that freshman calculus course, is mostly about calculating the right answer to a problem. Math is a much bigger subject; calculations are a trivially small part of the whole.

On the other hand, "The much bigger and real tragedy of math education in the US is the very large percentage of kids who have been labeled as 'not good at math'" is also very true. And many of those people might not be good at calculations but would be fine with the rest of the subject.

I think many of you are misunderstanding my point. I'm not saying math education is good or that students are well prepared. I'm saying that most students who succeed in HS will continue to succeed in college in spite of everything else. I definitely agree most freshmen students don't understand calculus in any deep way but most good students will by the time the are done with their freshman series. So again, I'm not defending the system, I'm nitpicking the argument :)

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.

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.

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.

Are you me?

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.

> 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).

Mathematics is not a spectator sport. This is why lectures should be there to present, not to teach. You have to teach yourself.

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 my linear algebra course notes that said "most numbers are close enough to non-zero numbers that this method should work everywhere."

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.

I'd have to disagree at least in my case. We did a good amount of reasonable real science experiments in our normal classes.

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.

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.

I see what you mean. In my country (France) we have the same system; except for preparatory schools, which are rigth after high school.

And let me tell you those are two different worlds. My first year professor actually began the year by saying "What you have been doing so far is not math. I'll show you math". At the first test, he said: "I like you guys, you make a nice gaussian curve. Ahem, well, the curve is centered around 20%". The class was full of people who had been in the top 3 all years so far. It was not pretty. It was not surprising either, as his tests were designed so that when a student gets 90% or more, he makes it more difficult for the next year.

My second professor was a huge control freak. He had been in the same class as someone who is now a Fields medallist. He knew where you were in the understanding of his course. And he would be relentless in pushing you, in a gentle but unforgiving manner. Oh the self-guilt when you were not studying. I had nightmares/dreams about him or his tests up to three years later.

All in all, it was worth it. I met people I know I will respect and be respected by for the next 40 years.

Definitely what needs to be done is that teachers need to be given more respect for the work that they do. This includes not paying them below anything else a mathematician could make elsewhere, and in general making it a better environment (who cares if your math teacher is the best at math if s/he has a bunch of bullshit standards to get through). I'm a college student who would love to be a math/cs teacher but I don't see it as anything sustainable or worth it.

The article: http://www.theatlantic.com/education/archive/2014/03/5-year-...

HN Discussion: https://news.ycombinator.com/item?id=7333998