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.

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

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.

To continue the author's music analogy, what do you think music class would be like if it were taught by teachers who have never listened to an orchestra in their life, are barely aware of the existence of different music styles, and cannot play any instrument? That's right, it would consist of drawing treble clefs, memorizing note positions, and transposing music.

Many high school math teachers actually have no idea what math is about. It is illogical to expect them to teach what they don't know.

There is no incentive for people who would make great math teachers to go into teaching in America. No social recognition. Ridiculously low salaries. Internal pressure to conform and avoid making bad teachers look bad. Who in their right mind would willingly do that, when they could be doing mathematics?

The author has a strong point. There is a serious disconnect between "math" as it is taught up through high school and "math" as it is taught in college (and how it really is, as a subject).

"Math" in high school is about calculation. Math is about useful abstraction. Students are expected to jump that gap on their own, without any outside help. At the same time as being expected to learn some concepts that are actually fairly difficult in their own right.

God help any students that have a full math professor teaching freshman calculus---the lectures will be about proofs while the homework and tests will be about calculated answers.

So, who complies a list of subjects worth of being called Mathematics?

Is arithmetic not maths?

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.

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.

Right on! Believe it or not, I failed every single math class I ever took multiple times. Just so boring and pointless. It wasn't hard, just pointless. I would like to see the way it's taught updated. It shouldn't be about the semantics, it should be about critical thinking. That's interesting.

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.

Must admit. I hated math because it always felt impractical. I think I was misguided and really what I needed was a better teacher /well rounded understanding of math /problem solving

Every kid should do calculus in high school. period

If not going into the nitty gritty detail, but an overview of how derivatives/integrals are so applicable in the real world.

Okay. I only have a high school education. Where do I go from here? (not being snarky - honest question. Khan academy?!)

"You never did math in high school"

Yes I did. The claim is false.

"I was always good at math."

From the ninth grade on, yes. The main means of measurement were standardized tests of math ability and/or knowledge.

I'll compare 'math' aptitude, knowledge, and accomplishments with you any time, any day, for money, marbles, or chalk. I'll give you a head start and big odds, and I will totally blow you away.

F'get about my opinion. Instead, (1) in the ninth grade I was sent to a math tournament, (2) twice I was sent to NSF summer programs in math, (3) I was a math major in college and got 'Honors in Math' with a paper on group representation theory, (4) my MATH SAT score was over 750 both times (strong evidence of being "good at math"), (5) my CEEB math score was over 650, (6) I never took freshman calculus, taught it to myself, alone, started in college with sophomore calculus and made As, (7) got 800 on my Math GRE knowledge test (means I knew some math), (8) used the differential equation

y' = k y (b - y)

to save FedEx (a viral growth equation for revenue projections that pleased the Board and saved the company), i.e., an original application of math, (9) used the statistics of power spectral estimation of stochastic processes to 'educate' some customers and win a competitive software development contract, (10) did some original work in stochastic processes to answer a question for the US Navy on global nuclear war limited to sea, (11) studied solid geometry in high school and later used it, the law of cosines for spherical triangles, to find great circle distances in a program, I designed and wrote, to schedule the fleet at FedEx, a program that pleased the Board, enabled funding, and saved the company, (12) my Ph.D. research was in stochastic optimal control, complete with measurable selection, that is, 'math'.

The claim is false, badly false.

Finally, as you hint, we will end with original work done, and I will pull out two of my peer-reviewed published papers and my Ph.D. dissertation, and with what you wrote you won't have the prerequisites to read any of them. Then, you lose the bet.

I feel sorry for your students. Go back to teaching the quadratic equation and binomial coefficients and f'get about your broad views of 'math'.

I get what the author is frustrated with, although I don't necessarily agree with his conclusion that you aren't doing math - you are doing math, but you are not learning to use math the way it should be used (in his music analogy - you are learning music, but you are not learning to use music the way it should be used, which requires actually playing). It's a common, and valid complaint about American education - we are very good at stripping out all practical applications of a subject and simply teaching a method for solving a problem that is completely unrelated to anything in reality. In one of Richard Feynman's books, he writes about his experience editing textbooks. He notices this, and in his typically fantastic way, rips the textbooks to shreds. One of his examples went something like this:

Some problem, while accurate in that the process to get to the answer worked, was a word problem. Part of this problem read "The Earth has two suns. One is blue, the other is green." He stops this early in the problem, since even here, they have made the problem about something that is completely unrelated to reality. Think back to high school - you have not lived long enough (in most cases) to be able to mentally reapply a process to a problem you have had in your own life, so education should be going far out of its way to present problems in a way that the kids encountering them can understand. The Wire had a great example - using gambling to teach probability (there's a moral argument there that I won't touch - the point is the kids could understand why they were learning math, since it solved a problem or gave them an advantage that they could immediately relate to). Programming could help with this (I haven't taken a math class since my senior year of high school, and always did well in all of them, but I feel like I understand why I would use algebra better now than ever, since I can actually relate to the concept of variables in reality, rather than "make the numbers into letters"). Baseball has taught me more about statistics and data analysis than any class I ever took.

I remember learning about finding the slope of an equation in 7th grade. I had a huge argument with my teacher (I was a bit of a handful back then), because she actually could not give me a single example of why I would ever need to care about the slope of a line beyond future math classes. That's a problem.

TL;DR - The author is likely more frustrated with American education's tendency to remove all relatability from a subject (and then not arming teacher's with good examples of how to reapply the methods they learned to a problem that encourages more investigation) than he is strictly accurate about students not actually doing math. And I agree.

Does anyone else think this is a profoundly ignorant and destructive article whose anodyne plausibility underscores the absence of a consistent model of education in the cultural lexicon?

I'm not trying to be impolite here; I just want to know how far my views are from the mainstream.

Is this really an important question? We still suck at teaching whatever it is, and that's a problem. What you call it is irrelevant.