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The typical math education relies a lot on rote learning

This has been known and lamented for a century, and it long ago transitioned from being a radical idea to being conventional wisdom. It has been orthodoxy for decades (although recently there has been some reaction against the orthodoxy by a few experimental charter schools.) So, if current math education involves a lot of rote learning, it's not for lack of awareness or lack of attempts to fix the problem. In rebellion against facts and rote learning, teachers have elevated understanding above problem-solving as the highest goal of mathematics education, but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.

And yes, they do want kids to get to the point where they can "forget" that sin^2(x) + cos^2(x) = 1 and regenerate that knowledge from visualizing the unit circle and applying the Pythagorean Theorem. That ideal has also been orthodoxy for decades. It just turns out that in practice, deeper understanding emerges from practice, and practice requires competence. Rote learning is a way of bootstrapping competence so competence can be turned into understanding. Believe me, progressive, "kinder and gentler" teachers were the norm in all the schools I've gone to, and they all tried every trick they knew to help us skip straight from ignorance to deep understanding with as little dry, repetitive work as possible, but they could not get around the need to learn many facts and techniques by heart.

By the way, the route through calculus you describe reflects that you are basically the kind of kid that the schools don't worry about. I was the same kind of student. It might seem strange to us, but to most students, the history of a subject and the problems that originally motivated it are the very definition of dry and disconnected. Nobody gives a damn about the awesome intellectual journeys of some dead nerds except kids like you and me who will learn the subject regardless of how it's taught. Differentiation is introduced first precisely because it makes the subject less dry to most students, because it allows them to immediately apply what they learn to simple concrete problems involving rates.

I agree mostly with your comment (check out my sibling one).

>but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.

Might I suggest that they take the top-down approach instead of the bottom-up one? If you have a clearly defined real world problem (no matter how simplistic), it's easy to see what information is missing and needed, and so it's then clear why a certain proof or relation is required. Then it's more reasonable to delve into the theoretical and abstract, when it is required by the real world.

Then you can make the real world problems harder to delve deeper into the theoreticals. You're more grounded in this way.

When you take the opposite approach and teach sin^2(x) + cos^2(x) = 1 starting from lines, angles, triangle relations, even your best students say "so fucking what?". In fact, I'm fairly certain that the teachers I had who could keep the whole class' attention and who could teach even the weakest students consistently displayed the top-down approach.

Edit: As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class. You could move a crank to lower or heighten the part the rubber was attached to, effectively changing the angle and thus the shooting distance. My dad was showing me how to calculate the angle of the elastic to the ground by knowing the length of the two sides. I instantly "got" why we needed trigonometry and could visualise how changes in the lengths of sides would affect the angles. It seems intuitive now but it wasn't at the time. Of course, I wouldn't solve problems until much later but I feel like a lot of my mates never made that same leap.

Finding some connection between trigonometry and angles of rubber-powered trebuchets is incidental and quite frankly, this approach isn't going to take you very far. One simply can't find real world mappings to atleast 99% of what is properly called math. Even if you'd like to strictly operate within the confines of high school geometry , it is a study of points, lines, planes, angle relations, norms & distances motivated by axioms & theorems that follow by symbol pushing. For example, it so happens that if you push the right symbols, you'll eventually work out that the area of a triangle is related to its sides ( http://en.wikipedia.org/wiki/Herons_formula ). But this is a consequence of your definition of area and triangle and side and so forth. The relationship falls out of those definitions. Its not about constructing a real wooden triangle out of real sticks & then measuring its area and then the lengths of its sticks and saying - hey, I told you so! Those things ie. the real wood and the real sticks,...reality - is just a sideshow. Real world problems are at best a distraction. If you were to focus only on these sort of real-world mappings, you'd end up as an accountant - because that's were 1 would map to $1, 2to $2 and add would mean 1 + 2 = $3 and so forth. That's what corresponds to reality and the real world. That isn't math. Math is where you say 1 + 2 = 0 because the abelian group G3 with three elements {0,1,2} can satisfy the addition rule and have three distinct elements only if 1 + 2 was 0. For if 1 + 2 was 1, then that would mean 2 was equal to zero by cancelation laws. Similarly if 1+2 was 2 then 1 would equal 0. Since you do have 3 elements, it stands to reason that 1+2 must be 0, because it can't be anything else in that system. That's math.

You sound like someone from the Konishi polity as described in Diaspora. I don't agree with you. The brain uses an associative learning algorithm and learning difficulty is proportional to how easily a concept can be integrated with what is already known. And it's an iterative process, so new knowledge can be built on old knowledge that was once itself abstract.

So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.

>you can start with concrete problems and generalize from there

You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.

> I could compute the determinant of square matrices,

I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked.

It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.

I heartily disagree with your initial point that you can only get so far with real world examples. Heron's formula itself is not even unintuitive, and in fact, the way he proved it, requires a minimal amount of symbol pushing. Maybe my opinion is unpopular, but I believe that anything that is not applicable to the real world and does not form the immediate base of anything that is applicable to the real world has no place in highschool education.

Those of us who do math for math's sake are mathematicians. For the rest of us, math is a tool, no better or worse than any other tools we have in problem solving. There is nothing wrong with benefiting solely from the end conclusions of math, without engaging in it unless we have to. For the majority of people, this suffices and anything more has no benefit, and is promptly forgotten.

So why, then, do we insist that a high school program, tailored to the majority of people, teach people to be proficient at theorizing rather than applying? The majority are accountants ... and doctors, and lawyers, and delivery people, and repairmen, and so on. People can learn about abelian groups in the future, when they know what they want to study. Let's be realistic here. Right now, we're dealing with people not being able to do middle school material 2 years out of college. We're trying to prepare people for what they're in for. For the sake of the public good, abelian groups and math as a whole can wait.

>For the sake of the public good, abelian groups can wait.

Amazing. I can't wait to put that on a plaque. "Abelian groups can wait for the sake of public good." Holy cow! Dude, Niels Abel was 17 when he invented most of the machinery that goes by the name Group Theory. By the age of 19, he had proved quintics don't have a general solution by simply cranking up his machinery. And today we have 17 and 19 year olds who not only don't know what a group is, but don't particularly want to know. Because, like you say, they'd rather be real world accountants & repairmen, not ivory tower mathematicians. Wouldn't you rather have just 1 Abel and a whole generation of pissed off accountants than the other way around ? I would. You know, when we do eventually get the hell out of this planet and conquer other dimensions and populate new worlds, get beyond this ethereal realm so to speak, it would be purely due to the ideas of an Abel or a Gauss or a Riemann. Even an army of real-life accountants wouldn't get you off this planet - they'd be busy calculating the price of the spacecraft with their fancy spreadsheets.

I guess we have different views of what's important. There are maybe 2-3 Abel's per century. Gauss completed some of his greatest work in his early twenties, but that doesn't mean that everyone else did as well as he did. I'd love for everyone to be Gausses and Abels, but the truth is, they aren't. Neither do they want to be. Neither are they unhappy "pissed off accountants" because they aren't, like you describe. Neither are they idiots because they aren't math-savvy, like you imply.

I'd rather human development and quality of life come before space travel. If there are junior Abels among us, they are going to stand out regardless of whether you teach them subject X in highschool or not. Furthermore, they are going to pursue a specialized education (college) in said subject anyway. I don't think not recognizing geniuses is a problem we have since geniuses tend to be fairly resourceful. Thus, we can focus on improving the standard of education for everyone else. I'm not suggesting a ban on the study of mathematics altogether.

There is no causation link as you imagine. No amount of increased math in highschool is going to breed geniuses like Abels. Increased general skills like logic and problem solving is going to result in higher ability and better education for almost everyone.

As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class....

I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math?

The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class.

When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig."

I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)

> I could appreciate it a lot more when I understood what kind of problems were Newton and Leibniz trying to solve when they came up with the formalized notion of Calculus.

This always bothered me in many of my math classes. We'd go over a new formula or concept and just hammer it home until it stuck. Professors rarely gave real-world examples of the topics they were teaching or indicated to the students how these new theorem interacted with others. If I asked them explicitly I might get a more practical explanation, but it was rarely offered to me without inquiring first.

I used to have a lower opinion of people who'd complain about learning math because "it's so pointless, I'll never use any of this." It bothered me that they just didn't want to learn something new. In hindsight, even if that were partly true, I can't blame them for having that attitude because it may have been partially instilled in them by their professors.

I'll never forget the feelings I had after I connected the dots in my head and noticed the relationship between integrals. I knew that integration would give me the area underneath a curve, but now we were learning about double and triple integrals for surface area and volume. It dawned on me that I'm basically doing the same thing I was before, albeit in several planes with more curves. I remember a lot of different emotions: appreciation for the sensibility and beauty of mathematics; pride that I figured something out on my own; but also a tinge of frustration that this revelation was never encouraged by my professor.

Exactly! You were fortunate enough to realize this during your course itself. But a vast majority of students never stumble upon this revealation and treat Maths as another chore that needs to be done with.

The insight you described is what Math classes should be about, not practicing problem after problem without any knowing what or why.

Calculus was described by them using the concept of 'infinitesimals', not through Limits. Limits was a clever abstraction that was evolved later to better explain Calculus and keep it consistent.

It also isn't even necessary to use limits. Deriving calculus using the hyperreal number system (which has both infinites and infinitesimals) has been proved to be equivalent to calculus with limits (i.e., any theorem true in one is true in the other).

So if we want to, we could go back to using infinitesimals.

http://en.wikipedia.org/wiki/Hyperreal_number

Thanks for the link.

I wouldn't want to rigorously teach Calculus using *R unless the relevant proofs are reasonably accessible (which, given that we're talking Number Theory, seems like a remote possibility), but it's wonderful how much simpler they make the arguments if you work with them.

The relevant proofs are easily accessible, probably more so than the limit based proofs of regular calculus.

The only thing that isn't easily accessible is the proof of transference, namely that what's true for star-R (don't know how to make asterisks format nicely) is also true for R.

I mean the equivalence / transference proofs. Otherwise you're pulling a theorem out of the air and telling students to just accept it. And by "accessible" I mean easy enough to understand that they don't become a huge distraction (or worse).

If one starts with star-R to begin with, then why does one need the transference theorems?

You're then discussing math in terms of a set whose properties don't obviously match anything the student has dealt with. It needs to be established, one way or another, that adding infinitesimals etc. is not introducing new behavior.

I don't see why that can't just be stated. Students need somewhere to start from, and there's plenty already that we say "trust us for now, we'll prove it later" - and it's not like the reals actually match anything the student has dealt with, at the corner cases, either...

No, it means you are starting with a different (but equivalent) set of axioms.

Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".

You're working with strictly _more_ axioms, and the extra axioms don't seem justified unless you use the equivalence proof to show you haven't added undesired behavior.

(If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)

This is a good book. I found it preferable to regular epsilon delta when I was in highschool.

http://www.math.wisc.edu/~keisler/calc.html

After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.

I came across Silvanus Thompson's 1910 reprinted textbook Calculus Made Easy [1], and it was hands down the best primer on any topic I've delved into.

1. http://www.amazon.co.uk/Calculus-Made-Easy-Very-Simplest-Int...

Available on Project Gutenberg here: http://www.gutenberg.org/files/33283/33283-pdf.pdf

I don't have a link handy, but there is a very clear scanned pdf of the second edition of Calculus Made Easy from 1914 available on archive.org

I've Silvanus sitting in my shelf, but am yet to look into it yet.

A couple of recommendations (not specific to just Calculus):

- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)

- Calculus (Apostle http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...).

- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.

- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html

- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)

- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.

- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/

I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.

If you've only read the first few chapters of Godel Escher Bach, you should really set a goal to continue reading. The book is filled with so much good information presented in a digestible format. Topics are slowly revealed throughout the book until you just get it. It's a great experience.

Have to agree on Calculus Made Easy. Found it in high school and quickly learned intro calculus better than anyone in the AP Calculus class.

Apostle's textbook sounds interesting, not that I read math textbooks for fun (or any reason at all if I can avoid it).

We were taught (retaught) Calculus in first ("freshman") year, first semester Pure Math using Spivak's textbook which starts from a set of axioms that define the Real numbers. While this is anything but historical, I think this is better because you aren't "explaining away" stuff. At no point were we asked to simply accept something (once the axioms have been introduced and exercised and, I guess, setting aside the rigors of language and logic).

We were initially taught Calculus (in high school) the way you describe -- series and then limits -- and it royally sucked (no-one understood it and it provided no historical context).

Apostol starts similarly, with the first chapter mostly being really basic set theory (and discussion of the reals, couple of proofs about fields, etc).

Reading both volumes of Apostol is the sort of math introduction that any college student at all interested in math should have.

My google search for "apostle calculus" gave me back a "did you mean apostol calculus". I think they just confused the name.

Fair enough, although I'd say the same of Spivak's book.

For those who have been inspired by this post and want to check out Apostol's excellent book, note that it is spelled that way. (It is a shortening of a Greek surname--Apostolopoulus, I think.)

Thanks for the correction. This mistake and the correction would be a good memory tool for me in the future!

Bibliography: https://en.wikipedia.org/wiki/Tom_M._Apostol.

I think part of the problem is how we are tested. I fell into the same situation in high school where I'd be able to derive complex formulas on the fly that I required to solve a problem, but it took me so long to do that compared to knowing the equation that I'd never finish exams (time would run out) and I had to resort to rote memorization in the end.

One way to level this is to allow cheat sheets or open books. This is a bold way of saying: "If you truly understand the topic, you'll be able to solve this problem even though it requires a mental step ahead from what we've seen in class." What usually happens here is that students who don't have a good grasp and get by on pattern-matching exam questions to questions done in class, complain about not having seen that type of question before.

Guess what, in real life, you almost always hit a type of question you haven't seen before! The kinds that have been known already are easily findable in any textbook. So why are schools focusing so much on the rote memorization / pattern matching rather than focusing on raw problem-solving skills? Algebra in 9th and 10th grade, precalc in 11th grade and intro calculus in 12 doesn't do anyone any good.

It's worth pointing out that Boaler's work is fairly controversial. There's a review by Bishop, Milgram, and Clopton that refutes many of her claims. Her work is interesting, but it seems that much of its appeal is that it confirms people's suspicions that the way they learned math was wrong.

To be fair, I don't know what the right way is. A lot of very different approaches have produced successful mathematicians. How to teach math is a recurring controversy in mathematics, and a scan of the Bulletins of the AMS will convince one that mathematicians are working on the same gut level that everyone else is, unfortunately.

I'm not spending a lot on this story or comments, only skimmed yours too (sorry) but I notice that you object to an emphasis on rote learning.

although it's nice to have history and background the truth is that the math that's most useful to us is pure rote - things like the multiplication table, adding and subtracting numbers.

when people go into a subject, like 3d game design, and 'wish they paid more attention in (that part of) math' they don't wish they had a finer appreciation for the background: they wish they would just know the formulas when they need them so they don't have to stop and think.

it's like logical arithmetic. if it's second nature to you you can refactor or's and and's and xor's and not's into and out of parentheses in code faster. who cares about the finer stuff behind it.

I am not saying this as someone who has a great deal of rote in me. But the part that I do have has served me well. I bet if I had been forced to memorize a 100*100 multiplication table and addition, and subtraction table (10,000 members each) and did my multiplication, division, addition, and subtraction, two digits at a time instead of how I do them (like everyone else) it would have served me quite well in life.

not saying that that's a good use of the precious little time kids have between 6 and 14, but just saying that rote is extremely useful for a lot of key things. set-theoretical logic, probably not so much.