[matthewmacleod]

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.

[6d0debc071]

> 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.

[dTal]

Your circle area formula is incorrect - you square the radius first, then multiply by pi. It's quite easy to see why if you substitute tau/2 for pi - 1/2 tau r^2 is clearly the integral of tau * r, where tau radians is a complete circle. Mixing diameter and radius in the same formula and hiding the resulting factor of two in a constant is a pedagogical disaster.

[chowells]

    The mapping of the symbols onto reality was often missing.

Perhaps because it's not a part of math? Math is an art. It's totally unconcerned with things like "reality". If you're so concerned with reality, you've probably never done math.

[6d0debc071]

If that's your definition of art, then I can only consider myself fortunate to have been doing something else. Though, it's hard to see why anyone would gain anything in studying it were that the case. They could just make totally random shit up and claim, with as good a justification as any other, that it was as worthwhile as the work of any renowned mathematician.

However, I don't agree that art is unconcerned with reality, nor that maths is. We learn to draw by looking at things in the world, we get our rules about anatomy and so on from there; we learn maths based - at least initially - on physical examples; we tell stories based upon common themes and situations. The basic rules of these things are drawn from their correspondence to reality; with what people have experience with; and form the meaningful grammar of the system. Art is always a language, and a language is always representative. Even when - as arts - we may bend a few rules and simplify certain aspects to lend emphasis or make it better suited to approaching a particular problem.

To hold otherwise seems to me to deprive them of a foundation to communicate any common meaning. It would be no better to learn maths, under that definition, than to spend your life insanely scraping crayons across a page. (Indeed, given that the symbols would also be arbitrary in their meaning, it would be hard to tell the difference.)

Is maths an art, is it a science?

It's a language of enquiry. To claim that it proceeds solely by a 1-1 correspondence to reality is to picture mathematicians off somewhere counting things and deriving pi from taking increasingly precise measurements of actual circles. (And, really, even physics proceeds with the help of a healthy dollop of imagination that forms the foundation of hypothesis.) To claim that it has no correspondence to reality is to reduce the whole exercise to nonsense.

I do not see how either position taken as an absolute is tenable.

[eli_gottlieb]

This is what mathematicians actually believe.

[nbouscal]

Some do, others don't. There is an interesting dichotomy: much of mathematics does appear to be completely dissociated from reality, and yet in the other direction, reality appears to be entirely mathematics. I think to reach a unification, it is absolutely essential that we have people working in each direction. You aren't likely to develop a theory of (∞,n)-categories by working backward from reality, and yet it turns out to be useful to have done so once you start exploring e.g. topological quantum field theories.

[MaysonL]

It's totally unconcerned with things like "reality".

If you're believing this, you've probably never done physics.

[Jtsummers]

> 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.

[omegaham]

The thing is, I can't think of another way to teach the curriculum.

Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You've counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you've done that, you abstract all of arithmetic with algebra.

And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have "learned" everything up to calculus, but they don't actually understand what's going on. They just know the formulae and how to plug-and-chug.

How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood why the formula was used just as much as how it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they're fucked.

In any case, though, I think that the current curriculum is as good as it's going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren't going to learn calculus unless you understand algebra, and you aren't going to learn algebra unless you understand arithmetic.

[ef4]

> The thing is, I can't think of another way to teach the curriculum.

But there's already a rich literature of real-world results for better ways to teach mathematics. See for example Seymour Papert's "Mindstorms" as a starting point (and much has been done since it was written ~30 years ago).

All children already learn quite a lot of fairly deep mathematical intuitions. We just take them for granted because everybody learns them.

For example: conservation of volume, the concept of "integer", order independence of cardinality, projecting orientation onto other reference frames, the equivalence between ordinal and cardinal numbers.

Everybody learns these things because they're embedded in our environments, and we can learn them playfully as children. When we create environments that embed even richer concepts, children learn those concepts just as easily. This is the explicit design goal of LOGO, and the whole family of descendants it has inspired.

Teaching in this way requires a degree of freedom and play that normal schools generally don't tolerate, which is why these proven, powerful tools still haven't taken over the world.

[Jtsummers]

I don't disagree with the order of teaching, in general, but with its pacing. Arithmetic shouldn't be given 6 (K-5) years. Algebra shouldn't be 3-4 years (6-8 or 6-9). We bore students with the same material slightly stepped up each year for years at a time until they get to high school. Then we try and hit the accelerator and make them jump from the most rudimentary concepts in arithmetic and algebra and get them through trigonemetry or calculus in 3 more years. The concepts of trig, calculus, probability and stats, linear algebra can all be taught earlier. Students are capable of this, but the curricula aren't designed around it.

I didn't even realize until college that Algebra II was linear algebra. The notations used in college linear algebra would've been difficult for me to grasp fully at the time, but they make solving those systems of equations so much easier. And learning the notation [in high school], getting to the courses in college they'd be far less intimidating. We have 13 years with students before college, plenty of time to introduce notation and higher order concepts slowly rather than dumping it on them freshman year of college.

EDIT: Clarification on time of something

[nitrogen]

A few weeks ago there was an article here on HN that suggested kindergarten students might do better learning an intuitive form of calculus and algebra before arithmetic. Yes, math is built out of layered abstractions, but we can rotate the entire conceptual space to use a different foundation and still get a complete picture in the end.

[Jtsummers]

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

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

[jfarmer]

Kids deal with abstraction every day. The very idea of "color" and "number" are abstractions of concrete experience.

Teaching kids basic group theory is very possible. You can play games with shapes in the plane to learn about dihedral groups (without ever using those words). Graph theory, as the author says, is another avenue.

The problem is that what students are practicing isn't math, any more than running after the ball when you miss a swing in tennis is practicing tennis. And you improve at what you practice.

[prawks]

Where does, say, proof by induction fit into that linear set of progressing abstraction? Surely you don't need to understand calculus to understand inductive reasoning?

[omegaham]

It ends up being on its own. You get introduced to it during geometry, but I don't think that it really gets taught until college.

I took BC calculus as a junior, so we got left with about a month between the seniors' leaving and the end of the year. My teacher said, "Okay, we're gonna learn number theory." Oh fuck, that was hard. It was completely unlike anything that we'd done before, and it required the development of completely different skills. I wasn't bad at it, but I definitely wasn't good at it either.

I'm glad that I got a taste of it, as that thought process has helped me in countless situations, but you're definitely correct in that it doesn't fit with the rest of the traditional curriculum.

[bstamour]

And this is why I hate the term "mathematical induction". It's actually a form of deductive reasoning, not inductive reasoning.

[reboog711]

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.

[k__]

I guess most people build CRUD stuff without (knowingly) using functional principles.

Programming is inherent mathematical, but if you don't progam functional, the math "behind" your software gets more abstract ... or should I say obscure?

In the end your computer is a big function with input, calculation and output. But if you write your code with side effects you can't do a fine function based split-up of your code, you have to consider bigger chunks of code, consisting of many dependent functions as one "mathematical" function.

If the math is more complex and logical reasoning about it gets harder, people start to think about it as "not-math" but something different.

[rthomas6]

I take issue with this because this is only one way to look at programming, not the inherently correct way. Yes, functional programming is a useful abstraction, but at the hardware level your processor uses something closer to a procedural paradigm. So your computer is not just a big function, not literally. Or rather, the "functions" your computer uses don't correspond 1:1 with the functions you write in functional computer language abstractions.

[k__]

Why not?

You got input, output and calculations.

What else do you need?

[rthomas6]

Good question. Why aren't processors optimized for functional programming? I think one reason is because of the way hardware memory works. So to answer what else you need, one other thing you need is physical memory and a way to organize the data in it. Forgive me for what follows if I misunderstand some functional programming aspect, because I'm primarily a digital hardware person, and programming in general is not my forte. Also I apologize if I explain things you already knew.

In hardware, CPU instructions are read sequentially from memory. These instructions are pretty basic... add two numbers, load a data word from a certain memory address, jump to a new address if two numbers are equal, etc. Modern processors do have some pretty fancy instructions but what I said is still basically true. So those instructions are our primitives. The only way to make abstract procedures from those primitives, from the perspective of assembly programming, is to make a sequential series of these primitive instructions starting at a known address, and then branch to that address and read those instructions in order. When you abstract many times out, as is common in functional programming, there starts to be a lot to keep track of. When you evaluate a function that is very abstract, how it looks in hardware is a whole lot of branching and returning to and from different memory addresses. Not necessarily bad, but it's starting to look pretty different in code vs. in hardware. And if you branch to uncached ("unexpected") locations, you add latency when you have to fetch instructions from RAM. You also have to keep track of any data needed at a higher level of the function, which necessitates automated memory management, including garbage collection. These things can introduce a lot of overhead in the program, especially when you have things in it like deep recursion.

tl;dr: There's no such thing as stateless assembly programming.

[cellis]

side effects.

[NoMoreNicksLeft]

> 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.

[the_watcher]

>> 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.