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Comment by Koshkin | Hacker News Reader

Learning "how to think" is just one part of it. The other part - the one that makes it much more difficult for many, if not most, people to learn math - especially the more abstract branches of it - is learning to think about math specifically. The reason is that mathematics creates its own universe of concepts and ideas, and this universe, all these notions are so different from what we have to deal with every day that learning them takes a lot of training, years of intensive experience dealing with mathematical structures of one kind or another, so it should come as no surprise that people have difficulty learning math.

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"all these notions are so different from what we have to deal with every day that learning them takes a lot of training"

I disagree in the general sense. I think most of mathematics with which "normal people" (by that I mean people who don't get a masters or phd in mathematics) have problems is directly correlated to problems and situations in the real world. The problem is that we teach mathematics as if we were computers - which we are not. Its about formulas, recognizing patterns, precise answers and proofs instead of general understanding and intuitions about things - which is where humans excel and how we work and remember things.

For example, being able to solve quadratic equations is nice but IMHO what's much more important is knowing how the coefficients and their signs shape the plot of the lines and how that might effect the trajectory and hang-time of a projectile. Also being able to look at a projectile and give an approximate quadratic equation is again IMO more important than knowing all the steps in the quadratic equation.

But this is just my opinion.

Henri Poincaré once said: "Mathematics is the art of giving the same name to different things"

I know the temptation is great to give lots of examples while teaching math, but you get the risk of only teaching little examples to students and not the abstract vision that math education demands.

Take your quadratic equations, it's all nice to demonstrate how coefficients shape the plot. But when you start talking about projectiles, you end up ignoring that quadratic equations are used... heavily... everywhere.

You may see them in chemistry, crypto or nuclear physics. But if your brain only remembers projectiles, it doesn't have the ability to abstract new problems the right way.

As a counterpoint to that, I flat-out despised abstract Math until I got to college precisely because I found it useless. At best I knew I was probably going into engineering in college and I'd been told "engineers use calculus". That was my sole motivation to learn anything about abstract math, and it was pretty bad. It didn't help that the "examples" given in class were mundane toy problems (ex: "how long will this arbitrary tank take to empty?" Well gee I don't know and I don't care. Is this the last tank of water on the island or something?)

Now physics on the other hand, chemistry, computer science, any time I could apply math to a real-world situation got me motivated. I liked Math that could solve actual existent problems, and I learned said math better as a result. The first time I learned about projectile motion I literally started picturing vectors everywhere for the rest of the day.

It also didn't stop me from learning abstraction later in college. In basic signal analysis we learned about the Fourier transform, and then we learned about the Laplace transform and how it was simply a more general Fourier transform (or the reverse if you prefer). Mastering Fourier first gave the Laplace a relatable mental context, it made the abstractions more real.

By contrast I had a Math professor once who tried to teach us theoretical induction and work his way down to examples. I had no idea what the fuck he was talking about until he hit the examples, then everything clicked all the way back up.

Maybe that's just how I'm wired, but I would argue for more and more relevant examples. There are tons of historical uses of math that people would find interesting. Talk about the math used to build the Hoover dam, or telegraphs, fly satellites, or make medicines, or decode Nazi transmissions. And talk about it in real terms with character, if you must simplify for the sake of time then say that's what you're doing. Make people realize they're learning something that has uses beyond Math class or the rarified ivory towers of Math PhDs. That by learning the material they're learning how to do meaningful things. The original, wildly popular SimCity games were basically raw spreadsheets and formulae, but people learned those spreadsheets and formulae because the games had character.

I know to a lot of Math people the abstractions are the "real" part, but most people I know simply don't process the world that way.

This is especially a problem in college. Few math profs seem to enjoy teaching and seem reluctant to ground their subject using concrete illustrations. Without these, there are no tangible points of reference, aside from derivations from known equations. But mere transformations rarely offer insight without adding some sort of context.

I think the crux of the problem in teaching math is the need to illustrate each concept from MULTIPLE perspectives: theoretical, tangible, incremental, graphical, dynamic, etc. It's only by connecting the concepts to manifestations that most of us appreciate the meaning of math. Few are satisfied by disembodied equations or proofs. Most of us need some form of grounding using what we already know to make sense of new concepts. Otherwise math is just castles in the air.

If you have 45 minutes, watch this talk: http://blog.mrmeyer.com/2016/nctm16-beyond-relevance-real-wo...

There's also a hurdle in having to work through the wildly varying notational and structural views/tastes of the individuals who created their specific categories/theories (biggest problem at least for me - while programming languages are nice and structured and standardized, traditional math is highly impure/unstandardized and relies too much on how individuals feel like when they write things down)

Learning math is like learning a language. It simply takes a lot of time and effort. If you read the rules of German grammar cover to cover, that will not prepare you to speak German; in order to be fluent, you MUST practice, over and over again, using and interacting with German.

Likewise for math. There is no "new book" to learn math that is going to blow the cover open on everything. It's a problem of motivating people to become fluent in math. I would like to see discussion in those terms -- fluency, because that's what it is.

"The reason is that mathematics creates its own universe of concepts and ideas"

This also applies to reading music and understanding music theory. Written music is a map of events/time - which is totally different than any other math I studied. I've noticed that a lot of musicians pick up programming pretty easily. I've often wondered if learning more symbolic systems that are different from natural language makes it easier to pick up additional systems like programming languages.

Many other commenters have pointed out how great examples and ties to the real world are; the love of applications is well-represented in this discussion. But I'll throw out beauty as an underrated motivator!

I love math because it's beautiful, primarily, and this isn't represented often in elementary or high school education. The idea of "function" or the idea of "constructive proof of bijection" are beautiful concepts. The patterns of math are beautiful (so beautiful I created an adult coloring book called Math with Crayons that focuses on Aztec diamonds, fully packed loops, lozenge tilings of hexagons known as totally symmetric self-complementary plane partitions -- you know, the stuff people are actually doing research on today!). Numbers to me are kind of boring. But how many kids get to draw fractals in class and then talk about the concept of dimension, and then argue about how one could rigorously define dimension, with the example of trying to define dimension for fractals as an edge case? How many kids get to draw non-crossing partitions and Dyck (lattice) paths and then try to prove that they are both enumerated by the Catalan numbers? How many kids get to think about the mathematical structures you allude to, and play with the fundamentals of category theory? Well, very few. And why?

Their teachers don't know any of that stuff.

If teachers only know computation, that's all they can teach.

If we had teachers who really had breadth and depth in math teaching math, I think we could start teaching concepts, structures, and beauty very early. I've had conversations about the basic ideas of category theory (relations not objects!) with many kinds of people. This is not out of the grasp of a kid. Set theory -- not out of the grasp of kids. The very idea of what a number is: in high school I got to take a seminar with other high school students on this, looking at John Stuart Mills and Frege and Russell and Whitehead and all this thinking about what a number is. But this is extraordinarily unusual material for a high school or even college student. Of course it's difficult to deal with these notions later, then -- people have been trained out of their natural curiosity and into a view of math as computation at worst and manipulation to find x at best, rather than seeing it as a way of dealing with beautiful structures.

> Learning "how to think" is just one part of it. The other part [...] is learning to think about math specifically.

Do you feel like you were taught how to think about math specifically, or how to think at all? Like probably most people here on HN, I love math and programming, but I'm struggling to recall whether and how I was taught to think in school. We went through a series of problems, and talked about the syntax and mechanics of solving those problems, using various frameworks (geometry, calculus, diff. eqs) So it feels like we learn math by example. Is learning how to think directly part of education, or only indirectly something that happens by seeing what others did?

I think that people are rarely taught how to think; this basic skill usually comes (rather quickly) in the process of "learning by example" - part of which is struggling with concrete problems - whether it is in mathematics, in programming, or in life in general.

Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".

I find this sentiment somewhat surprising.

Limits is incredibly easy to link to real physical things: movement, speed, acceleration.

If speed is distance over time, what's the speed of a moving object at a very specific instant?

"Ma'am, you were driving at ninety miles an hour."

"But, sir, I have only traveled twenty miles!"

Are you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve.

Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.

Will, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.

Ah, good point. I forgot about that problem with instantaneous speed.

lol sorry for the typos! Wrote that from my mobile phone at night.

I think these things have already been discussed to death some time around the year 1700, when they were called something like "mathematical analysis of infinitely small magnitudes" or some such; the difference between an "infinitely small magnitude" and zero was already clearly understood. The notion of a "limit" came only later, in an attempt to put this stuff on a firm logical foundation and was (and still is) seen more of a formality than something aiding our intuition in this area.

Even simple objects, like vectors, for example, can very quickly become so abstract (higher dimensions; complex vector spaces) that the physical intuition stops being useful and can even lead to mistakes. This is the point when new kinds of mathematical intuition becomes necessary, and building them is very difficult and takes time.

I remember doing only simple (two-dimensional Euclidean) vectors before calculus -- length, scaling, addition, subtraction... Agreed that they obviously can get pretty hairy, but I don't think that side is exposed to people before calculus? Complex vector math was calculus III for myself.

I think the most important part, especially for people who do not intend to be mathematicians, is how to apply math to solve problems. The math courses I took in high school and college were all about theory and nothing about application. These courses were taught as if it were still the 19th century and computers don't exist.

I would rather see teachers accept that computers exist, and teach students how to solve problems, even very complicated ones, using software (e.g. Mathematica, R, or whatever). Keep the theory to an absolute minimum (leave the theory courses to people who are interested.)Really, so what if you don't learn how to do calculus manually?

I suppose one can learn how to think by doing math, or they can just learn how to follow rules...and perhaps some may want to think about match specifically. However, I think most people would be encouraged to learn math if they are shown that math is just a way to model the world they live in and it can tell them specifically about some of the assumptions they have about it. For instance, I might assume if you put some fuel in a rocket and point it straight up that it will go into space and be somewhere. Math can tell me where it will be and the path it will take to get there...people identify more with those concrete things.

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