Math education: It’s not about numbers, it’s about learning how to think

I indeed believe you are wrong. I started getting interested in STEM when I was 16 and now have a masters in math on the one hand and work for one of the most prestigious Tech firms in existence on the other. Meanwhile, when I was young, I was encouraged to enter creative fields (mostly writing in my case) which I did have enthusiasm for but then dropped as a teenager. So I switched at least twice, what I wanted to do with my life and still turned out fine.

Honestly, the whole "you need to get them when they are young" idea never convinced me very much. It's never too late to learn a thing and often you need a certain age to actually appreciate something. You don't have to be a childhood prodigy to be good at something.

I rather find the idea pretty harming; I've met a lot of people who did not pursue the career they wanted, because they bought into the myth, that you need to start programming at 6 to get a job at a company like Google or Facebook.

Let kids be kids for a while. Don't worry, they can always figure out what they love later and it's never too late to reconsider.

Later the curriculum only expands on these problems so it's even harder and harder to catch up.

I have been working with college freshmen struggling with basic linear equations and such. While the profession is marginally unrelated to Math (e.g. Graphical Designer), I still wonder to this day how people can have 20-30-40 years and not know how they could (with a calculator in their hand!!) calculate the price after the discount. How can someone feel ok with this?

Think of stuff like the quadratic equation, or formula for the surface area of a cone. How many of us were just taught these magic formulae without first deriving them? Why do we start talking about pi without first explaining how it's calculated? Where does the "four-thirds" part come from in calculating the volume of a sphere?

Not to mention that there's never enough history to go along with the mechanics - who discovered the quadratic formula? What was their life like? Why were they playing with quadratic equations in the first place? This make math seem less like magic incantations and more like something that was sort of cobbled together by flawed weirdos in order to solve real-life problems, and evolved over time.

So, here's the thing. Did I have no problems with math until uni because I'm smart? Because I have a specific type of brain? Because of my upbringing? Because I had very good education in my early years? Some combination of the above?

I struggle to believe that people can't do algebra. I am convinced that with some thought and help they'd find it trivial. However, is that my own experience overriding my ability to take their perspective as true? I dismiss "I can't" as "I don't think I can", but who am I to judge that better than them?

It seems like an especially American thing - being good at math makes someone a freak and so it's something kids actively avoid at the ages in which being a freak is not tolerable.

That kind of attitude would need to be actively confronted because what happens is that kids who previously were developing skills tend to drop out given the peer-pressure.

Edit: best documenting link I could find but Google book won't let me copy a quote:

https://books.google.com/books?id=CzmpDAAAQBAJ&lpg=PA74&ots=...

It's nice, vague, and populist. Everything thinks they know how to do critical thinking. Everyone thinks they know how it could be taught. I suspect most people think that they have above average critical thinking skills. As long as they don't have to agree on how or what critical thinking is, everyone can agree it's a good thing.

The other one that's dumb is "you learn best when you teach yourself". Sure, if you are really super-interested in something then you learn better, and tend to teach yourself a bit, but correlation is not causation.

It's more important to foster an positive learning environment and encourage/draw out curiosity and creativity at this age. Help them become curious about the world, fascinated with what they're learning and find enjoyment in learning.

In terms of subjects, aim for a strong reading focus at an early age as it will pay dividends in all other subjects (even math) if they can read/comprehend well.

I clearly recall memorizing the times tables when I was young. It suddenly dawned on me that "times" wasn't just a word, it meant adding the number so many times. I no longer needed to memorize the table, I just applied the rule.

Some university engineering classes present formulae and tell the students to just apply them. This is disastrous in my not-so-humble opinion. Where the formula comes from should always be taught, so students understand the formula and the assumptions it was derived from.

Recalling my own experiences with middle school and high school, I saw my peers frequently fall into the trap of trying to memorize pipelines of steps while we were learning algebra, geometry, trig, pre-calc, and then calculus. The ones who "didn't get it" would fall apart to varying degrees when confronted with problems where you had to think in terms of combining toolboxes of strategies to unfamiliar but analogous patterns, but I don't necessarily think it was their fault, because they didn't know better and rote memorization had worked consistently until this point.

I often feel a bit sad about it because I did a ton of proof based stuff outside of class (as a mathlete), and that was a HUGE part of my development in logical reasoning, intellectual rigor, and critical thinking.

Just anecdotal evidence, I know, but for me it was the exact contrary that happened. As a 5-year old kid I had learn to read all by myself (my dad had taught me to recognize the letters and how to pronounce them), but I had huge conceptual difficulties in teaching myself things like subtraction (what do you mean you can take/subtract a thing from another thing? what happens when the thing which you're taking/subtracting is bigger than the thing from which you're subtracting? you cannot do that, it would result in a non-thing, a thing that is below 0, and all things below 0 technically do not exist).

In elementary and middle-school I was passable in maths, while very good in humanities, but all this changed in high-school, mostly because of my teacher, who taught me (and my colleagues) how to think about math. He's one of the reasons why I decided to pursue a STEM education and why I'm currently a programmer.

To me the real important thing is the attitudes imparted by your parents and peers. As noted in the article, it's acceptable to fail at math. Parents who have their own history of math anxiety probably don't really push their kids to excel in that field. Just the notion that I was expected to "get it" and keep trying until I did (perhaps helped along by some natural stubbornness) may have made the difference in my case.

My conclusion is that every one will have strong and weak spots, and it's very important to stay alert to this and not beat around the bush thinking it's not important.

Took me more than a decade to get back into intuitive thinking, where you can play mentally with symbols and ideas just like LEGOs (as I was around high school).

This is the opposite of my experience, I did horribly in math and hated it until I took calculus (which, I guess I should have taken in highschool but I tried to avoid it since the rest of math sucked.)

EDIT: just want to clarify: I did have a reasonably intuitive understanding the whole time, the earlier stuff just felt like busy work.

I remember "hitting the wall" in Honors Algebra II, when I just couldn't keep up with all the new things I was expected to compute: radicals, synthetic division, logarithms, etc. It was a major dent in my math confidence, which up until that point had be reinforced by being told I was "good at math".

I didn't start regaining that confidence until Chemistry, when I learned what pH actually meant. That sparked the epiphany that scale was deeply important in nature, and with that motivation, logarithms seemed much simpler.

We so quickly forget that most math wasn't discovered as some abstract exercise. But we still teach it that way, with the concepts before the applications.

A really good art teacher or a really good math teacher can really change the kind of future you'll have.

I've taught Calculus close to a dozen times now. I talked about integrals. I talked about derivatives. I tried to get into the intuitive nature of rates of change and the limiting process of Riemann sums. I used analogies and real-world examples and illustrations and interactive computer demonstrations.

Students don't listen to that part of the lecture. They just don't. They listen to the part where I start doing problems on the board, because to them math is about doing things, not concepts. That's only reinforced when they're asked on tests to just do things rather than demonstrate understanding of concepts. If, in a college class, you try to change that up by asking concept questions, you get bad end-of-semester evaluations because you've challenged them in a way they didn't expect and their parents aren't paying good money for them to be challenged.

(And realistically it's hard to ask concept questions at that level of mathematical maturity anyway. As answers you get mostly word salad with a smattering of misused jargon. I've tried.)

If you try to mix it up and use techniques from inquiry-based learning (Moore-style, flipped classroom, etc.) then you get a stern talking to from administration.

> , or what "infinitesimal" means.

Probably because even defining infinitesimals with anything resembling rigor is tricky, let alone using them productively. I wouldn't even try it in less than a junior or senior class. There is no such real number that is smaller (in absolute value) than every other real number yet bigger than zero. Infinitesimals, at the calculus level, are a convenient hand-wavy fiction to mask limits and a notational artifact from simpler times.

> For me that's a big failure of our current "science" education system: too much focus on stupid application of equations and formulas, and too little focus on actually comprehending the abstract concepts behind them.

I tried. Every single time I taught the class I tried to teach concepts. It's really easy to fall to the temptation of just working problems on the board every day, because trying anything else will just give you grief. Grief from the students, grief from your colleagues, grief from the administration. Grief from people like you who constantly chastise us for not being able to explain to teenagers that they should pay attention for the whole class. (I love my adult students though. Non-traditional students are the best.)

It's not worth rocking the educational boat if you don't already have tenure. If you do rock the boat you get a reputation for being hard or weird and students avoid your class anyway.

You hit a nerve. I've left academia for this and other reasons, but as you can tell I'm bitter about it.

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.

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.

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.

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.

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 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'd go so far to disagree with you that people do well without learning math skill (those tools of thinking). If you can deconstruct a problem, then you certainly learned that skill somewhere, just not in a math class.

Abstraction and reductionism are unusual capabilities to acquire on your own. As I recall, they're almost nonexistent in pre-linguistic societies like hunter-gatherers. If you have no words for abstract concepts, your thinking will be strictly concrete.

I think that in a lot of disciplines, you have to become fluent at manipulations, and at seeing and thinking in higher level patterns. Being able to think your way through a more complex problem would seem to benefit from, if not require, seeing multiple steps ahead in a progression. At least this is my perception.

My experience in school math was that it wasn't enough to satisfy myself that I knew how to solve a problem. I then had to work my way through a whole bunch of similar problems until I could perform the manipulations quickly. This is also how I managed to commit the definitions, axioms, and theorems to memory. If I didn't do that stuff, then I got my arse handed to me on the exam. I gave my kids pretty much the same advice.

https://www.adaa.org/living-with-anxiety/children/test-anxie...

There's no shame in it (although high schoolers can be assholes), and it can help your school accommodate your needs eg extra time to take your exams.

Start here http://web.mit.edu/5.95/readings/bloom-two-sigma.pdf but then there is a vast literature (thousands of studies and other research papers) exploring the general topic of math pedagogy. Some approaches and some teachers are radically more successful than others, given comparable students.

If you hand me one of your 20th percentile students who is struggling but willing to learn, and give us the same amount of time which would otherwise be spent in a classroom for a year of one-on-one face-to-face time, with the student also spending a typical amount of time working independently, I can have them outperforming your 80th percentile students by the end of the year, without issue. The difficulty of course is that direct mentoring by an expert tutor is too expensive for society to be willing to pay for at scale.

The main reasons that your teacher friends are stuck is because (a) many if not most of their students are unprepared before they arrive in any particular course, which is largely down to structural social factors and school scheduling inflexibility, and (b) the combination of lectures and independent work with inadequate feedback are for the average student a terribly inefficient and ineffective way to learn. Neither of those has all too much to do with biological predestination or whatever.

> But how often do you have to do any sort of complicated problem-solving with mathematics?

There are a LOT of chances to do that kind of complicated problem solving, especially if you're shopping or comparison shopping on your own. It's not that people don't have the chances, it's that people avoid the work involved in doing those kinds of comparisons.

In middle school in the US you start getting teachers specialized in their teaching subject matter, but often tangentially. Any one with enough college math credits can teach middle school math (with some other certifications, usually). High school you may, finally, get a math teacher who is a mathematician. If you're lucky.

People ask me if they see the beauty in math when I'm complaining. I do, but I also see the other kids outside playing while I'm held back from recess until I can tell you what 7*8 equals. Fuck that so much.

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles."

Alan Turing

Really? There are many times where I thought "How am I supposed to think of that?" In fact, that's probably what would happen most of the time.

The problem with high school math is that it has no context. It's just a bunch of seemingly arbitrary rules.

It's as if when learning programming, you would learn fragments of various languages here and there but never connect the various pieces into anything meaningful.

Lesson 1: boolean variables in Java

Lesson 2: If statements in python

Lesson 3: Three-part for-loop in C

Mid-term exam:

Q1: In the C language, how do you separate the components of a for loop? (correct answer: semicolons)

Q2: What can you put in the right hand side of the '=' symbol in a boolean variable declaration in Java? (correct answer: an expression that evaluates to either true or false).

Q3: You are supplied a function that can take any number and determine if it's a Crastomarian number. We have 20 foot ball players each wearing a T-shirt with its own number. The numbers go from 5 to 24. We want to find how many players have Crastomarian numbers. The program to find the correct answer will use a for loop. Write the header part of the for loop. (correct answer: `for(i = 5; i < 25; i++)`)

...

And this is supposed to be how you learn programming.

Also note: there's almost no way to know the answers without rote memorization. Of course you will never absorb these concept by osmosis because your day to day activity will not involve writing programs that do things. Instead, you're just memorizing the rules given to you and doing exercises that help you recall which rule to use for which situation.

If you haven't memorized the for syntax and written 25 different variations of it, you will not be able to know it.

And when I say that you write 25 different variations, I mean you write them as part of pointless exercises of course because these are not in the context of a program.

It's more like just repeating these lines until the pattern somehow hopefully sinks into your mind.

for(x = y; x < z; x++)

for(y = 10; y < j; j--, y++)

.. etc

Too many students lack the ability to perform basic algebra. I mean simple stuff like simplification, factoring, working with exponents and even fractions. Faced with a typical problem in calculus, of course they're going to flounder and make errors when each calculus problem involves perhaps a dozen simple algebraic manipulations.

These basic things have to be drilled and practiced A LOT before moving on to more advanced topics. There is no way around that. No amount of explanation and reasoning will help the students and instructors get around the fact that they just have to spend the time to develop fluency with the basics.

All arithmetic is math, but not all math is arithmetic.

Except that if you dig deep enough, all math is arithmetic under the hood. Algebra is doing the arithmetic on variables before they're numbers. Calculus is just useful shorthand for compound arithmetic; the derivative is a divide and the integral is a sum. Exponentiation is repeated multiplication, which is repeated addition. Linear algebra is arithmetic on blocks of numbers.

So we abstract over the individual arithmetic operations for our own convenience, there are too many of them. But it's all still fancy arithmetic.

Maybe we should call math "fancy arithmetic" to help all the people who are intimidated by math...

I mean, there is theory & practice in every field, technique and discovery. A medical scientist studying the development of aberrant cells in a petry dish has a pretty different job to a pediatrician or radiologist. They generally seem to coexist happily under the "medicine" label. For first year undergraduates, they would probably need to take most of the same classes.

I love talking about both subjects I went to university for actually and both can be difficult to get people interested in. There are massive misconceptions about both.

- arithmetics (a.k.a. Number Theory) is a vast branch of modern pure mathematics;

- numeric calculations, i.e. how to do them efficiently, is a very important part of the (applied) mathematics.

College algebra: pi.

Calc 1/2: pi, theta, capital sigma for sums. epsilon and delta to state the definition of a limit (but epsilon-delta proofs have largely disappeared from mainstream calc courses, so those two letters actually don't appear much).

Linear algebra: pi, capital sigma. Some people like to use stuff like phi or psi for linear maps. I don't.

On the one hand, in writing my course notes I have moved away from Greek letters somewhat in courses like abstract algebra. On the other hand, there is at least one good reason to use them: It helps you distinguish different kinds of objects by using different kinds of symbols. So, for example, I can write "f(x)" for a group map f applied to a group element x. If you follow the rule of picking letters from disparate parts of the alphabet for different kinds of things, that's okay. But I think some people prefer "phi(x)" exactly because the function phi "stands out" from the argument x.

Also, if I need 4 functions "in parallel", I could use alpha-beta-gamma-delta. If I try to do this using Roman letters, I use f, g, h, ... now what? ("i" is sort of reserved for indexing; "e" might mean the constant, but in any case is not often used to name a function.) You can subscript (f_1, f_2, f_3, f_4) but that introduces another kind of notational density and possibility of confusion. There are tradeoffs both ways.

There are places where I'm definitely in sympathy with the your general point. I've sometimes read something and wondered why the writer didn't just use a Roman letter (particularly when I don't recognize the Greek letter - is that a capital Greek whatsis or an emoji?). Sometimes, people use calligraphic script caps (e.g. for ideals), not realizing that they will be a pain for handwriting or TeXing notes. And in some subfields of math, people seem to like Fraktur or Hebrew characters which are a real nuisance (it took me a few tries to learn how to handwrite a passable aleph), particuarly when they're handwritten and the person writing has shitty penmanship so you can't tell what letter they wrote.

I agree with the general point, though: Greek letters should be used like seasoning.

Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).

As I went through school I saw many students who were bilingual either do poorly or exceptionally well in math.

These were students who spoke Spanish, Tagalog, Chinese, Burmese, etc. at home for the most part and then spoke English at school. Some learned well and some didn't.

Math is definitely "harder" in the common thinking of people, but I suspect that there is often a generic underlying issue.

A number of students (I would say a large part) go to school or college not "to learn" but rather to "pass the exam" or to "get the degree", they are not interested in the subject but in what the consequences of having a good grade in that subject are or - sometimes - only do whatever is needed to get a passing grade.

I don't blame the students, mind you it is not their fault, it is society in general that somehow has shifted the values, and culture is not anymore valued.

In a course I teach, I have students reading Guesstimation and doing two blogs: 1) A Guesstimation problem similar to that in the book, 2) Watch a TED Video and come up with a mathematically based skepticism with doing a guesstimate if possible to get a feeling for the skeptics answer.

My students love that part of the course and it gives them way more value than the rest of the course covering standard mathematics. They start to incorporate the guesstimation ways into their work and slowly start to see the true fuzzy outlines of real math.

Reading all their blogs and making appropriate comments is quite a bit of work.

I think the real take-away is that both physics and mathematics can be taught poorly, and both can be taught well.