[Irregardless]

People who say "I can't do math" are lazy? Sorry, but I think that's a gross oversimplification of the issue (which is also evidenced by the comparison to physical exercise) and the two assumptions fundamental to that opinion are just plain wrong. Namely:

> So what do people actually mean when they say that they can’t “do” math? Usually they are really stating one of two things. First, that they don’t like mathematics. Secondly, that mathematics is more difficult for them than other subjects, and that it takes a great deal more effort on their part to learn it.

He conveniently chose the two reasons that are easiest to dismiss (irony, anyone?). What he fails to account for is the fact that math education is so poor that many people don't truly understand what math is. Beyond arithmetic and algebra, they think it's some really complicated stuff with big numbers and funny symbols that geeky people with glasses do -- it's practically a foreign language to them, except it has a reputation for being much harder.

Why is this? If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach. Those who attempt to do so often do it very poorly, which leads students to the false belief that math is extremely difficult. On the other hand, it's very easy to teach someone to memorize formulas and plug in numbers, so that's what we're most often taught in math class. That's good enough to get us through the standardized test so we can graduate from high school, but memorizing formulas and plugging in numbers is not "doing math". So I believe many people are completely justified in saying "I can't do math".

What's more, people who "can do math" should be taking the blame for those who say "I can't do math" rather than using pointless semantics to wag a finger at them.

[dkirkman]

> If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach.

I think it's even worse than that -- I suspect that many teachers in American public schools are terrified of math themselves, and they transmit that terror to their students. It's going to be very hard for a student to learn to view mathematics as reason if their teachers don't see it that way.

There was a point when I was in grade school and we were learning formulas for the area of different shapes. When a trapezoid came up and I noted that it could be decomposed into a square and two triangles, I was admonished to just use the formula from the handout. Don't get in the habit of trying to think while doing math, it'll just get you in trouble.

This was from an otherwise excellent teacher, but when it came to math we were to turn the thinking switch to "off". This fear of math seemed to not been exceptional, even among my high school math instructors. I may have had a bad run (public school in California in the 80s), but I've been told similar stories by most everyone I've met who eventually managed to figure math out on their own.

[bjhoops1]

Stories like this make me glad I was homeschooled. Mom handed me a Saxon math textbook and I just read it and did the problems. No fear, no classroom consensus that "math is haaaard".

[noAlchemy]

I can support this from the other side. Coming out of a maths degree at a reasonably good university, a lot of the mathematicians who became teachers were those who struggled and pretty much gave up at some point in undergrad. Pushing many of those who will become teachers to mathematical breaking point seems like a bad way of doing things, and I think contributes to this. It's hard to think of a good alternative that still produces enough teachers though.

[aidenn0]

Only thing I can think of is to pay maths teachers really well, which probably means increasing class-sizes, so is a no-go at least in the current political clime of the US.

[Kluny]

I recall a math teacher I had in grade 9 - she was normally a phys. ed. teacher, who had been voluntold to teach math when the usual math teacher had a nervous breakdown and took a year off. She freely admitted that she had studied phys. ed. because she "loved to teach", and wanted her students to have a better time than she did, but she had struggled in English and Math, so she went for one of the easier specialties. Not exactly a shining role model there. All the other students loved her, of course, but I thought it would be better if she were a babysitter, not a teacher.

[lenazegher]

    What he fails to account for is the fact that math 
    education is so poor that many people don't truly 
    understand what math is. Beyond arithmetic and algebra,
    they think it's some really complicated stuff with big 
    numbers and funny symbols that geeky people with glasses
    do -- it's practically a foreign language to them, 
    except it has a reputation for being much harder.

I can attest to this. Academically, I am a reasonably able person, but I found math simply baffling at school. Arithmetic and algebra were fine. Rudimentary geometry made sense. When we got to trigonometry, things just fell apart for me. We were taught sine, cosine and tangent in the context of how they could be used to derive angles from other angles, not what they were and how they worked. They were presented as tools that could be used in particular ways that had to be memorized. To me, it felt like trying to teach an alien from another dimension to use a hammer without the alien having any intrinsic understanding of mass or momentum or kinetic energy or friction.

In fact, if I'm totally honest, I'm not 100% I completely understand the sine function now. And it wasn't just math. In physics, current, voltage, resistance etc. were taught as inputs to formulas. I know it must be challenging to teach about these kinds of principles that lack concrete macroscopic analogs, but I can't help but feel they could have done a better job than they did. In chemistry too, I remember being taught about valency and how you could work out the valency of an element by its position on the periodic table. I asked what valency actually was, either didn't understand or wasn't satisfied with the answer, asked again, and the teacher brushed off my question and carried on the with the lesson. "Oh well," I thought, "I guess I don't understand chemistry." That was when I was about 12 years old, and I didn't study chemistry after that. I studied biology until I was 16 because I had a teacher who took the time to actually explain things.

The worst part is, I went to a pretty good school. It must be absolutely dreadful at bad schools.

Most of this happened before I had regular access to the internet and the chance to learn about these things for myself. I can't help but feel the whole course of my schooling and advanced education might have been different had I had better (or at least different) teachers of hard science and math at an early age.

[SiVal]

It's the phrase, "I'm just no good at math" that breaks my heart. Most of the time, the student is taking the blame that belongs to the education industry.

Last week, my son had a "Chapter 9" math test here in a top-ranked Silicon Valley public school. His teacher pointed us to an official study guide PDF, which we went over carefully. I was not at all surprised to find that it covered a random grab bag of unrelated topics: sorting a half-dozen fractions, each with different denominators, two different silly algorithms for multidigit multiplication, how many $2.30 widgets can you buy for $9.00, and a few others.

This incoherent, random presentation of unrelated topics within a single chapter is totally characteristic of the "reform math" so beloved by our "progressive educators." They despise the approach of methodically working through a small number of carefully sequenced topics, making sure that the foundation of layer N is solid before getting to work building the closely related layer N+1 on top of it. They call it, "drill and kill," "soul-crushing," and "creativity destroying."

Instead of mastering a few closely-related concepts each year and systematically building expertise, they prefer "exposing" kids briefly to lots of unrelated math ideas, trusting that some kids will get some of it, and telling the rest to "trust the spiral," meaning trust that when they hop, skip, and jump over multiple topics the following year and the year after that, most of them will eventually "get" most of the stuff.

The result is that many parents just teach their kids real math outside of school. Many in our neighborhood send them to Chinese school, which teaches them math in addition to Chinese. The Chinese school buses line up in front of all of our local elementary schools at the end of each school day. (A lot of blond kids board those buses.) Some send them to Kumon, which is getting to be as common a sight around here as McDonalds or Starbucks.

I teach mine myself, using non-US curricula (Chinese, Japanese, and Singaporean in my case.) I feel terrible for the kids who don't have parents doing the schools' job for them, whose math skills are limited to what they can pick up from their classmates in "group discovery" sessions, since the "professional educators" have now decided that kids learn best what they discover for themselves and now serve merely as "guides on the side" in edu-speak.

My son took his Chapter 9 test and reported to me that, with the exception of testing the two different, useless multiplication algorithms, the test was a DIFFERENT grab bag of unrelated math topics, bearing little resemblance to the study guide. Totally typical of "reform math." He did fine, but only because he had learned all of it outside school. His friends who rely on what they learn at school think he's a genius.

So kids go through this ridiculous joke of a math education and can't do math. The school points at their friends who did just fine (because--shh!--they learned math elsewhere), the school takes credit for having taught them so well and tells the others and their parents, "well, not all kids are equally good at math, but many of your classmates learned quite well," clearly implying that the kids who didn't are somehow defective.

The result is that those kids will soon be saying, "I'm just no good at math." What a disgrace.

[mikevm]

Any chance you could link me to the non-US curricula materials you use to teach your kid?

I found a lot of the things you talked about in this school's approach to education: http://www.russianschool.com/about-us/our-approach

[SiVal]

I can't link you to the Chinese or Japanese materials, because I bought them in Shanghai and Tokyo. Also, they aren't written in English. The Googlers next door use Russian materials from Moscow, also not written in English.

It's hard to do better than Singaporean materials, which are in English and modified (not in a bad way) for the US market, which you can find at SingaporeMath.com. Their Primary Mathematics series is superb. I use the Standards Edition, which is said to track the California State Math Standards. That sounds ominous, but actually the state standards are excellent. The districts essentially ignore them by using a ridiculous "reform" curriculum that, being "a mile wide and an inch deep," will always include a checkmark every year for any topic you can think of, thereby covering anything mentioned in the state standards (superficially and in random order).

Note that for these Asian curricula, you REALLY need to know how to teach the math. The textbooks only provide visual aids and example problems, not the tutorial text (paragraphs of explanation) typical in US books. If you go for Singapore Math, you should get the Home Instructors Guide (at least for a few levels), which teaches you how to teach it.

And DON'T start a kid at too high a level. Use the placement tests downloadable from singaporemath.com to decide where to start. It's all about carefully building up from the bottom, mastering each level before moving on.

[mikevm]

Thank you very much!

[peripetylabs]

We were taught sine, cosine and tangent in the context of how they could be used to derive angles from other angles, not what they were and how they worked. They were presented as tools that could be used in particular ways that had to be memorized.

This has to be the worst things you could do to a student in a math class. In engineering they call it "plug and chug" -- students must plug numbers into a formula they've memorized and come up with an answer.

By the way, we learned trigonometry with the unit circle. If we forgot a formula, we'd just draw a little circle and derive it. I'm always grateful for that teacher.

[dkd]

same here, though i learnt it during my university years not at the high school.

[Bognar]

> In fact, if I'm totally honest, I'm not 100% I completely understand the sine function now.

I've found this .gif does wonders for explaining sine and cosine to people:

http://www.butlercc.edu/mathematics/math_courses/ma140/SineC...

Sine is horizontal, cosine is vertical.

[lappet]

I share many of your thoughts here - we learn so much about using mathematical tools but we don't know why they were invented in the first place. I was wondering if it would be useful to understand the history and basis of concepts like the sine function - would that help me connect to them better?

[VLM]

Your post is insightful. Imagine trying to teach philosophy like American schools teach math. That would be unimaginably awful. Which makes me thankful that (most?) American high schools don't offer philosophy class. Superficially it would be good for the students if done right, but since it will not be done right...

Even worse imagine history taught that way. Whoops that's pretty much how they do teach history and (coincidentally?) that doesn't work too well either.

[joel_perl_prog]

Well... I don't comment often, but as someone with a B.A. in mathematics who minored in philosophy, let me follow up on your comment:

Would a philosophy class really be good for high school students? Broadly speaking, I mean, not those 1 out of 100 students who are reading Sartre or Nietzsche (or even Dostoyevsky or Kafka) on their own, already, anyway.

A survey course, I mean, a 101-type of course, like you would see at a University. My feeling is that a course in introductory logic is the #1 most useful course that's missing in high school right now.

And again, there are going to be a few students for whom this would be redundant, but something like 99/100 students do not understand the machinery of thinking. If there's one thing my university education taught me, it's the machinery of thinking, carefully and rigorously. Most people never learn this, and I feel like this then precludes any understanding of anything advanced and even slightly abstract--and that includes both philosophy and (of course!) mathematics.

[fyi80]

> that a course in introductory logic

This is barely touched in standard geometry text books. 2-column proofs and all that.

But most people don't notice it, and it deserves far more treatment. Many of the smartest people I know attended special gifted programs in K-12 that did teach formal logic and informal critical thinking skills.

(correlation != causation, though)

[doktrin]

Philosophy is mandatory as part of the IB curriculum. The course I'm thinking of is typically named "Theory of Knowledge", and while formally an epistemology course, it can be taught quite broadly.

I can't speak to anyone else's experience, but I found being exposed to Philosophy while still in high school to be quite meaningful, and went on to dual-major in it while in college.

[dagw]

As another IB student I can concur that I found the Theory of Knowledge class very rewarding. And the things I learnt in that class have probably given more than the stuff I learnt in math or physics class, despite going off to study physics at university and then switching to math.

That being said I also remember being one of the few students who felt the course interesting and worthwhile. Most of the other students saw it as a waste of time and focused mainly on how to get a good enough grade to not affect their overall average while doing the least amount of work possible.

[john_b]

> What's more, people who "can do math" should be taking the blame for those who say "I can't do math" rather than using pointless semantics to wag a finger at them.

You're invoking a false dichotomy here by assuming that one of the two groups (if they are even well defined at all) should be assigned blame and the other should be held blameless.

It's debatable whether blame should be assigned at all. Many people who do not suffer math phobia live lives where advanced mathematics is rarely, if ever, needed. These people "can do math" but simply find little practical need for it. If their lives are no worse in the absence of serious mathematics, I see no reason to intervene. That said, I do think we would be better off with a more mathematically literate society.

In recent years it seems like there has been a great deal of collective guilt and introspection by the technically literate. It probably has a lot to do with the rapidly increasing difference in one's quality of life that deep technical knowledge of various kinds can produce for individuals. It will never be productive to launch crusades with mottos like "everyone can program!" or "everyone can do math!" because these crusades presume that everyone who can do X should do X. A far more productive use of our time and energies is to expose children to these disciplines early in their lives and be honest with them about the potential rewards (practical, personal, and aesthetic) they can bring. There is no need to blame anyone or try to make anyone feel guilty.

[enraged_camel]

>>Many people who do not suffer math phobia live lives where advanced mathematics is rarely, if ever, needed. These people "can do math" but simply find little practical need for it. If their lives are no worse in the absence of serious mathematics, I see no reason to intervene.

This is highly debatable. We had an economic meltdown just a few years ago, and one of the (many) reasons for it was that people were taking loans that they could not afford to pay off later, given their income, assets and expenses. Many of those people were victims of predatory lending because their math knowledge was so poor.

[zecho]

I would be careful to assume poor math knowledge where greed could just as easily explain the housing bubble.

[enraged_camel]

Hence my phrasing. I said "many of those people were victims." Others may simply have been greedy.

[diroussel]

Commenters in this thread might not think compound interest is advanced mathematics. But you're right, the majority don't have enough confidence in maths to attempt basic finanicial planning, and this is an issue for society.

[thedufer]

I think you're using different definitions of math. He clearly shows that he includes basic arithmetic in his definition. And yet, you started off by saying that arithmetic and algebra don't count as "doing math". I'm not really surprised that you came to a different conclusion by defining words differently, but I'm unsure as to why that's relevant.

[Irregardless]

> He clearly shows that he includes basic arithmetic in his definition.

He mentioned arithmetic just to show that he's not being 100% literal about the "I can't do math" statement.

[zabraxias]

This reminds me of the comment PG made about people with the most negative rebuttal to the argument getting up-voted.

The argument wasn't entirely about how you perceive math and more about being socially accepted, even proud, of being able to say "I can't do math". It's a very North American perspective and it would be similar to saying "I can't read" in most of Europe.

[chadillac83]

There are also learning disabilities. I've struggled with Math my entire life, from simple arithmetic in elementary school all they way up into high school mathematics. I believe I suffer from "math anxiety", although I've never been diagnosed. When doing math problems it's almost like dyslexia for me, the numbers get jumbled in my head... I know the basic processes I need to come to an answer, but everything in between gets twisted and mixed up along the way.

[diminoten]

How is it my fault (as someone who "can do math") that someone else can't/won't do math? What do you believe I am morally obligated to do that I'm not doing with regard to someone else's preference/understanding of mathematics?

[Irregardless]

That's mainly directed at the person who wrote the essay (a math professor) and others like him who blame the students rather than themselves. I think he should be examining the state of math education in this country rather than chastizing the people it has failed.

[jaynos]

It definitely comes down to the educational issue. I'm a mechanical engineer and have gotten pretty good with "math tricks" (e.g. 103x7 is difficult, but I can compute 100x7+3*7 much easier and know to rewrite the problem to be done in my head).

A trick like that should be pretty simple to figure out, but grade school math is taught in such a rigid fashion that students (who later become full grown adults) don't think to try it. Think about the last time you went to dinner with friends. How many calculators did it take to figure out the bill? Here in New Jersey, tax is 7% and 18% gratuity is pretty standard. Add up your meal and add 25% which you should be able to do in your head since you just need to divide by 4. Yet, last time I went out to dinner, the lawyer, accountant, and two physical therapists (i.e. 3 years of grad school) all pulled out their iPhones and then looked at me with confusion when I tried to explain the 25% solution. I'm not sure they need to be able to do the math in their heads (part of my job involves doing quick math in my head, so I have more practice), but the logic behind it shouldn't confuse them.

I often hear people talk about the need for high school classes that teach people how to balance a checkbook and other questionably useful skills. If you have to teach someone to balance a checkbook, you've already failed them. You've missed the part of education that should teach and develop the logic to make the checkbook lesson take 1 minute.

[rtpg]

>Why is this? If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach. Those who attempt to do so often do it very poorly, which leads students to the false belief that math is extremely difficult. On the other hand, it's very easy to teach someone to memorize formulas and plug in numbers, so that's what we're most often taught in math class. That's good enough to get us through the standardized test so we can graduate from high school, but memorizing formulas and plugging in numbers is not "doing math"

This got me thinking a bit: What if it's the opposite?

I know I'm really bad at memorizing formula, and I need to really figure out everything for it to stick in my head. You say it's easier to get people to do that, but loads of people have bad experiences with math, so maybe it's because we teach it that way and not in spite of?

Trying to apply memorized formula to a problem is a form of pattern matching, and it might be harder because of all the doubts from the abstraction. This might end up being less efficient than we would originally think (oh I memorised all this, but I have no confisdence in using it...). In the end we have just displaced the difficulty to something a lot less tangible.

Maybe we should try lowering the scope of what is taught, but really try to make sure people can use what they learn, even if it's small.

[dragonwriter]

> If I had to guess, I'd say it's based on the fact that math involves a lot of critical thinking and critical thinking is very difficult to teach.

I don't think critical thinking is very difficult to teach.

I think critical thinking is, despite being foundational, not prioritized in most educational curricula (and, particularly, not in most of the high-stakes testing regimes which we use to evaluate students, schools, teachers, etc.), and consequently insufficient effort is put into teaching it.

[thetabyte]

And I'm going to go one step further:

Critical thinking is not prioritized because it is hard to evaluate.

Due to the demand for teacher accountability, the insane level of competition for college entry, and the political games surrounding education policy, modern public education is entirely centered around examination and evaluation.

Not only is critical thinking challenging to evaluate, but, more importantly, people—read, parents—do not accept evaluations that report bad critical thinking skills. If a child can't answer 2 + 2 or who President Washington was, then they clearly didn't know. But if you ask a question that truly challenges critical thinking skills, and the child receives a bad score, the parents will be marching into an administrator's office with complaints of "trick questions" and "unfair grading". And fear of parent backlash drives American public school administration's decision making.

[dragonwriter]

> Critical thinking is not prioritized because it is hard to evaluate.

I don't think that's the root cause for why its never been considered a core skill and treated (when treated at all) as sort of an optional additional skill usually addressed, if at all, late in schooling as part of the English curriculum.

But I do think that's an additional challenge to getting it treated as a core focus in today's testing-obsessed public education context.

[calibraxis]

I think critical thinking is the opposite of what most schools want to teach. Take Khan Academy founder's book [1] which describes the mainstream "Prussian Model":

"The idea was not to produce independent thinkers, but to churn out loyal and tractable citzens who would learn the value of submitting to the authority of parents, teachers, church, and ultimately, king. The Prussian philosopher and political theorist Johann Gottlieb Fichte, a key figure in the development of the system, was perfectly explicit about its aims. 'If you want to influence a person,' he wrote, 'you must do more than merely talk to him; you must fashion him, and fashion him in such a way that he simply cannot will otherwise than what you wish him to will.'"

Chomsky further discusses the important role of "stupidity" in the educational system (like stupid assignments), in teaching obedience: (http://www.youtube.com/watch?v=pFf6_0T2ZoI)

[1] Salman Khan, "The One World School House"

[bicknergseng]

"I can't do informal semantics."