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.
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.
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.
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.
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?
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.