Practical Answers: what background do kids need to compete in technical subjects in college?

Mark's question on this thread was: A practical question for you. What does a child need to have learned by 12th grade to compete in the
technical fields at a top-notch school? I don't think that the kids in other countries are inherently smarter than our kids -- or at least
not enough to matter.

The question is important, and I don't claim to know the whole answer. But I think it's a big enough discussion that it warrants its own post, and my answer was too big to really be a comment. My answer is in two parts, the first part a practical answer, and the second part the anecdotal experiences I've had that explain why I've come to the conclusions I did in the first part. But first, You're right. They aren't smarter. What they are is well educated in the basics.

I haven't read the other comments in detail, but I don't agree that the kids need to know calc and Differential Equations before college. But practically speaking, what entering college freshmen need is an understanding of algebra, geometry, trig, algebraic geometry, that is proper, thorough, and deep--deep enough that they understand (even if they cannot prove) why you can add polynomials, for example, and they could understand a proof by induction. Basically, they need mastery.

So, specifically to the practical question:

I've just begun reading a set of textbooks that Prof. Wu recommended to me. (I literally out of the blue emailed him and asked what to do for this startup high school I'm involved with, and he answered--right when he was working on the big NMAP paper, too! What a generous man.) They are Japanese texts for 7th- 9th grade, translated, and available from U Chicago (don't let that dissuade you!). They are written by a mathematician Kodaira. They are thin, and not appropriate to non-higher-math-literate teachers. That is, they need some real fleshing out to create lecture notes from, probably, even for a homeschooler.

They are terrific. A child, or a class of children that really had mastery (Singapore Math as background) could handle it, and they'd learn a great deal. The 7th grade book has 7 chapters. That's it: 7 chapters. They are: integers, positive and negative numbers, "letters" and expressions--i.e. variables, equations, functions, plane geometry, and solid geometry.

The chapter on integers begins with prime numbers and factoring, common divisors and multiples. It introduces exponents. The second chapter moves on to how to handle positive and negative signs, and how to understand the rule (for example, for multiplication of positives and negatives, they go through some really nice examples of water entering a cistern, and flowing out at a different rate, etc.)

The book goes on to simple algebra--how to use "letters" and quantities. It teaches how to write expressions, how to simplify them, how to turn sentences into expression. Moving on, they learn substitution, linear expression, and then solving linear equations. They on to functions. Then plane figures and geometric solids.

But everything presented is sophisticated in that everything goes through great length to teach the rule and *why* the rule is true--with excellent examples, but the examples are very sparse, and there are not enough problems presented for a student to gain mastery. A teacher needs to add to this skeleton, but the skeleton is terrific. Why? Because everything it does prepares for a deeper understanding later. Every word is chosen carefully to make sure that the right lesson is being taught. Their explanations of equality, for example, are based on a balance or scale, rather than immediately on "solve for X". They make explicit the following rules: if A = B then A + C = B + C; if A = B then A/C = B/C. This makes it easier to learn more sophisticated math later because the scaffolding is correct.

The Japanese kids don't seem to learn calc before 11th at the very earliest, and certainly some learn it later than 12th. They appear to take various electives in senior year: discrete math, statistics, number theory. It will not be the last time they take those courses.

This leads to the next issue: never teach your child to test out of any course. Teach your child to retake the "basics" at every possible chance. If your kids learn calc in high school, convince them to take the Honors Calc sequence as freshman in college, or if they've still got time, do it their senior year from the local college. Take the same "syllabus" worth of material in more rigorous forms over and over again, rather than skip ahead. Building mastery is so incredibly important that it just makes everything easier later.

And this leads to the last problem: not only do your kids need grounding, they need to learn how to WORK HARD. This is difficult if they are naturals. The most important way for them to learn this is to see that other kids are brighter/faster/better than they are at something, but then learn that hard work matters more in the end.

So find a way to make sure they aren't the best kid in the room. Send them to math camp, physics camp, or music camp. Before they are 15, send them somewhere where they are merely average, but the structure and discipline of learning to work hard brings them above average. I think 15 is the last time they can really learn to catch up on the discipline, esp, if they are far ahead of the typical student. They need a couple years of knowing about discipline before college strats.

Our naturals have never been challenged in grammar or high school. Even if they are gifted enough to ace college, to get into grad school they are competing against kids who took a national test against 100 million other kids, and ranked in the top 20. Not 200, 20. The idea that they will suddenly know how to work hard enough in college to compete with kids who have been working hard for years is unreasonable. It simply takes too long to acquire the skill of learning how to work a problem, how to have discipline, focus, etc.

On to the anecdote, now.

20 years ago now, I took Calc BC in my junior year of high school, and I was 14 at the time (I was 15 by the end of that school year). My senior year of high school, at 15 and 16, I went to UC San Diego and took 1 quarter of multivariable calc, one quarter of linear algebra, and one quarter of Differential Equations. I got As in all the courses.

I didn't understand anything about Linear Algebra or Differential Equations. wrt Differential equations, that's pretty normal, because the courses are almost always taught poorly, without any overarching coherence to the material. But wrt linear algebra, I managed to get the highest score on the final and I never grokked the concept of the "span" of the space. It was bewildering to me. I could do the methods they asked, and the course was simple enough that I never had to guess what method to ask, so I could still ace it. But I understood nothing.

This is not surprising--I'd never seen this stuff before, and intellectually, the leaps I was asked to make were huge. I wasn't going to get it in one quarter, period, even if I was a "natural". But the issue was my lack of preparation: in all of my math schooling, I'd never really been given a basic understanding of what a system of linear equations MEANT, even though I must have seen them in (Dolciani's) Algebra 2. Now I was being asked to make that leap, and it was too much.

Then I got to MIT. I was a "natural" but I wasn't any math prodigy there--I was well more advanced in coursework than most freshmen--nearly everyone there HAD to take calc (even if they'd passed AP Calc AB), and had never seen differential eqs or linear algebra, but I was probably as good at math as the average physics major (my original major) and better than the average engineer, and I was definitely completely outclassed by the math majors.

The dept gave me credit for all of the calculus, including multivariable, and for differential equations, but not linear algebra. I took it again, and I barely got a C, for one main reason: I still didn't get it, but I had never had to STUDY before, so i didn't know how, and my trivial knowledge tricked me into thinking I did get it, and I didn't really understand how far I was from knowing what I didn't know. So I didn't really get the span on the 2nd time around either. Again--not that surprising for an entirely new concept. But how come I'd made it this far before seeing this concept?

So there I was, completely new to learning how to learn something I didn't know, not knowing what I didn't know, and with no way to bridge the gap. The successful kids who hadn't had those courses were far better offthan I was, because they were forced to study and learn the material well. But some of them realized quickly that they too didn't know how to do that. The kids who succeeded weren't the ones who were far ahead on material; they were ones who had been so grounded that these concepts were not intellectually new to them. This was true in physics as well. The successful students hadn't done AP physics in high school with calculus, but they knew how to draw a free-body diagram, which is a way of writing down all the forces on a system. They KNEW what forces to write down. They weren't sophisticated mathematically, but they were well grounded.

The foreign students, and the well educated math students HAD seen these concepts before they reached linear algebra. They still hadn't taken the course, officially, the way I had, yet they knew more of the material. They'd seen them in high school--they'd been given enough mastery of algebra that they really did have some intuitive sense of what a system of linear equations meant. They'd been given enough mastery in calculus (something AP classes simply DO NOT give)so that they understood functions better than I did, and that meant that when they went to linear algebra, they understood how linear operators work as functions. Over and over again, they'd been prepared for this material since junior high.

The only place where I ever caught up to my counterparts in my physics major was in statistical mechanics--because it was the first course where NOE ONE (except the Russians) had taken the material before, no one had seen it before. I was on even footing. (The Russians had done every problem we did as lower division undergrads in high school. All of them. Apparently, though, they never had any labs. This is part of the reason they were phenomenal: they were required to build their intuition from SOLVING problems and DISSECTING what the answers meant. No "hands on" manipulatives for them!)

For the folks who don't know what I'm talking about in the specific math stuff, here's the analogy: if you want to be an American history major, you take history in high school, American history as a lower division undergrad, and you take it again as an upper division undergrad. Each time, you're supposed to be getting a better understanding of how various factors, events, etc. influenced each other. You know more so you can make more intellectual leaps.

But if you never actually learned any history in high school, and you don't have the faintest idea what the main wars are, who the presidents were, when American industrialization began, then the undergrad course is way over your head. You barely can keep track of what happens in any century, and any lecture that assumes you know the value of the Missouri compromise and what impact it had on the Civil War and how slavery was the crux of that, you can't possibly hold these new ideas in your head. So even if you're a natural for reading and grokking what you've read about history, it doesn't matter--because you're so far behind compared to those who DID learn all of those facts and figures, and now can recall them at will to use as evidence. And that's where most of our students are, mathematically. They have no mastery.

The honest-to-goodness math prodigies were a different bunch entirely. I thought I was one (given I was taking calc bc at 14) until I met them. Maybe I could have been one, but I'd missed the window. They were kids who'd all been to math camp before junior year of high school. And of them, they broke down into who was a "natural" and who wasn't--nearly all of them decided they weren't really naturals after all. I'll write about that in another post, but basically, they'd been exposed to more number theory in high school summer camp than I would learn at MIT by graduation day with a bachelor's in math.

But that was 15-20 years ago. What's changed now? The answer is that the naturals are MUCH WORSE OFF now than I EVER was. I know this from being a grad student at UC Berkeley 5-10 years ago, and TAing CS courses. The entering freshmen/sophs there use the same text I used 15 years earlier, and I remember which parts were intellectually too big for me to jump. But that's not their problem: their problem is the trigonometry and the algebra II. They can't learn the hard parts, because they can't do the simple problem set problems that require them to know some basic high school math. They are so muddled they can't write down an algorithm to do something because they can't even compute the angle of something given some other related angles. They don't know how to divide fractions. They don't know how to compute sines and cosines. They don't know how to solve a geometry problem that requires a proof. They have no idea what a system of linear equations means. They don't understand what functions are, or what composition of functions means. Because of this, they can't dive in and learn the material asked of them, because they don't understand what the homework problem is asking for, and they don't know what they don't know. Their foreign and well taught counterparts do know, and are moving forward every day.