You never did math in high school
j2kun.svbtle.com
j2kun.svbtle.com
when a freshman college student tells me that he was always good at math, it translates to “I was very good at following obscure steps to manipulate mysterious symbols, without any real understanding of what they mean.”
The most common complaint I hear about undergraduate students is that they aren't able to blindly manipulate symbols, and consequently get hopelessly confused when they have to deal with "unintuitive" concepts.
That's who the current system panders to, because they're the ones who scream loudest when it doesn't. And they find it morally offensive if you ever ask them to solve a problem that isn't extremely similar to an example they've already seen.
And they're the ones who think they're "good at math". That's who highschool caters to because that's what standardized testing caters to and that's what the noisiest parents and students want... and teachers that have higher ideals are swimming up-stream if they want to actually challenge the kids.
"The trick to flying is to be able to throw yourself at the ground, and miss."
The quote does appear in HHGTG.
[edit: I don't have a copy of the original radio scripts on hand, but from the book:
"The Guide says there is an art to flying," said Ford,
"or rather a knack. The knack lies in learning how to
throw yourself at the ground and miss." He smiled
weakly. He pointed at the knees of his trousers and
held his arms up to show the elbows. They were all
torn and worn through.
"I haven't done very well so far," he said. He stuck
out his hand. "I'm very glad to see you again, Arthur,"
he added.
-- Life, the Universe and Everything
High school teachers, please make your students manipulate mysterious symbols more.
It's funny because when that article started I was totally on board with him, then we got there and I just laughed.
Isolating a variable is, in some sense, a blind manipulation, but so is, "I see a plus sign, that means I have to subtract". As a first thought about a problem, they represent very different understandings.
Math education is not homogenous. People are not homogenous. Everyone experiences math in different ways both in school and in everyday life. Some people will simply memorize the steps necessary to solve problems. Some will actually grasp the theory behind the problem solving. Some will experience very little of what is considered "classic" high school math. (At my school, those who are not tracked into calculus or statistics as Seniors take a course that deals with things like formal logic. It actually seems really interesting.) The OP's blanket assertions are simply wrong.
Moreover, math is a massive field. The OP's claim that calculus and algebra are not math but that "pattern recognition" is is just absurd.
As someone who moved to the states at 13 I have to agree with him. Most teachers in America are forced to prep their students for exams instead of teaching them how to think critically. Instead of building an intuition, students are forced to memorize rules and equations so that they can finish their standardized exams in time. The problems done in class and for hw are of the same format as the exams, with the numbers used being the only difference. I've seen plenty of college students get offended when a professor puts a problem on an exam that requires some critical thinking.
I did my math degree at a school that had a special program for future math teachers and they were consistently the worst performing students in my classes. Abstract Algebra was a disaster for them, most couldn't put together even the simplest proofs. With exams approaching, they all begged the professor to tell them which proofs they were required to memorize. Somehow I got by with perfect scores by understanding the content while they had a tough time pulling off anything above 20%.
Put real problem solving on the exam and this goes away. Of course, then teachers who can't problem solve (most of them) will be up in arms...
Now, I've met my fair share of people who could not problem solve their way out of a paper bag with a pair of scissors. And I would love for generic problem solving to be a focus very early in education. But high school is likely way too late for that, and so I think all this anger with high school math is being misdirected.
Math101 was the only paper I ever withdrew from. One lecturer was an Indian who barely spoke English, the other with a stutter who was constantly asked by fellow students to speak up. I later took a more complicated Mathematics paper, Math170, which I blazed through. It was simply because the lecturers were more engaged. They were far less 'experienced', but simply had more people skills.
Oh, you should watch CS people. Mathematicians are social butterflies in comparison. They mostly write proofs for other people, not machines.
High School:
Classes/Week = 5
Time/Class = 45 min
HW Frequency = Every Day or Every few Days
Exam Frequency = Every Week or Every Couple Weeks
University:
Classes/Week = 2 or 3 + Discussion
Time/Class = 90 or 120 min
HW Frequency = Every Week or Every Couple Weeks
Exam Frequency = Every Month or 3 per Semester
My sister-in-law is in high school, at a school that has fewer courses per term, and fewer per day, and going more in-depth in each class; I'm really glad she gets that experience.
More context switches only hurts the student's ability to get into a subject.
More interesting is his link near the end, about how he teaches graph theory to college students: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...
An A4 sheet of paper, when folded in half, becomes an A3 sheet of paper with the exact same ratio of long side to short side. Given this information, can you figure out the what the ratio is?
I must of spent 3 hours of that car ride doodling on a piece of paper trying to nut out that problem with no additional information or tools, ultimately unsuccessfully.
To this day, I think that was when the light switch went off for me about math as a creative problem solving endeavor and not just a rote series of calculations.
If an A3 sheet is X units by Y units (with X the long side) and A4 is 2Y units by X units then their ratios are X/Y and 2Y/X, respectively. If these are equal we have
2Y/X = X/Y
cross multiplying gives
2Y^2 = X^2
rearranging and taking a square root (X and Y are positive) gives
X/Y = sqrt(2).
[1] https://www.hobbyking.com/hobbyking/forum/uploads/56/A_size_...
Wait!
If somebody tell me before that I can use it for get out of a jungle... I could have listen better.
The class could have started like this:
"You were traveling in plane, when suddenly, it crashed. Nobody else survived. You don't know where you are. It look like a jungle. Not civilization around, no cellphone, nothing. You will die in 1 week if not reach civilization. How can you escape?"
And if in history (at the same time that in maths) we talk about the man (Pythagoras), and about the compass. Then in social about the problems in traveling in the ancient cultures. Geography about maps. In artist class ("Art" class was more about technical diagrams for me, we never ever do oleo or similar stuff), how draw maps and in spanish build histories about it. Then all the clases related to each other. Probably the math class must be the last of it, to make this build-up effective.
In short, all the classes connected around the theme of the week or what are we doing at the moment.
Even the old O-levels, taken at age 16, were equivalent to university level courses today. This fact is often used as proof that kids today aren't as smart at kids in the 60s, but in fact most kids in the 60s didn't take any exams whatsoever and just left school at age 15 with no qualifications.
"A certain well known mathematican, we'll call him Professor P.T. (these are not his initials...), upon his arrival at Harvard University, was scheduled to teach Math 1a (the first semester of freshman calculus.) He asked his fellow faculty members what he was supposed to teach in this course, and they told him: limits, continuity, differentiability, and a little bit of indefinite integration.
The next day he came back and asked: What am I supposed to cover in the second lecture?"
(from Spiro Karigiannis at http://mathoverflow.net/a/53238 - comments below refer to a version of the story where the professor is from the USSR specifically)
It's actually telling of the quality of education, given that I did better in Math in college than in high school (in North America), and I still found it not anywhere rigorous enough.
With that said, I've generally heard foreign students equate the junior year of a math degree at college to their senior year at high school. I don't dispute your general point.
I went to public school in southeast Ohio. Quadratic equations (for me) were 8th grade. Calculus started midway through 10th, but didn't really get moving until 11th grade.
"The Standards set grade-specific standards but do not define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations."
I.e., this defines what constitutes "grade-level" skill sets, but lots of better schools will exceed those standards by one or more years - their 10th graders may be operating at the 12th-grade standard, etc.
Every year, our students spend the budget of a James Cameron film on failing to understand calculus...
Honestly, college math wasn't much better. Rather than confronting them with critical thought, we simply replaced the old hack with a new one:
1. Recognize the "form" of the problem, corresponding to a section in the textbook.
2. Plug the parameters of the problem into an equation solvable by the method of that section.
3. Apply the method and write the answer.
When I realized this, I told my students about it.
There was a chapter on maxima and minima. But the only function they had learned was the quadratic, so all optimization problems boiled down to arranging things into a quadratic. The text had them graphing each quadratic. I decided it would be more interesting for the kids to see how we make our own formulas, so we derived one for the optimum of a quadratic function, and memorized it.
Now is this math or not? Well, I think there's a place in math that involves classifying the forms of expressions and equations, sort of like taxonomy in biology. But it shouldn't be the only thing.
I mean this is an old refrain, and not one I disagree with, but it's missing the positive side of the argument.
Foundations of Higher Mathematics: http://www.amazon.com/Foundations-Higher-Mathematics-Peter-F...
How to Solve It: http://en.wikipedia.org/wiki/How_to_Solve_It
EDIT:
Mathematical Reasoning: Writing and Proof: https://sites.google.com/site/mathematicalreasoning3ed/
Inquiry based learning is the way math should be taught. Teachers should pose interesting problems and guide the students to a solution instead of frantically rewriting equations from their notepads onto the blackboard.
For example, he attacks geometric proofs. Obviously, geometric proofs are useless and pointless, except that they represent a very simple and intuitive sandbox for learning proofs. Which is fantastic. How better would you teach kids to understand what math means beyond arithmetic? You give them a bunch of simple rules that make obvious sense, and then show them how you can use those to prove non-obvious things.
Our curriculum has all but dropped geometric proofs and the kids are poorer for it.
I think our curriculum here in North America (I'm in Ontario, but I imagine Americans have very similar classes) is fine - the problem is that people are afraid to make the problems really hard and force the students to really stretch their skills beyond just "same as the example but change the numbers". That's what's missing, not some esoteric subject matter, just taking the existing tools in the existing toolbox and pushing the kids to build higher instead of more.
And I did fine with math later on, in including my Ph.D. in applied math (stochastic optimal control) and peer reviewed publications in applied math -- with theorems and proofs.
The OP has a small point but takes it way, way too far.
People learn how to drive a car by doing it. They aren't encouraged to discover the laws of combustion or shown the magic of kinematics. They just learn how to press the petals.
I always see articles that claim I didn't do math in high school, and then people talk about things like this. I did this in high school. In the US. In 2001.
Literally, one of our test questions was: (a) prove the product of two even numbers is even (b) prove the product of two odd numbers is odd (c) prove the product of an odd and even number is even
And this wasn't regurgitating a proof we learned in class or had on homework, it was a question we had not seen before.
Blanket statements about what people see and don't see in high school are worthless.
This is why I think programming should be taught starting in elementary school. It gives you an application for the math you need to learn. I never cared about learning algebra, calculus, or trigonometry until I started programming. Now I wish I had paid attention and now I am going back and self educating on the subjects in order to empower my programming.
And yes, my school did mathematics in high school. We went through much of Euclid's Elements.
In my school and country, we take calculus 1 & 2 in grade 11; that's two years before we go to college. In grade 12, we take differential equations.
For example, in my high school, in a fairly small rural school, we were taught calculus in senior year. I was bored, and did it sooner, but that was the general rule. We also had things like advanced placement biology, physics, and so on. Generally speaking, at the end of those courses you can take the optional AP tests, which allow you to skip the Physics I, Calc I&II, etc., that are the normal fare for freshmen.
I can't recall anyone who had diff eq taught in high school, though we really did end up using it anyway in advanced physics. To roughly sketch the limits of what we did, I remember needing Green's theorem, and having to prove things using elliptic integrals, but nothing much beyond that.
I'm not trying to be cynical or discouraging, but dial back the certainty in your genius methods a couple of notches until you've done more than just gotten kids interested for an hour.
I think this is reasonable, as "math" is essentially just another name for "meta-arithmetic", and we draw similar boundaries between science and engineering, for example.