A Mathematician's Lament: An essay on mathematics education

[marcusesses]

Remember schools are to education what lawyers are to justice.

Schools are there to look after the kids while parents go to work, to give local politicians something good and wholesome to support and to give teacher's unions a reason to exist.

If the children learn anything it's pretty much a happy accident.

How would you suggest students learn mathematics?

I'm not disagreeing with you, but it seems that if schools are not (or are unwilling to) educate students, that job would fall to the parents, who may not have the requisite knowledge to teach math.

Are you suggesting a less-rigorous, more informal curriculum would work (i.e. learn to solve problems as they come up, but allow students more freedom to learn what they want)? If so, I lean towards agreeing with this, although it would cause problems for teacher's unions, universities, working families...

 

[Mu naught]

It's an old complaint, in 1900 British scientists were complaining that British education in science,particularly chemistry and engineering was so far behind Germany's that British industry would never be able to compete.

A century later it's clear that British chemical and car industries have nothing to fear from Germany's. America is in the same boat - it was able to invent the atomic bomb and go to the moon without relying on foreign education.

I wonder how many of the people who made those things possible were "pure" Americans?

 

[bignum]

I think the fundamental problem is not the students (on some level, it is), but it is the teacher's fault and the media.

Media says doing Math is for geeks, delinquents thinks it is true, and they hate math.

Incompetent Math teacher discouraging a talented Math student, Math student hates teacher, Math student hates Math.

 

[thrill3rnit3]

http://www.stmarksschool.org/academics/mathinstitute.aspx

Take a look. I think his method is pretty interesting.

 

[eof]

I remember having read Lockhart's essay a while back. I think my biggest issue with math is that we do not tell kids why the boring stuff we are learning should actually be appreciated. With this I mean Alfred Whitehead's famous quote:

Civilization advances by extending the number of important operations which we can perform without thinking about them.

A case in point would be addition. It would have been great if in later classes someone would have e.g. told me how this boring procedure that was done over and over might not have always been so trivial. With this I mean the following observations:

1. The reason why addition is simple is because of the base-10 system in which we write numbers (any other base would do too, so the key idea was the invention of based numbers).

2. Try adding by using e.g. Roman numerals.

3. When doing (2) you will actually cheat, because in your head you add in base-10.

4. The name of numbers in our language are in base-10 which makes them easy to add. Some ancient languages had random names for different numbers and could only count easily to some limit (i.e. up to where they had words for these numbers). We can easily add e.g. 2+3 but if the numbers where e.g. 27+35 and all our numbers had distinct names without a pattern say up to 100, could you do it?

 

[Astronuc]

Those exercises are visual training for kids to prepare for algebra; they are not 'instead' of algebra.

If you ever teach math to children, you will find physical blocks ('math manipulatives') to be very helpful in conveying concepts. This is just a paper version of making the question out of blocks. Remember the age level of the kids this is aimed at.

http://en.wikipedia.org/wiki/Cuisinaire_rods

Once kids can do the algebra, you don't need stuff like this any more. If a teacher is saying that you 'can't use algebra,' that is a problem with the teacher.

Why not just teach algebra, wouldn't that be easier - ie. why not teach the easy way right from the start, instead of teaching them the hard way first? Wouldn't a kid pick up bad habits by thinking in blocks rather than algebraically? (Or is this preparation for powerful diagrammatic methods like Feynman diagrams and graphical models?)

I loved the Cuisinaire rod set. My school in Australia introduced students in grade 2, possibly grade 1. That was back in 1964-1965, when I was about 6-7. It was helpful to visualize addition, multiplicatio, subtraction, division, factorizing, equivalence of 3x2 and 2x3. My parents even bought a set for my sister and younger brother.

I was surprised when I came to the US that they were essentially unknown, at least where my family was living. I found the US method of teaching math somewhat archaic and unspiring. I skipped half of 3rd grade because of the move, and found 4th grade quite easy, especially when it came to arithmetic as it was called. I had my math workbook confiscated because I preferred to do math problems during music lessons, and the 4th grade teacher was a bit upset that I was way ahead of the class. I also got marks off because I did the math in my head, and I didn't have to cross out numbers when carrying.

On the other hand, I was terrible in English (literature, writing, . . . ) and especially poetry. I found reading fiction stories excruciating because my mind was elsewhere. I got into trouble for not reading fiction books, which I didn't because I wanted to read books on geography, rocks and minerals, rockets and spacecraft , . . . . , and occasionally history, or other non-fiction.

And I got D's in hand writing because I just didn't write well. Writing cursive was painfully slow and my hand couldn't keep up with my mind.

 

[Sankaku]

And I got D's in hand writing because I just didn't write well. Writing cursive was painfully slow and my hand couldn't keep up with my mind.

I decided, for some reason, that I hated cursive writing. I stopped suddenly in something like grade 5 and refused to do it for ever after. I print to this day...

Just an update to the algebra vs blocks idea. My daughter is now in Grade 4 and starting to grasp basic algebraic concepts. However, making something very abstract (choose R for the red apples and G for the green apples and then manipulate the symbols around) is still too much of a leap for her.

There is an interesting algebra game we play that is building some of the concepts, but going straight to paper is not quite happening yet. Maybe within a year - each child is a bit different.

This is the kind of hands-on algebra game...

http://www.borenson.com/AboutHandsOnEquations/WhatIsHandsOnEquations/tabid/1003/Default.aspx

Except we just made our own with some counters (for x), different coloured dice (positive and negative numbers) and a hand-drawn balance. There is a lot of stuff you can do with the basic idea of "do the same thing to each side of the balance" that young kids get fairly intuitively.

I like games that have a mathematical connection. The card-game "Set" is one of my favorites and kids pick it up very quickly.

http://www.setgame.com/set/index.html

 

[Chubigans]

I've been thinking this since I began my mathematics education. I'm glad this is a known problem. Thanks for the enlightening, inspiring, and fun paper!

 

[SonyAD]

The "lyceum" (a high school with math & comp. science curriculum) I went to didn't teach me a lot of stuff I wanted to know (rotation equations, rotation around an arbitrary point, translation, the math of texture mapping, fish eye projection, camera tracking etc.) which I found out on my own. I was taught some stuff that's served me well, though, some stuff I already knew, some that I've never needed (integration and derivation, abstract algebra).

The math manual my teacher chose didn't even touch on rotation equations. She was, is, rather, a calculus and abstract algebra nut. I had gotten quite good at the algebra, not because I liked it, and have since forgot everything because it was all abstract silliness with no cool/immediately apparent application. You didn't need to remember as much clutter as with calculus, though.

I barely, barely passed calculus before the year was out. I didn't have internet access at the time and my folks hired a tutor. Even so, I barely scraped by. I didn't have any interest in learning all the calculus tables by heart and I would pretend to study while researching what I liked: geometry.

Specifically triangles, geometrical and numerical relationships of triangles. I built up a catalogue of formulas for lots of stuff, from the radius of the inscribed and circumscribed circles to the length of a bisector and the projection of a vertex on the opposite side, fish eye projection

$$Screen_{x} = CenterScreen_{x} + FOV{const} \cdot \frac{ R_{x} }{\sqrt{ R_{ x }^{2} + R_{ y }^{2} + R_{ z }^{2}}}$$

$$Screen_{y} = CenterScreen_{y} + FOV{const} \cdot \frac{ R_{y} }{\sqrt{ R_{ x }^{2} + R_{ y }^{2} + R_{ z }^{2}}}$$

, etc. Lost most of them over time, though.

In 10th grade I coded a software wireframe engine and made some scenes in Excel for it.

I showed the math teacher and she was like: You made that yourself? Bye.

You bet I would have liked a say in the curriculum and educational process. Or choosing whose class I was in.

On the other hand, my computer science teacher was brilliant. She used to train the olympic lot. But her great sin was she was in the bad habit of flunking kids who refused to think (some of whom did really good at math, wouldn't you know), which is all you really need to program, and grading kids fairly. So, naturally, the parents rallied with pitchforks and torches and pressured the head master to have have her replaced so in came the dumb airhead.

Most math teachers I've come across have seemed somewhat dull. They're good at regurgitating and applying arid, abstract math but can't develop new stuff on their own and crickets chirp if I ask them to help with something of practical value. To be fair, crickets chirping whom ever and wherever I ask to help me with something math related.

 

[tete-a-tete]

Where did you get this information? How many American high school students know how to express tan (x/2) in terms of sinx and cosx? Be honest: do you know? How many know the half-angle formulas ? All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads. I don't know about the situation in Japan, but I assume it's similar.

Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

The difference between China's math education and that of the West is that there's very little plug-and-chug in Chinese math classes: all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?

memorizing equation seems is the only way i do maths, i try to understand materials in western textbook, usually have large extend of texts combine with graphs, equations. The eastern way of doing maths require lot of hard work and practices. As long as you are keen on the subject and you know how much effort you willing to spend on the subject, that's all it about.And surely you will find the most comfortable way for your study afterwards.

 

[Diffy]

Also problems about piles of wood are pretty stupid. No one gives a damn about piles of wood, and if you talk to a professional logger they'll tell you that the problem is bogus anyway. Since I've taught at the University of Phoenix if I have to come up with a word problem it would be something like.

1) You just lost 40% in your 401(K) last year. Assuming that your employer doesn't/does match contributions this year, how much do you have to contribute to reach your retirement goals assuming the Dow goes to 12000, 10000, 8000, 5000?

2) If you were to get laid off tomorrow, how much money in the bank do you need to survive for X months?

3) What increase in salary do you need to make your tuition in UoP a positive investment?

4) How much money will you save if you pay off your credit cards?

All you have to do is to mention three or four of these sorts of questions, and then Algebra 101 is no longer boring or montonous, and at that point you focus on concepts so that people have the skills to answer those questions, and other questions which life throws at you. Again, if the US had a decent math education system, my students would have learned all of this in the 7th grade, but better late than never.

I sometimes wonder if the reason that Chinese are huge savers is that most educated Chinese in China can do basic algebra whereas a huge fraction of Americans can't. If you can't do math, you are going to have to rely on the bank to do the math for you, and you are likely to get screwed since the guy at the other end of the table knows how much money he can squeeze out of you, and you don't.

(How much is this adjustable rate mortgage going to cost me, if interest rates go up to 8%? How much do house prices have to go down before I'm underwater?)

I'm not sure about University Algebra, but that is not really the point of an article that addresses math education in K-12.

The author addresses the attempt of math teachers trying to make math questions relevant to students and explains why this makes math boring for them.

Anyone who has been in front of a classroom of 9th or 10th graders would tell you these types of questions would immediately put your class to sleep. These students are simply too apathetic.