I remember doing sets and Venn diagrams, but I also remember rote memorisation through multiplication tables. Through all of the '70s long division and carrying the 1 in multiplication were the norm. I think I must have just caught the end of the New Math thing.
Long division is not New Math. They’ve been teaching it that way for centuries.
The idea behind new math, IMHO, was a good one but flawed. The idea was that the vast majority of students were learning how to do something with no freakin clue as to why they were doing it. This concerned many people because of what use is it to know a process but have no idea what the process is doing and what it can be used for? An additional danger was that students who would be great in math would be turned away because of the grinding/nonthinking nature of it.
{Side note - Math is a weird subject. You aren’t REALLY exposed to it until Junior/even Senior year in college so you get the tragedy of people thinking the ‘like math’…majoring in it and when the truely meet it (almost when they are done with college) they hate it. Heck, I’ve met some math majors who graduated and had never even done ‘true’ mathematics. Grad school can come as a huge shock.}
Anyway…:)…the problem is that, again IMHO, Math is like English. You cannot move students to writing masterpieces without them learning about the alphabet, grammer, sentence structure etc etc etc. There are huge amounts of nitty-gritty details that one needs to master before one can do these things. Math is the same way.
When I taught math, the students did the grinding…but I did make a serious effort to explain WHY/WHAT they were doing…often. Unfortunately, many math teachers (IMO MOST math teachers) really don’t know themselves.
Ah, the new math. I always understood it to be a move toward explaining the “why” of math and a move away from rote memorization. The math wasn’t different, just the way it was presented. Since I only went through school once it’s the only way I know.
I managed to get a master degree in math and I still don’t understand some of what they were trying to say. Right here on my desk is my 11th grade “Modern Introductory Analysis” book. Some of the stuff in there still makes no sense at all and I definitely can’t see why they approach the topics the way the do.
Compare this to the current debate in English - “whole language” vs phonics - to see the politics involved. In first grade (1962-3) my teacher didn’t “believe” in phonics so I actually never encountered the concept until college. A few years before we were given the new math, we had an experiment in new English (or something like that, late 1960s) that involved new ways to diagram sentences.
Is that what they were called? I remember those suckers, and Venn diagrams, different numbering systems (binary, octal, Egyptian, Hebrew, etc.), and four different ways to do long division (well, I remember one of them, and not very well). I see what they were getting at, that if you learn the concepts, techniques such as integral calculus will be a piece of cake. And everyone will be a scientist.
The problem is that I still have problems remembering my multiplication table (I can never remember the result of 7 X 8, for instance), and calculus was still a bitch. If you’re a “math” person, you’re going to get it more easily, and if you’re not a “math” person (like me), well, you’re not going to even be able to multiply or divide without a calculator very well.
Thanks for that link.
The last bit, with the made up “examples” from different eras of math education, doesn’t seem to belong in an SD staff report IMO. It’s just a joke, of course, but it seems likely to distort people’s understanding of the issue.
I was a near-victm of this fraud. Basically, the “traditional” math curriculum was arithmetic> algebra>plane geometry>solid geometry>trigonometry>advanced algebra>calculus. “New Math” put in all kinds of topics , at the the-5th-8th grade level-set theory, boolean logic, modulo arithmetic, etc. These topics were poorly taught by teachers who didn’t understand most of the material as well. I was so confused by this (7th grade) that I thought I was losing my mind…until I went to a parochial HS with the traditional math-then it all made sense.
Basically a Cold war effort by the education industry, to sell new textbooks and a lot of Ed.D theses.
You might want to bookmark this post.
I guess you’re not a Death Cab fan, then?
“New Math” is one of my pet peeves. I was among the first to be subjected to it. We would have groups of educators come in and watch us do set theory crap. For 5 years in a row we spend the first several weeks of the year doing unions, intersections, etc. Then we ignored it all and went on to study real stuff. It was boring, repetitive, and useless. Like aother posters, I never used set theory til college, and the basics are just not that hard to understand. On the other hand, my early exposure did not help me with the hard stuff.
But more importantly, this was just theory gone crazy. There is no real reason to teach kids that zero is the same as the null set. To me, you want to teach math as a progression of concepts needed to solve classe of problems. First there is counting/enumeration (I have one peice of candy and two donuts), then addition and subtraction (I have 12 donuts and eat three, what is left?), then you get into things like calculating what is bigger, a 40’x60’ plot of land or a 50’x50’ one.
Teaching kids that a set of one and a set of two equals a set of three is just too abstract. On the other hand, I think teaching number bases fairly early is a good thing, especially base two and base 16 which are so important in computers.
What really kills me though, is that New Math had an aura of being “scientific” without it being subject to double-blind studies as to effectiveness. Some assholes just sat around and made it up out of whole cloth. Same with “whole language” and other trends.
Add to this the fact that teacher pay sucks, and the administration makes one miserable, and you just don’t have the caliber of teaching we need.
DanBlather: I thought whole language was the traditional method and phonics the theoretical new upstart crap. (Of course, if you go back far enough you get to thrashing as an educational method, so ‘new’ and ‘old’ are relative anyway.)
Not after seeing that!
Yes, I know. That’s why I said don’t get me started on it. The New Math way was to turn division into a process of subtraction, basically, that took forever and wasted a lot of paper. The only thing really useful I got out of it was a decent explanation of why division by zero is undefined: in other words, given 4/0, how many times would 0 be subtracted from 4 before nothing remained? Not even an infinite number of times would suffice. Therefore, it is left undefined.
In the Tom Lehrer song, he starts calculating 342-176 by going “3 from 2 is nine carry the 1, 7 from 3 is six” for the 10s and 1s digits. That makes no sense to me to do it that way. I always did it the (apparently) “new math” way, actually understanding what it means to borrow a “1” from a higher power. Does anyone actually do it the first way?
I remember seeing that the basic problem was a confusion of logically more fundamental with easier to understand. Sets logically precede numbers but set theory historically comes far towards the end. The Wikipedia article refers as well to inequalities (by which I suppose it means ‘best-fit’ algorithms) and matrices (used to work the inequalities after making them equations, among other things). I did all that at college as part of a computer degree but I can’t see how any of it could be possible without a strong foundation in basic arithmetic first. There are ways to write these things down as summations with superscripts and subscripts (often double) and I’ve never been able to make sense of those. They are supposed to ‘explain’ what’s going on but the 2-D layout is far simpler. I wish we had used them for ‘simultaneous equations’ at school. At the same time, I have never found anything in life that ever looked like a simultaneous equation - or even inequality for that matter.
What I did not realise then, but should have been obvious, is that matrix multiplication and complex numbers between them can describe just about any kind of other operation you care to invent. That is how and why they came to the fore in quantum mechanics and nuclear physics in the 1920s - call it a premultiplying Psi matrix, and from knowing what it was operating on and what the result was, discover what it is without needing thousands of different special operators.
I like all that stuff. I could never handle or see the point of calculus and abstract trigonometry. The Great Idea was that computers would be able to find ideal routes and distributions but it hit two problems in reality.
The first was that real systems are enormous and computers of the time just could not handle the number of points involved. I’ve worked on one system that did something of the sort and it was oil flow analysis using time on a couple of Cybers and a Cray. The input alone was a serious computer problem in itself (the one I was stuck with).
The second problem was that except for physical observations like that, data beyond traditional human assessment were usually far too variable and messy to fit any programmed minimisation systems. Some airlines use this kind of route analysis because flight distances and fuel costs are not influenced to the same extent as land routes by other variables. Shipping might use it too. It’s no use determining your cheapest overland distribution and routes between cities if they ignore getting through the cities at rush hour, unload times, union conditions on hours, need to lay up overnight, overtime rates and everything else that actually matters.
The main thing I remember from Venn Diagrams is that when it comes to putting actual numbers in them and solving the puzzle of how many like ice-cream and peaches but not coke, they don’t work and you are back to ‘ordinary’ arithmetic remembering that (A OR B) means (A+B-(A AND B)) (because you’ve already counted the (A AND B) as part of the A). It’s very handy as a visual representation of complicated descriptions - but again, try just drawing a Venn for more than three or four interleaving sets!
What they did, in a way, was to forget Hindu numerals and go back to something like Roman: I+I=II, +I=III. They might have wondered why it was that ‘real’ mathematics started with the Hindus and Moslems and not with the Romans.
My brother had the rods when I was a teenager and since they were not actually marked with lengths they didn’t really teach anything except that some things are bigger than others. It’s hard enough for an adult to estimate proportions by eye, just about impossible for a child. I’ve just looked it up and I’d find it hard to distinguish 3/4 from 2/3 by sight. Come to think of it, both those lead to further fractions as proportions of ten that aren’t in the system. No wonder the poor bugger never learnt anything!
I can’t figure out the meaning of the line you quoted. I can’t see where you would subtract a two from a three when subtracting 342-176.
Actually, the way I learned it, there’s not “carrying” in subtraction. You “borrow.” In my head, it goes:
Two is smaller than six, so I borrow one from the four, making 12 instead of two. 12 minus six is six. Six goes in the ones place in the answer. Next, three is smaller than seven, so I borrow one from the three, making 13 instead of three. Thirteen minus seven is (eight nine ten eleven twelve, thirteen*) six. Six in the tens place. And two minus one is one, so one in the hundreds. One hundred sixty six.
That’s the traditional way, right?
*As a kid, I stumbled on a method for doing subtractions that I don’t immediately remember the answer to. I count upwards from the smaller number, in a sort of sing-songy voice (in my head, not out loud) using the song to keep track of groups of five numbers, making it easy to keep track of how many numbers I counted.
Pffft, Gödel luck with that.
I should point out that vector and matrix math (which I can verify was taught to me, but not my dad, and hence would confuse him greatly when he helped me with homework) is the only kind of maths I use day-in day-out in my career as a graphics programmer.
Well played!
Apropos for this thread, it’s a typo.
It’s supposed to be 342-173.