What exactly was the "new math"?

Cecil's Columns/Staff Reports
41

Which is pretty slick considering there was no Department of Education for a decade and a half at least after “New Math” came out and even HEW was launched almost simultaneously.

Like Yllaria mentioned, its adoption was heavily fostered by the Space Race’s most enduring legacy, the usually simmmering and ocassionally briefly flaring furor that exists in American education systems since 1957 to this day about falling behind in maths and science. Say what you will, there was nothing leftist about New Math per se.

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All of those were taught to me in ordinary US elementary schools in the 50s, except, perhaps, number lines, which have always been so perfectly obvious to me that I can’t clearly remember whether I was taught about them or not. I even remember learning some basic set theory; I think I worked out for myself the isomorphism between set theory and formal logic. What I certainly did not have, until I was a senior in high school taking college math at night, was the exploration of the roots of arithmetic in terms of groups, rings, and fields. And I think that’s where “New Math” failed. Little children simply aren’t prepared for postulate-theorem reasoning. Not out of the blue. I suspect the historic approach was best—introducing formal theorem proving in the form of Euclidean geometry, somewhere around tenth grade. (I had it in eighth grade, but I shouldn’t be taken as normal.)

43

What is “casting”?

44

Possibly the same as carrying: to transfer from one place (as a column) to another <carry a number in adding>

45

No, it’s a method of checking answers in arithmetic.

46

“Casting out nines” is an historic error-checking method. For addition, it works something like this:

1573 + 5862 = 7435
Add up the digits of each number; repeat until just one digit is left:
1 + 5 + 7 + 3 = 16; 1 + 6 = 7
5 + 8 + 6 + 2 = 21; 2 + 1 = 3
7 and 3 are your check digits

Add up the check digits:
7 + 3 = 10

Add up the digits of the sum as before:
1 + 0 = 1
This is your check result

Now add up the digits of the original sum, repeating as before
7 + 4 + 3 + 5 = 19; 1 + 9 = 10; 1 + 0 = 1

This result (1) is the same as the check result, so the answer 7435 is probably correct.

The process of adding up digits is the same as dividing by 9 to get the remainder until one digit is left, except that a result of 9 by adding up digits is equivalent to a result of 0 when dividing.

1573 / 9 = 174, remainder 7
5862 / 9 = 651, remainder 3
7435 / 9 = 826, remainder 1

It also works for multiplication:
23 * 47 = 1081
First check digit: 2 + 3 = 5
Second check digit: 4 + 7 = 11; 1 + 1 = 2
Multiply and get the check result: 5 * 2 = 10; 1 + 0 = 1
Compare to the result: 1 + 0 + 8 + 1 = 10; 1 + 0 = 1

It can be made to work with subtraction, but may give a negative result; in that case, add 9:
67 - 32 = 35
First check digit: 6 + 7 = 13; 1 + 3 = 4
Second check digit: 3 + 2 = 5
Subtract and get the check result: 4 - 5 = -1
Add 9 to make it positive: -1 + 9 = 8
Compare to the result: 3 + 5 = 8

For many years, this was taught to all students, but, as calculating machines became more common (calculating machines, as opposed to computers, started being mass produced in Victorian days), it became less used, until, by the time I was in school, it was filed under “quaint and curious”—something to amuse the brighter kids with if they had extra time.

47

It’s also near-impossible to teach mathematical theory to kids who are just beyond the “learn to count to 10” stage. Teaching different bases to 1st graders (and I was in that age range and personally experienced it) doesn’t help anything. I think we learned binary, base 8 and…maybe base 12?

48

And it was probably just the opposite. Fear of the Soviet Union following the launch of Sputnik.

49

Negative numbers confused me terribly. How could you possibly have less than one? The teacher had us drawing number lines with negative numbers on the left, positive on the right, and I still couldn’t get it. Nobody else asked about it, so I was too scared to ask. Finally I asked the teacher privately, “How can you have less than nothing?”

She answered, “It’s like owing money.”

Then I got it, but for months afterward, I had to mentally rephrase every problem of
-7 + 5 as “If Jane owes seven dollars but pays back five, then how much does she owe?”

50

A somewhat related article on Slate this week:

Powers &8^]

51

Is this why I’m terrible at math?

52

That’s how I explain it to my students. It’s the only thing that seems to make sense. (Same with double negatives. In English, double negatives make a positive.)

53

Here’s a picture of Abraham Lincoln’s homework.

If you look at each long division problem, you see a big “X” next to it surrounded by four numbers. That’s how you cast nines in division.

I don’t quite remember the entire procedure, but it went something like this:

[ul]
[li]You add up the digits of the numerator and put it on the left. [/li][li]You add up the digits of the denominator and put it on the bottom. [/li][li]You take the answer, cast the nines and put it on the right. [/li][li]You then multiply the left side by the right side, add in the remainder, cast nines, and put it on the top.[/li][/ul]
If the top and bottom agree, you probably have the right answer.

54

Historically, you’re quite correct. Negative numbers were once a big debate. How can you have less than zero? Negative numbers don’t really exist, but they prove to be useful in math.

I’ve had a similar argument with my wife about imaginary numbers These are numbers based upon the square root of -1. She kept telling me "They’re not REAL numbers! They’re IMAGINARY! Ironically, she was absolutely correct. Imaginary numbers are not in the realm of real numbers.

Like negative numbers, they don’t exist in the real world, but as any electrical engineer can tell you, they’re very useful mathematically. Fractals wouldn’t exist without them.

55

Actually, imaginary numbers are quite common in reality. Magnetism is imaginary electricity, time is imaginary space, and, if tachyons exist, they have imaginary mass.

56

My one class in alternating current circuits used imaginary numbers (well, complex numbers, which couple imaginary numbers with real numbers).

57

Explain, please. In 500 words or less. =)
Powers &8^]

58

Can’t explain in 500 words or less, but any appropriate textbook will explain it, albeit in rather more words. I will say that the actual equations used in all three cases make it obvious.

(In electricity, because i was already in use to mean “current”, the square root of -1 is generally represented by j, instead.)

59

Imagine someone with a philosophical objection to negative numbers. As far as he’s concerned, there’s no such thing. He can still calculate income and outgo in his checkbook - but he does it in a roundabout way, by adding up all the outgo, and adding up all the income, and using a special rule to combine the two numbers - a rule that only works if income is greater than outgo, since it subtracts outgo from income. It’s a cumbersome system, but it gets the right answer. However, it’s a lot more convenient to invent negative numbers, and just treat income as positive, and outgo as negative. Now you can keep track of your total funds at all times, and avoid the pain of keeping track of two separate numbers in order to do so.

Imaginary numbers are the same kind of thing. You could calculate electrical field quantities and magnetic field quantities using separate rules - it would be terribly complicated, because they feed back into each other, but it would be possible, but instead, it was noticed that by inventing “complex numbers” you could combine the electrical and magnetic quantities into one number, and simplify your bookkeeping considerably.

60

It occurred to me that I’m paraphrasing something I read a while back on USENET, so let me link to that http://groups.google.com/group/sci.physics/msg/79e260889f566c0f . The author of the article points out that if you have an objection to fractions you can get rid of them too, by suitable new rules.

"Who needs fractions, when you can use an ordered pair of integers instead?
And apply the rules of fractions to that ordered pair, without ever
using the dreaded “f” word. "