The Old Math?

[vakkotaur]

I've heard references to "the new math" from time to time and from what I can gather it was a change in the teaching of mathematics that happened as a reaction to the launch of Sputnik. Since Sputnik I was launched in 1957 and I was born rather later, I never encountered whatever the "old math" was. It seemed strange that were "old" and "new" versions of math. Didn't numbers work the same, after all?

I've heard Tom Lehrer's song, New Math which has this in the introduction:

Consider the following subtraction problem, which I will put up here: 342 - 173.




 342
-173
----




Now remember how we used to do that. Three from two is nine; carry the one, and if you're under 35 or went to a private school you say seven from three is six, but if you're over 35 and went to a public school you say eight from four is six; carry the one so we have 169, but in the new approach, as you know, the important thing is to understand what you're doing rather than to get the right answer. Here's how they do it now.

Three from two is nine? Seven from three is six? Eight from four is six? These are alien ways of describing things to someone who got taught the "New Math" method which the song describes. I didn't think much about it until recently as I wondered if the New Math really is an impediment to getting a problem solved. I decided to see if I could make sense of the odd, to me, lines in the introduction.

Here's an addition table, something that is simple enough. Take the numbers you want to add from the top row and the left column and where their respective columns and rows intersect, there's the sum. The nice thing about addition is that order doesn't matter. 2+5 is the same as 5+2.




+ | 0  1  2  3  4  5  6  7  8  9 10
-----------------------------------
 0| 0  1  2  3  4  5  6  7  8  9 10
 1| 1  2  3  4  5  6  7  8  9 10 11
 2| 2  3  4  5  6  7  8  9 10 11 12
 3| 3  4  5  6  7  8  9 10 11 12 13
 4| 4  5  6  7  8  9 10 11 12 13 14
 5| 5  6  7  8  9 10 11 12 13 14 15
 6| 6  7  8  9 10 11 12 13 14 15 16
 7| 7  8  9 10 11 12 13 14 15 16 17
 8| 8  9 10 11 12 13 14 15 16 17 18
 9| 9 10 11 12 13 14 15 16 17 18 19
10|10 11 12 13 14 15 16 17 18 19 20




It's straightforward and not too hard for a person to memorize. It's something where rote learning makes sense and works.

Subtraction isn't quite so easy because order is important. 5-2 is not the same as 2-5. I've never seen a subtraction table used in school, and the order problem is probably why. But suppose you wanted to make one and would always subtract the numbers in the left column from the numbers in the top row?

It would start out looking like this:




- | 0  1  2  3  4  5  6  7  8  9 10
-----------------------------------
 0| 0  1  2  3  4  5  6  7  8  9 10
 1|    0  1  2  3  4  5  6  7  8  9
 2|       0  1  2  3  4  5  6  7  8
 3|          0  1  2  3  4  5  6  7
 4|             0  1  2  3  4  5  6
 5|                0  1  2  3  4  5
 6|                   0  1  2  3  4
 7|                      0  1  2  3
 8|                         0  1  2
 9|                            0  1
10|                               0




But that's only about half of it. How to fill in the other half? The first thought I would normally have would be to complete the table with negative numbers, like so:




- |  0  1  2  3  4  5  6  7  8  9 10
------------------------------------
 0|  0  1  2  3  4  5  6  7  8  9 10
 1| -1  0  1  2  3  4  5  6  7  8  9
 2| -2 -1  0  1  2  3  4  5  6  7  8
 3| -3 -2 -1  0  1  2  3  4  5  6  7
 4| -4 -3 -2 -1  0  1  2  3  4  5  6
 5| -5 -4 -3 -2 -1  0  1  2  3  4  5
 6| -6 -5 -4 -3 -2 -1  0  1  2  3  4
 7| -7 -6 -5 -4 -3 -2 -1  0  1  2  3
 8| -8 -7 -6 -5 -4 -3 -2 -1  0  1  2
 9| -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1
10|-10 -9 -8 -7 -6 -5 -4 -3 -2 -1  0




It's true enough, but doesn't show what is going on in the Lehrer introduction. So, just what is going on there?

"Three from two is nine."

Now, using 2-3 alone isn't going to work for this, as the answer to that is -1. Ah, but lets say it's 42-3. Then the New Math way says that the 4 in 42 is forty ones and ten of those can be borrowed by the two and this happens: 30 + (12-3) which becomes 30 + 9 (there's the nine!) which is 39.

Or the -1 (2-3=-1) was added to the borrowed 10 and the result was 9. The -1 was carried and added to the 4, and the answer is still 39. "Three from two is nine; carry the one."

Skipping the negative result intermediate step would speed things up a bit:




- |  0  1  2  3  4  5  6  7  8  9
----------------------------------
 0|  0  1  2  3  4  5  6  7  8  9 
 1| (9) 0  1  2  3  4  5  6  7  8 
 2| (8)(9) 0  1  2  3  4  5  6  7 
 3| (7)(8)(9) 0  1  2  3  4  5  6 
 4| (6)(7)(8)(9) 0  1  2  3  4  5 
 5| (5)(6)(7)(8)(9) 0  1  2  3  4 
 6| (4)(5)(6)(7)(8)(9) 0  1  2  3 
 7| (3)(4)(5)(6)(7)(8)(9) 0  1  2 
 8| (2)(3)(4)(5)(6)(7)(8)(9) 0  1 
 9| (1)(2)(3)(4)(5)(6)(7)(8)(9) 0 




All the numbers in parenthesis would have a -1 to carry. That only works if there's a number to carry a -1 to, so the table isn't universal. But if memorized, it would perhaps speed things up. It eliminates the negative result step and, compared to the New Math way, skips the borrowing and re-grouping.

There might be another advantage. In the New Math method, even after a borrow and regroup, you have things like 12-3=9. It's possible to memorize all the cases needed to deal with things like that, but the table might be simpler, at least once memorized.

I'm not going to make a big effort to memorize this table, but now I have an idea of what was going with things like "seven from three is six." The "eight from four is six" is another bit of making things just a bit easier. It's matter of just how you "carry the one." Using the example in the song:




 342
-173
----




"Three from two is nine, carry the one." can be done two ways. Here's his "if you're under 35 or went to a private school you say seven from three is six" where the negative one is carried and added to the top number:




      -1       -1
 342  342  332  332  232
-173 -173 -173 -173 -173
---- ---- ---- ---- ----
 __9   _9  _69  _69  169




Adding a negative is, of course, subtraction. And subtraction is supposedly just a bit harder, if only people's minds, than addition. So there might be a slight advantage if the subtraction step was replaced with addition. There's already enough subtraction going on, so why make it worse?

Here's the "if you're over 35 and went to a public school you say eight from four is six; carry the one" version of the same problem:




 
 342   342  342  342  342
-173  -173 -183 -183 -283
----   +1  ---- +1    ---
 __9  ----  _69 ----  169
       __9       _69




Instead of adding a negative one (subtracting one) from the top number, the same result can be had by adding one to the bottom number. This also makes "carry the one" mean the same thing, to the person doing it, as it does in addition.

Comments

[foolscap001.livejournal.com]

i.e. 2 - 3 is congruent to 9 modulo 10, etc. (C and assembly language programmers beware; we're using division and remainder as defined by the division theorem, so that -1 / 10 is -1, with remainder 9.)

[altivo.livejournal.com]

"New math" was logical but not a sensible way to teach the subject, IMHO.

However, I don't know what the method was that Lehrer described, either. When doing subtraction, we didn't use the wording "carry the one" at all, but rather "borrow one" from the column to the left.

"Three is larger than two so borrow one from the four, making it a three. Three from twelve leaves nine (the borrowed one becomes a ten, of course)..."

This stuff may well be regional. I am 56 years old. I was taught the so-called "old math". My younger sister was taught the "new math" and though she is in accounting today it is in spite of, rather than because of that teaching. She had a terrible time with it until junior high or thereabouts.

Math itself didn't change. It was the teaching methodology that was changed, and the theory was that you could try to teach whole concepts at once. Overall, it didn't work too well. Set theory for seven year olds just isn't practical, and neither parents nor the students cared much for it.

It was like teaching the physics and mechanics of clock construction in order to get around to how to tell time. There's a logical argument for that, but in actuality it doesn't make much sense.

[vakkotaur.livejournal.com]

I suspect the same change in teaching method also affected chemistry. The high school chemistry I went through seemed like toned down version of college chemistry with its emphasis on theory. I think that turned off a lot of people who wouldn't have had much trouble had they been taught "descriptive chemistry" as I'd earlier learned it from an older textbook. It was lighter on theory, at least at first, and went through the various groups (alkali metals, alkali earth metals, halogens, inert/noble gasses, etc.) and gave a feel of how things worked. The theory was there, but only enough to help explain why things were the way they were, not as the primary focus.

I wonder what else got messed up in a bad overreaction to Sputnik and such.

[altivo.livejournal.com]

Physics. I'd forgotten. Probably your chemistry was like mine, because they changed the chemistry and physics just before I got to them. Those are usually high school subjects, where math starts in primary school. PSSC was the acronym for the secondary school science program changes, but I don't remember what it meant.

In the case of physics, we called it "skateboard" physics because we spent a whole semester rolling blocks of wood on wheels, with "ticker tapes" attached to them, then measuring the gaps between the dots on the tapes and doing math to figure out how velocity, acceleration, and mass were related. Many people found the math extremely difficult, but that was because they approached it graphically and tried to reduce it to figures and formulas by looking at the graphs.

I can understand the desire to teach the underlying concepts, but I'm not sure this worked for most.

[nefaria.livejournal.com]

I'd just solve the problem as 12 - 3 = 9, 13 - 7 = 6, 2 - 1 = 1, 169 is the answer. You only need to remember two things at each step: whether you carried a 1 in the previous step and whether you carred a 1 in the current step. And all my intermediate subtractions are mathematically correct, no false statements like "8 from 4 is 6", 8 from 4 is -4.

[vakkotaur.livejournal.com]

I suspect you've hit upon the big deal between Old Math and New Math, there. The Old Math folks were interested in the answer alone, really, and didn't care too much how they got there so long as they got there easily and reliably. The New Math instruction was (is) about making everything MC, Mathematically Correct.

[electronin.livejournal.com]

I can't carry any more.

342
-173
----

173+100=273
273+60=333
333+9=342

[jcw-da-dmg.livejournal.com]

Oh No! He just proved that the universe doesn't exis . . .