Dang, but that seems to be a popular year for Dopers.
Not sure if I learned “new math” or “old math,” but that’s the way I learned it too. Except when borrowing, I’d cross out the 4, write a 3 over it, and stick the borrowed 1 next to the 2. Then the problem becomes 12 - 3 = 9, 3 - 1 = 2.
Well yeah, but that’s really hard to do on a message board. 
OK, I just had another blast-from-the-past long-dormant memory. Something they made us do pertaining to division.
Let’s see if I can do it justice with the code tag…
Here's what I think of as the "normal" way to do division.
_22.09___
9.4 )207.7
- 188
----------
19.7
- 18.8
----------
0.900
- 0.846
-----------
0.054
(etc)
For at least one year, they forced us to write it this way on paper instead:
_________
9.4 )207.7 | 20.
20 x 9.4 - 188 |
---------- |
19.7 |+ 2.
2 x 9.4 - 18.8 |
---------- |
0.9 |+ 0.0
0 x 9.4 - 0.0 |
---------- |
0.900 |+ 0.09
0.9 x 9.4 - 0.846 |
-------------|--------
0.054 = 20.09
I hadn’t thought about terms like associative and distributive in years, and certainly never realized that was considered new math. But I always use the distributive (I think) for doing multiplication in my head.
Instead of working out a problem like 112 x 15 using columns and such (this is a simplified example), I automatically break it down into:
(100 x 15) + (12 x 10) + (6 x 10)
You know, writing that out makes it seem like a longer way of doing things, but I assure you that it goes pretty quickly in my head. If that’s new math, then I find it extraordinary useful in daily life.
Actually, looking at my example again, I’d have been more likely to do it this way:
(112 x 10) = x, then add 0.5x to get answer. Either way works.
Allow me to suck the joy out of Tom Lehrer (who really is wonderful).
To paraphrase him:
Take a problem,
342
-173
The old way to do it is to start in the right-hand column: 2-3=9, carry a one [from the next column].
You take the 1 from the 4 in the 342 to get 3.[he makes some comment about how older people would instead add the one to the 7 in 173, but I’ve never seen that.
Now you’ve got 3-7 in the middle column. You borrow a 1 from the 3 in 342, to turn the 3-7 into 13-7. That gives you 6. So far you’ve got 69.
You’ve borrowed a 1 from the 3 in 342, turning it into a 2 instead. The last thing you do is to subtract the 1 in 173 from this 2, to get 1. Presto, your answer is 169!
The problem with this method is what one of my professors called the “Trained Seal approach”: students can do it without having any idea what they’re doing. It’s like a magic spell they’re casting on the numbers. And when students don’t know why they’re doing something, they think it’s stupid, and they get it confused with other magic spells they have to cast (it’s very common to see kids adding a one instead of borrowing a 1, getting it confused with the magic spell for addition), and they have trouble applying principles flexibly and powerfully.
Let’s look at the same problem in New Math. As he says, the important thing to do is to know what you’re doing, rather than to get the right answer. That’s satire: you are supposed to get the right answer, but you’re also supposed to understand why it’s the right answer. His song explains why it’s the right answer in the most confusing possible language.
To start with, you look at the right-hand column again, and see 2-3. You can’t do this, so again, you look at the next column. However, instead of just saying “carry the 1,” you recognize that the 4 in 342 actually represents 40, or “four tens”. You can, of course, turn that into three tens and ten ones: this is called regrouping. Note that “borrowing” and “carrying” are very out of vogue in education circles: word is that these terms confuse kids much more than “regrouping” does.
So you regroup the four tens into three tens and one ten. Add that one ten to your 2 to get 12; you can subtract 3 from that and get 9. That’s your first digit.
Now you go to the next column–but you call it the tens place, to reinforce understanding of place value and what the numerals really represent. Because of the regrouping you just did, you’ve got a three in the tens place, from which you need to subtract a seven. Since you can’t do this, you look ahead to the 3 in the hundreds place: this is really thirty tens, which you can regroup into 2 hundreds and ten tens. (he says you turn it into “ten ones,” but this is both incorrect and confusing. He may have said it this way to improve scansion, or to further confuse the audience).
So regroup, and add your ten tens to the 3 tens you currently have to get 13 tens. Now you can subtract your 7 tens from those 13 tens to get six tens; that’s the number that goes in your tens place in the answer.
Finally, go to the hundreds place: you need to subtract 1 hundred from 2 hundred. That gives you 1 for your hundreds place, and your answer is 69.
Again, the idea is that at each point in the operation, you know exactly what you’re doing. Good teachers can explain problems like this with all sorts of manipulatives. Play money (dollars, dimes, and pennies) can work very well for money-savvy kids. You can use bundles of straws so that kids can feel what it means to regroup, as they dismantle a bundle of 10 straws to get 10 ones. You can use a place-value chart, with pockets for each place value, and have kids talk through what it means to move chits between the different pockets and how they need to transform the chits in the move (e.g., they need to gather 10 chits together if they want to remove them from the 1s pocket and put a single chit in the 10s pocket). You can use base-10 blocks so that kids can easily see how a 10-rod is equivalent to ten 1-cubes, and that ten 10-rods is equivalent to one 100-square.
Bad teachers, of course, do it seriously like Lehrer does it flippantly. That, I think, is why New Math has such a bad reputation. The underlying idea–that students should never be made to perform an algorithm that they don’t understand–is fantastic, IMO.
Daniel
But they don’t call it ‘borrowing’ anymore - it’s called ‘regrouping.’
We had a night once with my daughter when she learned this task, and I was helping her with her homework. I kept saying, “You need to borrow” and she’d say, “I don’t know what that is. We learned to regroup.”
I kept saying, “No, you have to borrow” and tried to show her how to do it. We just went around in circles - I kept saying ‘borrow’ and she kept saying ‘regroup’ but neither of us realized we meant the same thing.
The reasoning is that ‘borrowing’ implies giving back what you borrow, and you never give back what you borrow from the other column. :rolleyes:
That’s how the teacher explained it to me.
I’m not sure I buy your teacher’s explanation, but I do prefer the term “regroup”: it more accurately describes what you’re doing. “Borrow” doesn’t really bear any relationship to the act of changing three tens to two tens and ten ones.
Daniel
1962 here.
I’m a 40-something math major, and the semester I took my first upper-division class involving proofs we reviewed set theory, relationships and functions. I distinctly remember doing my set theory assignment and thinking “holy moley, we did this kind of stuff in grade school… no wonder I understand it!” It was certainly easier than multivariate calculus to figure out, that’s for sure. Later on, stuff like number bases and arrays turned out to be useful in discrete math and abstract algebra. Go figure.
There’s arithmetic, and then there’s mathematics. Old math=arithmetic. New math = abstract mathematics. Kids really should learn both and how they’re explicitly connected.
I remember teachers using both “regroup” and “borrow.” I also remember that learning concepts always makes math a lot easier for me–or any subject, for that matter. Everything up to and including pre-algebra was easy once I got the concept down.
So I think the idea of New Math was probably a good one, but like most educational “programs,” it was only as helpful as every individual teacher in every class room could make it. It always amazes me when pundits or Newsweek or Time will proclaim that such and such educational technique is “failing.” The fact is that no one can ever really draw such a conclusion, because nobody can be in every classroom of a school or a district for every hour of the school day. Mr. Top Administrator may say, “We’re going back to basic phonics, because whole language has failed,” but in reality the teachers were using a mixture of both all along, and will continue to do so, and without even paying much mind to Mr. Top Administrator, because they know the last place he can be found is in a real classroom.
Devoted (and competent) teachers are very practical, and they’ll do whatever works. When they’re suddenly expected to do something which they aren’t very familiar with, then things can go wrong. Maybe that’s what happened to New Math.
I should perhaps clarify something: I have absolutely no resentment about teachers teaching the concepts behind the manipulations. Almost all my rage at “new math” is the insistence on writing the cumbersome lengthy explanatory bullshit down on paper. More work for me as a student. More drudgery. I hear “show your work” one more time I’m up in the tower with a 30-06, I swear to god.
I hate, hate, hate “show your work”. I used to be a mental math machine, pumping out solutions in my head and converting miles to kilometers to centimeters to inches to feet to miles without pausing to look at my calculator or my paper. But my high school chem teacher forced me to write out this long string of multiplications and divisions every time I did a conversion, like so:
3 ft 12 in 2.54 cm 1 m
4 yd * ----- * ------- * ----------- * -----------
1 yd 1 ft 1 in 100 cm
Now I can’t convert anything without writing that crap down, and I’m in my fourth semester of college.
The problem with not showing your work is that for every student who can accurately convert things in their head there are tens or hundreds of them who can’t do that accurately. And then the teacher, or TA, is left with a distraught student asking, “Where did I go wrong?” Without the work being shown on the paper answering that very legitimate question can become nearly theological.
Similarly, always including units is also important for any problem in physics, or chemistry. (After all, there is at least one rocket scientist who thought they were superfluous - and found out otherwise, after his/her calculations were used to send one of the Mars probes into the surface of the planet.)
I almost stripped a gear when I looked at my nephew’s math book. IMO it was flawed in a number of ways. First, it was a laid out like a Peter Max cartoon. Lines and colors and boxes (oh my). A person’s attention was drawn all over the page making it a nightmare for kids with dyslexia or attention problems. The second flaw was not that it used new math, it used a half dozen or more methods for each type of problem.
I’m not so arrogant as to insist there is only 1 method of teaching math but I can say 2 things with a degree of certainty:
- my depression-era parents (and classmates) could all do math with a high degree of competency.
- teaching multiple methods of the same concept reduces the amount of time spent on a single method which reduces the potential for mastering a single task.
While I’m on a soapbox, there is no place for math questions that involve estimation UNTIL the task is mastered. Estimation is useful as a check to see that you’re in the ballpark. I was appalled that some of my nephew’s homework and tests were entirely estimated.
Interesting discussion…only I’m still puzzled why I find the math textbooks of the 1930’s-40’s so much clearer than the modern texts. If you take away all the pretty colors, visuals, etc., I think the writing styles of modern texts leave a lot to be desired. Anyway, I think the trditional way of teaching math has its merits-throwing a lot of unrelated concepts at kids is a good way to confuse a lot of them!
I had New Math in the early 1970s. I was a very bright kid but this shit just totally escaped me. I remember hitting sixth grade and, just as other posters have written, I was suddenly supposed to know algebra. Not a fucking clue then, not a fucking clue now at age 40.
NM curriculum really crippled me.
Or it might increase it, for those kids who don’t understand whichever single method is chosen.
My kids had tests and homework that were entirely estimated. Made perfect sense to me, because estimation was a unit of its own. When they did a unit on decimals, all the tests and homework involved decimals.
I survived New Math during 1960-1966. My dad took the evening classes to learn about it, as others mentioned.
I hated Set Theory, and at the time hated Bases. I understood them but they seemed stupid to me. I wanted them to give me regular problems to solve! So I’d go home and make up my own problems to solve. Geek-in-embryo I guess.
I survived New Math with an enjoyment of math intact. I took Calculus in college for the fun of it, even though it wasn’t required for my social science major.
If in a technical field, rarely will you regret being anal about units. Fear the engineer who is not anal about units. (Okay, straight up units conversion is one thing, but in a more complex formula, especially if those giving you inputs aren’t giving you the most friendly or compatible units, then being thorough is necessary)
Can I not do a “dry run” of the equation, units-only, after finishing the calculation in my head?