# What exactly was the "new math"? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [November 5, 2011, 5:42am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936 "2011-11-05T05:42:25Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![JRDelirious](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jrdelirious/32/9531_2.png) [@JRDelirious](https://boards.straightdope.com/u/JRDelirious)\ **Post date:** [November 9, 2011, 1:19am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/41 "2011-11-09T01:19:29Z")
> [@drewder](#): > > So they invent “new math” and lobby the DOE to force classrooms to teach it. 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. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 9, 2011, 3:08am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/42 "2011-11-09T03:08:12Z")
> [@Yllaria](#): > > My experience with New Math was from a child’s perspective, so I’m not sure what changes stuck. Did previous elementary teaching systems introduce the following: number lines, greater than/less than, what a percentage IS rather than how to calculate one, and the notation of different base systems (binary is an example)? 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.) ---
**Author:** ![Irishman](https://avatars.discourse-cdn.com/v4/letter/i/b487fb/32.png) [@Irishman](https://boards.straightdope.com/u/Irishman)\ **Post date:** [November 9, 2011, 4:06pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/43 "2011-11-09T16:06:35Z")
> [@qazwart](#): > > (For example, in the third grade, I _proved_ why casting 9s works and that you can cast 6s in Base 7 and 15s in Base 16). What is “casting”? ---
**Author:** ![Floater](https://avatars.discourse-cdn.com/v4/letter/f/dc4da7/32.png) [@Floater](https://boards.straightdope.com/u/Floater)\ **Post date:** [November 9, 2011, 4:42pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/44 "2011-11-09T16:42:43Z")
Possibly the same as carrying: to transfer from one place (as a column) to another \ > **[Definition of CARRY](https://www.merriam-webster.com/dictionary/carry)** > > to move while supporting : transport; to convey by direct communication; conduct, escort… See the full definition ---
**Author:** ![President\_Johnny\_Gentle](https://avatars.discourse-cdn.com/v4/letter/p/5f9b8f/32.png) [@President\_Johnny\_Gentle](https://boards.straightdope.com/u/President_Johnny_Gentle)\ **Post date:** [November 9, 2011, 6:43pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/45 "2011-11-09T18:43:02Z")
No, it’s a method of [checking answers](http://en.wikipedia.org/wiki/Casting_out_nines) in arithmetic. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 9, 2011, 7:04pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/46 "2011-11-09T19:04:53Z")
“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. ---
**Author:** ![Fenris](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/fenris/32/498_2.png) [@Fenris](https://boards.straightdope.com/u/Fenris)\ **Post date:** [November 9, 2011, 7:31pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/47 "2011-11-09T19:31:03Z")
> [@Little\_Nemo](#): > > A > The idea was that instead of teaching math by having kids memorize a bunch of procedures, they would be taught mathematical theory. And once they understood the theory behind math, the procedures would be obvious. > > In the abstract, it wasn’t a bad idea. A person who understands the theoretical basis of a subject is going to have a better understanding than somebody who’s just memorized the facts. > > But in practice, it’s very difficult to learn mathematical theory if you don’t already have a solid base of mathematical facts. 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? ---
**Author:** ![fumster](https://avatars.discourse-cdn.com/v4/letter/f/e9c0ed/32.png) [@fumster](https://boards.straightdope.com/u/fumster)\ **Post date:** [November 10, 2011, 3:52pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/48 "2011-11-10T15:52:42Z")
> [@JRDelirious](#): > > 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. And it was probably just the opposite. Fear of the Soviet Union following the launch of Sputnik. ---
**Author:** ![Dendarii\_Dame](https://avatars.discourse-cdn.com/v4/letter/d/bbce88/32.png) [@Dendarii\_Dame](https://boards.straightdope.com/u/Dendarii_Dame)\ **Post date:** [November 10, 2011, 6:43pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/49 "2011-11-10T18:43:49Z")
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?” ---
**Author:** ![Powers](https://avatars.discourse-cdn.com/v4/letter/p/edb3f5/32.png) [@Powers](https://boards.straightdope.com/u/Powers)\ **Post date:** [November 13, 2011, 4:16pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/50 "2011-11-13T16:16:00Z")
A somewhat related article on _Slate_ this week: > **[How To Fix Math Education in High School and College](https://slate.com/technology/2011/11/math-education-how-colleges-and-high-schools-can-fix-it.html)** > > This week, Future Tense is south of the border to attend Ciudad de las Ideas in Puebla, Mexico. The name means “City of Ideas,” and accordingly, Puebla... Powers &8^] ---
**Author:** ![Farmer\_Jane](https://avatars.discourse-cdn.com/v4/letter/f/bbe5ce/32.png) [@Farmer\_Jane](https://boards.straightdope.com/u/Farmer_Jane)\ **Post date:** [November 13, 2011, 4:24pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/51 "2011-11-13T16:24:58Z")
Is this why I’m terrible at math? ---
**Author:** ![Farmer\_Jane](https://avatars.discourse-cdn.com/v4/letter/f/bbe5ce/32.png) [@Farmer\_Jane](https://boards.straightdope.com/u/Farmer_Jane)\ **Post date:** [November 13, 2011, 4:27pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/52 "2011-11-13T16:27:01Z")
> [@Dendarii\_Dame](#): > > 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?” 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.) ---
**Author:** ![qazwart](https://avatars.discourse-cdn.com/v4/letter/q/5fc32e/32.png) [@qazwart](https://boards.straightdope.com/u/qazwart)\ **Post date:** [November 15, 2011, 6:13pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/53 "2011-11-15T18:13:29Z")
> [@John\_W.Kennedy](#): > > “Casting out nines” is an historic error-checking method. Here’s a picture of Abraham Lincoln’s [homework](http://www.alincolnlearning.us/Abesmathpaper.html). 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. ---
**Author:** ![qazwart](https://avatars.discourse-cdn.com/v4/letter/q/5fc32e/32.png) [@qazwart](https://boards.straightdope.com/u/qazwart)\ **Post date:** [November 15, 2011, 6:23pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/54 "2011-11-15T18:23:25Z")
> [@Dendarii\_Dame](#): > > 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?” 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](http://en.wikipedia.org/wiki/Fractal) wouldn’t exist without them. ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 16, 2011, 4:12am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/55 "2011-11-16T04:12:18Z")
Actually, imaginary numbers are quite common in reality. Magnetism is imaginary electricity, time is imaginary space, and, if tachyons exist, they have imaginary mass. ---
**Author:** ![Irishman](https://avatars.discourse-cdn.com/v4/letter/i/b487fb/32.png) [@Irishman](https://boards.straightdope.com/u/Irishman)\ **Post date:** [November 16, 2011, 8:05am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/56 "2011-11-16T08:05:32Z")
My one class in alternating current circuits used imaginary numbers (well, complex numbers, which couple imaginary numbers with real numbers). ---
**Author:** ![Powers](https://avatars.discourse-cdn.com/v4/letter/p/edb3f5/32.png) [@Powers](https://boards.straightdope.com/u/Powers)\ **Post date:** [November 16, 2011, 2:47pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/57 "2011-11-16T14:47:50Z")
> [@John\_W.Kennedy](#): > > Actually, imaginary numbers are quite common in reality. Magnetism is imaginary electricity, time is imaginary space, and, if tachyons exist, they have imaginary mass. Explain, please. In 500 words or less. =) Powers &8^] ---
**Author:** ![John\_W.Kennedy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_w.kennedy/32/1031_2.png) [@John\_W.Kennedy](https://boards.straightdope.com/u/John_W.Kennedy)\ **Post date:** [November 16, 2011, 11:07pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/58 "2011-11-16T23:07:22Z")
> [@Powers](#): > > Explain, please. In 500 words or less. =) 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.) ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [November 16, 2011, 11:22pm UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/59 "2011-11-16T23:22:56Z")
> [@Powers](#): > > Explain, please. In 500 words or less. =) > Powers &8^] 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. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [November 17, 2011, 12:20am UTC](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936/60 "2011-11-17T00:20:56Z")
> [@Andy\_L](#): > > 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. 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](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. " [Previous page](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936.md?page=2) [Next page](https://boards.straightdope.com/t/what-exactly-was-the-new-math/601936.md?page=4)