# What exactly was the "new math"?

Dear Straight Dope:

I'm old enough to have heard of the "new math" but as far as I can tell that was what I was taught in school. I have asked people older than myself what exactly the "old math" was. They were definitely sure I must have been exposed to the newfangled stuff, but seemed puzzled when I asked them what the differences were. Personally, I really couldn't find any math that I was taught in high school or college that didn't have solid foundations before this century.

In the fifteenth century, when German parents wanted their kids to learn addition and subtraction, they sent them to local universities. To get them to learn multiplication and division, however, they needed to send their kids to Italy for graduate school. The new math that arrived in Europe soon after, which transformed CCLXIV x MDCCCIV into a problem that we can teach to sixth graders, was truly revolutionary. The new math of the sixties was, well, like many other movements of the sixties, disruptive, despised, and moderately beneficial, and is now still around, but incognito.

After Sputnik was launched, Americans felt the schools were in crisis. The National Science Foundation (NSF), created in 1950 to promote basic scientific research, was expanded in 1957 and began to examine and promote change in secondary school education in math, biology, chemistry, and social sciences. The changes in the curricula and texts had a filter-down effect on the primary schools as well. The main thrust of these changes was a switch from teacher "telling" and student recitation to "inquiry" and "discovery," with the hope that students would be more likely to retain information they found out themselves than what was just told to them in lecture form and memorized. In the hard sciences, and to a lesser extent the social sciences, this was described as "hands-on learning." It's a teaching technique still held in high regard by educators and parents today.

In the more abstract mathematics, however, the 'hands-on' connotation was disturbing to teachers and parents who had learned the addition facts and multiplication tables by rote. One focus of the new math was set theory, where students were encouraged to think of numbers in a new, hopefully more concrete way. Students would take a set of four items, and add it to another set of five. Yes, the result was still nine, but the emphasis was on the concept of addition, rather than the answer per se. Using this technique, students were hoped to discover that the sets would yield the same number regardless of their order (the commutative property), and that taking one original set from the combined set would yield the other original set, thereby discovering subtraction, the inverse of addition. Other aspects of the new math including using number bases other than base-10 and introducing more abstract number theory concepts such as prime numbers earlier in the students' careers. As you say, none of these concepts were newly discovered in the 20th century; the shift was purely in teaching technique, not in basic concepts.

Teachers were quite resistant to this, noting that instruction of the class as a whole was less uniform, and that the possibility of some students falling too far behind was greatly increased. Parents were more vocal in their opposition, claiming that they couldn't help their third-graders with their homework anymore, and pointed to a noticeable decline in the more concrete skills such as computation. New math was derided in the public forum, such as in Tom Lehrer's song New Math: "It's so simple / So very simple, / That only a child can do it!" By 1976, only 9% of school districts were using the NSF-proscribed curriculum in their math programs. Morris Kline, in *Why Johnny Can't Add: The Failure of the New Math*, wrote that "with near perfect regularity, [teachers] applaud the return to tradition content [and] instructional methods, and higher standards of student performance."

The textbooks were only dominated by the new math system for about 10 years. However, elements such as set theory and base-n computation are still retained to this day, albeit with less emphasis and as a much smaller portion of the overall curriculum. Don't even get me started on the so-called "New New Math", which is a debate that remains to be answered in the classrooms, school board and PTA meetings, and maybe the courts, and promises to become a hot-button topic in the next couple of years. Stay tuned.

SDSTAFF Jill adds:

Here's my dad's answer:

"I remember the introduction of 'new math' way back in Scotts Valley. I really don't remember what it was except my impression is that it consisted more of talking about math rather than doing it. Most of the parents were worried sick, because they didn't understand it.

"As a help I can remember my youth in Dudley Elementary School (K, 1-7). It was called arithmetic not math. The first ten or fifteen minutes of every day for seven years consisted of excercises in adding, subtracting, multiplying, and dividing numbers. We were each handed a card with the day's problems on it. We each had a tablet of thin, translucent paper. The card went under the top page. We wrote the answers on the paper. Everybody hated this. On the other hand everybody who went through seven years of this could handle numbers without a calculator.

"Of course we also did the usual things in arithmetic class. One thing different was we learned logarithms in the seventh grade. Logarithms are not used anymore because of the computer and electronic calculator. I still love them, because they are real neat. ('Cool' is the new math word for 'neat.')

"The basic difference is that we did arithmetic, which is about numbers, and new math is more about ideas and concepts. Another is, 'This is the problem. What is the answer?' as opposed to, 'This is the concept. What does it mean?' Another attempt to making learning fun instead of work."

SDSTAFF Dex adds:

The following examples may help to clarify the difference between the new and old math.

1960: A logger sells a truckload of lumber for $100. His cost of production is 4/5 of this price. What is his profit?

1970 (Traditional math): A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?

1975 (New Math): A logger exchanges a set L of lumber for a set M of money. The cardinality of set M is 100 and each element is worth $1.

(a) make 100 dots representing the elements of the set M

(b) The set C representing costs of production contains 20 fewer points than set M. Represent the set C as a subset of the set M.

(c) What is the cardinality of the set P of profits?

1990 (Dumbed-down math): A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Underline the number 20.

1997 (Whole Math): By cutting down a forest full of beautiful trees, a logger makes $20.

(a) What do you think of this way of making money?

(b) How did the forest birds and squirrels feel?

(c) Draw a picture of the forest as you'd like it to look.