"new math"

If you spend any time at all watching 70s-era sitcom reruns, you will inevitably run into someone (usually in a punch line for when something doesn’t add up, literally or figuratively) saying “That must be that new math I’ve heard about.” Granted, much time has passed, and it’s not new anymore, but what does this refer to? I know it’s just a joke, but it has to be referring to something real, or it wouldn’t be so common to see in these old TV shows. Is it something that we just know as “math” but that was new then? Or is it something that faded away?

It took not one, not two, but three members of the SDSAB to answer this question. See the Staff Report What exactly was the “new math”?

Apparently, the math I learned in school was new math, but they never called it that in class. One thing I distinctly remember was the introduction of set theory in the fifth grade. What a freaking waste of time. I didn’t really have any use for set theory until I got to college.

In 1963 or so, I was part of a group in my school that studied new math, or SMSG (School Mathematics Study Group). No textbook, just a big white paper booklet with typewritten text. They singled out the “smart” kids for this idiotic program and the rest of the class studied traditional arithmetic. Guess what? I never learned math basics, like how to subtract from zero. I didn’t learn math from the SMSG, either, it was all crazy theory talk about “the empty set” that confounded me. Being in that program hurt me for life and turned me against math. I never caught up because I had no grasp of the basic facts. Even the most challenged student in my class learned the basics; I was simply never exposed to basic traditional math. To this day I struggle with every basic math fact you can think of. It enrages me.

As a child going to elementary school in the 70’s, I remember it being called “New Math” only in 1st grade. I also remember spending time on grouping things. Two sets being in “Union” with each other, or not. Totally confusing to a 1st grader. It never really turned me off to math… that didn’t happen until 9th grade when I had they psycho algebra 2 teacher.

Some background: Back around the turn of the century, the big thing in academia was trying to find a “foundation” of math, one field of mathematics from which, given a few simple postulates and rules, you could build up all the others. For a while, set theory was considered the most likely contender, until Bertrand Russel proved that it was insufficient. (Russel’s attempts, and those of all other mathematicians, also proved futile, but that’s another story).

Fast forward to the middle of the century: We (the U. S.) got the idea that our students were dreadfully behind the rest of the world. Somebody in power got the idea that it would help to start teaching math with the “foundations”, right from the get-go. Somebody else (or maybe the same thrice-damned individual) apparently saw somewhere in an old book that set theory was believed to be a foundation for all of mathematics. He, or yet a third “educator”, then tried to figure out from scatch how to get arithmatic from set theory, and got it hopelessly wrong[sup]*[/sup] in the process. This mistaken interpretation of a failed attempt at a practically useless foundation of math was then forced upon students nationwide.

*An example of the way that the New Math got it wrong: To illustrate, say, two plus three equals five, the New Math texts would tell you that a set of two elements, plus a set of three elements, would yield a set of five elements. Not necessarily true. {1,2} + {2,3,4} = {1,2,3,4}: A set of two elements plus a set of three elements here equals a set of four elements.

Imposter. You’re not Chronos.

Yeah, I’m going to have to agree.

“until Bertrand Russel proved that it was insufficient”???

BR busted his lobes trying to get it to work. It was Godel that pointed out that it wouldn’t work.

I think Tom Lehrer figured it out.

These characterizations seem unfair, or at least representative only of those for whom “new math” did not work out well. And the 2+3=5 example of a “new math” error is obviously simple misunderstanding - there are no mathematicians who are confused by this.

I went through grade school in the mid 60s and remember “new math” being taught, specifically set theory and also base 10 versus other bases as possible numerical systems. This gave me more to think about, I am sure - but we certainly covered subtraction and other basics, too.

In my case math stayed somewhat interesting and challenging and eventually I got a degree in physics with all the calculus, differential equations, matrix algebra and so forth that that implies. I think this more inspired approach to teaching more about what’s going on inside math, rather than just the painful rote methods by themselves, helped. I am still working as a scientist, and loving it, and can’t help but think “new math” is a small part of the reason.

Russel showed that set theory was insufficient, and therefore attempted to build a foundation out of something else, namely symbolic logic. Gödel then showed that symbolic logic was insufficient.

Russell still didn’t show that set theory was insufficient; he showed that a set theory where any old collection of objects was called a “set” was insufficient. Saying that certain collections aren’t sets works just fine.

And there’s still hope for a workaround for Gödel’s result, but that would require redefining notions like “truth” and “satisfiability”.

I learned the basics of mathematics from 1978 to 1980, so I guess I was part of the “last wave” of New Math-ers. And it seems as though we encountered a more refined version than the 1960s kids. I don’t remember ever hearing the word “set” or “cardinality”. I just remember our math books showing pictures of little toy boats…2 boats on one side of the line–Three boats on the other side. (we were shown sets) We then had to draw the total number of boats. This made us learn to visualize what the concept “2+3” actually meant. It sounds pretty hokey, but you have to admit, its a big improvement on the old method of repeating over and over again “1+1 is 2, 1+2 is 3, 1+3 is 4”.

I worked in an afterschool program a couple of years ago, so I got to see a lot of semi-current textbooks…it looks like they’re still teaching basic concepts this way (at least up to multiplication & division). Of course no-one ever calls it “New Math” anymore because of how much the concept was ridiculed in the past.