Day 5: CM, Page v

The course title "Concrete Mathematics" was originally intended as an antidote to "Abstract Mathematics," since concrete classical results were rapidly being swept out of the modern mathematical curiculum by a new wave of abstract ideas popularly called the "New Math".
- Concrete Mathematics, page v

Day 5: The first page of the preface of CM. From here on out there won't be much more biographical information. We'll be getting into the meat of the book. But before that, we have a bit more honeymoon with the preface.

The first page of the preface discusses some of the changes that were happening in the educational system when the book was written in the 60s and 70s. The quote above claims that "Concrete Mathematics" was an "antidote" to "abstract mathematics" or "new math". So what's the difference between them?

According to wikipedia, pure mathematics is "motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction and beauty." New math was an educational movement in the 1960s, introduced after the Spunik crisis, that attempted to provide students with a better mathematical foundation. It focused on abstract and pure mathematical concepts:

New Math emphasized mathematical structure through abstract concepts like set theory and number bases other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world.... For example, in some cases pupils were taught axiomatic set theory at an early age. The idea behind this was that if the axiomatic foundations of mathematics were introduced to children, they could easily cope with the theorems of the mathematical system later.
- wikipedia

Not surprisingly, there was some backlash against new math. Parents and teachers were concerned that the material was too abstract and that children weren't learning the basics. For example, "Professor George F. Simmons wrote that the New Math produced students who had 'heard of the commutative law, but did not know the multiplication table'" (wikipedia), and Morris Kline wrote a book called "Why Johnny Can't Add: the Failure of the New Math".

Concrete Mathematics was also a part of the backlash. It eschews abstract mathematics and attempts to provide a pragmatic foundation for computer scientists and programmers.

The end-users of mathematics studies were at that time mostly in the physical sciences and engineering; and they expected manipulative skill in calculus, rather than more abstract ideas. Some compromises have since been required, given that discrete mathematics is the basic language of computing.
- wikipedia

New math is even mocked in popular culture. It was mentioned in a Simpson episode (Dog of Death) when when "Principal Skinner describes the new textbooks he'd like to use: 'History books that know how the Korean War came out… math books that don't have that base six crap in them…'" (wikipedia).



Tom Lehrer wrote a song entitle "New Math" that captures the frustration people felt. It "centered around the process of subtracting 173 from 342 in base 10 and in base 8. The song is in the style of a lecture about the general concept of subtraction in arbitrary number systems, illustrated by two simple calculations, and highlights the emphasis on insight and abstract concepts of the New Math approach."

Excerpt:

So you've got thirteen 10s and you take away seven

And that leaves five ... well, six, actually.

But the idea is the important thing.

Chorus:

Hooray for new math,

New-hoo-hoo-math,

It won't do you a bit of good to review math.

It's so simple,

So very simple,

That only a child can do it!

Concrete Mathematics was developed as an "antitode" to more abstract mathematics and the new math movement. So we can look forward to lots of pragmatic, non-abstract math! But first the rest of the preface...