Thursday, December 29, 2011

145: Why Johnny Couldn't Add

It's school board election season again here in Oregon, and while I'm not running this time, it got me thinking about educational issues.  In particular, one topic I've been meaning to cover for a while is New Math.  Those of you old enough to have learned elementary school math in the 1960's or early 1970's, or a little younger but geeky enough (like me) to have browsed math textbooks in used bookstores in the later 1970s, will probably recognize the term.  New Math was a revolutionary change in math education, spurred by a reaction to the Soviets beating the US into space.  Naturally, the solution to that dilemma was to have a team of academic math professors, led by Ed Begle from Yale, come up with a new curriculum totally divorced from any experience educating young children.  What could go wrong?

Let's look at what the New Math was.  Basically, before this movement, elementary school math consisted of lots and lots of drilling of arithmetic problems.  While this wasn't very exciting, it did result in most children getting a solid 'number sense' and becoming very comfortable with doing basic addition, subtraction, multiplication, and division without calculators.   The theory behind the New Math was that children were unsatisfied with this because they want to understand the real logical foundations of what they are doing.  Thus concepts were introduced like understanding the difference between the concept of numbers and the written symbols known as numerals, set theory, converting numbers between different bases, modular arithmetic, and similar areas.

In the vast majority of cases, the net effect of all the time spent on these advanced concepts at the expense of gaining basic arithmetic competency and number sense was that kids could mimic some advanced mathematical terms, but were severely lacking in the ability to do everyday calculations.  Tom Lehrer famously made fun of this situation in his song 'New Math':


We should point out that some of these ideas had some level of usefulness as illustrations.  For example, suppose you want to illustrate that 4+3 equals 3+4.  If you have a set of four elements, written as a circle around 4 dots, and add it to a set of 3 elements, you can see easily that the order doesn't matter.  And look at how we deal with multi-digit numbers:  for example, 123 is equal to 1 times 100, plus 2 times 10, plus 3 times 1-- each place is a new power of the base of your number system, in this case 10.   You were probably doing 'carrying' and 'borrowing' when adding and subtracting multi-digit numbers for a long time before you understood that it was this place-based system that allowed you to do this-- and maybe learning to represent numbers in other bases would make this clearer.  

Some studies did show that talented teachers who fully understood these concepts, and used them appropriately as part of an arithmetic class, could indeed enable kids to better understand basic arithmetic.  They would have to be carefully guided to show how, for example, set theory led to the actual operations like addition and subtraction, or how the algorithms for working with multi-digit numbers originated in our base 10 number system.   But the vast majority of teachers didn't understand these advanced concepts much better than the kids, and instead turned them into a new set of random stuff to be memorized.  Rather than concrete and useful numerical operations, students were now engaging in rote repetition of complex ideas that were simply not useful to them at their level of mathematical sophistication.

An online article at 'Straight Dope' illustrates the change in education nicely with an example problem.
Traditional math: A logger sells a truckload of lumber for $100. His cost of production is $80. What is his profit?
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?
OK, there is probably a bit of exaggeration in this example, as I doubt kids actually were asked to draw sets of size 100.  But you get the general flavor.  After phrasing this basic profit problem in terms of set theory, has anything of value really been added, or has a simple, concrete concept been obscured?

The death knell of New Math was sounded in 1973, when Morris Kline published his famous book "Why Johnny Can't Add".   Kline rightfully pointed out that, "abstraction is not the first stage but the last stage in a mathematical development".   People need to understand numbers in a concrete way first, to the point where they have a natural instinct for them, and then maybe it's worth talking about abstractions like set theory and alternate bases.  The vast majority of kids are not little Bertrand Russells and Ludvig Wittgensteins, demanding a full axiomatic justification of what they are learning.   Repeated arithmetic drills may not be glamorous, but they get the job done.
I'd like to say that after New Math, teaching in the U.S. returned to an emphasis on sensible, back-to-basics methods that actually work.  In some parts of the country this was true.  But in other areas, New Math has been followed by various further educational fads such as calculator mania, radical multiculturalism, "Discovery Math", and "New New Math".  You may recall that back in episode 70, I discussed my frustration that a fast food cashier could not figure out 2+2+1 is less than 7 in her head.  And I'm afraid that the current talk in the news about our need to improve math & science education will result in the creation of yet more new fads instead of a return to common sense.  But at least I'll never lack for topics to make fun of in this podcast.

And this has been your math mutation for today.

  • New Math at Wikipedia
  • Another article
  • One more article
  • No comments:

    Post a Comment