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LET’S begin paradoxically with a summary: Old Math is rote memory and recitation of facts. New Math is rigor describing what numbers are. Common Core Math is intuition describing what arithmetic operations are.
A brief example of Old Math: Learn the “Twos Table” and parrot it back on demand. 2 x 1 = 2, 2 x 2 = 4, 2 x 3 = 6, …, 2 x 9 = 18, 2 x 10 = 20.
Old Math’s characteristics: It can become a litany of facts, boring to learn and unrelated to anything in the real world.
“You know 2 x 3 = 6, so tell me about 3 x 2?” “Hey, 3 x 2 sounds like the Threes Table, and I haven’t learned that yet.”
And, worse yet, “Johnny has 2 bags, each containing 3 apples. How many apples does he have?” “I can’t do word problems….”
Back in the 1960s, math educators identified these shortcomings of Old Math and decided to replace rote memory with rigor. As Nicolas Bourbaki realized, numbers have their foundation in set theory. See http://wp.me/p2ETap-1Od. Please don’t be put off by Bourbaki being a phantom math guy, and French to boot. Zut alors!
New Math’s basics: Some collections of things have two objects. Others do not.
This commonality of sets is fundamental to a number’s “twoness.”
In fact (here comes a mind blower), there is a set with nothing in it at all. We’ll call it the “empty set” and write it Ø.
New Math replaced boring litanies with arcane concepts that even the teacher didn’t seem to understand. (She may have been learning set theory just a bit ahead of her students.) Parents understood it even less.
New Math answered the 3 x 2 query, but in an obscurely technical manner: “3 x 2 must be 6, because 2 x 3 = 6 and multiplication is a commutative operation. It’s associative and distributive too.” “Say wha?!?”
“Besides, I still can’t do word problems….”
For a brilliant musical essay on this composed in 1965, see Tom Lehrer’s song “New Math,” http://goo.gl/Vwb4LG.
Beginning in 1972, Lehrer taught both mathematics and musical theater at The University of California, Santa Cruz. He’s retired now, age 86, with a musical legacy available as The Remains of Tom Lehrer.
By the mid-1970s, New Math was phased out. The strongest criticism was that its abstractions were not sensibly a first stage, but rather a final stage. Hence, many curricula returned to the basics of rote memory and recitation.
Common Core State Standards are the 21st-century response to “Why can’t Johnny read or do arithmetic?” Depending on points of view, this premise may or may not be meaningful.
Some argue that, internationally, U.S. kids rank 26th in math and 21st in science, behind kids in countries like Singapore, Japan and Canada. Others challenge this by noting U.S. supremacy in science and technology: Why didn’t Amazon, Boeing, Google, Microsoft and other world names arise in these “smarter” countries? Educational standards worldwide are difficult to compare.
Common Core Math replaces rote memory and rigor with a methodology of intuitive tally.
At its basic level, Common Core’s intuition is understood by all. However, it may become bewildering to parents as the level advances.
“There are three ponds. Each has four ducks in it. How many ducks all together?”
“Gee, mom, I can do a word problem!”
Things get a bit foolish when the task is 134 – 71.
This tally technique reminds me of a graphical version of Tom Lehrer’s song.
On the other hand, intuition at its best is exemplified by the Trinomial Cube (see http://wp.me/p2ETap-qG).
Common Core ideas don’t have to replace rote memorization of mathematical facts. Rather, they can reinforce rigor in a practical way.
That is, optimal education in mathematics would seem to include aspects of all three methodologies: Old Math’s rote memory, New Math’s rigor and Common Core’s intuition. Each has beneficial things to offer.
Last, I decry and lament the politicizing of any of these methodologies. Mathematics and science are neither liberal nor conservative. ds
© Dennis Simanaitis, SimanaitisSays.com, 2014