Simanaitis Says
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COMMON CORE MATH
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.
Tom Lehrer, mathematician, pianist, satirist extraordinaire. Image c. 1960.
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?”
How many ducks in total?
“Gee, mom, I can do a word problem!”
Things get a bit foolish when the task is 134 – 71.
A subtraction example in Common Core.
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).
The Trinomial Cube: a meaningful example of intuition in math education.
Common Core ideas don’t have to replace rote memorization of mathematical facts. Rather, they can reinforce rigor in a practical way.
The top surfaces of these Trinomial Cube elements show that (A+B)2 = A2 + 2AB + B2.
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
Simanaitis Says 
This is coincidental in that my daughter starts 7th grade today (and her school is going to use the “core”, I guess). And, of yes, I am afraid that my head will explode!!!
A wonderful example of Core Math in action is the old mechanical calculator that my father used to use in his business until the early seventies, the Odhner Arithmometer (pictures can be seen at the Wikipedia page on mechanical calculators).
Multiplication was accomplished by entering a number (including decimals, with the decimal being marked by a sliding marker) in the main register, and then the crank was used to add the number as many times as desired, with the number of times showing up in the bottom left register, and the result in the lower right. Decimal location was always based on old math, for example two decimals in the upper register, and three in the lower left, translated into six in the answer register (2×3). The bottom carriage was shiftable so that 10, 100, 1000 times was easily accomplished without having to crank that many times.
To multiply say 150.25 by 20, you’d enter 150.25, shift the carriage by ten (one click), and then crank the device twice. Bottom left register would show 20, and the bottom right would show 3005.
Addition was easy. Enter first number, crank once. Enter second number, crank once, and the result would show in the answer register. Subtraction was accomplished by turning the crank in the reverse direction. Division was a bit of a trial and error process. Enter the dividend and crank once. Then enter the divisor and subtract multiple times until a bell rang and answer display would show 999999999.9(999 etc ). Crank back once, then shift carriage and crank until the bell rang again, and so on till the answer was found to a reasonable number of decimal places.
This was a great way to visualize numbers in action. Machine currently resides in my calculator museum at home, along with my old HP25 and TI59 programmable calculators, and an old Casio Basic computer that had a single line display, that worked fine until one day I hit the display on a chair back as I was swinging my jacket to put it on and destroyed the display. The one item sadly missing from my museum is my old 6″ pocket sized slide rule. I know I would not have thrown it out, but just cannot find it anywhere.
What a neat gizmo ( and collection)! Check out my slide rule item for the Thatcher slide rule.
I have an early Olivetti laptop (sold in the U.S. as a Tandy). I shall unearth it, fit new batteries and fire it up–if possible. I recall it had a three-line display (?).
The TI-59 had a little magnetic card reader slot, and I have about 60 little programs, some that I wrote and some that I got/bought through the TI exchange service. You’d get one free program for each one you submitted (that was approved, some got rejected for not providing sufficient functionality in TI’ts opinion), and others you paid a small fee to get. You’d get a key stroke listing with instructions on how to use it, and perhaps a sample data run. You’d then have to key it in, step by step and save to mag cards. I had a printer base that would output a cash register sized thermal strip, and indeed many programs were written that only really worked with the printer, as input/output was too complex for just the one line display. The last time I checked the machine still worked, but many of the program strips have faded (used it regularly at work in the early 1980s).
The HP25 had no storage, and of course you’d have to input all your program steps (fortunately only 49, so not too much to key in, but also limiting on what you could do). It unfortunately no longer works, but was used at grad school in the mid 70s and at work, even with the TI-59 in service. It used the great RPN notation with a 4 number stack. My favourite calculator on my iPhone is a generic HP style RPN clone.
The Casio also had a dock (much more compact than the TI-59), that had a thermal printer, and a dictation device sized mini-cassette recorder for storing programs. Never developed too many programs for it, but used it as my calculator for years.
My current calculator collection (in addition to the iPhone “HP”), includes a non-RPN HP that I bought after I destroyed the Casio, and a tiny four function, plus memory unit that I bought at K-mart for $5 while out of town for a site visit 19 years ago, and still running strong on the original batteries.
Oh yes, and aviation style bezel slide rule calculators on 5 of my watches, not that I use those too much (kind of tough these days with my 60 year old eyes).