November 29, 2007

Wherein we excoriate Everyday Mathematics and also demonstrate the plausibility of letting secondary-grade students use calculators

As Joanne Jacobs notes (hat tip), some of the questions on the NYC-used and Texas-rejected Everyday Mathematics series are just absurd: if math were a color, a food, a type of weather, or a political party, ... oh, wait. We have a mashup: if your political party were a color, what would it be?

I've never seen any of that particular series, but it was mentioned in a comment thread on an entry about communicating math standards (a post from two months ago). I wonder if the most vociferous ideological complaints about Everyday Mathematics are by folks who would disagree with letting kids use calculators on tests. I'm very sympathetic to that argument from one perspective: children should learn fluency in tasks such as multiplication. (We have a copy of Bill Handley's Speed Mathematics book in our house, and I absorbed a few ideas from Jakow Trachtenberg's book when I was a child.)

But at the same time, not having calculators leaves multiple-choice problems vulnerable to testwise strategies.  I don't know which states have exams with two- and three-digit multiplication problems, but the following is a fairly easy example of finding the right answer without doing the problem.

Consider an extreme example: 47,583 x 97,621. We know three facts about the answer:

  • The last digit of the answer is 3. (Multiply last digits.)
  • The answer is a multiple of 9. (Cast out nines from the two numbers.)
  • The first digit of the answer is 4. (Estimating 4.7*0.97.)

With that information, I probably don't have to perform any calculations other than addition and single-digit multiplication (1*3, 0*7, and 4*1). 

I wrote all of the above before calling up my computer's calculator. For those who are curious, the answer is 4,645,100,043. That happens to be 9*516,122,227, no remainder.

Are these really the type of skills such tests are designed to measure? I'm not saying the skills are bad to have: estimation is very important, and casting out nines is an excellent check on answers. But there is a rather romantic notion floating around that somehow, if we buckle down and remove calculators from the hands of kids in all situations, men will be real men, women will be real women, and international math and science scores will be real international math and science scores (apologies to Douglas Adams fans).

Somewhere between Everyday Mathematics and macho attitudes towards calculators, there must be sanity.

Addendum/explanation of why casting out nines works as a check on multiplication. Let X=9x+a and Y=9y+b, where |x| and |y| are the largest possible for a and b to be integers as well. (I.e., a and b are the ordinary remainders when you divide X and Y by 9.)

X*Y = (9x + a)*(9y + b) = 81xy +9(ay+bx) + ab. Since the first two terms are multiples of 9, the remainder of X*Y when divided by 9 will be ab. This works with any chosen number to divide everything by, but since we normally work in base 10, 9 is the easiest numeral to work with. (If your species generally had Z fingers and therefore used a base Z system, you'd probably be casting out Z-1's.)

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Posted in Accountability Frankenstein on November 29, 2007 2:05 PM |