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## Friday, November 21, 2014

### FOILed again?

The statement in red here is one I came across while taking another look at Common Core standards. I decided to Google the statement. I did discover that Google will not let you use more than 32 words between quotes, so I Googled just the 1st sentence in quotes (so Google looks for an exact match), and got 14,100 hits. Listed below are just a few of them.

There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

It's a bit misleading. The second sentence says "may have a better chance". Anyone who really and truly understands what binomial multiplication is and why it is, WILL have a better chance. A much much better chance.

It's also interesting that the statements refer to "a mnemonic device" and gives it a bad flavor. I suspect the device the authors had in mind was the old FOIL method for binomial multiplication, and the impetus behind the statements was the knowledge that there are teachers who taught only FOIL, and nothing more. (Could it be in use here only because F-O-I-L actually spells a word in English? Could this be why US math is falling behind?!? Imagine!?!) Do you know the meaning of FACE? Does it help or hinder in music studies?

I find it odd that a statement which embodies such a major thrust in educational philosophy is phrased in the context of binomial multiplication.  Multiplying binomials is something most people never do in their lives. However, if binomial (and polynomial) multiplication where actually dealt with carefully and fully they would be recognized as the foundation behind most algorithms for multiplication. Multiplying a pair of two digit numbers is actually a binomial multiplication. Multiplying 564 and 37 is actually multiplying a trinomial and a binomial. (I find it interesting that my spell-checker just marked "trinomial" as a misspelled word. For the record, it does not acknowledge "pf" as a possible error.)

The 4th grade standards include this: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

The arrays and area models referred to here are exactly the same devices that are, or could be, used to describe the distributive process applied when multiplying polynomials. It would be nice if our educational system would recognize this and capitalize this, and perhaps gather 4th grade and high school teachers together to carefully comb out more connections. I an willing to bet that a substantial number of 4th grade teachers are not tuned in to the distributive rule as it relates to polynomial multiplication. A few visuals are here.

Whatever happens with Common Core, let's at least recognize that it doesn't have all the right answers, but it could be spurring people into asking the right questions. I hope.