Friday, March 29, 2013

Math problems of the week: multiplying polynomials in traditional vs. Reform algebra

I. The only problems involving polynomial multiplication in the Integrated Mathematics 2 textbook (p. 517 and p. 642)



II. The first problem set in the Special Rules of Multiplication chapter of Wentworth's New School Algebra (published in 1898) (pp. 76-77):


III. Extra Credit: 

1. Reform Math is obsessed with patterns, but not at the level of abstraction seen in Wentworth's New School Algebra. Here we have a discussion of how to turn certain products of trinomial pairs into products of binomial pairs of the form (A+B)(A-B). Write a personal reflection on how practice recognizing abstract mathematical patterns and conforming expressions to certain special abstract patterns might prepare you for higher level math.

2. Integrated Mathematics is one of the rare Reform Math texts that provides an index. I used it to locate the polynomial multiplication problems given above. In the process, I noticed that the "Multicultural connections" entry is about ten times as long as the "Multiplication" entry. (see below). Discuss.

3 comments:

Auntie Ann said...

What level/grade is IM2 aimed at?

Barry Garelick said...

You picked up on one of my biggest peeves with reform math; the notion put forth that math is about patterns or algebra is about patterns. Well, yes, math is full of patterns, so what? What do you make of the patterns. You said it nicely above. Some patterns have uses, and recognizing how one form/pattern can be re-expressed in a form that allows for an easier analysis of a problem is one of the things mathematics is about. So, it is important that students can recognize that a^2 - x^2 +2by - y^2 is made up in part by a perfect square trinomial that can be obtained by rewriting the expression as a^2 - (x^2 - 2by + y^2), and further recognizing that this can be written as a^2 - (x-y)^2, which can then be factored into (a - x + y)(a + x - y).



Henderson and Pingry (the authors of my algebra books that I used in school) were aware of this, and that's how they escalated the problems in a problem set so that students would have challenges, rather than being presented with . "exercises"


I show my students a magic trick in which someone thinks of a number from 1 through 31. I show them five cards with numbers written on them, and ask them which cards their number is on. Once they tell me, I then tell them the number they were thinking of. I don't tell them how it's done right away. The first card is all the odd numbers from 1 through 31. The second also has some patterns to it that students pick up on, as to all of them, actually. (The last card contains the numbers from 16 through 31). But these patterns do them no good in figuring out either how the trick is done, or why mathematically it works. It's based on binary numbers. I won't go into the math of it here, but whatever cards they say they saw their number on, you take the first number of each such card and add up the numbers.


Now the pattern that's really important to notice is that the first such number on each card is a power of 2. So on card 1, it's 1; card 2 it's 2; card 3 it's 4; and so on. That's a pattern that if you know what binaries are and how they work will allow you to see the math of the trick.


Fuzzies would think it's wonderful that students see that the first card is odd numbers, and all the other patterns they see that really do nothing to get at the math of the trick.

Katharine Beals said...

Auntie Anne, I've been having trouble tracking down a specific answer to your question, as Reform Math textbooks are notoriously coy about indicating grade/level within the textbook itself. I did stumble across a document that indicates that it is correlated to the 2007 Pennsylvania Math Assessment Anchors: https://docs.google.com/viewer?a=v&q=cache:NhKKzE5tVgYJ:holtmcdougal.hmhco.com/hm_data/pdf/states/PA/PA140-IM2_Gr11.pdf+&hl=en&gl=us&pid=bl&srcid=ADGEESjDQt4GOk_OzjPbbVuE_yOtU6Hio0Y2QMzEOsrdVlhPpgI8bC_jgK_6kLfQ7yzK9X36oVRj89L4NAxIaX0GtRfM6FJpqU95rruyD8oOezWGF9SPPJ6HVn7TImhMe6ZzozUCRdKX&sig=AHIEtbRENnZOo3KZMfLOJsFIrgptfMI6kQ