## Friday, February 22, 2013

### Math problems of the week: Reform vs. traditional algebra

Introducing the Quadratic Formula I. In Integrated Math 2, Unit 4, "Quadratic Formula and Graphs," p. 215 [click to enlarge]:

II. In The University of Chicago Mathematics Project Algebra, Chapter 9, "Quadratic Equations and Square Roots," p. 574 [click to enlarge]:

III. In Wentworth's New School Algebra, Chapter XIX, "Quadratic Equations," p. 278 [click to enlarge]:

III. Extra Credit:

The Reform Math texts don't show students where the quadratic formula comes from; the second text simply tells students that it's "famous" and to "memorize it today." Wentworth provides a complete derivation.  All three texts show students how to apply the formula to quadratic equations, but only Wentworth assumes that students already know how to turn a quadratic equation into an equation of the form ax2 + bx + c.

Given this, consider one of the good points made by Keith Devlin in the piece I blogged about earlier this week:
If mention of the word algebra automatically conjures up memorizing the use of the formula for solving a quadratic equation, chances are you had this kind of deficient school math education. For one thing, that’s not algebra but arithmetic; for another, it’s not at all representative of what algebra is, namely, thinking and reasoning about entire classes of numbers, using logic rather than arithmetic.
Question 1:

Should Devlin be focusing his efforts on creating computer games that somehow teach math non-symbolically, and, as reported in the following excerpt from an article in Inside Higher Education, defending Jo Boaler's highly flawed study of Reform vs. traditional high school math programs:

Keith Devlin, director of the Human Sciences and Technologies Advanced Research Institute at Stanford, said that he has "enormous respect" for Boaler, although he characterized himself as someone who doesn't know her well, but has read her work and is sympathetic to it. He said that he shares her views, but that he does so "based on my own experience and from reading the work of others," not from his own research. So he said that while he has also faced "unprofessional" attacks when he has expressed those views, he hasn't attracted the same level of criticism as has Boaler.
Of her critics, Devlin said that "I suspect they fear her because she brings hard data that threatens their view of how children should be taught mathematics." He said that the criticisms of Boaler reach "the point of character assassination."
Or should he be critiquing today's Reform Math texts for teaching algebra largely as the mindless arithmetic application of formulas rather than as the logical sequences of algebraic manipulations taught in traditional texts? And, as Barry Garelick discusses in EdNews, for teaching geometry as "the application of theorems rather than the proving of propositions"?

Question 2:

Should Devlin happen upon this blog post and/or my previous one about him, will he characterize what I write as an "unprofessional" attack and/or as a "character assassination" that ignores "hard data that threatens my view of how children should be taught mathematics"?

Anonymous said...

Almost certainly - but you'll have to trust him on the "hard data" part, because revealing it would be a breach of confidentiality with the subjects. And it's scandalous that you should even ask someone to do that.

Barry Garelick said...

The hard data are available publicly. Test scores are aggregated and available to the public. And that's how Jim Milgram and others identified the names of the schools. They matched up the test scores that Boaler reported, with the test scores reported for schools on the public data base. There is nothing that prohibits the release of such public data. The only thing prohibited is releasing the individual student test scores. So it isn't a breach of confidentiality nor is it scandalous to ask that someone be responsible in reporting what is available on public data bases.

Anonymous said...

Aw, you know I'm just teasing you, Barry.

Lsquared said...

It's very disappointing. Reformers talk a good line, but when you look at the details of the textbooks, you find that they really aren't living up to their purported ideals. Great example of that today. (PS--you don't have to go back nearly as far as Wentworth to find better treatments of the quadratic formula)