Saturday, November 8, 2014

My daughter can now date Barry's daughter!

Barry Garelick once told me that anyone who wishes to date his daughter must first successfully derive the Quadratic Formula. A few days ago my daughter proved up to the task.

In fact, she was able to derive it on her own the first time around, with minimal assistance: without having first had it demonstrated to her. In Constructivist parlance, she "discovered" it! But only after lots of cumulative, guided practice placed it squarely within her Zone of Proximal Development. (Cumulative, guided practice of the sort that, incidentally, is entirely missing from Reform Math Algebra, which, if it asks the students to learn the formula at all, has them do so not via conceptual understanding, but via rote memorization).

My daughter's cumulative, guided practice included solving dozens of quadratic equations of varying complexity (including so-called "literal equations" in which the coefficients themselves are variables): first by factoring, and then by completing the square. I credit in particular her honing the technique of multiplying the equation by four times the co-efficient of the squared term before completing the square. Not only does this simplify the process by eliminating the need for fractions; it also makes the Quadratic Formula derivation a tad more elegant.

It's the difference between this:

And this:



Or, in her own hand:
 


Of course, either way works--whether for handling quadratics, or for dating Barry's daughter.

Having to resort to rote memorization, on the other hand, substantially limits your future prospects: both mathematical and romantic.

4 comments:

C T said...

What curriculum has she been using?

Katharine Beals said...

It's the Wentworth's "New School Algebra" that I've used in many of the math comparison problems. Great book, published in 1919!

Auntie Ann said...

I have a copy and it's nice and straightforward.

Does she actually work from the book? Ours is in good shape, but I wonder how well it would hold up to full use.

Niels Henrik Abel said...

I must have an earlier edition (_A School Algebra_ by Wentworth, 1895). I have been going through it lately to outline it in preparation for creating an online introductory algebra class. I've been comparing some of its problems to the remedial algebra I've seen as a tutor, and I am amazed at the depth and complexity of the problems from the older text. Why can't they make textbooks like that anymore? Instead, modern publishers insist on dressing up their pig with two-inch margins and four-color printing. Have to justify the triple-digit price somehow, I guess ~