“But, why?”
And back to the drawing-board I go yet again.
Yesterday was for the most part a day of mommy-learning. I was teaching my kids how to expand certain algebraic fractions in to their infinite series, a process which requires long division of polynomials. As I began describing the process to them yesterday over lunch, I realized that the first time we passed through long division of polynomials my kids understood the mechanics, or the “how”, but not the “why”. And so my daughter told me over the dining table that I could explain the “how” until I was blue in the face but that neither she nor her brother would ever remember how to do this type of algebraic math unless I could explain the “why.”
I sat back and asking what they meant, watched as they both picked up bits of chalk and began drawing on our chalk board. My daughter began talking about Vi Hart and how well she described infinite series and elephants and drew pictures that she remembered having seen. My son agreed with her and piped up with an added endorsement of Murderous Maths’ description and drew a picture of a spiral based on 1×2 rectangles.
Then came the twin discussion between them about the downsides of the various math curricula and why the “why” was so important.
And so while they finished eating their lunch, I grabbed all the algebra texts we have in the house. I scanned through the pile: Saxon, a smattering of college texts, Cocker, Ray, and even a book designed to help college teachers teach math. And there was nothing but a clinical description of how to go about the process of polynomial division. Not even a small amount of explanation of why the steps are taken and certainly no description of what the process relates to in the real world.
I looked over the table and began telling my kids that we would have to table the lesson for a day while I did further research. I told them that they were right to ask why and it only exposed my own lack of deep understanding when it came to polynomial division.
The answer to my reply was unanimous: you should ask Daddy when he gets home from work because he always explains why, even though his explanations sometimes have so much “why” that they go on too long and get boring.
So later when the Hub got home from work, I explained our math speedbump to him. He thought for a little bit and then began to work out the “why” for me. Together we designed a series of “whys” that ranged from “polynomial long division is a methodical way to cancel out the common factors between the two polynomials,” to graphs of two polynomial functions and the graph of the final division that led to a discussion of calculus.
I took these explanations back to my kids and discovered that each child responded and understood the “why” of polynomial long division at a different explanation. I started each child with the same discussion, and ended in two very, very different places. I think that it must be absolutely impossible for every child in a classroom to fully understand and master a subject when they are taught en masse. When my kids are following along and understanding, I can teach them together, but when one or both do not deeply understand, I really have to talk with them seperately to get the concept into their head sucessfully.
Regardless, we conquered the “why” of polynomial long division and the “why” of expanding infinite series from the division of two polynomials yesterday. And I am certain that having achieved the “why” my kids will not, ever forget the “how.”
