I’m teaching three sections of algebra next year, and I’ve been thinking a lot about the essential questions at play, the ones that can tie it all together. I have lots of little ideas about it, but one I really like is the notion of “dependence”.
How long would it take to drive to LA? It depends. Mostly it depends on where I would leave from and how fast I would drive. How much longer would it take to drive to another city? It depends. What’s the area of a square? This only depends on the side length. You could say it depends on the perimeter, but the perimeter depends on the side length, so by transitivity… This ties really nicely into my “Algebra 2: functions and abstract algebra” course, because a function’s output is uniquely dependent on the input.
The dimension of a problem is simply how many pieces of information the answer depends upon. What is the equation of a line in the plane? (classic high school math question) This has dimension two, because it relies on the slope and intercept of the line. Or it has dimension two, because it depends on the selection of any two points. This idea seems to wind its way through a lot of mathematics, so I’m hoping my students will watch it closely. Almost every mathematical structure can be seen through the lens of functions of several variables.
* * *
“What’s the best way to educate kids? The search for the answer to this question only leads to more questions: Who are the kids? Where are they from? How old are they? What do they love to do? What is their home situation? . . .”
The one-size-fits-all industrial model of education views all of its students roughly and demands they fit school as it is. Human minds, particularly for children, differ wildly, and so should the experiences that develop and result from these minds. “The best way to educate kids” depends on far more than our school system currently handles.
When it comes to teaching kids, I suspect the dimensions are many!