In a blog post a while back, I threw down the gauntlet to algebra teachers: identify four big ideas that could ground the course and make it more intellectually worthy. My new virtual friend Patrick Honner took me up on it here. The 4 ideas he identifies are:
1) Algebraic Structure
2) Binary Relations
3) The Cartesian Plane
Here is my reply:
Touché, Patrick! At first blush, these are fine candidates for big ideas. However, before we can say for sure, of course, we need to define a big idea – and warn about a common misconception as to what a big idea is.
Let’s start with the misconception. A big idea is not an overarching category that merely relates many pieces of content together. So, equation or number are not big ideas to me; they are just names for a vast category of entities. Similarly in English: word is not a big idea. In fact, even a little kid already brings that idea to the class.
What is a big idea? It is an idea that provides new insight, order, and usefulness for otherwise confusing, discrete-and-isolated-seeming elements. A big idea has intellectual power, in other words: it helps you make sense of things and enables transfer and new connections. The periodic table is a big idea. So is axiomatic system.
Another way to say it: number and word are foundational ideas but not “big” ideas. Once you get what they categorize, their power is over. Force is not a big idea but F = ma is one of the biggest ideas ever in intellectual history. The same is true of democracy in history. It establishes a category, a kind of governance, but beyond that you gain no intellectual power beyond the initial concept. Inalienable rights, however is a big idea because we keep using, refining, and questioning the idea to advance freedom.
I am inclined, therefore, to say that the Cartesian plane is a big idea. Indeed, it was a big idea historically, as you know. Suddenly, seemingly difficult to solve geometry problems could be solved more easily by analytical means, and soon thereafter math was no longer limited to equations to the 3rd power.
But Cartesian plane as a phrase leaves me a bit cold. It doesn’t capture the power of the idea. I think we need a phrase like “we can map all possible 2-dimensional figures and relationships using a simple coordinate system” or some such thing (with less ugliness).
Similarly, binary relations is a powerful idea because once you get it, many seemingly disparate and random algebraic problem solving moves make sense. However, again, I am inclined to state the idea somewhat differently to stress the “power” angle. I think the more helpful way to state it is: binary relations provide useful equivalences, based on complete reciprocity because the equivalences are key to problem finding and problem solving.
I realize that you may mean all of my phrasings to be implied; I just want to stress the “bigness” aspect a bit more.
By this reasoning, I am less sure about the idea of algebraic structure. I am unclear on just what the power of the idea is or how algebraic structure differs from any other structure. For example, what is then the difference between logical or linguistic structure and algebraic structure? How is that structure helpful for insight and problem solving? That’s almost like saying the structure of our bodies is a big idea. For that matter, isn’t the structure the binary and reversible relations? If not, what’s the difference?
Function is, on the face of it, the most obvious big idea, given that it dominates instruction, as you mention. And it certainly is one of the first times that students are expected to grasp the idea that a function models a relationship involving variability. But the more I think about it, I am less sure of it. Consider that the drawn-out concept is “showing that something is a function of something else.” But once you get that f(x) is just shorthand for that idea, then what? It seems more “foundational” than “big” to just use the word function. Is there another way to phrase what the power of the idea is?
Ah: lurking in the background is a familiar question for us. Does the idea remain big once you are no longer a novice? “Big” according to whom?
I am harkening back to our conversation about rigor. Ultimately, we run into the same problem: is bigness a relative term? A big idea to an expert is likely to be inscrutable to most novices, and a big idea to a novice may be trivial to an expert.
So, I wish to up the ante. To me a big idea is big for both: I am looking for those ideas that are big – powerful and fecund – for both novice and expert.
I always return to this simple example from soccer: Create dangerous space on offense; collapse dangerous space on defense is a big idea at every level of the game, from kid to pro. And it is transferrable to all space-conquest sports like lacrosse, hockey, and basketball. Truly big.
Do your 4 ideas have that same power, if perhaps implicitly? I remain unsure on some of them. And part of my lack of certainty is that I remain unsure about algebra as a whole. Is it, as a whole, a big idea? Or is it just a useful set of tools? Part of my criticism of algebra courses as currently designed is that they provide no intellectual priorities or direction – the courses are actually framed, taught, and assessed as merely a set of tools (like teaching soccer only via drills and never playing games). So, until and unless we see algebra courses framed more like courses in Philosophy, C++, Economics, or The Short Story (which arguably have more of an overall intellectual purpose than algebra courses) I am unclear on what the best big ideas for algebra are.
Jumping back to arithmetic makes my clack of clarity clearer, perhaps. Many texts and standards identify place value as a big idea. I don’t see that as a big idea at all. Place value is a derivative idea from the iterative structure of a number system. (If anything, place value confuses kids rather than helping them!) I’d be more likely, therefore, to offer the associative, commutative, and distributive properties as big ideas in arithmetic since they help you understand and use the structure to solve problems.
But I am not a mathematician. Thoughts from the experts?
PS: This back and forth has generated a lively and friendly exchange with a number of people online. But the best response came here. Read it and then read my reply – particularly if you like the soccer angle as much as the math angle!