Learning Strategies

According to the tenets of standards-based education, any teacher should focus primarily on what is “essential to know” and only secondarily on what is “nice to know.” It’s hard to disagree with this idea.

But I’m going to try.

The basic idea, actually, is pretty much unassailable. Suppose, for example, you determine that all high-school graduates need to understand the difference between a mean and a median and need to be able to compute basic probabilities. Those are probably reasonable goals, and in that case you really do want to distinguish the essential-to-know (measures of central tendency, definition of probability) from the nice-to-know (measures of dispersion, permutations vs. combinations). I feel confident that standards-based learning, which says that we should “begin with the end in mind,” is a lot more appropriate than older approaches that are based either on a long inventory of skills or perhaps on a few fluffy concepts.

But...there are a couple of big hidden assumptions here. One is that we (the experts) know what’s essential to know, and that we know it for all students. The trouble is that there are many students for whom measures of dispersion and distinguishing permutations from combinations are essential knowledge. Doesn’t the “essential to know” philosophy fly in the face of differentiated instruction, which recognizes that different students have different needs? I suppose you could say that there’s no real conflict, since one issue is talking about ends and the other is talking about means, but that would only be true in an ideal world in which everyone had all the time and all the motivation that one might want.

So we have some workarounds that provide partial solutions to this dilemma. The big one is the recognition that there are different courses. What’s essential for a college-prep Algebra II student is know is simply not the same as what’s essential for an AP Statistics student to know. So let’s focus on a single course: what’s essential for a college-prep Algebra II student to know?

Our math curriculum at Weston focuses on big ideas, which students explore in a small number of units, rather than focusing on dozens of small skills. In Algebra II the four units are as follows:

1. Quadratic Functions


2. Exponential and Logarithmic Functions


3. Systems of Equations and Inequalities


4. Number Theory with Cryptography

And I won’t bore you with the big ideas, but you can check them here. Within these four units and 19 associated big ideas — not to mention several dozen key concepts — there’s a lot of room for disagreement. Is it essential to memorize the quadratic formula? Do college-prep students need to know the natural logarithm? Does one have to be able to solve a system of three equations without technology? Does number theory drive the crypto applications or vice versa?

And that’s just one question from each unit. I could easily list twenty.

And I can’t even differentiate my answers by distinguishing among specific students. Since I don’t know what your needs are going to be in the future, if you’re a student of mine I can’t predict whether some particular math concept will become essential to you in some future course or even in some future political decision that you’ll be called upon to make. So the only solution is to use our best judgment about what will leave the most doors open to you. But essential? I have no idea what will be essential.

I mentioned above that there are two hidden assumptions. The other one is that students will have a more appropriate learning experience if the nice-to-know is subordinated to the essential-to-know. The origin of this assumption is reasonably clear: without it, too many teachers would make decisions that “seemed like a good idea at the time.” But a course that is limited to what somebody considers essential is an impoverished course. The true tragedy of No Child Left Behind and MCAS is that too many schools and teachers have knocked half of the life out of their curricula. Although I suppose one could argue that everything in a rich curriculum is essential to know, it would be tough to sustain such an argument; the more honest approach is to admit that much of it is merely “nice to know” but so what? If teachers teach what they love and can get a significant number of students to do likewise, it’s going to be an improved experience for those students. I’ve often written about The Saturday Course in this blog; there are a great many reasons why this program is so astonishingly successful, but surely the major explanation is that talented teachers have the freedom to teach topics that they’re passionate about to students are motivated to learn. Yes, what I teach in The Saturday Course may not be essential to know, but it’s an essential part of my students’ education.

Labels: math, teaching and learning, Weston