I’ve worked this year to refine the learning cycle in my class. I try to use an intro activity to set the groundwork, some explicit instruction to clarify misconceptions and consolidate big ideas, and a whole bunch of formative assessment and deliberate practice that dives deeper and deeper into a topic over time.
I’ve found that the amount of explicit instruction I give in this model can actually be quite small. I don’t need to model every single application of exponential functions. Instead, I can explicate some key points about the structure of these functions, connect them to logarithms, and clarify how the function is connected to a graph. Just a few minutes, well planned and well placed, goes a long way. That’s not to say that everything students learn happens in those few minutes — but the less explicit instruction I give, the more students pay attention to it, and I can then make deliberate decisions based on formative assessment around what needs more instruction.
So now I’m looking at a whole bunch of class time spent watching my students do math, and a whole lot of choices around what to highlight, what to discuss, and what to emphasize. Here I’ve looked for guidance from How People Learn, and in particular the second chapter, “How Experts Differ from Novices”. One of the goals of math education has to be to move students from being novices toward being experts. I have no illusion that all students will leave my class with broad and transferable expertise in mathematics, but I can still use research to move students in that direction.
Two differences between novices and experts have caught my eye, and informed the instruction I choose to emphasize.
First, from How People Learn: “Experts notice features and meaningful patterns of information that are not noticed by novices” (31). One hallmark of expertise is sifting the signal from the noise. Students who build fluency with a concept and quickly recognize patterns and connections between representations are able to draw insights about novel solution methods and strategies. But students who aren’t able to pick out the essential elements of a problem become overwhelmed and often resort to blindly plugging numbers into equations, losing valuable opportunities for metacognitive thinking about the structure of a problem.
When I’m making a point about a problem or choosing a student to share a strategy, I want to emphasize what that student noticed about the problem — and why that piece of information was worth noticing. It could be seeing a u and a du in an integral — because looking for functions and their derivatives is a useful thought pattern when integrating; or realizing that values in a function are increasing by a constant ratio — because looking at how values in a function change is a useful tool to decide what type of function it is; or noticing that dropping an altitude creates a path to an answer — because there are more tools to find missing pieces of right triangles than other geometric figures.
Second: “Experts’ knowledge cannot be reduced to sets of isolated facts or propositions but instead reflects contexts of applicability; that is, the knowledge is ‘conditionalized’ on a set of circumstances” (31). Experts don’t just have lots of well-understood math concepts sitting around in their brains; they have a set of heuristics for figuring out when to apply a certain concept, and also know when a rule doesn’t apply. This goes well beyond a textbook understanding — graphing a bunch of functions in the “polynomials” chapter, and then graphing a bunch of functions in the “rational functions” chapter, without thinking critically about the similarities and differences in structure.
To teach conditionalizing, I am asking questions to follow up on student sharing — what about that problem made you decide to use integration by parts? Why was an exponential model appropriate for this situation? Why was factoring useful to solve this equation? If I’m doing anything right, students are building knowledge beyond how to execute a mathematical tool, to understand when and where it is applicable.
I believe in the value of deliberate explicit instruction — and I also believe that what students learn from it is inversely correlated with the amount of explicit instruction I give. If I’m going to be deliberate about what I say and when I choose to say it, noticing and conditionalizing are useful patterns of thinking to watch out for, and emphasize when I have the opportunity.