Something fun for a change

In which I continue reading the references

I’ve been writing a lot about the ed psych literature and needed a break, so today I am going to write about a synthesis of the math ed literature, The effects of classroom mathematics teaching on students’ learning by Hiebert and Grouws, 2007, a chapter in the second NCTM research handbook.¹ I love this chapter because it is balanced, careful in its conclusions, and well written. In fact it is so well written that this post will consist mostly of quotations.

Definitions of skill and understanding

By skill efficiency, we mean the accurate, smooth, and rapid execution of mathematical procedures . . . We do not include the flexible use of skills or their adaptation to fit new situations. By conceptual understanding, we mean mental connections among mathematical facts, procedures, and ideas.

There are two things to note here. First, the exclusion of flexibility from the definition of skill makes a clean cut between the two notions. I’ve always thought that flexibility belongs more under the conceptual understanding column. “Accurate, smooth, and rapid execution” slots well into findings from the cognitive load literature. Second, the definition of conceptual understanding is about making connections. That’s something you can observe and measure more easily than some nebulous notion of whether the student possesses the concept.

Defined this narrowly and this precisely, the two goals are not in tension. You can study the effectiveness of various teaching practices for one or the other or both.

The article proceeds to discuss features of teaching that promote these goals, but first issues a caveat.

we do not expect the features of teaching that facilitate skill efficiency and conceptual understanding to fall neatly into categories such as “expository” or “discovery.” In fact, the features of teaching we describe do not fit easily into any of the categories frequently used to describe teaching: direct instruction versus inquiry-based teaching, student-centered versus teacher-centered teaching, traditional versus reform-based teaching, and so on. . . . we will argue that most of these categories, distinctions, and labels are now more confusing than helpful, and further advances in research as well as clarity of policy recommendations will benefit from abandoning these labels.

Hear hear!

Supporting skill efficiency

The authors review a number of studies and summarize them as follows:

teaching that facilitates skill efficiency is rapidly paced, includes teacher modeling with many teacher-directed product-type questions, and displays a smooth transition from demonstration to substantial amounts of error free practice. Noteworthy in this set of features is the central role played by the teacher in organizing, pacing, and presenting information to meet well-defined learning goals.

Notice the precision here: “teacher-directed product-type questions”—as opposed to process questions, I assume—and “error free practice”—not just any old practice, but practice designed to be within the capabilities of the student. Notice also the central role of the teacher, so refreshing in these days.

I like that the authors followed through on the commitment to avoid labels here. The package of teaching practices that they describe here looks a lot like explicit instruction, but it’s presented simply as a package, possibly one that could be combined with the principles for conceptual understanding that follow.

Supporting conceptual understanding

The authors identify two key features of teaching that support conceptual understanding. The first of these is that “teachers and students attend explicitly to concepts.”

By attending to concepts we mean treating mathematical connections in an explicit and public way. . . . This could include discussing the mathematical meaning underlying procedures, asking questions about how different solution strategies are similar to and different from each other, considering the ways in which mathematical problems build on each other or are special (or general) cases of each other, attending to the relationships among mathematical ideas, and reminding students about the main point of the lesson and how this point fits within the current sequence of lessons and ideas.

Well, duh, but it “becomes more interesting when one discovers that the claim is supported across a wide range of research designs and holds true across different instructional treatments or systems.” This decoupling of the feature from specific instructional approaches points to a basic truth.

Especially important here are the words “explicit” and “public”. This echoes the practicing connections framework I talked about here. You can’t assume that the connections will follow implicitly from students doing the math. You have to name them and design activities that lead to them. Here again the authors are agnostic on the instructional model:

We believe the evidence does not justify a single or “best” method of instruction to facilitate conceptual understanding. But we believe the data do support a feature of instruction that might be part of many methods: explicit attention to conceptual development of the mathematics.

The second feature that supports conceptual understanding is that “students struggle with important mathematics.” The authors are careful to say what sort of struggle they mean.

We use the word struggle to mean that students expend effort to make sense of mathematics, to figure something out that is not immediately apparent. We do not use struggle to mean needless frustration or extreme levels of challenge created by nonsensical or overly difficult problems. We do not mean the feelings of despair that some students can experience when little of the material makes sense. The struggle we have in mind comes from solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed (Hiebert et al., 1996). By struggling with important mathematics we mean the opposite of simply being presented information to be memorized or being asked only to practice what has been demonstrated. [Bold added]

I think everybody arguing about productive struggle today, on both sides of the debate, should just memorize this paragraph. The bolded phrase captures perfectly what I had in mind in What if the struggle isn’t productive? when I talked about the implications of limited working memory and suggested that problems should be designed at the edge of students’ knowledge and designed to activate that knowledge. This is Vygotsky’s zone of proximal development, which the authors refer to a couple of paragraphs later. The authors repeat their no-labels refrain:

Struggle is usually associated with student-centered or student inquiry approaches. But we can imagine teacher-centered approaches that provide targeted and highly structured activities during which students are asked to solve challenging problems and work through challenging ideas. . . . the possibility that appropriate struggle can be built into teacher-centered approaches is one reason why we think the old labels of student-centered versus teacher-centered instruction can be so misleading.

I’ve never liked the terms “student-centered” and “teacher-centered.” The center is the classroom, where students and teachers interact, and there are different ways of organizing that interaction.

From efficiency to fluency

Having separated skill efficiency from conceptual understanding for the purposes of analysis, the authors bring them back together in a section entitled “Teaching Features that Promote Conceptual Understanding also Promote Skill Fluency.” Following Adding It Up, they define fluency as possessing, in addition to skill efficiency, the ability “to adapt . . . skills to solve new kinds of tasks.” They conclude with a “plausible conjecture” that the two features of teaching they have identified as promoting conceptual understanding—explicit attention to connections and engaging students in struggling with the mathematics—also promote procedural fluency.

Apparently, it is not the case that only one set of teaching features facilitates skill learning and another set facilitates conceptual learning. In this case, two quite different kinds of features both seem to promote skill learning.

I’ll conclude with a quotation from earlier in the chapter.

Ausubel (1963) proposed a 2 × 2 matrix with rote versus meaningful learning on one axis and discovery versus expository teaching on the other axis. Ausubel contended that these dimensions were independent. Expository teaching, said Ausubel, does not necessarily produce rote learning, and discovery teaching does not necessarily produce meaningful learning.

Amen.

1. If you don’t think that sounds like fun, stay with me.