Written by Sarah Stecher published 10 months ago

The Math Medic lessons are designed to build deep, conceptual understanding but we also know that procedural fluency is one of the five strands of mathematical proficiency. Historically, procedural fluency has been the primary, or even sole, focus in many math classrooms, at the expense of other math competencies like adaptive reasoning, strategic competence, and conceptual understanding. We do want students to be able to select an appropriate procedure and use it flexibly and efficiently, but the essential question is *when*. Our lessons build conceptual understanding first and work towards procedural fluency over time. Certain procedures or algorithms or strategies should become automatic, but only because students have a deep understanding of what’s going on in the background. For a more general overview on how we assign homework and why, see this blog post.

### Do procedural questions actually assess "basic" understanding?

A typical textbook homework set usually starts with procedural questions then moves toward the infamous “word problems” and sometimes culminates with an out-of-the-box conceptual question. The assumption is that if a student can do a procedural problem, they have the basic skills down and if they can do the conceptual problems or applied problems, they have a deep understanding. But while procedural problems may help reinforce some of the mechanics, they often do little to demonstrate mastery of the learning targets. Furthermore, we want to argue that conceptual questions shouldn’t be saved until the end and don’t always have to be extraordinarily challenging; their defining feature is that that they encourage and even require making connections to underlying ideas and can still be used to establish procedural fluency.

One of the things we should rethink is the difficulty level of a procedural question. We tend to think that the problems where you only have to plug numbers into a formula to get an answer (here’s the radius of a sphere, find its volume) are the easiest, because they’re a straight-shot, plug-and-chug kind of question. They also most closely resemble a worked example, which we think makes it accessible, but actually tends to reinforce mimicking. Procedural problems are actually quite abstract because they strip away much of the context that normally helps students make sense of problems. In this framework, procedural questions could be considered some of the hardest, because they don’t offer a lot of clues as to how to make sense of an answer. Although a student can be "successful" at these types of problems when they've followed all the steps, we will see that such success does not always correlate with understanding.

Let’s look at how we might start a homework set after students have just learned slope.

Both options assess the same learning target—finding a slope given two ordered pairs. The first option is procedural. Students will most likely rely on the slope formula of (y2-y1)/(x2-x1) . Even for students who understand slope as rise over run, incorrect answers abound because many students still struggle with finding differences with negative numbers. Upon seeing that this student answered the very first question in the homework incorrectly, we might conclude that this student does not understand slope. But is this really the case?

In Option 2, students are finding slope but in a more conceptual way that supports their sense making. They can see and mark the vertical distance and horizontal distance between the two points. Students could of course still use the formula, but there are multiple ways they can check the reasonableness of their answer. The graph clearly shows that the final answer should be negative since the second point is lower than the first (from left to right). Additionally, students can count squares to find the rise and run of this line and can clearly see that the horizontal distance is NOT 2 (a common mistake would be to calculate 5-3=2 using the formula). This is an appropriate entry-level question about slope. If a student answers this question incorrectly, I can say fairly confidently that this student is still missing some key understanding about the meaning of slope and can make plans to address the gap.

### When can we expect procedural fluency?

Sometimes we are tempted to think that if a student “discovered” a certain formula, procedure, or algorithm, then they are ready to use that generalized formula from here on out and that they are necessarily connecting that formula back to the process for discovering it. But in most cases, students need multiple opportunities to see the connection between their strategy (often informal and concrete) and the generalized formula. In the example above, students might need to plot points, count squares or use a number line multiple times before they see why subtracting the x- and y-coordinates is the same thing as finding the horizonal and vertical distances, and why division is necessary. If we want students to build conceptual understanding, then we must be okay with math being inefficient for a while. Rushing to the algorithm short circuits the learning cycle.

**Is there a way to build both conceptual understanding and procedural fluency at the same time?** We think so. Let’s say that our learning target is for students to be able to determine the volume of a sphere and they discovered the formula V=4/3πr^3 in today’s activity. Note that this learning target is pretty procedural--finding a volume--but even a volume formula rests on an understanding of the relationship between a sphere's radius and its volume. Take a look at the following four options.

Option 1 would be a pretty standard way of starting a homework assignment, quiz, or test. What do we learn about a student’s understanding if they get this problem correct?

- S/he copied the margin notes down correctly
- S/he is adept at using a calculator and uses parentheses appropriately.

It’s not that we should never use a question like this, nor that careful note taking and precision with tools is not important, but we can’t assume a correct answer here implies understanding of volume of spheres.

Option 2 is more open-ended. Although it might seem like this question addresses an entirely different learning target around solving algebraically for a missing dimension, note that we are not solving for one specific radius. We would expect a student to pick a radius, substitute it into the formula and check whether this volume satisfies the condition. If it doesn’t, they have to repeat this process. Note that this question actually gives students more practice applying the formula than option 1. Since two possible radii are needed and a range of possible volumes is given, students might glean something about how quickly growth happens when a variable is being cubed. Some students might consider a range of possible radii that would meet this condition.

Option 3 connects learning to previous learning, but it still directly tied to our learning target of evaluating the volume of a solid using formulas. We can see how students are making sense of the formulas based on if they actually calculate both volumes (a likely solution path) or if they can see that r^2*h is the same as r^3 when the radius is the same as the height, and thus the volume of the sphere must be bigger since πr^3 is multiplied by 4/3, a number bigger than 1. How students approach this question gives us helpful insight into how they are thinking about the math and what they understand about how the formulas work.

Option 4 asks students to compare three different volumes. A rudimentary understanding of the volume formula of a sphere would acknowledge that a bigger radius will always produce a bigger volume. The first two are thus easy to compare, given that a student knows a diameter is double the radius (which they should at this point). The hemisphere adds an interesting dimension but again, this problem can be easily solved by applying the formula (yet another opportunity to build fluency). However, the more interesting question becomes whether a sphere with a radius of 3 is bigger or smaller than a hemisphere with a radius of 6. A student might quickly assert that they are the same since both include the idea of halving, but an understanding of the volume formula would indicate that the effects of doubling the radius and then halving the volume do not cancel each other out (they do not need to be able to say that the hemisphere of radius 6 has 4 times the volume as the sphere of radius 3, that would be a different learning target).

Options 2, 3, and 4 all enhance procedural fluency while still assessing conceptual understanding. They are differentiated in that they allow for multiple solution paths and give us more valuable information about what students understand and can do with respect to this learning target. (These three options don't represent a progression, they are just three possible approaches to practicing this learning target)

**To summarize:** before giving a procedural question, we have to ask ourselves what the purpose of the question is and what we will learn about the student’s understanding. We need to consider when we give a procedural question and if students are ready for that level of abstraction. Finally, we need to distinguish between students that can mimic a worked example line by line and those that have the conceptual background knowledge to select from a variety of strategies, choosing the algorithm or procedure when it is most efficient, not just because it’s what they’ve been told to do.

For more information on writing and modifying questions, see this post from NCTM.