Procedural fluency vs. conceptual understanding

I use a standards-based assessment system in my classroom. Assessments are marked based on evidence of conceptual understanding using a pretty extensive rubric. Most of my students are new to this idea, and it’s a struggle for them to understand.

The students who struggle the most are often the ones who have had some success in traditional math classes, where grades are based more on the ability to perform algorithms, procedures, and calculations fluently. Assessing for understanding requires that they not only are able to perform procedures correctly (which is of course still important), but also show evidence that they understand the underlying concepts. This is incredibly frustrating for students who are able to learn procedures without understanding why they work. 
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It’s also tricky for me, as a teacher, to illustrate why this is so important. On a recent precalculus assessment over vectors, some students produced interesting responses that, sometimes within the same student’s test, made me think a great deal about this difference, and provided me with some great fodder for explaining it.

​Context:
The learning target for this assessment is N-VM.A and B: Represent and model with vector quantities and perform operations on vectors. I knew the students needed some scaffolding for the process of adding vectors. It is complicated, with many opportunities for errors. It’s also a great example of the kind of problem where you can do the calculations absolutely correctly without having any understanding of what you’re doing: just give me some formulas for r and theta, and I’m golden, right? Well, not really, especially when you have to figure out the angle at the end. There might be a bunch of “rules” to teach students about when to subtract from 180, add to 180, ditto for 360, but I don’t use ‘em, or know ‘em: in my opinion, you really have to understand what you're doing to reach valid solutions on these problems. I don't know a better way to show this understanding than visual models.

The week before the test, we went through this problem together as a class. Students worked it out, and put it in their notebooks for reference. My assessments are open-notes, so the intention of this exercise was to give students a thorough walk-through of one vector addition problem to use to solve problems on the assessment.

The point I made repeatedly while working this through with three sections of students was this:

• It would be very difficult to confidently answer the last part (the angle) without a visual model.
• It would be nearly impossible to show evidence of conceptual understanding without a visual model.
• You should use a visual model when answering these questions so that you can be confident your answer makes sense.

So, I had an assessment with three different vector addition problems. One was a “naked numbers” problem (here are two vectors; add them together), one was a classic word problem (two tractors pulling on a tree stump in different directions with different forces), and one was a bit more of a problem (navigating a course for a boat that accounts for the current of the water). I suggested that students focus on one of these and do it well (I only give them 25-30 minutes to complete these assessments).

Student A’s response to the boating problem made me really hopeful: precise use of notation, clear reasoning, good calculations, and a visual model that, while not very accurate, at least shows that A has a reasonable idea of where the boat’s going. Then I saw the angle at the end, and said to myself, “Dangit! A is just blindly following the procedure we did in class from his notebook. What a bummer!”

Then I looked at the word problem, and it appeared to be the same story, so I was like, “clear evidence that A is a robot who blindly follows procedures without understanding. What a bummer!”

But I looked a little closer, and saw this diagram at the end. A knows the resultant doesn’t go at a 110 degree angle. A just needs a slight correction. In my opinion, that one diagram (plus the one at the beginning) tips the scale from a robot student to a student showing proficient understanding.

Here’s student B, who, I would argue, is doing exactly what I was accusing A of. B answers with the 110 degrees, draws it, walks away. B is missing the initial combined vector diagram, and the confirmation diagram at the end. B is close, but not quite proficient.

Student C used COLORS!!!, which they know makes me biased. But hey! C screwed up the calculation for the angle at the end. WRONG, RIGHT? Well, yeah, until C did this super-sweet confirmation to check if the answer made sense. C gets it, just made a calculation error. Proficient conceptual understanding, needs work on procedures and showing reasoning.

Here’s student D, who’s a little further from the goal than student B, but using some correct procedures. What’s missing? Well, there are quite a few things missing, but most of all, it’s sense-making. Getting negative x and y values for a vector in the first quadrant should be a red flag for a student who understands what s/he is doing. D’s calculator is in radian mode.

Finally, we have student E, which is just a model of a correct, clearly communicated, unquestionably proficient response. Are there things E could do to improve? Definitely (like that check that C did), but this is a great start, and should serve as a good model for others.

​These are just a few examples of how interesting assessing responses can be when you look for understanding and reasoning rather than right or wrong. I’d love to hear opinions from anyone who would like to discuss.
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But back to the students. I’m only giving them written feedback on this assessment, no grade (we take two assessments over every topic, and this is just the first). I’m hoping that sharing these responses with the students next week and having them do a little assessment or comparison of their own work will help make the point clear: using the right algorithms, even if you do it well, isn’t enough to prove that you understand the concepts.