An Assessment Conundrum

On Twitter, Nick Corley writes:

Students were SUPPOSED to use the distance formula, but look at the response in the pic. How do you grade?????

This is a great discussion starter! Here are my thoughts.

Assessment

Should the student be penalized for finding the correct answer, using a correct method? That would be particularly egregious if classmates who applied the formula without understanding would get full credit. Such a practice would prioritize compliance over understanding. Is the purpose of assessment to measure the extent to which students are obedient?

Unlike the mechanical application of the formula, this student’s work shows a solid understanding of where the formula comes from:

• The student understood that subtracting the coordinates yields the two legs of the right triangle. Many students who blindly apply the formula do not know that.
• The student knew to use the Pythagorean theorem, which of course is what the formula is about.

Some people thought they should have “shown their work”. As far as I’m concerned, the purpose of a student showing their work is for me to see how they got to the answer. These sketches do a great job of that! (It would have been a nice touch to identify the right angle, but as far as I’m concerned that’s not a deal-breaker.) 

In some cases, showing work is to make clear one didn’t copy someone else’s answer. If a neighboring student had the same diagrams, then that would be a concern, but that’s a whole nother topic, and I don’t think that’s what Nick was asking about.

So in my view, the student should get full credit.

But there’s more to talk about!

Formulas

You ask: if they get full credit, how will they be motivated to learn the distance formula?

In my view, the distance formula is only important if and when you want to create a spreadsheet or computer program to compute distances. Or if and when you want to generalize to three dimensions. If the idea is just to use the formula to find the distance between two specific points, this student’s approach has many advantages over the formula:

• It breaks up the calculation into small chunks. For many students, this reduces the risk of error.
• It is one less thing to memorize. Once you understand this approach, it is pretty much impossible to forget it.

For those reasons, this approach should be discussed with the class. Such a discussion would of course review and reinforce the logic underlying the formula.

In some cases formulas provide a useful shortcut. Once they are understood, there is no reason not to use them. If they are being used without understanding, that may be better than nothing, but it would be important to start a conversation about why they work. In fact, a great question on a test might be: explain why the distance formula works.

As a general rule, formulas should encapsulate understanding, not substitute for it. 

Policy

Certainly, if students were going to be penalized for solving the problem without using the distance formula, they should be warned about that in the wording of the question. But in general, is it a good policy to require a particular path to the answer?

In my view, the default should be to let students solve problems in whatever way makes sense to them. This will help build their confidence, and is likely to be more durable than trying to parrot an ill-understood technique. Sometimes their approach will be different from yours, or from their classmates. If it is mathematically correct, it is fine! Again, the idea is to show understanding, not compliance.

But like everything else in our line of work, I see this as a guideline, not an inflexible rule. Systems of linear equations are a classic example of something that can be solved in different ways. Different strategies involve different understandings — graphing, algebraic symbol manipulation, matrices. We may want to know which of those the student is comfortable with, and thus we may specify a solution method, or more than one, on a quiz. (Even then, I would give partial credit for a student who solves the problem correctly but in a different way, and I would discuss which method seems best suited to different problems.)

Another sort of argument for challenging students’ approaches is about efficiency. I think that deserves a case by case discussion, in full awareness of the availability of calculators and computer algebra systems, the importance of mental arithmetic, and so on. But that’s not today’s topic.

— Henri