Proof of Procedural Learning

One of the many reasons for assessments in the classroom is to check for student understanding. Ideally, we are taught to teach conceptually and not so heavily on procedures.

 

While grading a recent exam I decided to tally up which problems students got wrong to see which problems seemed to stump most of the students. At the end I noticed a particular question had fooled nearly half the class. This was the question:

If an 8.2 oz bag of candy costs $2.82, how much does one ounce, or one unit cost?

It was multiple choice with the choices being A) $0.34 B) $2.91 C) $3.09 D) $0.27

Any one with conceptual understanding of a unit cost would know that B and C can not be correct since they are greater than $2.82 which is the cost of 8.2 units. However, most of the incorrect responses were in fact B) $2.91 while the correct answer is A) $0.34 because if you see it has a ratio of ounces to cost being 8.2 : 2.82, then dividing both sides by 8.2 would give you 1 : 0.34 which shows the cost of one ounce as being $0.34.

However, if a student has only learned the procedural way to solve this problem, they might realize this is a division problem and then simply start dividing numbers. Since 8.2 is listed first in the question, and 2.82 second, they might simply enter 8.2 and divide that by 2.82. This would give us an incorrect answer of $2.91 per ounce.

Now here is where student understanding is shown to be either procedural or conceptual. A student with conceptual understanding would notice immediately that this answer is wrong. A unit cost should be less than the cost of 8.2 units, so this answer must be wrong. However, the exams I graded showed that these students didn’t notice the contradiction implying that they merely understand the procedures for such problems.

However, this failure to notice an answer that doesn’t make sense could have surfaced for many reasons. Perhaps one student guessed wrong and everyone else managed to copy their answer, or the students typed in some numbers into their calculator and happened to have 2.91 pop out as an answer. But I fear the answer is that the students don’t know the concept behind finding unit costs. Their procedure was simply to divide some numbers until you see your quotient is one of the multiple choices.

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