I’m returning to a theme from an earlier post, misreading of instructions on quizzes and exams. This week’s quiz in my precalculus class included the following question.

Q1. Create a piecewise function that represents the graph shown below. Assume the pieces of the graph are built from the functions listed below, with appropriate transformations:

y = x

y = x^2

y = ln(x)

The graph shows a continuous, piecewise function from 0 to 3 that was build from three joined segments of subfunctions,

y = x from 0<=x<=1

y = (x-2)^2 from 1<x<2

y = ln (x-1) from 2<=x<=3

Those three lines are, essentially, the answer I was looking for.

Most of the students answered the question with transformations, not always correct transformations, but in the spirit of my intention. About 1/3 of the students, however, answered as follows (some with mistakes in the inequalities, but let’s ignore that),

y = x from 0=<x<=1

y = x^2 from 1<x<2

y= ln (x) from 2<=x<=3

From the reading of the instructions, they seemed to have assumed that ‘with appropriate transformations’ meant that the transformations did not need to be shown in the list of piecewise functions. Apparently they thought that I was testing for the correct partitioning of the domain, and that I didn’t really care about an accurate representation of the graph for each partition. Since more than just one or two students chose this interpretation, I am wondering if perhaps the question is indeed ambiguous. Therefore, I have a question to those of you who feel that you can put aside what I’ve already explained and can read the question as if you were taking the quiz along with my students. Do you believe that the question, as worded, is ambiguous and might have misled you into replying without transformations? Do you have any suggestions for a better re-wording of the question? Comments at all?

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Your question is more than ambiguous to me. It is incomprehensible. Not because you are not clear but rather because I have never taken calculus and my last math class was over 20 years ago. However, I can relate to a class not following instructions on an assignment or quiz. And I agree with you that it is wise to reevaluate the quiz or assignment if more than two students appear to follow an instruction but not the intended instruction. I was in a similar situation as you, although it was a counseling class, not a math class. There was a group of students that completed the assignment similarly but inaccurately. Knowing the students and their limited contact in class, I did not suspect cheating. I suspected my instructions. The next semester when I taught the same class, before I sent the students home with the assignment, I did a mini version of a similar assignment in class so I could coach them along their process. This time around I saw the results I was looking for.

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Wow! I didn’t even see the ambiguity until you pointed it out. How about this response?

y = x with appropriate transformations from 0=<x<=1

y = x^2 with appropriate transformations from 1<x<2

y= ln (x) with appropriate transformations from 2<=x<=3

Or this?

Not possible. No transformation of the given functions will produce the graph for 0 <= x <=1

The first is a more literal (mis)reading of the prompt, the second is even more literal and overlooks the identity transformation as students often do. Yet each response reveals quite a bit of knowledge, and each would score poorly on an automated grading system. Fortunately, we're not yet automated…

This happens to me ALL THE TIME. As carefully as I phrase an assessment prompt, as many drafts as I go through in response to unanticipated interpretations, it still happens that a student will answer a question I didn't intend to ask that is contained within my very words. I've come to view such responses as fundamentally unpredictable, and come to see that they often demonstrate competence that I would otherwise not have seen. It's not "partial credit," it's credit where credit is due. The response you cited from 1/3 of your students could be something like B+ quality. They partitioned the domain correctly, and correctly identified the linear, quadratic, and logarithmic segments. Even though it's wrong, I couldn't call that response an F.

As instructors, we're completely versed in the language and assumptions of our subject. The hard part, as you say, is to "put aside" what we've already learned and see our subject with fresh eyes. What is transparent to us is often opaque to our students. On good days, I get to translucent.

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