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The Question: Does combining two or more topics into one question on a mathematics examination when the topics have not notably been combined in the course lecture, homework, or other assessments provide a good measure for any of the following points?

  • Ability to combine knowledge quickly in a new situation.
  • Demonstrate knowledge of all topics incorporated.
  • Passing correctly demonstrates working knowledge of the topics more than independent questions on each topic.
  • Failure correctly shows that the student might not posses the skills to succeed at later courses or does not have a working knowledge of the topics.

Any educational research included would be appreciated.

Notes: The exam is for undergraduate calculus students (first or second year university) who are not all assumed to be math or science students to provide a summative assessment of their knowledge thus far in the course (see ME.SE Exam Philosophy answer by Benjamin Dickman). We may assume that this is not a multiple choice exam.

A Not Particularly Hard Example: Topics: Separable Differential Equations, Conic Sections

Sketch several solution curves to $\dfrac{dy}{dx}= -\dfrac{x}{2y}$.

Removed: My department uses multiple choice for machine grading purposes (mandatory for this course), so particular answers with this in mind would be grateful but not fully necessary.

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  • $\begingroup$ Would passing this question be required to pass the course or just for those wanting maximum marks? With multiple choice, the validity of questions involving multiple steps or combining multiple topics is problematic. It is unclear exactly what failure would demonstrate. Of course, with multiple choice, success does not even demonstrate anything! It would seem best used as a formative assessment freed from your departments exam policy. $\endgroup$ – Richard Jan 22 '15 at 14:26
  • $\begingroup$ That's my issue with multiple choice as well, but there is no chance to get around that. That's why I left it more open to broader answers. And correctly answering the question for maximal marks, but on the exam there would be several of such questions forcing the student to correctly answer many to get high marks. $\endgroup$ – Chris C Jan 22 '15 at 14:36
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    $\begingroup$ i would say that if you haven't discussed similar problems with combinations of topics in class that it would be fruitless to give them during an exam, likely just frustrating students. That being said, i would highly recommend going over these types of problems in class because it allows students to make connections between topics which makes the learning stick much better. $\endgroup$ – celeriko Jan 22 '15 at 14:36
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    $\begingroup$ After the first year or so of writing my own exams, I concluded that it was (for me) not a good idea to have questions combining two or more nontrivial things. Not even combinations we had done in class. $\endgroup$ – Gerald Edgar Jan 22 '15 at 15:22
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    $\begingroup$ @GeraldEdgar I don't think that combining topics is necessarily a problem, as long as each step carries marks and errors are followed through. There are classes where ability to combine knowledge is a stated aim - leading to an hour long exam question. Students expect this and practice for it. Assessors then search students' solutions for evidence of specific skills. The correct final solution isn't required to pass the exam, and the exam is just one part of the course. I don't know how this could work in multi-choice exams. $\endgroup$ – Richard Jan 22 '15 at 22:18
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Still a comment, but expanded. I wrote

After the first year or so of writing my own exams, I concluded that it was (for me) not a good idea to have questions combining two or more nontrivial things. Not even combinations we had done in class.

For example: in integral calculus, computing an area in the plane. I give them some equations describing a region in the plane, then ask them to compute the area.
BAD IDEA
There are two non-trivial steps: (1) find the integral to compute, then (2) evaluate the integral. The problem for the grader (me): if there is an error in the first part, then the second part may become much easier, or much harder. There is no sensible way to assign partial credit for that second part.

Instead of that, I would ask two questions. One in which they find the integral to compute, but do not evaluate it. Another in which they evaluate an integral.

I am talking about in-class time-limited exams. Take-home problems (which we call "homework", and may or may not be graded) of course will include two- or three-part problems.

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