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I'm teaching a math course which covers several mathematics topics including differential equations (DE). In a recent exam I asked the students a question in which they are given a DE and are asked to solve it using at least two of the different solution techniques they learned in class. My intention was that asking for different solution techniques reflects fundamentally different skills than a single solution. Do you think this is true?

In other words, are there fundamental differences in the learning outcomes measured between the following two questions:

  1. Solve the differential equation dy/dx = y/x
  2. Solve the differential equation dy/dx = y/x using at least two different solution techniques.
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    This would be a useful question for matheducators.SE. – Chris C Mar 11 '15 at 3:25
  • It most certainly would be. I searched for "education" to see if a suitable SE exist but didn't notice the matheducators one. – jak123 Mar 11 '15 at 6:17
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Given the subject you are teaching, I'd say that question 2 only adds a little information about "learning outcomes", over question 1. Students need to learn to recognize when a given technique is even viable, and then to choose the method that is most appropriate (simple, straight forward) if multiple methods are viable. Question 2 is somewhat more informative about this than Question 1.

I say "somewhat" because there are other possible questions and approaches that could be much more informative, if you really think it is important to measure other "learning outcomes" (i.e. what students know, and what they can do with that knowledge). For example, you could ask them to explain why the chose the method they used, and to explain why they chose not to use other methods. You could ask them to compare and contrast the alternative methods. I realize that it's rare that math tests include essay questions, but that is only a cultural norm. There's no reason that they could be utilized more. (Too often, math students simply "plug in the formula" to get the answer and have little idea why it works or what they should do in some uncommon circumstance.)

For another example, what if you gave them a test question where none of the existing methods was an ideal or even good match. The question might be: "How would you modify this differential equation so that it is solvable, and also that the solution is approximately correct for the original equation?" I realize that this question may be beyond the reach of students who are just starting to learn differential equations. But I offer this as an example of a test question that really would measure a learning outcome different from the skill of identifying the appropriate method and then following the solution steps.

  • Interesting suggestions. By learning outcomes I mean a set of skills (not topics) students are supposed to learn. For example, ability to solve differential equations can be classified within the skill of application of basic science knowledge to the field (that is how we classify here at my institution). However, I was thinking that adding another layer to the question (by requesting more than one technique) would shift the question (at least partially) more in the direction of critical thinking skills, by contrasting the two solutions and realizing the uniqueness of the solution. – jak123 Mar 11 '15 at 6:16

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