A Question of Prerequisites

The initial idea was simple: come up with a hypothetical question about how performance in a prerequisite course should correlate with performance in a course with a goal towards identifying whether or not we inherently limited students achievements.  The problem, as always, was in the details.

Here’s the initial version of the question (more or less): For a course, Math 2, which has Math 1 as a prerequisite, what are the probability that a student who made an A in Math 1 will make an A in Math 2. Repeat for all the possible grades in Math 1 and in Math 2. Limiting to only grades of A, B, C for each, why aren’t all the probabilities 1/3?

Then the problems began.

  1. What does it mean that Math 1 is a prerequisite for Math 2?  How much of Math 1 actually appears in Math 2, and is not (re)covered in Math 2? For example, say Math 1 is an algebra course and Math 2 a calculus course. How many of the algebra topics in Math 1 show up in any detail in calculus? At what point does not being fluent or knowledgeable about algebra start interfering with learning calculus, or is the problem just in the algebra portion of the class, i.e. algebraic simplification of results? In our hypothetical situation we can assume that the answer is ‘all of Math 1 is needed in Math 2 to understand Math 2’, but that’s pretty unrealistic.
  2. What does A, B, C mean? We can change this to a numerical score and base it on a set of specific learning objectives (think of a 100 question critical concept final). So we can say A means 90+% knowledge, but then how do we know that is current.  And then it gets more complicated for multiple grades averaging to a C.  Is a student who made 70% on every assignment the ‘same’ as a student that made 100s mixed with 50s on the various assignments, or even the student that made 60s on tests but pulled out a C on homework or other ‘friendly’ assignment.
  3. Then we get closer to the real issue. If a student makes a C in Math 1, can they make an A in Math 2? I don’t mean numerically, or what’s possible, I mean does whatever led to the student making a C in Math 1, limit them so that it is very unlikely they could make and A in Math 2. For example, save Math 1 is Differential Calculus and the student made a C because they struggled with the algebra and/or trigonometry. If they don’t increase their skills in those areas, do they really have a chance for an A in Math 2 (Integral Calculus).

So, although I never came up with a good form of the question, I did form an opinion: I think we too often curse students who performed poorly in prior courses, well beyond how that performance is relevant to the current course. I’d suggest that forming and being faithful to some carefully formulated learning objectives would help out, but that’s for another post.

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Some Random Thoughts

I’m back, and so here’s a list of some topics that I’ve been thinking about and might expand in later posts:

  1. The transition from basic conceptual understanding to fluent understanding expressed through action can come through imitation or through true growth. Most of the time we can’t tell the difference, but we might be able to tell by the next topic that is learned.
  2. There are two dimensions of learning (at least): (1) learning it (whatever it is) and (2) learning to learn better. Every course should address both of these and should expect that prior courses have done so also.
  3. A challenge: addressing students mis-knowledge, by unteaching them that which they must unlearn. Two big components: moving them from bad or incomplete schemas to higher level and more effective ones and adding recognition and verify to their problem solving methods.
  4. Another challenge: helping students move from extrinsic to intrinsic motivation.
  5. We are good at helping students learn what to do (or at least, how to do it once they know it needs to be done), but not so good at helping them value the why we do it. For us (teacher/practitioners) this is a natural part of our being, so we need to think about how to make this invisible part of our work more visible to students.

That’s it for now.