What is transference and does it really happen?

If there’s one major theme in my postings or at least my I-should-post-about-that-ings, it is my struggle with the idea of prerequisites. In a broader context, this becomes a struggle with one motivation for much of the higher-ed curriculum: transference. By this I mean the idea that learning about method M in course C will prepare a student to apply method M’ in life-scenario C’ where the allowed differences between M and M’, and C and C’ are subject to debate. For example, in a math class we study word problems, using whatever course relevant math methods we wish to reinforce (e.g. implicit differentiation in related rates problems) and when we connect this course to the university-wide general education outcomes, we check of the box for ‘Problem Solving’. The implication being, I think, that having passed our course, which required some (specific) problems to be solved, that the student has grown in his or her ability to solve problems in general.

Here’s my take: I don’t think transference happens much, if at all. I think when it is observed, it comes from very deliberate and possibly long-term practice in learning to identify or look for similar contexts and in generalizing methods. It also shows when there’s an underlying skill that gives the student either an advantage in carrying over a method to very similar problems (e.g. factoring quadratics to factoring cubics with no constant term), or a skill that is both basic and has a natural, commonly occuring context (e.g. addition).


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.