Mathematics tools

Oh, I suppose the OU must have a subscription to that Journal that just magically makes the Read the Full Text link work. I did not realise that it was not publicly available. Sorry.

The article does not seem to have an abstract, but the first three paragraphs summarise what they are trying to do:

Abstract knowledge, such as mathematical knowledge, is often difficult to acquire and even more difficult to apply to novel situations. It is widely believed that a successful approach to this challenge is to present the learner with multiple concrete and highly familiar examples of the to-be-learned concept. For instance, a mathematics instructor teaching simple probability theory may present probabilities by randomly choosing a red marble from a bag containing red and blue marbles and by rolling a six-sided die. These concrete, familiar examples instantiate the concept of probability and may facilitate learning by connecting the learner's existing knowledge with new, to-be-learned knowledge. Alternatively, the concept can be instantiated in a more abstract manner as the probability of choosing one of n things from a larger set of m things.

The belief in the effectiveness of multiple concrete instantiations is reasonable: A student who sees a variety of instantiations of a concept may be more likely to recognize a novel analogous situation and apply what was learned. Learning multiple instantiations of a concept may result in an abstract, schematic knowledge representation, which, in turn, promotes knowledge transfer, or application of the learned concept to novel situations. However, concrete information may compete for attention with deep to-be-learned structure. Specifically, transfer of conceptual knowledge is more likely to occur after learning a generic instantiation than after learning a concrete one.

Therefore, we ask: Is learning multiple concrete instantiations the most efficient route to promoting transfer of mathematical knowledge? Here, we tested a hypothesis that learning a single generic instantiation (that is, one that communicates minimal extraneous information) may result in better knowledge transfer than learning multiple concrete, contextualized instantiations.

They do some experiments with (quite small) groups of students. The example they use is teaching the rules for the cyclic group of order 3. Some students get taught via 1, 2 or 3 concrete realisations of the group, while others get taught using just abstract symbols. Then they are given a test where they need to apply what they learned to a new concrete realisation. The group that had been taught using abstract symbols were better at applying the knowledge to this situation. There are then some follow-up experiments to explore this finding.

There is another summary of this article at http://www.ams.org/mathmedia/#three. Google also finds http://www.mathchique.com/2008/05/lessons-on-how-to-misinterpret-research.html by searching on the article title.