We know that students learn well from studying relevant examples. We also know that new knowledge tends to be locked into the context in which it was learnt because it is hard for novices to differentiate between the surface features of a problem and its deeper structure. Indeed, expert physicists have been shown to classify physics problems on the basis of their deep structure – whether a question involves the conservation of energy or the application of Newton’s second law – whilst novices focus on the surface features – problems involving springs, problems involving slopes and so on.

One way of making the deep structure more apparent is to vary the kinds of examples used. We could, for instance, select examples that have the same deep structure but different surface features. The problem with such an approach is that it will increase the cognitive load; several examples all set in the same context would have a lower load than a variety of different ones.

An interesting recent study perhaps shed’s some light on the issues. Students were give a very basic ‘selection with replacement’ strategy to learn. Imagine that four people may each choose from a six-item menu; how many possible different sets of selections are there? This is an example of a ‘selection with replacement’ problem.

In the experiment, there were two ways that the students could learn; sets of problems in a similar context or sets of problems in varying contexts. As we might predict, the researchers found that the students with more prior knowledge benefited most from the varied problems whereas those without much prior knowledge benefited most from the similar contexts.

And it’s not just about managing cognitive load. According to the theory of ‘progressive alignment’, problems set in similar context can actually *facilitate *novice students in noticing the deep structure. The researchers found evidence for this because the advantages of similar examples for novices were greatest on problems set in new contexts i.e. the similar examples were better at enabling students to see the deep structure and transfer their knowledge.

Interestingly, the researchers then took things a step further. In their second experiment, they decided to look at what would happen if they *explicitly instructed the students in the deep structure* of the problems (up until this point the students were effectively discovering it for themselves). They gave additional graphical or verbal representations of the deep structure by stating, for example, that “For each of… ONE of… was chosen.”

The students who had this training then performed like the knowledgeable students in the first experiment (with the verbal representations of the deep structure faring better than the graphical ones). In other words, novice students now learnt more from varied examples than from similar ones.

If these results are robust and replicable then I think there are the following implications for our teaching

- Similar examples can be useful early in training. We should not necessarily try to cycle students through diverse examples straight away, thinking that this will aid transfer. The reverse might be true.
- We should seek to make explicit the deep structure of the examples that we select.

Which is why problems in high school algebra break down into types: coin and number problems, work problems, mixture problems, and distance/rate problems. The deep structure of these problems lend themselves to variants that occur in many different disciplines.

Problem solving builds from a foundation of problem types and in so doing create schemas for these types and their variants. Reformers think that throwing new and totally different problems at students builds a “problem solving schema”, that is stimulated by metacognitive exercises. No; it doesn’t work that way.

See also: http://www.ams.org/notices/201310/rnoti-p1340.pdf

Reblogged this on The Echo Chamber.