Kaminsky, J. et al. Science. (2008)
When learning new mathematical concepts, for example the concept of mathematical groups of three objects, is it better to start with a concrete example or with a generic representation? This is the question Kaminsky and colleagues at Ohio State University investigated in their 2008 article.
80 college students were divided in four groups, one of which received an abstract introduction while the others received one, two or three concrete examples (one of which is illustrated below as "Concrete A").
The dependent variable was transfer: whether or not participants could answer multiple-choice questions (isomorphic to the questions used during learning of the examples) regarding the exercise below:
(This picture is actually from a different Kaminsky article,
"Do Children Need Concrete Instantiations to Learn an Abstract Concept?",
but she refers to it in the present article.)
The results were that just learning the one generic illustration gave superior learning to learning one, two, or three concrete examples.
Kaminsky et al. then conducted three more experiments. In experiment two, students were explicitly helped to find the analogies between the concrete examples. This did not affect transfer.
In experiment three, students were asked to reflect on any similarities in the concrete examples and write their reflections down. The results were bi-modal. About half the group achieved high transfer, while half performed as before.
In experiment four, half the students were taught only the generic representation, and half were given first a concrete example and then the generic representation. Transfer was higher for both groups compared to the students in the previous experiments, but even here the one generic representation gave better transfer than concrete-then-generic.
The results of experiment 1 and 4 are illustrated in the figure below:
"Do Children Need Concrete Instantiations to Learn an Abstract Concept?"
Kaminsky et al., available here
These results have been mentioned in popular media, the general public has reacted strongly and mostly negatively. It seems that the belief that concrete is better than abstract is very important to many people.
So what does this imply for teachers? Kaminsky et al. argue that giving generic representations is superior to giving concrete examples to introduce new concepts and methods. However, it is conceivable that transfer would be even better if the sequence was generic-then-concrete. The main idea seems to be that introducing concrete examples before the generic representation somehow locks the students into a restricted way of looking at the concept. Also, it may happen that the extraneous information in concrete examples distract students from focusing on the core concept. Most teachers have experienced that concrete representations help students understand a problem or concept, as is supported in the Kaminsky study on children. But we need to be careful and evaluate not just the direct and immediate gains in understanding, but also more long term mastery and transfer.
The original article, published in Science, is available here.
Challenge: Help me create a good task which tests abstract vs. concrete and I'll test it, on 40 students or so, and do all the rigorous stats and everything. I actually need to do an experiment anyways for my psychology course-work and this is the most interesting idea I've had so far.