Fractals of Learning

I've been doing some data analysis on responses of students to a comparative judgement exercise I did with them last year. Basically, they were presented with pairs of documents on various topics in science, technology and society, and asked "Which do you find more interesting and why?"

The responses collected over two weeks from about 150 students were surprisingly rich, and I've become interested in drawing distinctions between them. Some students clearly are transformed by many of the things which they read about (and this was in the context of a face-to-face course which also gravitated around these topics), and their answers reflect an emerging understanding. Other students, while they might also appear to engage with the process, are a bit more shallow in their response. 

To look at this, I've looked at a number of dimensions of their engagement and plotted the shifts in entropy in each dimension. So, we can look at the variety of documents or topics they talk about: some students stick to the same topic (so there is continually low entropy), while others choose a wide variety (so entropy jumps around). The amount of text they write also has an entropy over time, as does the entropy of the text itself. This last one is interesting because it can reveal key words in the same way that a word cloud might: key concepts get repeated, so the entropy gets reduced. 

What then would we expect to see of a student gradually discovering some new concept which helps them connect many topics? Perhaps an initial phase of high entropy in document choice, high entropy in concepts used and low entropy in the amount of text (responses might be a similar length). As time goes on, a concept might assert itself as dominant in a number of responses. The concept entropy goes down, while the document entropy might continue to oscillate. 

The overall pattern is counterpoint, rather like this graph below:

The graphical figure above is a representation of the positive and negative shifts in entropy of the main variables (going across the top), followed by the positive and negative shifts in the relative entropy of variables to one another. The further over to the right when patterns change is an indication of increasing "counterpoint" between the different variables. The further to the left is a sign of particular change in particular variables. From top to bottom is time, measured in slots where responses were made.

Not all the graphs are so rich in their counterpoint. This one (admittedly with fewer comparisons) is much more synchronous. There's a "wobble" in the middle where things are shifted in different directions, while at the end, the comments on the documents, the type of documents, and the type of topics all vary at once. If there was a common concept that had been found here, one would expect to see that the entropy of the comments would really be lower. But the graph and the diagram provide a frame for asking questions about it.

This one is more rich. It has a definite structure of entropies shifting up and down, and at the end there is a kind of unity which is produced. Looking at the student comments, it was quite apparent that there were a number of concepts which had an impact.

It doesn't always work as a technique, but there does appear to be a correlation between the shape of these graphs and the ways in which the students developed their ideas in their writing which merits further study.

More interestingly, this one (below) produced a richly contrapuntal picture, but when I looked at the data, it was collected over a very short period of time, meaning that this was the result of a one-off concentrated effort, rather that a longitudinal process. But that is interesting too, because there is a fractal structure to this stuff. A small sample can be observed to display a pattern which can then be contextualised within a larger context where that pattern might be repeated (for example, with a different set of concepts), or it might be shown to be an isolated island within a larger pattern which is in fact quite different.

Either way, the potential is there to use these graphs as a way of getting students to reflect on their own activities. I'm not sure I would go so far as to say "your graph should look like this", but awareness of the correlations between intellectual engagement and patterns of entropy is an interesting way of engaging learners in thinking about their own learning processes. Actually, it also might be possible to produce a 3d landscape from these diagrams, and from that a "google map" of personal learning: now that is interesting, isn't it?