‘Understanding’ is used in a myriad of ways in, for example, educational standards and in assessment grids, more often than not in fuzzy ways without any acceptable theoretical underpinnings. Expect the straw man of ‘rote knowledge’ to be not far away. A simple question then might clear up the mist: Exactly what is it that is called ‘understanding’ here? Is it not only knowing A and B, but also the relation between A and B? Etcetera. Translate the pseudo-psychology into psychology, or even better into epistemology.
I will not be done with this subject in only one blog, of course. In this blog I’d like to share a personal anecdote, and try to connect it to the literature, and to experiences that all of you surely will have had.
One of the mathematics subjects in my 11th and 12th year was analytical geometry, a somewhat mysterious geometry emphasizing polar coordinates and geometric places. I never quite ‘understood’ this math; I was able to solve its problems however. In the Netherlands pupils have to sit exit examinations. The geometry examination 1962 was a viva voce exam by one’s own teacher and a mathematician from one of the Dutch universities. I was worried, being perfectly aware that my ‘not understanding’ would be a big risk. The day before this exam was Ascension Day, a perfect opportunity to exercise the whole stuff once more. Lo and behold, there was that moment I saw the light, everything fell on its place, I understood this geometry! And so it proved to be: the viva voce examination was kind of hilarious, I had no difficulty whatever to answer the questions. No, that is not exactly what happened; what happened was a fluid conversation in the language of analytical geometry. Got the highest grade possible. Never after was there an opportunity to cash in on my knowledge of analytical geometry, however, having opted for the study of psychology in Utrecht.
Noel Entwistle studied this kind of phenomenon: cramming before the exam, as a result of which the subject-matter might (suddenly) become totally transparent to the pupil or student. He dubbed this phenomenon a ‘knowledge object’. I’d like to call it perfect mastery of a circumscribed domain.
As reported experiences backed up by perfect exam scores, there is nothing psychological implied in such a knowledge object. One might speculate, of course. What surely is happening in intensive exam preparation is that as a result much of the relevant domain knowledge in declarative memory becomes easily available, even more or less simultaneously. Therefore, these are very fertile hours as regards possibilities for all kinds of links and cross-references to get established, if they weren’t already. A well connected body of knowledge is what many people in fact mean with the term ‘understanding’. Of course, limited capacity working memory will stay the bottleneck it always was. Therefore, cramming without a solid preparation that resulted in much knowledge being automated or linked together tightly (in ‘chunks’) will end in disaster. On this concept of ‘chunks’ see, for example, Taatgen, Lebiere & Anderson; it is highly technical, though.
My experience was about knowing one’s way in a complicated subject—analytical geometry—while being aware of not really understanding it. It is well known among (at least some) teachers of physics (in secondary education as well as in college courses) that pupils or students may be perfectly capable of solving the usual problems, while their understanding of physics is still of the aristotelian kind, not the newtonian one.
- To this day every student of elementary physics has to struggle with the same errors and misconceptions which then had to be overcome, and on a reduced scale, in the teaching of this branch of knowledge in schools, history repeats itself every year. The reason is obvious: Aristotle merely formulated the most commonplace experiences in the matter of motion as universal scientific propositions, whereas classical mechanics, with its principle of inertia and its proportionality of force and acceleration, makes assertions which not only are never confirmed by everyday experience, but whose direct experimental verification is impossible …. (p. 30).
Champagne, Gunstone and Klopfer (1985, p. 62), citing from E. J. Dijksterhuis (1951/1969). The mechanization of the world picture. Oxford University Press.
Audrey B. Champagne, Richard F. Gunstone and Leopold E. Klopfer (1985). Instructional consequences of students’ knowledge about physical phenomena. In Leo H. T. West and A. Leon Lines: Cognitive structure and conceptual change (pp. 61-90). Academic Press.
A recent research article on misconceptions on the sinking of heavier/lighter balls in water (Potvin, Masson, Lafortune & Cyr 2015) offers a nice illustration as well as a clever technique to assess the influence of misconceptions on question answering.
Now it seems I have saddled you with more questions than answers. True, more blogs to come, that’s for sure. I have tried to show that the experience of understanding or not understanding might be a true feeling. Nevertheless, understanding is a strange concept that had better be replaced by descriptions that are less laden with folk psychological. Understanding might be a quality of the network of knowledge chunks: simply much better connected, or pruned from faulty connections to, for example, aristotelian concepts of physics. If understanding is about concepts and relations between concepts, then its assessment should be assessment of knowledge of those concepts and their relations.
‘A network of knowledge chunks’ is a description borrowed from cognitive psychological theories. That is not bad: there is something to learn, even from sophisticated models, on pitfalls and misconceptions in the design of assessment test items. It must be possible to do it without psychology, sticking to epistemologal descriptions of course content that may be up for assessment.
I take no questions analytical geometry 😉 I’ve lost my understanding of it completely. Might be interesting to find out how much time it would take for me to bring my knowledge and understanding of analytical geometry back to level 1962 (see Willingham 2014 on what one takes away from school).
Noel Entwistle & Dorothy Entwistle (2003). Preparing for Examinations: The interplay of memorising and understanding, and the development of knowledge objects. Higher Education Research & Development, 22, 19-41. pdf
- The analysis showed that the distinction between “understanding” and “memorising” is not easy to delineate, with “committing to memory” and “rote learning of details” both contributing to the production of a knowledge object.
- While examinations have been increasingly castigated for encouraging memorisation and reproduction, our analysis has shown that the process of revision for essay examinations can involve deep reflection on topics and an active search for understanding, at least where conceptual understanding is being expected.
Patrice Potvin, Steve Masson, Stéphanie Lafortune & Guillaume Cyr (2015). Persistence of the intuitive conception that heavier objects sink more: A reaction time study with different levels of interference. International Journal of Science and Mathematics Education, 13 open access
Niels Taatgen, Christian Lebiere & John Anderson (2006). Modeling paradigms in ACT-R. In R. Sun (Ed.): Cognition and multi-agent interaction: from cognitive modeling to social simulation. (29-52) Cambridge University Press. concept or download a scan here.
Daniel Willingham (Aug 12, 2016). Why knowledge is unforgettable. Adults remember more of what they learned in school than they think they do—thanks to an aspect of education that doesn’t get much attention in policy debates. The Atlantic blog