Saturday 22 February 2020

Levels of Ability and "Gradus ad Parnassum": a Pedagogy of Constructing Nothing

In education, levels are everywhere. There are levels of skill, stages of accomplishment, grades, competencies and so on. Arguments rage as to whether levels are "real" or not. But obviously there is a difference between someone at Level 1, and someone at Level 8 (for example). Throughout the history of education, attempts have been made to create pedagogies which follow a staged approach to the acquisition of skill. The formalities of these approaches, and arguments about the true nature of levels (for example, whether one might naturally acquire high levels of skill without the formalities of a levelled pedagogy), have been a key battleground in education, from an almost dogmatic insistence that "things must be done in this way" to a open "inquiry-based" approach.  It surprises me that in all of these debates, which remain unresolved, little thought has gone into what actually constitutes a level.

Partly this may be because levels are seen as specific things which relate to a discipline. And yet, there are fundamental similarities between pedagogical approaches from learning Latin, music, or maths to astrophysics and medicine. There are stages, outcomes, assessments, and so on. One might think that these things are the products of the institutional structures around which we organise education. That might be true. But "levels" are nevertheless demonstrable irrespective of what assessment technique might be in operation, and their means of establishment have at the very least a family resemblance.

The most interesting and ancient of the levelled approaches is the "Gradus ad Parnassum". This refers to a range of different pedagogical approaches in different subjects. I became familiar with it through musical education, because it was the name of a treatise on counterpoint by Johann Fux. Fux's approach to counterpoint was to present learners with progressively complex exercises for them to complete. Because music is very abstract, these exercises are interesting because they present an almost paradigmatic case of the differences between one level and the next.

The basic idea is to write countermelodies to a given melody written in very long notes (called a Cantus Firmus). First, each long note is accompanied by one other long note which harmonises with it and whose construction must obey simple rules which form the foundation of the rules for the rest of the exercises. Secondly, each long note is accompanied by two shorter notes in half the rhythm. Then it is done by fours, and so on. Gradually the students learns the fundamental rules and how to mix combinations of shorter and longer notes over the original melody. The resulting music sounds like Palastrina. The technique was used by generations of composers who followed.

Fux's Gradus is interesting because each level has a certain completeness. The completeness of one level leads through the expansion and complexification of the technique to the next level. It would be very interesting to explore language pedagogies, and maths pedagogies for similar patterns of completeness. But I'm particularly interested in what this "completeness" at each level is.

It is not, I think, a construction of a particular accomplishment. That I think is an epiphenomenon. Somehow, by the performance of one level, a kind of 3-dimensional construction is made which eventually determines that the particular level is exhausted in possibilities. In other words, at a certain point, what happens next must be to stop at this stage, prepared to move on to the next stage. It may not be so different from a level in Space Invaders - and there there is a clue. What marks the end of a level, but the construction of Nothing. The invaders have gone, and so we begin again.

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