Three Levels of Understanding

I taught the same course for about a ten-year period, sometimes twice a year. My recent reading about “threshold concepts” in different disciplines has made me rethink my own evolution in understanding the material in the course I taught. I now perceive three stages of my learning.

First, as a graduate student, I conquered all the major concepts and their interrelationships. This meant I could answer any question that an instructor could pose to me; I could “solve all the problems at the end of chapters;” I could read the research literature in the field. In retrospect, this was the level of understanding that only made me a good student in an advanced course in the field. I now realize how superficial such understanding really was.

Second, post-PhD, after teaching the course a few times, I realized a different level of understanding. Upon reading just a few words of a question, I could classify it into its proper context among all the concepts in the field. I could mix and match concepts in creating new questions, to test students’ ability to synthesize them. Part of this understanding arose from lecturing on the different components of the field and inventing progressively complicated constructions of pieces of knowledge as the course proceeded. Concepts built upon concepts, creating more possible combinations. Teaching led me to relearning the material, but in a new way.

As I assembled class examinations over the years, I forced myself to construct different versions of questions, each of which tested the same subdomains. The questions were different enough from one another that clever students who acquired prior examinations could not profit from knowing prior exam questions. Indeed, in later years of the course, I would give out to students many of the questions of prior examinations in order to help them study. My hope was that their exposure to hundreds of questions related to the field could indeed give them this same second level of understanding. They too would be able to quickly recognize what combination of knowledge a given question was seeking.

The third level of understanding came later in my teaching of the course. After grading thousands of examinations, I was exposed to all the possible wrong answers to a given type of question. I had seen every possible mistake. Further, in talking with students after exams, I learned what errors in logic led to the wrong answers. When I was at my best helping a student, I was able to identify the problem in their logic, go back to the material that troubled them and correct it. These episodes were often transformative for the student; everything became clear. A roadblock had been removed. In some sense, I felt that was my best teaching. Many of the causes of errors stemmed from their lack of understanding at the first level mentioned above. They never really “got” one of the early concepts in the course, leading to confusion with diagnosing how best to approach a given question much later in the course.

This problem is quite close to the notion of “threshold concepts” made popular by David Pace and his colleagues. They make the argument that there are key concepts in every field that are gateways to more sophisticated understanding. The more structured a given field, the more likely that there are key central notions that are building blocks to the entire field. One of the surprising findings of the research is that some sophisticated experts in a field cannot easily identify these threshold concepts. They find it difficult to remember the time when they learned the material for the first time. They look upon the field with completely different perspectives than that of a novice. Hence, the relative difficulty of key concepts to comprehend upon first exposure is something they have forgotten (or maybe never perceived).

One of the ongoing challenges of teaching, it seems, is remembering the perspective of the learner, not the expert in the field. The learner is constructing the conceptual structure, brick by brick. The expert sees from above the entire, integrated structure, in all its beauty.