“Science may be described as the art of systematic over-simplification…”
– Karl Popper
I suppose one can hardly have a view on what teaching is, without having a view on learning! Obviously teaching speaks to learning. As before, my goal here is to keep this simple; learning is:
- The accumulation of knowledge
- Knowledge of the connections between the knowledge
By this point you either agree entirely, are concerned about other subtleties to learning that I’ve left out, or have thrown your arms up in outrage and declared this ‘mere transmission teaching.’ I should probably elaborate a little.
Can ‘learning’ be modelled as the acquisition of knowledge, and connections between knowledge?
Simplification I’ve written before, and will no doubt write again, on the power of simplicity. I’ve heard Popper’s quote above to be taken or read as his disparaging science in some way; the full form of the quote, often forgotten, is: “Science may be described as the art of systematic over-simplification, the art of discerning what we may with advantage omit.” All of science produces simplified models of reality, yet in their simplification lays their power. In searching for the exact form of Popper’s words, I came across several other interesting quotes on simplification: “Knowledge is a process of piling up facts; wisdom lies in their simplification.” – Martin H. Fischer “Order and simplification are the first steps toward the mastery of a subject.” – Thomas Mann “A good teacher is a master of simplification and an enemy of simplism.” – Louis Berman “In science complexity is considered a cost, which must be justified by a sufficiently rich set of new (and preferably) interesting predictions of facts that the existing theory cannot explain.” – Daniel Kahneman
There is a non-trival distinction between ‘simplistic’ and ‘simplicity’
Where there was criticism, the typical critique to my post on ‘what is teaching’ was to say ‘but there’s so much more to teaching!’ One can’t help but feel those critics didn’t properly read the post, given that it starts by stating its attempt to cut through the pervasive over-complication of most contemporary views of teaching, while acknowledging that of course there will always be something more to what a teacher has to do, either by necessity of the job or to create the right conditions for learning in a school environment to take place. In other words, those critics missed the point, leaving me with little else to say in response. But, what might my model of learning above overlook? But what about understanding? I would assert that we cannot ‘teach understanding,’ we can only teach knowledge. This is why the second bullet is important; it speaks, I think, to understanding. We need to know what ‘understanding’ is before we can talk about it. Here’s an example I love: in the earlier post on simplicity I showed this way of conceptualising the area of 2D shapes, and placed the triangle and trapezium together like this:
The triangle and trapezium are related in many ways
A friend commented that, for the first time, she realised that a triangle is a trapezium, where a = 0 (a being the length of one of the two parallel sides – the top one, in my image.) Without going into detail about the maths, the point is that my friend, who was also a maths teacher, and me, and any maths teacher I’ve told this to since, saw something new about the relationship between a triangle and a trapezium; in that sense, our understanding of these two shapes, and of mathematics in general, grew in some small way. I take from this, two lessons: First – Understanding is never complete. There is never a moment when you can say ‘I completely understand this now and there’s nothing more I can learn; I’m done.’ This is important, since it means while we can aspire to understanding at school and through our teaching, we can never make it a simple time-bound objective per se; we should and can never have an objective of a lesson be ‘To understand X.’ We may seek to progress towards greater understanding, but little more. Second – Our understanding came from either the realisation (in the case of my friend) or the reception of (in the case of myself and others) a new piece of knowledge – that a triangle can be viewed as a trapezium where side length a is 0. So it was knowledge connecting knowledge we already had that spoke to greater understanding. Across any subject, understanding can be conceptualised in this way. You may know that the first world war started in 1914; knowing why is started then requires lots more knowledge of the ‘that’ type, between which causal connections are then formed – ‘because of this, then that happened.’ Those connections are what we would refer to as understanding; surely a person who can articulate not just what happened but why it happened has a greater depth of understanding? It might be that there’s something special in the moment where a person accepts the why – do they accept that this leading to that makes sense? For example, I can easily accept how a trapezium of a = 0 is a triangle, while others would need time and possibly guidance to see why that is the case, and some may continue to struggle despite expert help. It could be that it’s the acceptance that we refer to as ‘understanding,’ and I would acknowledge that in that moment lies something special. We could next speak about people who challenge the causal connection they’re being given; those who can imagine other valid outcomes: “But couldn’t something else have happened because of that instead?” or “Couldn’t it be instead that this is what caused that to happen?” Perhaps we would argue that those people are exhibiting an even greater depth of understanding, one which could then translate into an ability to effectively write critical essays. For them to be able to do this, however, they must therefore have an even greater breadth of factual knowledge, as well as knowledge about how to connect those facts! However we look at it, for my part using knowledge as a bedrock of deep, complex understanding is a model that always stands up to scrutiny (by all means, let me know of any examples where you think it might not!) Not only that, but it is a model that is readily understood by a mortal teacher – “I need to teach these facts, make sure they’re known and retained, and then I need to start drawing the connections between them.”
Knowledge is the foundation of understanding, not ‘the lower league’
In this model, what we perceive on the surface as ‘understanding’ is a manifestation of a broad body of deeply-connected and interrelated knowledge. We can ‘promote’ understanding by teaching knowledge and its interconnections, but we cannot teach it directly. Yes, it is a simplified model of learning, but therein lays its power. Part 2 Tomorrow I’ll look at how this model deals with the risks of rote learning, building an inquiring mind, and development of skills.