Recently, I have written about the importance of frequent problem solving for physics students. Thomas Kuhn, the philosopher of science, wrote that:
Students of physics regularly report that they have read through a chapter of their text, understood it perfectly, but nonetheless had difficulty solving the problems at the end of the chapter. Almost invariably their difficulty is in setting up the appropriate equations, in relating the words and examples given in the text to the particular problems they are asked to solve. Ordinarily, also, those difficulties dissolve in the same way. The student discovers a way to see his problem as like a problem he has already encountered.
Second Thoughts on Paradigms, Thomas Kuhn
This post explores this idea, demonstrating the importance of two types of knowledge: subject knowledge and procedural knowledge in learning physics. I have used an example to demonstrate the knowledge involved.
This example is adapted from a lovely physics book: 200 Puzzling Physics Problems (with hints and solutions) by Gnadig, Honyek and Riley (2001). The puzzles (without the hints or solutions) can be found here.
A bottle of water is suspended from a fixed point by a inextensible rope. The bottle is set in motion and the system swings as a pendulum. However, the bottle leaks and the water slowly flows out of the bottom of it. How does the period of the swinging motion change as the water is lost?
There is a lot of knowledge hidden needed to solve this problem: Continue reading
The more abstract a concept is, the less useful learners find definitions or explanations. There is no lightbulb moment. Rather, our understanding grows by a gradual accumulation of experience, of problems solved.
Energy is a perfect example of this:
“It is important to realize that in physics today, we have no knowledge of what energy “is.” We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. It is an abstract thing in that it does not tell us the mechanism or the reason for the various formulas.“
Richard Feynman, in The Feynman Lectures on Physics (1964) Vol I, 4-1
Conservation of Energy
We do not know energy what energy “is.” We cannot model it or visualise it. The equations highlight this: energy must always be calculated from more accessible quantities. We use mass and velocity for kinetic energy; mass and height for gravitational potential energy; the spring constant and the displacement for elastic energy; the mass and change in temperature for heat energy and the change in mass and the speed of light for nuclear energy. Continue reading
In my last flurry of blogs (here, here and here), I wrote about the limits of definitions for learning . Learning the definition does incredibly little to develop understanding. For example: Electrical current is the rate of flow of charge.
Even if you know what rate means and you have a clear understanding of charge, you still don’t really know what current is. You don’t know how to use it in calculations or how to use it in writing or discussion. You don’t have a proper ‘feel’ for current.
Thomas Kuhn, the philosopher of science, argues that we learn scientific concepts by ‘acquiring an arsenal of exemplars‘ – often the bank of questions at the end of each chapter in textbooks (Kuhn, second thoughts on paradigms, 1977).
So spending learning time to memorising the definition is not a great use of time. Spend that time on learning exemplars instead. One effective way of doing this is by using worked examples. Hattie (in Visible Learning, 2009, p172/3) describes the worked example cycle as typically:
- exposure to the example question
- a training phase
- a testing phase.
Variations include: matching text to diagrams; fading (gradually removing steps in the example) and self-explanation of the stages.
In addition to learning to solve the exemplar questions, I would add: talk about models and practical work and reading and writing sentences containing the target concepts.
This leades to a far richer undrestanding of a concept – I know my definitions now, though I didn’t when I was using them as part of my degree. I didn’t need to know them – I understood them instead.