In my article on understanding difficult concepts I mention using analogies as a powerful method for understanding difficult and abstract concepts. Indeed some people, such as the cognitive scientist Douglas R. Hofstadter suggest that analogy is how we think:
“every concept we have is essentially nothing but a tightly packaged bundle of analogies, …all we do when we think is to move fluidly from concept to concept — in other words, to leap from one analogy-bundle to another — ..lastly, that such concept-to-concept leaps are themselves made via analogical connection.”1
You can see Douglas Hofstadter speak about analogy in this Stanford lecture. Another prominent researcher in the field of analogy is Keith Holyoak. Keith talks about analogy and the brain in this Lecture.
Let's explore the power of analogies through trying to find one for the derivative. The derivative is a concept that is often difficult to comprehend for students. I myself remember struggling with dy/dx notation during school: ‘The derivative gives the rate of change of a function for any value of x' – now that's an abstract concept if ever I've heard one!
In the following 4 videos I use the analogy of a space ship to try to facilitate some intuition for viewers. Hopefully by the end of the 4 videos the link between the ‘rate of change' and the derivative will be a bit clearer. I love how the spaceship is used in this analogy, because what both spaceships and analogies help us do is to explore new territory, and to move from something concrete (earth, in the case of the spaceship) into something more abstract.
I'd love any feedback on this analogy as it's still very much in development, please let me know how it's helped or hindered your understanding.
Video 1 introduces how the derivative can be related to a Space Ship.
Video 2 applies the Space Ship analogy to 3 graphs to show how it can be used to calculate the derivative.
Video 3 takes explores how the derivative (found mathematically) can be related to our Space Ship analogy.
Video 4 links the spaceship analogy to the Rate of Change of a function.
- Gentner, D., Holyoak, K. and Kokinov, B. (2001). The analogical mind. 1st ed. Cambridge, Mass.: MIT Press, p.499.