Definitions: The cover up

In school mathematics the emphasis is on techniques and methods (find the roots of this quadratic, solve this simultaneous equation, use trig to find how high this clock tower is, etc). At an advanced level definitions become much more central and this school emphasis causes students to miss their significance* — they are abstract concepts to be skimmed over.

But definitions do need to be learned precisely. How can you apply the definition of continuous function or abelian group in a theorem or show something is an example if you don’t know the exact details of the definition?

To help students learn the precise definition, I teach what I call the cover up technique. My aim is show my students a better way to deal with definitions rather than skimming over or repeatedly reading it in the hope that it will stick.

The cover up technique is simple: Read the definition while consciously attempting to observe what is written there. Next cover it up and attempt to write out the definition in full from memory. No peeking. Once done, check the written down version against the precise version.

Repeating this a few times works better than just rereading until it supposedly goes in. The reason for this is that retrieving facts from memory is better for learning than mere repeated exposure. There is much evidence for the power of retrieval. For a good summary (and introduction) see James Lang‘s book Small Teaching: Everyday lessons from the science of learning by that I mentioned in an earlier post.

* Please note, I’m not suggesting that there is necessarily anything wrong with this emphasis or that it should change. I’m happy for it to be a problem that teachers at advanced levels need to deal with.