The title of this posting is the title of a chapter in “Making Learning Whole”, by David Perkins (2009). Of the books on education I have read to date, this is the first that resonated completely with me. He describes the way I try to teach, and more – thus giving me much to reflect upon. I recommend it highly.
The list of skills related to “the game of learning” I see as being most important for math and science students to acquire, and therefore worth devoting some time to teaching explicitly over the course of the school year (since they are also more generally applicable) are:
- What is it you need to learn: a concept, a skill, or a fact? Concepts can often require thought, and time spent discussing them with others while being watchful for subtleties. Skills often require repetition and varying levels of difficulty. Facts can sometimes be obvious if they are based on an underlying concept; if the facts are not obvious, search for a way to link them to one or more concepts or themes, then practice retrieving them along with related information.
- Frustration is a normal part of the learning process, one which can often lead to greater understanding and retention once you have worked your way through it. Expect to become frustrated at one or more points in learning new material, and expect to work your way through it successfully… having such expectations from the outset somehow makes the process less frustrating!
- Beware of memorizing. While it may help you pass the test tomorrow, if you do not grasp the concepts behind what is being presented, the material you memorized will probably fade from memory fairly quickly. Math is a subject that builds upon previous knowledge. You cannot afford to have it fade. While it may sometimes be more difficult to grasp a concept than it is to memorize a formula, it is well worth the effort and is likely to result in better grades over the long term than a memorization approach.
- Reading a textbook is not the same as reading a novel for pleasure. Read every sentence closely to ensure there is no chance you could have mis-interpreted the author’s words. Look for, and think about ideas/concepts that are being developed and linked. Cover up each sample problem as you come to it, work to solve it yourself from the information provided so far (this will often take some time), then compare your solution to the author’s. Look for and think about connections and similarities with other procedures or concepts you have already learned.
- “Why?” is a great question. Ask it often, both of yourself, and of those you are learning from. Why am I learning this? Why I am solving this problem this way? Why am I NOT solving it another way? Why does the solution make sense as an answer to the problem?
- Learn how you best master new topics. Possibilities include: by listening to others, by talking with others, by writing, by illustrating the situation in some way, by watching others solve problems, and/or by solving problems yourself. Perhaps some combination of these work for you. If your usual approach isn’t working for a topic, try a different one.
- Use multiple approaches to learning new material: read a different textbook’s description (one that takes a different approach) of the same material, ask a friend about how they perceive the material, and/or ask a different teacher to explain it.
- Figure out where this new material appears or will be relevant in the world around you. Immerse yourself in an experience that the new material applies to so that you have first-hand knowledge of things, and questions to go with it.
- Watching someone else solve a problem is not AT ALL the same as solving it yourself. You must master solving all related problem types reasonably efficiently by yourself.
- Being able to solve all related problem types does not necessarily mean you understand the underlying concepts. Both are very important.
- Find and solve “uglier” problems than are likely to be on the test. After accomplishing this, all the types of problems you had been doing will feel much easier.
I recently came across a wonderful description of the techniques an MIT student used to master material four times faster than the normal rate. I highly recommend it: Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning.
Another good resource is Stan Brown’s web site, and in particular this page: How to Read a Math Book.
Thanks for these. I think I’ll put them up on the VLE for my A level students.
Good post. The “Beware of Memorizing” particularly resontes with me.
See my post Math Isn’t Memory Work