Developmental Mathematics Revival!

Memorize This!

Slope formula? Area of triangle? Quadratic formula? Basic number facts?

What is the proper role of ‘memorization’ in learning? Specifically, is memorization needed in a math class?

Let me start with a short anecdote from today’s class. A student needed to divide 16 by 8; he pulled out his graphing calculator. Another student needed to know what the common factor was between 16 and 40; he also got out his calculator, and tried dividing each by 4 …

We have been stuck on a rejection of ‘rote’ learning, with a poor association of ‘memorizing’ with ‘rote’. Now, there are actually times when rote learning is fine — though most of us (myself included) do not use this very often, in favor of more active learning.

This has gone so far as to result in students being told to NOT memorize; one of my colleagues tells students that they can always look it up (in the ‘real’ world). On balance, this has harmed far more students than it has helped. Let me explain why this might be true.

First of all, the human brain ‘wants’ to remember things (including formulas and facts) — you dial an arbitrary sequence of 10 digits more than once, and your brain is likely to work on remembering that phone number. Telling somebody to ‘not memorize’ comes very close to telling them ‘turn your brain off!’. We can’t condemn memorizing and condemn lack of learning; they go together.

Secondly, the progression from novice to more expert states involves a process called ‘chunking’. We, as mathematicians, have a very large chunk size in domains where we have practiced and thought; while a student sees 15 steps in a series, we see 2 collections of steps. When faced with a novel problem, we bring these chunks and our understanding of their connections to the problem. Experts in any field have a large chunk size, often numbering 10 to 15 specifics in a chunk. Telling a student to ‘not memorize’ is telling them “it is okay to have a chunk size of 1 (one)” — which means that they are likely to appear as a novice in that field, no matter how much they work. [Memory, especially clusters of connected memories, seem to be a critical building material for our ‘chunks’; some writers call these ‘schema’ instead of ‘chunks’.]

Thirdly, and fortunately, it does not work to tell somebody to not memorize (see the first point). The bad part is that some students actually listen to us, and they remember less because of it. Basically, this is saying that our advice has damage that is limited by accident, not design.

In all my reading of learning theory, over a period of decades, I have yet to find a cognitive scientist say that ‘memorizing is bad’. From a learning point of view, it is all just learning. If a person memorizes a formula, without having practiced in varying contexts and without connecting it to other information, then they will be limited in how they can apply this formula; if a person does not memorize a formula, they have to organize their learning around other information — not connected to a formula. We see students who have a vague notion that area is length times width, and connect all ‘area’ information to this; this incomplete learning creates unnecessary barriers. If students know multiple formulas for area, as an example, they connect all of these to their understanding of ‘area’; they become better problem solvers … and transfer of learning is much more likely with this more complicated mental map of ‘area’. The best situation is one where students have several area formulae available from memory, all connected to a concept of ‘product of two measurements, and perhaps a constant.

Mathematics is not the only domain with an interest in (not-)memorizing. Language learning has also dealt with this, as well as others (see http://scottthornbury.wordpress.com/tag/cognitivist-learning-theory/, and you might also enjoy http://thankyoubrain.com/Files/What%20Good%20Is%20Learning.pdf).

The part that actually bothers me the most, however, is the attitude resulting from students not remembering basic information. As long as he has to get a calculator for ’16 divided by 8′, he is going to feel dumb about math. A sense of proficiency and competence goes a long way towards persevering. Our students do not need a barrier added to their challenges, a barrier constructed out of our good intentions when we say DO NOT MEMORIZE!

Memorizing does not need to be ‘rote’, as we all know from personal experience. Memorizing happens due to time on task, with a little reflection on the learning involved. Memorizing is a natural process for a human brain; we need to take advantage of this capacity.

Memorizing alone will not be enough, and never has been. However, without memorizing we limit the long term mathematical development of our students; we reinforce negative attitudes, and we create learners who have trouble transferring their learning. Let’s keep a healthy balance — some memorizing, a lot of understanding, building connections, and enough practice to build competence.