The Myth of Memorization

When I think back to my experience in math classes in elementary school, one of my strongest memories is attempting to get perfect on math drills. These drills focused on memorization and speed, which are two topics that are greatly discussed and debated in today's teaching world. But where do I stand on these topics?

Personally, I am not a very fast mathematical thinker. Math drills always frustrated me as a child, because I knew that I was able to answer all of the correct answers, but I would so often be forced to leave the last four or five questions blank because of the arbitrary time limit set on the math drill. If I hadn't known the answer, I wouldn't have felt so badly about the blank questions, but I absolutely hated it when I wanted to show my knowledge but was blocked from doing so. In order to answer all of the questions, I rushed and became more reckless, then would receive back my drill and discover that I had wrong answers to questions that I did actually know correctly.

Receiving a low grade in a subject not because you answered questions incorrectly, but because you weren't given enough time to complete the work and show your capability is a humiliating experience, and causes students to believe that they aren't capable of doing math even though they are perfectly capable of doing so. I do not think that timed math drills are a beneficial teaching technique, however, I do think that the repetition that certain exercises provide can be useful to develop math fluency.

Being mathematically fluent in basic multiplication enables a student to use this base knowledge for performing mental math, and also for quickly realizing when the solution to a math problem doesn't seem right. Mathematical fluency is closely partnered with mathematical intuition. This fluency is far beyond shallow memorization, but contains a deep understanding of math that, like a muscle, has been strengthened to work more efficiently. What's dangerous is when we confuse memorization and deep understanding, believing that a student knows a concept better than they actually do, since mathematical concepts build on each other and require firm prior understandings. One of the huge problems with memorization without understanding is that the information, such as a specific formula, that we are using is not stored permanently for future use, but is disposed of after we have no immediate need for it. Another issue is the fact that while our brain is holding these temporarily memorized facts, it has less processing power left to use for tasks like problem solving.

For students to develop a lasting understanding of math, we need to encourage their mathematical intuition. We need to guide students towards internalizing the big ideas in math without telling them the answers, but giving them the space to struggle through problems and develop their own solutions. Sebastian Thrun, the former VP of Google, believes that we should be "only willing to manipulate equations [we] understand" and avoid "fixed rule manipulations [that] are devoid of an intuitive interpretation." Formulas in and of themselves are not problematic, but we should only use formulas when we understand why and how they work, and "expose students to as many methods and means of representation in math as possible so they can decide for themselves what suits their brain the most" (Bernadette James). For some students, this deeper understanding will not come through formulas, but through drawing.

To celebrate alternative methods to solving math problems, here is an interesting video that shows how to do multiplication using lines.