The “Memorize or Melt” posts are a series of posts focused on the generally poor pedagogy of having students memorize formulae, techniques, etc. with the misguided belief that somehow the student will be better for it. This is the “kick off” post that will explain the author’s overall views.
Practically everyone who has gone through any type of formal schooling has had to do some form of memorization either through repetition or remembering cute aphorisms or mnemonics. Memorize the alphabet. Memorize your social security number. Memorize your address. “Look both ways before you cross.” Memorize the times tables. Memorize your PIN (Personal Identification Number). And the list can go on.
Some of these things make sense to memorize. A secret, ninja-spy is probably going to have to commit to memory her mission, the layout of the area, her target, etc. (American) Football players need to have memorized their playbook (and if they somehow got hold of it, their opponent’s playbook).
Having a good memory is an asset, for sure. It is pretty much the singular reason why human civilization exists. Knowledge is retained and passed down over the generations. Progress is made through experimentation, which requires the experimenter to “learn” from the failures of his experiments; this requires memory. Having a great recall can be an asset, and maybe even in some sense, a necessity, in professions like medicine or law. But what is worth remembering?
- Does a programmer memorize the library he imports? Does a programmer memorize syntax for every language in which she writes?
- Does an artist memorize every brushstroke to his paintings?
- Does a cook memorize every recipe?
Clearly not! So, do math students need to memorize trig identities? Do they need to memorize integration tables? Do they need to memorize the “quadratic formula”? Here too, the answer is “No!”. Does it help if these things are memorized? Maybe. What is meant by help? Does it help to solve a textbook problem faster than if the information weren’t memorized. Yes, in that case it does. Does it help students to understand the material any better? No. Does it help to keep students interested, motivated, curious? No.
So why do it? Why are so many students subject to memorization drills? The blame can be spread in lots of ways, but this author will put the blame squarely on instructors. The instructor is 100% in charge of his class. If, for whatever reason, the instructor feels that there is some information that students must know but is beyond the scope of the course to have students “prove” or otherwise re-derive, then the instructor has to pick from one of two choices:
- Make the students memorize the information.
- Allow students to have a handy reference whenever such information is needed.
If the instructor takes the route of memorization, then he is almost surely (not in a measure-theoretic sense, but in a colloquial use of the phrase sense) guilty of bad teaching and guilty of causing “math trauma.” What is math trauma? It’s the thing that causes things like this: “I hate math!”; “All I remember from math class was negative \(b\) plus or minus the square root of … shoot me now”; “I was never good at math.”
When finding out that I was a mathematician, a history professor said to me, holding his two index fingers up to form a cross, “You stay away from me! I have nothing but miserable memories about you, your kind, and your profession.” Eek. True story.
Now, not all memorization is bad in mathematics. It is probably helpful to know that two comes after one and three comes after two or to know one’s times tables, for example. But this type of memorization is more a learned familiarity rather than something that is recited over and over again without context or soul.
An instructor should think long and hard and ask, “Is this absolutely necessary for the student to have to memorize? or are they just as well-served to have a reference since explaining the derivation or demonstrating the proof is out of scope?” before going down the “memorize or melt” route. For the vast majority of the cases, the instructor should find that memorization is not the answer.
Finally, mnemonic devices, while handy, can be another source of math trauma. Mnemonic devices are typically introduced by expressing to students “Here’s a simple way to remember this.” For example, “Ok, now we have our four quadrants. Where is sine positive? Where is cosine positive? A simple way to remember this is CAST.” @#!% Don’t do this!! How does CAST make any sense? Students are told that the convention is that quadrant \(\mathrm{I}\) is the “upper right quadrant.” But CAST requires that the student begin in quadrant \(\mathrm{IV}\) (the positive portion of the horizontal axis and the negative portion of the vertical axis), and this now means that the student has to somehow, somewhere keep in memory this useless factoid.
So, then why not ASTC? Well, what in the world is ASTC? “All Students Take Calculus”. More memorization. Useless. Stop this. Something like “CAST” should never be taught. Songs about the quadratic formula shouldn’t have to be imposed onto students. Sing with me: ♪ sine, cosine, cosine, sine, cosine, cosine, sine, sine ♫ — apparently a song to help remember that $$\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$$ and $$\cos(a + b) = \cos(a)\cos(b) – \sin(a)\sin(b)$$
Imagine a professor of Measure Theory saying to his students, “Here’s a great way to remember Lebesgue’s Dominated Convergence Theorem …” That doesn’t happen because it doesn’t make sense!
Learning takes time and math has to marinate. Rebel a little or a lot against the math curriculum imposed on to you — the qualified, caring, passionate instructor — if it is forcing you in a position to have to impose memorization gimmicks to get the students through. Memorization is not education in the slightest degree. Mathematicians should summarily reject any pedagogy that insists on meaningless memorization. Teach the why and the how. Skills and drills have value, but not when it turns the student into a mindless automaton. Find a way to teach without making it agony for the students. Give students reference sheets and assess them on their understanding. Teach the mechanics that are appropriate for the course. Talk to your colleagues. Brainstorm.
With all that said, future posts in the “Memorize or Melt” series will have “Memorize or Melt” in the title. The purpose will be to discuss popular memorization techniques for “learning” new material and how the instructor can get away from it. If you have some great ways to connect to students about any topic, feel free to comment! If you are looking for suggestions, feel free to ask in the comments section below! And of course, reasonable disagreement is a good thing too, so please share your thoughts.