The Least Memorization Possible
I’ve found that there are two general groups of people when it comes to subjects like mathematics and physics. There are those who memorize, and those who internalize the material. Both can bring understanding to the student, but they are much different.
One of my mathematics professors illustrated this when he said, “As a mathematician, I like to do the least memorization possible.”
At first, this struck me as a little ironic, because a staple of mathematics exams is just remembering the truckloads of formulas for various situations. I know that at least in my classes in university, we get no formulas, no regular expressions (such as the trigonometric identities), no unit circle, or anything else. Everything needs to stay in our head, which means we have to memorize some things.
However, the deeper point I think he was trying to get at was that mathematics isn’t about remembering formulas and knowing when to use them. Sure, that’s what happens when we work on these ideas for a long time and get used to doing them, but the point is that these steps and procedures we take shouldn’t necessarily feel foreign. At the very least, they need to be logical and consistent. Doing a double integration by parts with say components $x^2$ and $e^x$ by choosing the former as $u$ and the latter as $dv$ but then doing the opposite after the first integration by parts isn’t logical.
I try to keep this in mind when working on both improving my skills in a mathematics or physics class and while trying to tie everything together for the end of the semester. I don’t want to remember a thousand different formulas. Instead, I want to remember the intuitive and powerful principles that I learned throughout the semester and be able to apply them when I get to problems, without necessarily memorizing everything.
However, I do want to point out one final thing: a lot of the teachers are being a bit disingenuous when telling you, “It’s not the end of the world if you don’t remember a formula. You can easily re-derive it.” Sure, that’s true and you can do that, but most students do not have the time to re-derive a formula and then answer the question that was troubling them on a test. This is particularly true if the test has a short duration (as mine were) or if there are multiple questions in which you have to do this. Given enough time, I’m sure I could get the formulas I needed, but that kind of time isn’t typically available during tests. That’s why it bugs me when teachers say this, because it’s true but not practical.
In a broader sense though, there’s something nice about being able to remember a few principles and working from there. I’m not saying you have to reinvent calculus for your test, but it might not hurt to try and “compress” the number of things you need to remember into more broad categories that can adapt to your specific situation.
Since I just finished up my exams for the semester, I’ve noticed a funny trend that most students seem to engage in. I call it “last-minute stuffing”, and it just refers to the minutes before a test where students quiz themselves and try to remember a bunch of information that they memorized.
At my university, the exams are mostly held in our sports complex, since there’s a bunch of room. Right before the exams, all the students taking the exam are bunched in the hallway outside, and this is where I hear the muttering and recitation going on. I can almost hear the desperation in some of their voices as they try to explain a concept to their friends but can’t quite get it, and how they try to memorize the way to do certain problems.
I don’t know if it’s just because I’m faking it to seem more in control than I really am, but this has almost never been something I did. The only time I remember doing this is when I was trying to remember a the different scenarios for trig substitution when I was about to enter a test that wasn’t integral calculus, but differential equations, where it could pop up. The reason I did that was because we never really worked with trig substitution in a while, and I needed to remember at least the first step.
Barring that, I won’t engage in this type of last-minute rush. The reason is simple: it’s not necessary. I reminded my friends of this as they too looked around with me and saw a bunch of people trying to recite things about the test. I said, “What you look at in the next five minutes isn’t going to change your mark to a drastic degree (if at all). The fact is that we’ve looked at the subject for three months, so all the work that we’ve done has prepared us for the test.”
That’s what I find people seem to forget during exam season. Tests aren’t aced by studying a bunch at the last minute (at least, I’d say that this isn’t the case for most people). Rather, it’s through learning and working throughout the semester.
You’ve done the work all semester, so now you just have to show that.
When I look at a student who knows what they’re doing while solving a physics or mathematics problem versus someone who has no clue what they are doing, there’s an enormous difference in their confidence. The former can usually zero-in on the objective of the problem and knows the strategy needed to tackle it while the latter will try to remember useful formulas or try to do something more or less random.
This is relevant with almost all the students I tutor. I can tell the difference almost right off, because the person that doesn’t know what they are doing is usually unsure of the appropriate strategy and seems lost. This is only further exemplified when they pick the correct strategy to answer the question, but then ask me if what they are doing is correct.
I try to rarely give encouragement like this. The reason is simple: actually solving a problem (unless it’s a real hairy differential equation or integration) is relatively straightforward. It’s the logic of going from one step to the next that is more difficult, and the point of learning. Therefore, by answering the student’s question when they have to make a decision of what to do, I am taking away the most difficult part of the problem.
Instead, I do my best to let them choose the path to follow. Consequently, they get to figure out what works for them and what doesn’t make sense to do. If I just told them which steps to do, the crux of most questions would be gone.
As a tutor, my goal is to give the students I help the confidence they need to figure out the steps to a problem on their own. That is also why I try to stop them from referring to a memory aid or their notes during every problem, because I want them to feel comfortable with saying, “This is the next step. Not because my notes say so, but because it’s the next logical thing to do if we want to answer the question.” In my eyes, my job is only to do this, and definitely not to give them the procedure during each question. There’s a time for refining a procedure for a certain kind of problem later, but when they are first having problems it tends to be due to something conceptual.
When I think back to my own experiences in tests where I’ve felt confident on some questions and not so much on others, the reason I did not feel confident at some points was because I didn’t know what kind of steps to take. But when I was confident, it was due to practicing many times and internalizing the process. That’s confidence, and it’s what I want the students I tutor to have when I’m done working with them.
If you want to work on a particular quality for a student, work on their confidence, and the rest should follow.
When I was in secondary school, I really hated going to French class (I still wouldn’t particularly enjoy it). It wasn’t that the teacher was horrible or anything. Instead, it was simply because I didn’t like taking language classes other than English. I really excelled in English, and the huge chasm in my abilities in French versus English weren’t something I liked being reminded of. In English, not only was I well read and could write, I could also speak well. Conversely, speaking was my weakest link in French. It’s frustrating being able to know all the words that you want to say in your head (and even being able to think them), but not be able to actually express them. As such, I didn’t participate in class at all, preferring to only listen.
The result of this was that I came close to getting a horrible mark in the class for never participating. Since I didn’t want that, I changed my habits about halfway through the term and participated as needed. I even found that participating made me more engaged in the class, rather than just waiting for it to end.
The thing I see over and over again in a science or mathematics class now that I am out of secondary school and into higher education is the lack of participation in class. And I know it’s not because the students aren’t listening or are bored. Rather, no one participates because we are scared to be wrong. Almost every time a professor asks a question in my classes, they are met with silence, because we never want to answer a question and be wrong. I’m definitely a part of this group, and I struggle with this all the time.
Rationally, there’s no better place to be wrong than when giving answers to teachers. If the teacher is doing their job correctly, a mistake in an answer is an opportunity to expand someone’s mind and teach them a way to think. Done wrong, it makes a student feel unwelcome to share their answers again, most likely closing the door on future participation. I feel this all the time in class, and it’s why I don’t answer all that often. However, I’d almost wager that I am one of the most frequent contributors, showing just how infrequent we participate in class.
Being on the other end of the situation, I know that it’s not a fun feeling when no one wants to answer the question you pose in class. As such, I struggle not answering in class. After all, it makes no difference if I get something wrong in class. Who cares if others think I’m an idiot for asking the question? Who cares if it’s obvious to everyone else, but not me?
Personally, I’ve always struggled with this to some degree. I’m not actually shy, but I’m always so obsessed with being “good” at subjects that I don’t want to appear like I’m a novice. Therefore, I try to hide my lack of knowledge by doing the “rough work” at home, so I can appear to be knowledgeable during class. The truth, of course, is that the class is there for me to learn, not to know everything before we start.
This is why I encourage participation in mathematics and science class. We need to stop being worried about what others will think of our hypotheses and ideas. Yes, people will judge them, but who really cares? On an intellectual level, I know that asking questions and giving answers in class (even if I’m not sure about them) is a way to help me, not hinder me. As such, it’s one of my personal goals to work on, and I hope you can reflect on this as well. Do you participate in school, even when you aren’t sure about your answer? Do you ask questions, no matter how dumb? If the answer is “no”, I encourage you to work on this with me.
Participation in class is a key component to learning. Do what most people aren’t, and use it.