A slightly cheerier entry for you today, my friends. Although, as I said last time, I think I’ve reached the limit of my mathematical abilities, that limit’s quite a bit higher than I thought it was. I never thought I could get through even pre-calc, but I’ve managed to survive–not excel, but at least survive–through basic calculus. So if I did better than I thought I could, do I have any hints or tips that helped me succeed? As a matter of fact, I very much do, and here I’ll describe them!
The first, and arguably most important factor IMO in improving your skills if you’re not a “math person” is a matter of attitude. Thus, the first general attitude adjustment I found that really helped me get through this semester’s challenges was…
Rule 1: Don’t tell yourself, “Man, I suck at math!” Instead, say, “Man, I suck at these types of problems. That’s the first step towards not sucking!
Looking back at myself when I was a kid, back in middle school and high school, I now realize that the biggest problem I had with math (and math-heavy science) was my attitude towards it. Or, perhaps more specifically, my attitude towards my relation with the subject.
See, in middle and high school, whenever I did poorly on a test or got a question wrong, I’d get frustrated and say to myself, “God damn it, I suck at math! This stuff just isn’t for me.” Over time, I started to define myself as a “writer” rather than a math guy–I’d generally do much better in English and History than my math courses. So every time I failed to improve at math, I chalked it up to my *identity* rather than any specific lack of skill or understanding.
This way of thinking made it so I *couldn’t* improve, even if I wanted to. It was essentially a form of giving up–and not giving up in the good sense, after you’ve recognized your limitations, but giving up when you don’t even understand why you’re giving up. If you define yourself as “someone who’s bad at math,” any failure becomes, by necessity, inevitable, instead of explicable in terms of things you can improve on, or at the very least, explicable in terms of intellectual weaknesses you can at least make some effort to mitigate.
Once I got to college–the second time around, that is–I had a much healthier outlook, at least in my humble estimation. When things started, even with the basic limit stuff I had a bit of trouble–I kept getting points taken off because I didn’t write some of the limit notation properly (long story short, in limit problems where you actually plug in numbers, my prof took off points if you kept the little “limit as x approaches a” symbol). Back in middle or highschool, I would have just thrown up my hands and said “Meh, I suck at math, I’ll never get those points.” Now, however, you know what I said?
“Man, I’m sucking at this limit notation stuff.”
I no longer chalked it down to some immutable aspect of my personality or makeup. And this allowed me much more flexibility and much more *rationality* when it came to addressing the issue. When I said to myself, “I’m sucking at *this particular problem,*” it gave me at least a start in figuring out either ways to improve or some specific reason I was failing. In this case, after thinking about it, I started writing down “PLUG IN NUMBERS, REMOVE LIMIT SIGN” on every quiz with those types of problems, and that was enough to jog my memory every time I saw it and keep me from getting points deducted. Now, my prof thought this was kinda weird, but didn’t care if it helped me, so my grades improved–far more than they would have if I was still in my teenage mentality!
Now, at some point you will run into a brick wall in terms of your abilities. As I said in my previous entry, my memory and attention to detail just aren’t what they used to be, so I’d have a lot of trouble in more advanced math courses. Still, even this is a healthier outlook than just saying “I’m not a math person.” As I also mentioned in this entry, if you have a specific idea of what your weaknesses are, you can mitigate them to some extent. So realizing that I have problems with memory and detail *specifically* instead of just a vague “sucking at math” attitude allows me to take specific actions in some cases (like reminding myself of the remove-limit-sign step) or get a better idea of things I might actually be good at, in this case, any fields of study which don’t require *both* mass memorization *and* a lot of focus on small details. Or, in other words, saying you’re “not a math person” doesn’t give you any insight into what kind of person you actually *are.* But soberly assessing your weaknesses can tell you your strengths, which makes it much easier to find better matches for your abilities and more constructive paths to follow down the line.
So, with a better general attitude established, here’s another more specific attitude adjustment that also helped me improve in math.
Rule 2: If you find yourself getting frustrated, try a different approach to solving a problem.
The funny thing is, taking a stance similar to the one I used a lot *with videogames* helped me do a bit better on a lot of math problems. Sound weird? Let me explain.
I know, it sounds hokey and dumb, but if you define “gamer” as “anyone who plays videogames a lot,” I’ve definitely been a gamer since I was a kid, and as you can tell by how much I play Monster Hunter Iceborne, I still am. And oddly enough, I’ve found one strategy I often use when playing games to come in handy in math as well. It all boils down to this: Rather than getting frustrated when you find a challenge that you can’t seem to meet, try a new strategy and see if it’s much easier.
See, in a lot of the action games I play–Monster Hunter, Bloodborne, Blasphemous, whatever–sometimes you can run up against a tough boss that seems impossible for you. You just keep dying over and over and getting nowhere. When that happens, I find that taking a step back and trying a new strategy can make even a really tough boss trivially easy. For instance, in Monster Hunter, when I was having trouble fighting monsters with the Gunlance, I just equipped the Evade Extender skill and things went MUCH easier. In layman’s terms, rather than wearing myself out with one approach to a problem, I used other tools available to me to find a way around it and triumph.
The same applies to math. If you’re trying one way to solve a problem and it’s just not working for you–you keep making mistakes, you just don’t understand it, whatever–try another way and you might do much better. For instance, early on in the course we learned about finding certain kinds of limits in division problems that end up equalling 0/0 or infinity/infinity. Doing those requires some irksome factoring, but later on we learned a quick solution, L’Hopital’s Rule, that gives us the answer if we just take the derivative of the numerator and denominator, which is MUCH easier as long as you’re just dealing with simple derivative rules. So rather than getting frustrated trying to do a limit problem the long and hard way, L’Hopital’s Rule offers a quicker and easier way to get around it, just like the Evade Extender skill in Monhun offers a quick and easy way to fight monsters if you have a Gunlance.
With general attitudes out of the way, here are three specific tips I used to do a little better
Trick 1: If you can’t understand a certain concept, Look at how your teacher or instructors you see online solve problems, and follow their methodology step-by-step, even if you don’t get the logic or higher ideas behind what they’re doing. Once you figure out the mechanical process behind doing the problem, you’ll be better able to figure out the mathematical reasoning behind it.
As I mentioned previously, the antidifferentiation stuff was really difficult for me. I barely understood it when I took the course, and I still don’t really understand it. However, I couldn’t just leave it at that, or else I’d totally lose all those points. So what I did was I watched my prof’s videos along with a couple of Khan Academy ones, and without trying to understand the rationale behind what they were doing, I just watched how, exactly, they were manipulating the numbers and symbols and wrote it down step-by step, so I could at least follow some sort of method to get me the right answer. Here’s how it looked, for example, on the study sheet I made:
Example Problem: Find f if the derivative of f is some given equation and f of some specific number is another specific number.
STEP A: The objective of these problems is to find the exact value of the constant you’re always given. So the first thing you do is FIND THE ANTIDERIVATIVE OF THE GIVEN DERIVATIVE FUNCTION.
STEP B: ADD A +C TO THAT.
STEP C: Take your antiderivative equation and set it equal to whatever number they gave you in the original problem, so it looks like “something something + C = whatever they said f(x) equals.”
STEP D: Turn all the xs in that equation to the number in the parentheses of the original equation they gave you.
STEP D: solve for C
STEP E BE SURE TO WRITE OUT THE WHOLE ANTIDIFFERENTIATED EQUATION THAT YOU FOUND! SOLVING FOR C IS ONLY THE SECOND LAST PART OF THE PROBLEM!
Yeah, it’s not elegant, but it did allow me to get those questions right, which is good enough. And once you can do that, you can take a step back and think about *why* the approach is working, which will allow you to really understand what’s going on down the line! Now, understanding is the ultimate goal, so this sort of “following steps without really knowing why” isn’t ideal forever, but it can give you a good start, which might be just what the doctor ordered.
Trick 2: Related to trick one, if you see a lot of weird, confusing, or scary-looking symbols, first try to relate them to the particular ways you keep seeing problems involving them get solved. Once you do that, it’ll be easier for you to figure out what they actually mean.
Another problem I had to do involved “Riemann Sums.” They were always marked by this funny symbol, the Greek letter sigma (Σ). Now, one of the things that intimidates people in higher-level math is seeing a lot of strange letters and symbols they haven’t seen before and almost never see at all outside of the math classroom. Unlike regular numbers, plus signs, division signs, and all that we see in regular life about every day, most people don’t encounter Greek letters or some of the other thingies (like the antidifferention symbol) at all and can easily get confused! What made things a lot easier for me was watching how my teacher solved problems involving those symbols, and then associating the symbol with the basic technique he used, so it no longer seemed weird and alien. For instance, long story short, in problems involving that sigma letter, what happened was that we just had to add (following a certain rule) a bunch of times. That’s why it’s called a “Riemann *sum*!” So after I figured out what we were doing, every time I saw the funny Greek letter I just said to myself, “Oh, time to just add over and over again, that’s not too hard.” So that way, the peculiar notation became less intimidating.
Trick 3: You have to memorize, but try to memorize two-for-ones when you can.
There’s just no getting around it, memorization is key in math. But sometimes you can make it a little easier on yourself. For instance, you just have to memorize the differentiation and antidifferentiation rules. However, since antidifferentiation is (sorta) just differentiation-in-reverse, that makes learning some of the antidifferentiation rules much easier, at least for trigonometric equations. I won’t bore you with too many details, but essentially, the derivatives for some simple trig equations go like this:
If you differentiate sin(x), you get cos(x).
If you differentiate cos(x), you get negative sin(x)
And so on. Now, the thing is, these map perfectly onto the antidifferentiation rules. Like, if you’re asked to find the antiderivative of cos(x), the answer is sin(x), and if you’re asked to find the antiderivative of sin(x), you get negative cos(x) (which makes sense, remember that making something negative is just reversing its sign). So if you have a chart that lists the derivatives of each of the trig equations, you *also* know their antiderivatives! It’s easy-peasy and it might make memorization a *little* easier if you don’t have to try re-memorizing two separate charts (one for trig derivatives and one for trig anti-derivatives) and instead can memorize just one, so long as you know the relationship between them, of course!
Well, those are all the tips n’ tricks I found this semester–if any of you have any of your own, dear readers, feel free to let me know in the comments, I may find them useful! Till then, see ya next time 😀