Being caught up with the prerequisites is a huge issue. If someone is weak in arithmetic, they'll struggle with algebra. If weak in algebra, they'll struggle in calculus.
If they learned by "guess and try," they'll simply hit a wall when problems cease to have a number as an answer.
Re:Calculus Learning Guide
by InclinedPlane on Wednesday April 13, 2016 @07:48PM
(#11493293)
Also, if it takes you, say, 2 minutes to struggle through a simple multiple choice algebra problem that you're able to overcome through lots of brute force methods but you still get the right answer, you can pass algebra tests easily, but you're going to struggle when you hit calculus. You need to be able to solve some of those problems mostly in your head in seconds, and the methods need to be intuitive and familiar to you. Moreover, the rest of your work depends on having the right answer to that problem. If you can get the right answer 90% of the time with a lot of effort that might be good enough to pass a class, but it's not actually good enough to use those skills practically. In an easy calculus problem you might have to work through dozens of little simple algebra problems, and if it takes minutes to do each one, and you get the wrong answer 10% or more of the time, then calculus problems are going to take hours and you're going to get the wrong answer most of the time.
trigonometry and advanced algebra is really a problem to me when studing calculus, any tips to improve that?
Re:Calculus Learning Guide
by InclinedPlane on Wednesday April 13, 2016 @07:41PM
(#11493266)
I wish I had a better answer. Find a good book or set of resources and go through it again. A good method might be to go through chapter by chapter and skim through the subjects and the problem sets, if you see something that isn't "slam dunk" easy then consider reviewing that material and just reading through that chapter and doing the problem sets until it makes sense. Similarly, if you're working through something in calculus and you hit a speed bump, try to figure out exactly what you're struggling with and then go back to your algebra/trig materials and focus in on that until you've learned it.
Khan Academy has a ton of videos on algebra and trig on youtube, so that's also another good resource. If you have the money, consider hiring a tutor for a few short sessions, they can be remarkably affordable. Also, if you're in college there are study groups and office hours, so consider taking advantage of those.
The most important thing in math is to make sure you're actually learning the material rather than just skimming over it and passing the tests.
You can also try various games, though sadly I don't have any recommendations.
One is that mathematics in general is taught very poorly (well, all subjects really, but mathematics bears the brunt even more so). There's a great concentration on "teaching to the test", rote memorization, "plug and chug", churn through as much "material" as possible and then forget it over summer, that sort of thing. Cargo cult education, basically. This is very damaging to the teaching of calculus because it is very much a subject that is dependent on some very fundamental core concepts. If you don't "get" those core concepts, then calculus will be forever opaque to you. Worse, this pattern of "education" applies all the way down the line to every math subject taught in school, including the pre-requisites for calculus: advanced algebra and trigonometry.
And calculus is very dependent on actually knowing that pre-requisite math. Not being dependent on plug-and-chug, guess-and-check, "eliminate 2 wrong answers from the multiple choice question and then guess", or "cram for the test and then forget it". All of those techniques work well at gaming the educational system, but they leave you weak in the actual skills of applying trigonometry and advanced algebra to problems, and using them as tools while learning other, more advanced, mathematics. So you have a typical calculus course trying to lay down enormous concrete slabs of new concepts and "material" but it just sinks into the bog rather than building up a structure sitting on a solid foundation.
Additionally, calculus is often used as a filter course. Calculus has a potentially infinite amount of "trivia" associated with it. All manner of rules, substitutions, identities, definitions, theorems, and so forth. As such, calculus effectively has a "difficulty dial" than any teacher can set arbitrarily depending on how much memorization and busy work they want their students to slog through. And many teachers decide to set this at just high enough of a level to make calculus a real pain in the ass for most students. There's always a million justifications for doing this, but most of them are pretty weak. There's a justification for acquiring familiarity with all of these things, and proving you can walk through the steps to do harder problems, but in practice it's mostly useless. People who actually use calculus day to day generally just keep a cheat sheet around of integration and derivation rules and substitutions and whatnot, or they employ the use of a computer to do the heavy lifting. Once you understand the concepts there's not much value in slogging through many mindless examples just for bragging rights.
All of this leads to people getting the impression that Calculus requires superhuman skills to learn (no, it just requires being actually caught up with the pre-requisites before hand) and also that the application of calculus tends to be boring and monotonous. If education did a better job of ensuring the pre-calculus fundamentals were thoroughly in place, and concentrated more on concepts and applications of calculus (like physics, engineering, geometry, control systems, computer graphics, and so much more) would, I think, lead to not only a lot more students succeeding at learning calculus well but also enjoying it.