A common refrain I often hear from students who are new to Calculus when they seek out a tutor is that they have some homework problems that they do not know how to solve because their teacher/instructor/professor did not show them how to do it. Often times, I also see these students being overly dependent on memorizing solutions to examples they see in class in hopes that this is all they need to do to is repeat these solutions on their homework and exams. My best guess is that this is how they made it through high school algebra.
I also sense this sort of culture shock in students who:
- are always locked in an endless cycle of “How should I start?” and “What should I do next?” questions,
- seem generally concerned about what they are supposed to do as if there is only one correct way to solve a problem,
- complain that the exam was nothing like the homework, even though the exam covered the same concepts.
Anybody who has seen my comments on /r/calculus over the last year or two may already know my thoughts on the topic, but they do bear repeating again once more in a pinned post. I post my thoughts again, in hopes they reach new Calculus students who come here for help on their homework, mainly due to the situation I am posting about.
Having a second job where I also tutor high school students in algebra, I often find that some algebra classes are set up so that students only need to memorize, memorize, memorize what the teacher does.
Then they get to Calculus, often in a college setting, and are smacked in the face with the reality that memorization alone is not going to get them through Calculus. This is because it is a common expectation among Calculus instructors and professors that students apply problem-solving skills.
How are we supposed to solve problems if we aren’t shown how to solve them?
That’s the entire point of solving problems. That you are supposed to figure it out for yourself. There are two kinds of math questions that appear on homework and exams: Exercises and problems.
What is the difference? An exercise is a question where the solution process is already known to the person answering the question. Your instructor shows you how to evaluate a limit of a rational function by factoring and cancelling factors. Then you are asked to do the same thing on the homework, probably several times, and then once again on your first midterm. This is a situation where memorizing what the instructor does in class is perfectly viable.
A problem, on the other hand, is a situation requiring you to devise a process to come to a solution, not just simply applying a process you have seen before. If you rely on someone to give/tell you a process to solve a problem, you aren’t solving a problem. You are simply implementing someone else’s solution.
This is one reason why instructors do not show you how to solve literally every problem you will encounter on the homework and exams. It’s not because your instructor is being lazy, it’s because you are expected to apply problem-solving skills. A second reason, of course, is that there are far too many different problem situations that require different processes (even if they differ by one minor difference), and so it is just plain impractical for an instructor to cover every single problem situation, not to mention it being impractical to try to memorize all of them.
My third personal reason, a reason I suspect is shared by many other instructors, is that I have an interest in assessing whether or not you understand Calculus concepts. Giving you an exam where you can get away with regurgitating what you saw in class does not do this. I would not be able to distinguish a student who understands Calculus concepts from one who is really good at memorizing solutions. No, memorizing a solution you see in class does not mean you understand the material. What does help me see whether or not you understand the material is if you are able to adapt to new situations.
So then how do I figure things out if I am not told how to solve a problem?
If you are one of these students, and you are seeing a tutor, or coming to /r/calculus for help, instead of focusing on trying to slog through your homework assignment, please use it as an opportunity to improve upon your problem-solving habits. As much I enjoy helping students, I would rather devote my energy helping them become more independent rather than them continuing to depend on help. Don’t just learn how to do your homework, learn how to be a more effective and independent problem-solver.
Discard the mindset that problem-solving is about doing what you think you should do. This is a rather defeating mindset when it comes to solving problems. Avoid the ”How should I start?” and “What should I do next?” The word “should” implies you are expecting to memorize yet another solution so that you can regurgitate it on the exam.
Instead, ask yourself, “What can I do?” And in answering this question, you will review what you already know, which includes any mathematical knowledge you bring into Calculus from previous math classes (*cough*algebra*cough*trigonometry*cough*). Take all those prerequisites seriously. Really. Either by mental recall, or by keeping your own notebook (maybe you even kept your notes from high school algebra), make sure you keep a grip on prerequisites. Because the more prerequisite knowledge you can recall, the more like you you are going to find an answer to “What can I do?”
Next, when it comes to learning new concepts in Calculus, you want to keep these three things in mind:
- When can the concept be applied.
- What the concept is good for (i.e., what kind of information can you get with it)?
- How to properly utilize the concept.
When reviewing what you know to solve a problem, you are looking for concepts that apply to the problem situation you are facing, whether at the beginning, or partway through (1). You may also have an idea which direction you want to take, so you would keep (2) in mind as well.
Sometimes, however, more than one concept applies, and failing to choose one based on (2), you may have to just try one anyways. Sometimes, you may have more than one way to apply a concept, and you are not sure what choice to make. Never be afraid to try something. Don’t be afraid of running into a dead end. This is the reality of problem-solving. A moment of realization happens when you simply try something without an expectation of a result.
Furthermore, when learning new concepts, and your teacher shows examples applying these new concepts, resist the urge to try to memorize the entire solution. The entire point of an example is to showcase a new concept, not to give you another solution to memorize.
If you can put an end to your “What should I do?” questions and instead ask “Should I try XYZ concept/tool?” that is an improvement, but even better is to try it out anyway. You don’t need anybody’s permission, not even your instructor’s, to try something out. Try it, and if you are not sure if you did it correctly, or if you went in the right direction, then we are still here and can give you feedback on your attempt.
Other miscellaneous study advice:
Don’t wait until the last minute to get a start on your homework that you have a whole week to work on. Furthermore, s p a c e o u t your studying. Chip away a little bit at your homework each night instead of trying to get it done all in one sitting. That way, the concepts stay consistently fresh in your mind instead of having to remember what your teacher taught you a week ago.
If you are lost or confused, please do your best to try to explain how it is you are lost or confused. Just throwing up your hands and saying “I’m lost” without any further clarification is useless to anybody who is attempting to help you because we need to know what it is you do know. We need to know where your understanding ends and confusion begins. Ultimately, any new instruction you receive must be tied to knowledge you already have.
Sometimes, when learning a new concept, it may be a good idea to separate mastering the new concept from using the concept to solve a problem. A favorite example of mine is integration by substitution. Often times, I find students learning how to perform a substitution at the same time as when they are attempting to use substitution to evaluate an integral. I personally think it is better to first learn how to perform substitution first, including all the nuances involved, before worrying about whether or not you are choosing the right substitution to solve an integral. Spend some time just practicing substitution for its own sake. The same applies to other concepts. Practice concepts so that you can learn how to do it correctly before you start using it to solve problems.
Finally, in a teacher-student relationship, both the student and the teacher have responsibilities. The teacher has the responsibility to teach, but the student also has the responsibility to learn, and mutual cooperation is absolutely necessary. The teacher is not there to do all of the work. You are now in college (or an AP class in high school) and now need to put more effort into your learning than you have previously made.
(Thanks to /u/You_dont_care_anyway for some suggestions.)
