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How to retain concepts from math and physics?

  1. Jan 1, 2016 #1
    Hey everyone,

    I am beginning calculus 2 and physics this coming semester. I am not very anxious about learning the material, as it seems concepts typically develop overtime. However, I have a huge issue with retaining material. Previously, I've basically worked multiple practice problems until the ideas stuck and solving problems became intuitive. But, this last semester, I realized while this is effective short-term, it is not long-term, and it isn't a very efficient way of studying. I could spend 20+ hrs/wk on Calculus 1 homework just to reinforce basic ideas. Conceptually, I seem to get the material, it's just with REMEMBERING that I have trouble with.

    This tends to be a big issue for me with more memorization-based classes like softer sciences. I actually can end up spending more time on these courses than my conceptual course loads (chemistry, math, etc.) solely because I can't retain the information.

    I have a few weeks before classes begin, but the anxiety is already kicking in. Do you have any general techniques that you apply which work for you? For memorization, I typically just make flash cards and use rote memorization. With math, I typically practice, practice, practice. But, I've noticed that OVER practicing makes me kind of use an algorithm, or recollecting a similar problem in order to solve new problems. That is, if I over practice it is no longer conceptual, but just purely computational.

    Thanks =/
  2. jcsd
  3. Jan 1, 2016 #2
    I tend to use older textbooks. I noticed that older textbooks are not as flashy(lack many unnecessary pictures), are a bit more formal, and provide both rigor and intuition. I make it a habit to use at least 2 textbooks. The main text the class requires and a supplement. Sometimes I have ditched the required book all together and used it only as a problem book, and just bought another book. Read the text, annotated it, ask questions while reading, ask the instructor, and never use a solutions manual.

    Rote memorization is bad in the long run. As you are aware, it is hard to recall this information in the long term. There is a difference between memorizing and actually wanting to understand something.

    Our school required Stewart Calculus, i ditched it and used 2 book instead for the whole calculus series. The books were Serge Lang: Calculus and Thomas Calculus with Analytical Geometry 3rd ed.

    Taking harder professors then easy A professors helps in the long run. Sure you may get a c, b, or even repeat. However, the knowledge gained while being intellectually challenged is vastly superior to a superficial A.

    I found Calculus 2 trivial after having instructors from hell in all my previous math courses. Not that much to learn conceptually in calculus 2, just a few proofs, what it means to parameterize a function, more integration techniques, and sequences/series.

    Calculus 1 is a lot more difficult, because you are introduced to more abstract ideas, i.e., limits, continuity, integrals.

    Cal 3 is the easiest of the calculus classes, however the drawing portions can be hard. I really sucked at drawing. Need to practice this section more.

    I would review all the calculus 1 material from page 1, solving the problems, and never looking at solutions manual. That is your best chance for success.
  4. Jan 1, 2016 #3
    Also, what an instructor told me during calculus 1. Was do not do every problem in the book. There is to many and not enough time, thanks to these frankenstein 1300 page books. Do a few problems of each type, which vary in difficulty. If you struggle with some of these, go back and read attempt again. Then tackle the interesting problems. It is better to spend your time solving a few good problems then spending the same time solving 40 problems that are similar and only differ by numbers given.

    Although, some sections require you to do every problem. Ie., when you learn techniques that are computational. For example, differentiation, integration, and Laplace Transformation ( you will see this in differential equations).
  5. Jan 2, 2016 #4


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    By "remembering", what're you referring to? Remember steps needed to solve a problem? Most intro calculus courses are computation based, i.e. whats the derivative/integral of "f".

    Don't have any advice here, and I won't lie, I used a certain website to find the easiest GE professors and took them. I couldn't care less about "Art of the Renaissance Period." I'm just not cultured enough.

    Not a whole lot wrong with that when you're starting off. Practicing computation is important for these classes. As long as you've retained what the general ideas are, delta method, fundamental theorems, etc, or at least could reference them quickly and say "Oh now I remember." Then you're probably at the same level as your peers in the same classes. The only sure fire way to retain something long term is to keep using it.

    That said, I've always compartmentalized the math in my head under three levels:

    1. Pure computation, using "rules" and "tricks" to solve problems. Algorithm based as you've stated.
    2. The significance of whats happening, why is ##\int \sin{(x)}dx = -cos(x) + C##, why're there multiple solutions for some problems, what is the "C", whats continuous actually mean, etc. This is the area you'll find most useful for intro physics, and setting up word problems (related rates, moments, yaddayadda) - in my opinion. You're looking more at the physical significance, if there is any, and whats going on in the problem.
    3. Abstract thought and logic. Proving that a function is continuous, that the fundamental theorem of calculus is valid, epsilon delta proofs, etc.

    Obviously, this is just a gross over simplification.

    If in your exercises you aren't stopping to think about 2, maybe you should stop for a second and analyze why the result you get is valid. What's the significance, what is it telling you, etc. I think 2 became much clearer after taking intro physics, you really start to apply the mathematics. You've probably not even been introduced to 3. yet, don't worry, it's coming. =)

    Do problems until you're blue in the face, take a look at the higher numbered problems. Most of the defacto intro calculus books put the interesting problems near the end.
  6. Jan 2, 2016 #5


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    It sounds like you have a studying mindset as opposed to a teaching mindset. Think about how differently you'd approach these tasks: (1) getting an A on a test and (2) teaching the material to a class. Try to approach the material using the latter mindset. It could help to work with others, so you can explain concepts to each other to test your understanding.
  7. Jan 2, 2016 #6


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    The simplest way to answer the topic question is , "Use it or lose it".

    Add to that, review the topics or subjects every few months or every few years.
  8. Jan 2, 2016 #7
    It may just be your mentality when you're practicing the problems, because the main idea of practicing them is so that the general procedure and computations are ingrained to the point where you can most importantly recognize that the problem calls for such a concept that through sheer practice and dedication to the material, you can almost instantly pull from the back of your mind and apply it to the task at hand. I had this problem initially in the semester where I would solve problems with my TA mainly to see that I can "get all my points" but I always felt annoyed when he would immediately ask after I solve a problem these two questions: (I had to answer these verbally; if anything he only commented on my board work if I was doing a algebra heavy problem)

    - what did you do?
    - why did you do that?

    At first, I just answered his questions over time mainly so I can see if I was doing the problems correctly. Over time however, I realized that my problem solving skills under anxiety heavy situations (our calculus exams were always at least 10-15 problems to be fully done in 50 minutes) had immensely improved.

    I realized that it wasn't just doing problems and checking to see if I got the right answer that helped me improve my overall knowledge(and fascination) with the material. It was me applying all of my attention and focus on each problem I worked through that made my brain retain - not the formulas, or 1-2-3 algorithms to problems - but the entire concept, the intuition behind it, and it's capabilities/applications to solving problems in general. It didn't just help me know how to solve tough integrals or tricky looking derivatives, but it taught my brain how to problem solve.

    So here's my two cents:
    Work through your practice problems at a good pace: not to fast that you can't remember what you actually did in that previous problem to solve the current problem, but not too slow that you're wasting your time on something you truly understand in its current entirety.

    Talk about/Work through these concepts with peers! Working with people always helps you understand what you need to be able to do, and what you may still be slacking at. By forcing yourself to be able to explain to someone how to solve a problem, you're showing yourself/reinforcing the concept into your brain rigorously enough where memorization won't really be the issue anymore; it'll just be the method of application.

    For stuff where you have no choice BUT to memorize them(stuff like volume formulas and stuff you never really cared to remember), use flash cards like you're already doing but think about something that you can connect that difficult term with so that when you have to come up with it, you can instantly remember that quick shortcut you're bound to remember and use that to recall that formula.

    Lastly, you can't learn anything if you aren't interested in the slightest detail of it! I'm not saying you can only learn the material of stuff you specifically like to learn about/find interesting as a whole, but you have to make it interesting to yourself in some way unique (or just FAMILIAR (too lazy to bold haha)) to yourself. Ask yourself this:
    Would you rather learn/remember something interesting about -insert a hobby of yours here- ?
    Or would you rather learn/want to remember something about one of your grandmas really lame hobbies?
    I could go on and on about this, but I came to realize this once I read the preface of a intro to physics book online from the free physics book thread by a professor at Duke University. It's a great read in general if you're interested in some of the subject matter presented in his textbooks.
    Our brain works through connections; retaining information calls for the need of strong connections.

    TL;DR: I've had this same exact experience(took intro to Calculus, Physics: Mechanics, and Intro to Chemistry all at once:nb) for my first semester of University) recently, so read this chapter of text when you have some time to sit down and indulge your brain with some introspection :).
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