Maximizing Learning: Top-Down Approach to Studying Concepts and Problems

[tronter]

Do any of you do the following: First study all of the concepts of a particular subject, and then do all the problems afterwards. For example, you first study all of the concepts of introductory topology. After doing this, you go back and do problems that you find interesting. I would think that this would maximize learning in the least amount of time? Opinions?

 

[tronter]

any opinions?

 

[Equilibrium]

good idea
how many problems have you done per leeson?

 

[Younglearner]

that is how I study for every one of my high school courses

 

[Younglearner]

has worked so far

 

[mathwonk]

i wrote my algebra book a little differently, on the assumption that it is hard to learn too many concepts without working with them.

so i sprinkled the problems throughout the section. i.e. i would introduce a concept, illustrate it, and then give some easy problems to reinforce the concept.

then i would prove a theorem about the concept and then give some harder problems requiring you to use the result just proved, and the to extend it using the arguments from the proof.

this is recommended for deep learning, but high school usually does not require much deep understanding, just trivial computations. depends somewhat on the high school though.

but AP level courses for example are usually pretty shallow, since they are aimed at a rather shallow test.

although my book was written for grad students, it actually may be accessible to good high school students, although it does assume you know what matrices are in the beginning and determinants, defining and treating them thoroughly later.

they are free on my webpage, as are my lower level algebra notes, you might take a look and see how they go. the method you are describing, learning all the concepts before doing any problems, seems hard to imagine being sufficient, if there a lot of concepts or difficult ones.

 

[tronter]

yeah, I guess your right. Its just that there are so many mathematical topics(probability theory, algebraic geometry, mathematical physics, etc..), and I want to study them all. But the approach I was describing would probably not be sufficient as you were saying.

 

[kdinser]

In any kind of problem solving class, math, physics, etc... I look briefly at the section headings, then I go right to the problems and try to solve them. I start at the most basic equations I know, and try to applie it to the problem. Once I understand the problem and I convince myself that I don't know enough to solve it, then I go back and read the chapter. I find that I can't get interested in the material until I know how it can be applied.

 

[mathwonk]

i am curious what you think of my graduate algebra book. give it a whirl, even if you are not a grad student.

 

[tronter]

mathwonk, your graduate algebra book is excellent. I see now what you mean by putting exercises as the reader reads along. What I am doing is reading through your whole book, as well as looking at the exercises. I think about the proofs/exercises and the methods to use to solve/prove them. I don't formally write down anything yet. After finishing your whole book, I have an idea to do many of the problems/proofs as well as learning all the concepts (maybe not an expert yet). Then I go back and formally do the proofs/problems. In this way, I have maximized learning in a least amount of time, and the problems/proofs because easier to solve (e.g. ex 7 page 10). If my grandmother asks me about a concept introduced later on in the book, I won't have go and study it, because I have studied the whole book. I have some idea as to go about solving it. So by studying math in this way, I can study more topics in the same amount of time. In other words, I separate studying concepts (still look and think about problems) and doing problems/proofs. Is this method still inefficient?

 

[neurocomp2003]

i like to study that way ...have a quick reference...and then go through the problems ...unfortunately most text are written so that you have to go through the proofs and don't have all the theory on one page.
Then again it could be designed to make the student do it themselve.s

 

[Pythagorean]

i wrote my algebra book a little differently, on the assumption that it is hard to learn too many concepts without working with them.

so i sprinkled the problems throughout the section. i.e. i would introduce a concept, illustrate it, and then give some easy problems to reinforce the concept.

then i would prove a theorem about the concept and then give some harder problems requiring you to use the result just proved, and the to extend it using the arguments from the proof.

this is recommended for deep learning, but high school usually does not require much deep understanding, just trivial computations. depends somewhat on the high school though.

but AP level courses for example are usually pretty shallow, since they are aimed at a rather shallow test.

although my book was written for grad students, it actually may be accessible to good high school students, although it does assume you know what matrices are in the beginning and determinants, defining and treating them thoroughly later.

they are free on my webpage, as are my lower level algebra notes, you might take a look and see how they go. the method you are describing, learning all the concepts before doing any problems, seems hard to imagine being sufficient, if there a lot of concepts or difficult ones.

This reminds me of how Griffiths did Electrodynamics and it was the easiest book for digestion I've had so far (though one could always use more variety of examples, but there's only finite space in a textbook).

 

[tronter]

mathwonk, do you think my learning method is feasible for studying your book?

 

[ice109]

i am curious what you think of my graduate algebra book. give it a whirl, even if you are not a grad student.

graduate algebra book? as in undergrad or actually graduate