Proven Studying Habits for a Limited Memory?

Question

I learned Calculus many years ago and at the time I thought that I knew it well. (i.e. got good grades, was employed as a tutor, etc) I am now going back and studying Calculus again with the hopes of moving onto to more difficult subjects such as Topology and Differential Geometry. I have a limited memory and average intelligence, but I want to understand Calculus very deeply and I am very motivated. By understanding I mean that I would like to be able to recall important definitions and concepts and be able to prove the theorems and answer >90% of the exercises in a text. I am using Apostle's Calculus and focusing on understanding and being able to reproduce the proofs in each section before moving on as well as at least attempting every problem at the end of the sections. What methods of studying are the most productive to achieve understanding as I have defined it above? I am curious because I am finding that I don't remember many of the central concepts from Calculus that I supposedly learned many years ago, and at the time I would have had great confidence that I would remember these ideas into the future. Has anyone had a similar experience? Thanks.

Answer 1

I'm going through Apostol's calculus also. I think his presentation does much more to promote deep understanding than many of the multi-color cookbooks that have taken over in American high schools.

Make it Stick is worth a read if you are concerned with your retention. It summarizes recent findings in learning research.

The most interesting part of the book was its suggestion that some traditional study practices are worse than useless because they give learners false confidence about what they know. The worst in this regard is the strategy of rereading highlighted passages to study. A runner-up is copying lecture notes by hand into a "study guide". Both study strategies, the book suggests, make learners mistake familiarity with material for understanding of it.

Much more effective than rereading or copying is engaging in "retrieval practice". This boils down to quizzing yourself informally on material that your brain has had some time to forget.

For mathematics, attempt a selection of the proofs and exercises that you completed a week ago. If you are in chapter 5 or 6, can you still explain the foundational concepts--like why is the archimedean property of the real numbers a consequence of the least upper-bound axiom?

Constant reviewing of older unrelated content takes much more effort and time than only doing exercises and recreating the proofs right after you finish reading the most recent section. But effortful learning is what will stick with you over the years.

I like this quote about the struggle of understanding mathematics from Greg Egan's novel Diaspora:

The only way to grasp a mathematical concept was to see it in a multitude of different contexts, think through dozens of specific examples, and find at least two or three metaphors to power intuitive speculations. Understanding an idea meant entangling it so thoroughly with all the other symbols in your mind that it changed the way you thought about everything.

Answer 2

It is natural that you have forgotten many of the central concepts.

I have been working for >10 years and i have just recently come back to the university and i have also forgotten many important concepts. But i found it quite easy to pick up calculus, linear algebra and the rest of the basic courses.

I spent some time studying the basics on my own and then joined more advanced courses. But i choose to just study one new course at the time and simultaneously keep rehearsing the basics.

In a couple of months i think im ready to study at full pace. Topology, like you mention, is my next course in a couple of weeks.

Answer 3

Knowing what you know (sometimes called metacognition) is important as you're trying to learn things.

I recommend working problems (such as those on this site) to get practice. If you want to test yourself, many college calculus instructors post old exams on their web page.

Answer 4

There is a limited amount of computational material that requires the use of memory until it is automated by practice. Things like basic estimation techniques and nonobvious algebraic rules for differentiation and integration. Also a few special functions and important infinite series that simply "are what they are", and whose properties one gets accustomed to over time. For those things, time, exposure and practice are the only road to internalizing the ideas, but some books are far better than others.

Other than that, calculus relies on a small number of concepts that are relatively easy to understand and accept without memorization, such as integral = area, the maximum of a function occuring at places where the graph is horizontal, and the inverse relation between differentiation and integration. This is obscured in practice by the huge number of calculation problems that expose students' level of background in algebra, geometry and trigonometry, as tripwires separate from the new calculus material.