Retain problems and combat regression in learning

Question

Regressive Learning

It's a really stressful situation. I can achieve but not retain expertise in maths problems.

---

History

6 months back, I studied integration in Calculus at college. I learnt it all by myself as I was didn't attended college for days on end. I learnt it with great level of comprehension. In the beginning I solved all the exercises one by one, dealing with all kinds of varied techniques required by the various questions.

Over 6 months, I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones). This mastery was achieved by reviewing and re-solving the exercises once a month for 5 months.

I used to do all this timed practice like a race-against-time using a 1-minute hourglass.

---

Amnesia

Now, there came a Gap of 3 months between my college ending and university starting. Now I can't do any of the questions of integration except the simplest ones and even they would take me well over a minute.

This proves the point that "practice" and "understanding" is not all to learning maths. There must be something else too that teachers and students are overlooking.

---

Future Concerns

Now, unlike college, there will be 80 exercises instead of 8 at the university-level. What do I do, I cannot just practice maths-problems again and again and again every month & then fall & then go from prince to pauper after a couple of months gap in review (in terms of exam preparation).There are a lot of twisted techniques, a lot of rules and methods in solving problems just like a toolbox.

My question is this: Why does this happen to me & What can I do ?

Any help in this regard is highly appreciated. Moreover, I believe I have come to the right place as the people here can surely help with their knowledge of mathematics, learning , education and pedagogy.

---

Edit

A Note on Problem-Solving

"Understanding" means "knowing what specific techniques or some combinations of them have to be applied in a specific orderly manner to solve the integrals".

Mathematical problem-solving is akin to programming, what really matters is proper, orderly application of logic and procedural operations.

It's not so much about "conceptual enlightenment" & "intuition" & "deep-learning".

Since we are humans, we cannot store all these rules of if X, then Y, else Z. Neither can we analyse all multiple possible approaches (brute-force method) without expending considerable time and effort. That, my friends, is The Problem.

Answer 1

"Solving (routine) problems fast" is a useful skill, but very far from what you really need. You have to be proficient in reading, understanding and criticizing proofs, come up with your own proofs. Be able to plan (and execute the plan) to attack ill-defined problems. Use tools, like a computer algebra system, Google or math.SE to ease over routine stuff/stuff you forgot/never learned. Learn to learn efficiently. How to select what to study, find material on your level that agrees with your style.

Answer 2

I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones).

It sounds like you were doing very simple, plug-and-chug problems. For example, evaluate the integral from x = -3 to x = 4 of (4x³+3x²)dx.

Problems that require detailed proofs, or examination of multiple aspects of a situation, require more than two minutes of work. A "solid understanding" is evidenced by the ability to:

1. explain and verify the inner workings of a technology,
2. notice the limitations of the technology, and
3. use the technology to solve problems one encounters.

Mathematical proofs are explanations and verifications. They also provide many opportunities to notice the limitations of various theorems. Can you re-construct a general proof for how a derivative is calculated? For the derivative of a polynomial term? For converting between various ways of expressing e? For the chain rule, product rule, and quotient rule? If you never were able to do these things, I doubt that you had a "solid understanding" of calculus.

Applications of mathematics to physics problems, engineering problems, and various real-world problems also provide opportunities to notice the limitations of various mathematical techniques.