How to Ace Number Theory (and Other Proof-Based Classes)
This Summer I took a course on number theory. The course was challenging but with the right study techniques, I was able to land an A in the class (>= 94%). More broadly, I want to discuss the correct study approach for performing well in proof based math classes as they differ significantly from application based math courses. The American primary school system focuses almost exclusively on applied math concepts. This leaves most college students unprepared for the absolute onslaught of proofs involved in an undergraduate math degree. A typical course in number theory covers
- prime numbers
- Diophantine equations
- Pythagorean triplets
- linear and quadratic congruences
- theorems of Euclid, Euler, Fermat, and Wilson
- continued fractions
- basic cryptography
The Four Stages of Competence
There are four stages of competence in psychology:
- Unconscious Incompetence: you don’t know what you don’t know
- Conscious Incompetence: you know what you don’t know
- Conscious Competence: you somewhat know the material
- Unconscious Competence: you effortlessly know the material
To perform well in a math course, you must reach the unconscious competence stage. You know you have reached unconscious competence when solving any problem from the course is trivial. I want to emphasize that the key to reaching this level of mastery does not lie in increasing your study time, but concentrating your time on the correct tasks in the correct order. In all likelihood, implementing these study strategies will decrease your total time spent studying as well as your stress around exam time. The rest of this post will focus on how to efficiently reach unconscious competence.
For any math class, taking notes is pretty straightforward. Simply copy everything the professor writes on the board. When you have questions write the question next to the related notes, then raise your hand and ask it. When you receive the answer, write the answer close to where you wrote the question. Your notes will now look like an annotated version of the professor’s lecture.
Strategy 0: Review Fundamentals In The First Week
At the beginning of a proof-based math course it pays dividends to review fundamental concepts. These include: quadratic expansions, factorizations, cubic expansions, common sequences, series, sums and the binomial theorem. Having these concepts fresh in your mind will accelerate the speed at which you digest the information covered in the course and aid in solving tricky proofs.
Strategy 1: Daily Lecture-Note Worship
One of the toughest components of taking proof based math courses is the sheer volume of information you have to process. In applied math classes, you are commonly shown a specific problem solving algorithm, and are merely tasked with repeating it across a wide variety of cases. In proof based classes, you are introduced to a massive library of axioms, theorems, and definitions; all of which you will need to have mastered and on hand so you can solve increasingly complex problems. Oftentimes you will need to devote significant thought to a problem before a solution becomes apparent.
After each lecture, go through your notes. Assure that if you were tasked with giving the same lecture, you would be able to do so flawlessly. This exercise will expose subtle misconceptions acquired during class. Clearing these doubts immediately will save you time and stress. In practice, this looks like glancing/scanning a page from your notes, re-writing it from memory while explaining the concepts out-loud as if you were the professor. You will inevitably slip up on occasion. When you do; this is an opportunity to close a gap in your knowledge.
The brain can be deceiving. It is easy to read a math proof and think to oneself “this makes sense.” But, the true test of understanding is being able to replicate it from scratch.
You glance at a page of your lecture notes and start deriving all the theorems from the page. On one particular theorem you get stuck. If you spend a non-trivial time trying to finish the proof, you should refer back to your notes and then resume. You repeat this process until you can derive and explain every theorem from the page without pausing or referring back to your notes.
Use this checklist to make sure you’ve implemented this correctly.
- I can recall all definitions, theorems, axioms, and proofs covered from lecture
- I can solve any example problem covered in class effortlessly
- I can comfortably talk about the main ideas covered in the lecture
Completing this step should place you comfortably at the conscious competence level. The proofs from your homework assignments will involve similar techniques to the theorems and examples introduced in class. By reviewing your lecture before starting your homework, you save yourself time spent flipping back and forth between notes while trying to solve problems. Additionally, you are reducing your exam preparation time down the road.
Strategy 2: Chunking
The idea of chunking is that by mastering simpler problem solving patterns first, you free up working memory to focus on more complex patterns. This concept is intuitive. Perhaps at some point in your math journey, solving a simple linear equation was extremely challenging. Then shortly after mastering these patterns, you moved onto systems of equations, then quadratic equations, and systems of quadratic equations. Using the same techniques required to solve simple linear equations but chaining them together with additional concepts.
Had you only vaguely learned solving linear equations, the subsequent topics would require far more working memory, and the total cognitive load of the task at hand would be overwhelming. How can you chunk in practice?
I use an acronym: DRA (derive, recite, apply). When you learn a new theorem; derive it from scratch. Once you understand the intuition behind it; repeat the proof/theorem until you’ve memorized it. Finally, use the theorem in specific applications (typically given in homework problems). This will ensure you have ‘chunked’ the concept so that it does not require additional working memory and becomes second nature. Undoubtedly, there will be further theorems introduced that rely on more earlier theorems taught in class. Such classes often build an axiomatic pyramid that can lose students who have not invested the time to strengthen their foundation.
Strategy 3: Two-Pass Technique
You will be given homework assignments. During the first pass-through, you will devote significant chunks of time to thinking through each problem carefully. Once you complete the assignment, quickly solve every problem once more. You will find that the second pass-through is faster. Additionally, you will make the transition from conscious competence to unconscious competence.
- I can solve any question from the assignment at lightning speed
- I have a good understanding of each problem and how it relates to the topics covered in class
- I can recall all of the definitions, theorems, and axioms the homework has covered
Strategy 4: Hard Questions Only (exam prep)
There are two variables that will effect your exam performance: accuracy and speed. You have already done every homework assignment twice; mastered all the content in your lectures as you learned it. There isn’t much left to do. Most students study lightly throughout the semester and study intensively around exam time. We have done the inverse: mastered every topic as we learned it. Before the exam, pick a few of the hardest questions from your homework assignments and lectures. Make sure you are able to solve these problems at lightning speed. Studying base-case or easy problems is a waste of time. The time spent before exams should be focused on making sure there aren’t any subtle misconceptions lingering in your mind.
Strategy 5: Exam Performance
Even if you prepare rigorously for the exam, there is still a chance you will see problems you cannot immediately solve. This is normal as some of the problems might be tricky and require deep thought/insight. If you come across such a problem, do not linger on it. Read it, think about it briefly, and proceed to solve other problems. Once all questions you are able to immediately solve are complete, return to the harder questions.
As long as you follow the steps covered in this article, there shouldn’t be any reason why you can’t land an A in number theory or any proof-based class for that matter. It is important to note that cramming is not an option for these classes. In the past you may have succeeded in cramming for Calculus, but proof-based classes are insight driven. As such, the concepts require time for rumination.