Advice from A+ Students

MATH 155, Calculus II for the Biological Sciences, Spring 2012

“I was asked by Professor Jedwab to provide some advice and my perspective on how to succeed in MATH 155. Unfortunately, I don't have any study strategies that all of you are not already familiar with. Mathematics courses are not intended to be easy A's, but this being said, with persistent effort and care throughout the semester, you can feel confident walking into the final.

Throughout the entire semester, I dedicated an hour everyday during the week towards calculus. I reviewed my lecture notes and worked through all in-class examples without looking at the solutions. With the remaining time, I worked through the textbook questions that were relevant to the lecture material. I cannot stress how important it is to work through these questions. By working through these questions, this confirmed that I had a comprehensive understanding of the material, and not just the ability to memorize the solution methods that we went over in class. Many students neglect to do these questions, but keep in mind that there was reason those specific questions were selected! Every weekend, I spent an hour to start on my instructor assignments early. I ALWAYS made my best attempt at the assignment questions and wrote out an attempt at each question BEFORE stepping foot into the workshop. It was too tempting to be able to ask the TAs the correct method without first trying to figure it out yourself. I believe that we all learn the best by understanding our mistakes. Consistent hard work throughout the semester, with just one hour a day, will be must more effective than the five consecutive nights of cramming before the final exam. For approaching the course material, I found describing theorems by drawing a graph demonstrating the definition prove to be most helpful. It forced me to think critically about the theorem, its restrictions and when they do or do not apply. I was able to understand the theorem as opposed to simply reciting them word for word. Throughout the semester, after every new topic, I would spent a few minutes trying to organize and link all the topics of the course together. I made summary lists, that I added to throughout the semester, that drew connections between the topics. As we learn new theorems and methods, they are just mechanics that can be integrated together. And when you are able to form these connections, all the material throughout the semester, cumulatively, will seem less daunting.

There are no secrets to succeeding in this course. I definitely do not consider myself to be any brighter than the average calculus student in your class. It is due to consistent study habits and efforts, that we are all capable of, that enables success. Best of luck this semester!”

MACM 201, Discrete Mathematics II, Spring 2011

“Like any math course, the key to avoid significant amounts of cramming before exams is to distribute the studying. That is, treat the weekly quizzes seriously. Set aside time to answer each assigned question as completely and concisely as possible. You'll find that you can answer all but around 3 questions by yourself without difficulty. These 3 questions are almost certainly going to be on the quiz. Make sure you take these questions to the Algebra Workshop and get a TA to explain it to you completely. The TA's are there for a reason; don't be afraid to spend as much time as you need until you can explain the solution to someone else without any difficulty. As for lectures, it will help a lot if you read the corresponding chapter in the textbook beforehand. MACM 201 is a difficult course; trust me when I say that you won't understand the material the first time you look at it. This is especially true for learning the graph theory proofs. When it comes to memorizing definitions and proofs, try rewriting them in your own words and checking with a TA to see if they're still valid. The act of trying to simplify a definition or proof will help you understand the purpose of every word and sentence. It tells you which words and phrases are strictly necessary and which aren't.

As for the exam itself, the questions will be of two different types: mechanical questions and creative questions. Mechanical questions are the types of questions that you're used to doing from your homework. They can be answered without really understanding what you're doing; these questions simply test your ability to follow certain rules like a computer. Examples of this type include definitions of vocabulary, Principle of Inclusion-Exclusion, finding the coefficient of xn, solving recurrence relations once they've been set up, and applying Dijsktra/Kruskal/Prim's algorithms to a graph. You should be able to do these questions in your sleep; on an exam, when time is limited, these are the questions you do first. Creative questions, on the other hand, really test your understanding. Finding the answer for these types of questions is rarely straightforward; you'll have to play around with the question and try to solve it from different viewpoints. Examples of this type include setting up (but not solving) generating functions and recurrence relations, as well as pretty much all of graph theory. All you can really do to prepare for these questions is practice a lot of past MACM 201 exams and develop general heuristics for the questions. For example, although Hamilton paths don't have straightforward necessary/sufficient conditions like Euler circuits do, there are a set of general guidelines that you can follow in order to find a Hamilton path in a given graph. Although it can all seem overwhelming, making it through MACM 201 will be a really rewarding experience; good luck!”

MATH 341, Algebra III: Groups, Spring 2014

“Welcome to MATH 341! Abstract algebra is a super cool topic to study; you won't get bored. As an upper level math course, the class size is probably going to be fairly small compared to your usual introductory math courses (I'm looking at you, MATH 151 and MACM 101). Consequently, you should try to interact with Jonathan as much as possible during lecture. Jonathan's style of teaching is great because after introducing a new concept, he'll illustrate how to apply the concept with an exercise (often taken straight from the textbook). This is your opportunity to make some guesses and see if you've been paying attention to what's been going on in lecture so far! Don't be afraid to raise you hand and make mistakes; I promise Jonathan won't bite if you mess up. You paid a lot to take this course; you might as well get your money's worth by trying to get out as much as you can from the lectures. The same thing goes for the tutorials: show up for all of them and interact with the TA. Think out loud and ask questions! Discuss your ideas with some friends!

As for studying for the exams, I suggest that you practice a LOT of exercises. Gallian's textbook is great and filled with tons of interesting problems that, at first, seem really difficult. However, once you get the idea, these exercises often have a short, elegant proof. In fact (spoiler alert!), Jonathan takes most of the problems from his exams directly from either the chapter exercises or the supplementary exercises of the textbook (with perhaps some minor modifications). Furthermore, I recommend that you develop a library of example groups that satisfy as well as violate various properties as you encounter new groups/properties throughout the course. Can you name a group that is not abelian? Can you think of an infinite group such that each one of its elements has finite order? Can you think of two groups that have the same order but are not isomorphic? Can you name a group that doesn't satisfy the converse of Lagrange's Theorem? Can you give an example of a subgroup that is not normal in some bigger group? It is often very illuminating if you can construct an example that illustrates why some claim doesn't hold in general or why such a strong hypothesis is required in some theorem. It is especially useful for the final exam, where you are asked to either prove or disprove some claim; your first step should be to quickly check if the claim holds for some of the more standard example groups in your library.

Good luck with the course! You'll have loads of fun, I promise. =]”

MATH 480W (formerly MATH 370W), The Art and Craft of Problem Solving, Fall 2014

“Dear MATH 370W Student,

Welcome to the most satisfying course of your undergraduate career. I hope you will enjoy it as much as I did. The fact that you are enrolled in this course means that you are more than capable of achieving an excellent grade. Being that MATH370 is all about solving interesting problems; my advice is focused on the process of approaching and solving difficult problems, as well as tips I wish I had going into this course.

There is no concrete way to solve the problems posed in this course. Finding a solution is difficult, and the only way to improve is to solve more problems and work through examples. Many of the problems have their own quirks and require special “tricks” to solve, which you'll only understand by tackling more and more problems.

That being said, the real meat of this course is in writing and presenting your solutions in a coherent manner. Much like writing an essay, solving problems is an iterative process. The following is one way to approach problem solving:

  1. FInd a solution to the problem (usually a “trick” or lateral thinking).
  2. Sketch an overall strategy that will solve the problem, omitting details.
  3. Write a first draft. The important part is writing it, not having it 100% correct on the first try.
  4. Realize that almost everything you wrote was terrible, and rewrite it.
  5. GOTO 4.
The work is split 50/50 between seeing the “trick(s)” that solve(s) the problem, and convincing someone (i.e. the prof) that your solution is correct. Plan your work accordingly.

My advice on solving the problems focuses more on the way you approach the problem, arrive at the solution, and less on the actual solution itself (i.e. Was the problem secretly basic linear algebra? Or was the polynomial question really about counting?). The problems that you will see are so varied that often you will seldom not see the same “type” of problem twice. Gaining insight on the problem solving process is incredibly valuable, and ultimately the this will be the most important skill that you will undoubtedly want to take away from this course.

My next piece of advice involves always being in the zone. The assignments are on a weekly basis and are tough, so stay on top of them. Look at the problems ASAP. Even if you don't attempt to solve them immediately, just having them in the back of your mind will make all the difference when you want to write your final solution.

One last piece of advice is to look at old Putnam exams. Spending 3 hours at a time working through half a Putnam can really help the problem solving process sink in — don't look at the answers until you've finished!”