Title: How to Succeed in College Mathematics, by Richard M. Dahlke, PhD.
An excellent guidebook of interest to anyone studying mathematics at the post-secondary level. Written by Professor Richard M. Dahlke, a U.S.-based professor of mathematics, How to Succeed in College Mathematics gives useful advice on topics such as improving academic performance, keeping a healthy attitude towards studying, and how to improve problem-solving skills.
An illustration of a plane in 3D-space passing through the origin. A plane such as this can be recognized as a vector space, a topic commonly studied by mathematics students at the university level in courses on linear algebra.
The table of contents is quite long, with 27 chapters. Here is a breakdown of the main topics covered throughout the book:
- What mathematics is, and what to expect from studying it.
- How to manage time as a math student.
- What it means to understand math, and how to improve your personal learning.
- How to have a good attitude about learning, and your ability to succeed.
- Student vs. instructor responsibilities.
- How to increase problem-solving ability.
Get Better at Learning Mathematics
I found that some of the best, and most actionable, advice was on the subject of improving your academic performance. Something as simple as reading the textbook before class is often overlooked by hurried students, but Dahlke makes a strong case for making the time to do it. “If you read content before class that will be developed in class, you will gain far more when in class. You can focus on what you couldn’t understand from your reading by paying closer attention when it is presented in class, and by being prepared to ask questions,” Dahlke writes. He presents a simple framework, called SQ3R, to help you read a textbook more effectively:
- Survey (S): skim through the section, reading the titles, headings, introduction, charts, examples and other high-level features.
- Question (Q): write questions based on what you surveyed or what you’ve seen in class so far.
- Read (R): read carefully. Reading mathematics is not like reading a novel, Dahlke writes. Another point is made by Sheldon Axler, author of Linear Algebra Done Right: “If you zip through a page in less than an hour, you are probably going too fast. When you encounter the phrase ‘as you should verify’, you should indeed do the verification, which will usually require some writing on your part. When steps are left out, you need to supply the missing pieces. You should ponder and internalize each definition. For each theorem, you should seek examples to show why each hypothesis is necessary,” Axler writes.
- Recite (R): After finding the answer to one of the questions you wrote down, you should reinforce it by reciting the answer.
- Record (R): Writing down your answer will further help with remembering. “We often think we understand something by quickly going over it in our mind; but when we try to express our thoughts in writing we realize they are not as well formed as we thought,” Dahlke writes. The written answers should not be detailed, as extensive writing interrupts the reading process. Instead, Dahlke suggests writing cue words or phrases. (It is at this point that I personally would suggest following along on examples and proofs with pencil and paper, as this helps ensure you understand the fine details. It is all too easy to simply skim a proof and believe you understand it.)
- Review (R): You forget about half of what you’ve read very soon after reading it, so you should immediately review each question to see if you can answer it without looking at the cues.
This advice is very similar to that of a popular YouTube video by learning coach Matt DiMaio, which you can watch below.
Not only does Dahlke convince you that you should spend more time reading mathematics, he also makes the case that writing about mathematics can help you learn more. This is because writing about mathematics forces you to research, consolidate and internalize mathematical concepts in order to present them to others. This process can expose misunderstandings and strengthen higher-order thought processes. Dahlke states that writing exercises should be more regularly assigned in mathematics courses, and suggests the student make their own if their courses do not provide any. “Nothing I did in my teaching has equipped my students to understand mathematics as much as requiring them to write about mathematics,” Dahlke writes.
It is a well-known fact that you learn better when you have a good attitude about the subject and are interested in it. One way to maintain a good attitude, in my view, is to be aware of the benefits of studying mathematics. Even one course in the subject can be a valuable experience, Dahlke writes. “Mathematics is not the only discipline in which you can improve general learning skills. However, mathematics is viewed by many as the best discipline for improving certain ones. For example, it is second-to-none in improving analytical thinking skills.” Learning mathematics also equips you with skills that make it easier to learn practically anything else, too. Particularly at higher levels of mathematics, the focus shifts away from the instructor as a learning guide and more towards the student. Math texts become denser, and the student is expected to fill in many more details themselves (for example, in the textbook for one of my math courses this semester, more than half of the proofs were left to the reader to do as exercises). Therefore, a graduated mathematician is generally one who is adept at self-teaching. As Dahlke writes, math degrees don’t come with a personal professor you can pull out of your pocket every time you require help or guidance.
Make Problem-Solving The Focus
The focus of anyone’s study of mathematics ought to be on solving problems, Dahlke writes. “Mathematics will become more exciting to you … What motivates you to keep working on a problem is the anticipation that the solution will eventually evolve.” Solving problems helps you learn because you get to apply the theorems yourself, while gaining insight into how they work. This builds up your mathematical background, hence enabling you to tackle more difficult problems later down the road. An appropriate quote by René Descartes comes to mind: “Each problem that I solved became a rule, which served afterwards to solve other problems.”
Dahlke provides a general method for approaching problem-solving, but I will not summarize it here. In fact, he simply states in detail a method previously described by George Polya, who wrote How to Solve It in 1945. You can read about Polya’s method here…
The discussion on how to solve problems is useful, but may not be new to well-read mathematics students. How to Solve It is a famous and frequently-recommended book in the field, and Polya’s ideas are also presented in James Stewart’s Calculus, one of the most commonly-used calculus textbooks in North America. What Dahlke does bring to the discussion, however, is information on what traits a good problem-solver has. For example, Dahlke writes that problem-solvers understand it is difficult and takes time. Few problems will give way on your first try, so perseverence is important. There is nothing wrong with setting a problem aside to allow it some time to percolate in your mind. It is remarkable how helpful a night’s rest can be to finally cracking a tough problem. Of course, to be successful in all of this, you have to try. Andrew Wiles, the mathematician who proved Fermat’s Last Theorem, once said: “However impenetrable it seems, if you don’t try it, then you can never do it.”
How to Succeed in College Mathematics is worth reading if you are, or plan to become, a mathematics major at university or college. I liked this book in particular because many of the suggestions within are genuinely useful and not all obvious. Even for the topics Dahlke covers which you are familiar with, he casts new light on the matter in a way that makes it worth hearing about again.