The most difficult part of a math class is the beginning. The reason for this is that the challenges faced at the beginning of a class are more difficult to solve than those at the end of the course. Towards the middle and end of a course, the focus is mostly on completing assignments and studying for tests. These are certainly not always easy tasks, but they are clear objectives with straightforward responses. In order to get better results on assignments and tests the solution is often to apply more effort: more time set aside for doing the homework, more time working with groups and going to office hours, and more time spent studying the material. When a math class begins, we are prone get distracted, tune out, or get lost in a new framework. These are problems more or less rooted in our psychology, and not as easy to overcome through sheer brute force. The same could be said for reading through math textbooks, even for self-study.
At the beginning of a course, the students are often in many different places, academically speaking. Students will have a variety of backgrounds with different courses already taken and different levels of exposure to the material. If the course has a direct prerequisite course, some students may have taken it recently while others may have taken it years ago. This is especially true of upper division mathematics courses, which do not necessarily have an overall linear ordering to them. For this reason, most courses (and most textbooks) have some time and space set out in the beginning for review and a gentle introduction to the material.
For well prepared students, this introductory period is a prime risk for getting distracted. Distractions can some in many forms. There are external distractions, such as tv shows, video games, or other hobbies. There are also ‘internal’ distractions (meaning, in this case, math related) such as reading ahead in the textbook or spending more time on other math not related to the class at hand. Personally, I am often vulnerable to getting distracted by other, more exciting areas of math when one of my courses feels like it is going too slow with material I have already seen.
The problem with getting distracted, even when the material seems basic, is that eventually the material will transition to not being basic and that change can be hard to notice if you are not paying attention. The point in a course where the content changes from review to new material may not be pointed out by the professor, and it is different for every student. In fact, it may be wrong to think of a discrete change from introductory material to core material. If you have already seen some of the material before, there will be some that you know well, and some topics that you may have seen but still need to solidify. As an example, when I began my first abstract algebra course, I had already seen some group theory before. I was comfortable with the basic definitions of a group and had seen other topics like subgroups, cyclic groups, and even quotient groups but I was much less familiar with the latter topics. There was no single point where the course shifted from material that I knew to material that I did not. It was a gradual slide from content I knew well to topics I was not as familiar with. Had I just tuned out from the beginning, I might not have noticed where the course was catching up to and surpassing my previous knowledge until it was too late.
The consequence of being distracted and falling behind in the beginning is that you will spend the rest of the course playing catch-up. You do not every want to be playing catch-up in a math course, especially an upper division one. My main strategy to avoid this is to embrace procrastination rather than procrastination. Instead of putting off what I consider ‘easy’ material until later, I make a conscious effort to do easy tasks first and get them done. Then if they really were easy I have them done and out of the way and can work on harder things, guilt free. If I was wrong and the material was really more difficult than I anticipated, I get that reality check early on and have plenty of time to get it done. Adopting this mindset helps me to view being over-prepared as an opportunity for success rather than an exercise in tedium. Another way to avoid getting distracted when things seem easy is to focus on helping any other students who may not find the content easy. This is also a great way to stay engaged and probably even learn new things from people’s questions.
Aside from the risk of getting distracted during review material, there is another reason why the beginning of a math course is the most difficult part of a course: you are being introduced to a new and unfamiliar framework. Many classes at the undergraduate level have some sort of main idea that is used when talking about everything else in the course. For a proofs course, it is sets. For a first abstract algebra course, it is groups. For linear algebra, it is linear maps. And so on. This framework defines the flavor of the course and determines how concepts are approached.
The first few weeks of a course can often be a mad dash to cover some review material and introduce students to this new framework as soon as possible. By the later parts of a course this central idea has become more familiar and students have standard strategies for attacking proofs. Near the end of a group theory course the material may be new, but the students are comfortable dealing with groups and better able to assimilate the new information.
Students who struggle at the beginning (whether due to the difficulty of the material or because they were distracted and tuned out) may miss not properly pick up what they need to know about the central framework of the course and will be playing catch up for the entire rest of the course. Surely, the same could be said about almost any part of the course which forms the basis for future topics, but the beginning is by far the most crucial since everything else rests upon it.
What can be done about this? My advice to everyone is to focus on maintaining discipline at the beginning of a class. If the material seems basic, look at that as an opportunity to excel rather than and excuse to slack off or get distracted. If you are having trouble getting a handle on the central ideas of the course seek help as soon as possible, either from your peers, professor/TAs, or online resources. The beginning of the course is the time to lay a foundation for success, not later on when exams loom.