How to study Maths Effectively?

In my day-to-day interaction with the students I teach, both Maths majors and Science and Engineering majors offering Maths courses, I am often asked the question:”Sir you keep saying we are not studying hard enough, how does one really study maths effectively”? The answer to this question is not as straight forward as one may think, since it comes from students from different backgrounds. One main observation is that, most students don’t think they even need to attend the lectures, talkless of reading their maths lecture notes and textbooks. That is why they are always surprised when in an exam or a test I ask questions on definitions of some concepts like this: “What do you understand by the following concepts: a) a function, b) domain of definition of a function, c) a continuous function at a point, d) an open ball, e) a convex set, etc.” In such a question I want to know the students’ understanding of the concept studied. For me it’s important to know how to define those terms. If not, how will a student be looking for the domain of definition of a function without being able to tell what it actually means. How can one be looking for what he/she can’t even define. Many students don’t think they have to read their maths lecture notes as they would read their biology notes, and that is the first mistake they make in studying the subject. In the sequel, we will try to highlight some key study guidelines that work for maths.
As a student offering maths as a major or offering some engineering maths courses as a science and engineering major, you have to first and foremost attend the Maths lectures. Most students run away from lectures for various reasons and expect to do well – I wonder how. After attending the class, the second most important thing to do is to read your maths lecture notes as you would read your biology, history, geography and physics notes. That’s is the basic thing to do while studying maths. Doing that will give you an idea of the concepts introduced in the notes, their definitions, examples of such and how their computations are done.

1. Tips for Maths majors

As a Maths major, after attending the lecture in class (which is very important),

• You have to read your notes each time you have a lecture, make sure you read what you did in class the same day before going to bed. While reading don’t try to understand everything that’s there, just read. As you read you will be remembering the things that the lecturer said, the explanations he/she gave, even the fun stories he/she told you during the lecture. Doing that feeds your brain with information, and as you sleep, your brain will be working on those informations, so that when next you take the same note, this time around reading to understand, nothing will look new to you.
• For that second reading, you will have to pay attention to the Definitions, the Lemmas, the Propositions, the Theorems and their proofs. In each of these identify the hypotheses and the conclusions. Most students make that confusion during problem solving sessions, by not knowing exactly what they have to prove. Reading and understanding proofs of Lemmas, Propositions and Theorems, exposes you to the methods and techniques of proof. You get to see how to make use of the Lemmas to prove theorems, how to use the theorems you studied in a lower level to prove the ones of your current levels and so on. It builds your confidence to start solving exercises.

What to do when reading for the second time?

• Rewrite the proofs of the Lemmas, Propositions and Theorems, detailing every step. Where the lecturer used the famous expression:”Obviously, we have that…” Or something like ” it is easy to see that”, you have to make sure it is actually obvious and easy to you. Write out the details. When another Theorem from a different course has been used or cited, go back to your textbooks or lecture notes and check out that Theorem, its hypotheses and conclusions to make sure you understand while it is needed at that particular step of the proof.
• Consult other textbooks to see their approach of proving the same Theorems and more illustrative examples.
• Think of other ways/methods of proving those Theorems and do it. It helps build your critical thinking skills and abilities to solve exercises and figure out proofs – something very important in Research.
• Pay close attention to the examples in the notes. They give you an idea of how the concept just studied works, how the Theorem just seen is used to solve an exercise. So read them and redo them on your own. As in, after reading and trying to understand the solution, write the example out on a piece of paper, close your notes and redo that example by yourself without looking at the solution. When you are done, check out the solution to see if you got everything right or you missed out anything, and clarify what you missed out.

Now that you are done reading, it’s time for you to practice what you have learned by solving exercises. Most students skip the first and second reading steps above and jump straight to this last step, only to face serious difficulties and wander around for a while, looking for answers from the internet or from their colleagues – you have to try it by yourself first. The reading steps are fundamental, you can’t be trying to show that an open ball is a convex set if you don’t even know what a convex set is, and what property an element in an open ball satisfies. So it is very crucial to read and understand the concepts first. What to do when solving an exercise?

• Read the question at least twice.
• Identify and write down the hypotheses and the conclusions. What are you given and what are you asked to prove, show or compute?
• Then solve…. Write down your solution, making sure every single step is logical.
• Attend the tutorial class and be willing to go to the board to share your ideas on how you attempted the exercises. Also pay attention to the methods others used, they might be simpler or might contain new information you need to know.

With these I want to believe that any Maths major can do well. Always go back to the lecturer for more explanations on the examples you didn’t understand, as well as the concepts you didn’t understand. You can also show him the exercises you did and where you are stuck, and ask questions on the ones you didn’t get correctly.

2. Tips for Science and Engineering students offering Engineering Maths courses.

Unlike Maths majors, Science and Engineering students offering Engineering Maths are rarely expected to prove anything. Their Maths courses are mostly computational, as in they are taught how to solve Engineering Problems using maths, how to compute solutions. So for them the first reading and the second reading without the proofs is usually enough. When I teach engineering maths, most often I don’t even give the proofs of Theorems we use in the lecture. To my Engineering Maths students I recommend the reading and redoing of the examples while studying, before moving to the problem solving part in the following steps:

• Read the notes the same day you had the lecture before going to bed, for the reasons explained above.
• Read a second time, learning to formulate the definitions-understanding and saying it in your own words. You can even rewrite your reformulations.
• Look for the applications of the Maths concepts in your field of study. For example “applications of derivative in Civil Engineering, in Chemical Engineering, etc”. This helps motivate you to study your engineering maths course, as most students neglect the course for not seeing reasons why they need it.
• Redo all the examples as explained above.
• Attempt the exercises following the steps above.
• Always go back to the lecturer for more explanation on the examples you didn’t understand, as well as the concepts you didn’t understand. Ask your mates who seem to understand better.
• Attend the problem solving sessions and be willing to go to the board. Most students are afraid of the board, they are shy that their mates ( who most of the times don’t also understand) will laugh at them if they don’t get the answer correct. Don’t pay attention to that, be humble and learn.

I hope with this few ideas of mine, you will be able to improve your grades in Maths. These are tips I used when I was at your level and it worked for me. So I think if you apply them deligently it will also work for you.
Don’t hesitate to write to me if you need more tips and if you have any suggestion or something else to add. I will be waiting in the comments section. Cheers!