This article was originally written and published in the year 2015 in my expired and defunct blog I created for my Mathematics Department, and I have edited and updated it here. I believe that this is probably one of the most important articles I have written for learning Mathematics. If only students are willing to apply the principles here in their learning, they will excel in Mathematics. In fact, the principles that I have shared here can also be adapted and applied to studying any other subjects.
Table of Contents
Wrong Approach to Learning Mathematics
As a Mathematics teacher, I was often asked by students and parents, “How to improve on my Mathematics grades?” Almost without waiting for my reply, they would answer their own questions by saying, “To improve on my Mathematics grades, I have to practice a lot, right? That’s the only way.”
Somehow I beg to differ on that. Whenever someone says that, he/she usually means “practice makes perfect” and by “practicing” many mathematical problems you naturally get better at it. Did you notice that I put the word “practice” in quotation? It is because usually people subconsciously have a specific meaning attached to the word “practice”.
By practicing, people have in mind practicing Mathematics like practicing a musical instrument through repeated motion like honing a psychomotor skill. Reciting spellings for words and “multiplication tables” like little children, as if Mathematics at A-level is some kind of low level cognitive skill.
I will not deny that in secondary schools, most students learn Mathematics by “practicing” repeatedly similar kind of mathematical problems and thus the term “drill and practice” came into existence in education. Some books even use that as their title. That also explains why I find students being able to apply certain mathematical skills they learned in the past but not knowing the meaning and reasons behind the application of these skills.
This approach of “practicing” in learning Mathematics back during secondary school worked because many of the final examination are quite similar year after year without much variation. It is somewhat like if you have done a few, you have seen all of them.
However, if the examination questions were to deviate from the many repetitive questions that you have been practicing, those students who have practiced solving mathematical problems without a good understanding will hit a roadblock. This is usually the situation in the A-level Mathematics examination.
Worse still, they have learned to applied these skills with the wrong understanding and in some cases they have conjured up their own wrong idea while formulating patterns from their own observations. As a result, they could not relate old skills to new mathematical skills and concepts that they are to acquire at A-level.
Please do not get me wrong.
I am not saying here that you do not have to solve mathematical problems on your journey to learning Mathematics. Undoubtedly people who are skilful in Mathematics do solve many mathematical problems. The act of solving many mathematical problems is only an outward observable behaviour of an inward cognitive exercise but there is a difference between one who benefits from the act and one who does not.
An Approach to Scoring Good Grades in Mathematics
Let me share a two-step process taught to me by a friend of mine back in university who has done very well for all his examinations during our varsity days. He has done very well in his academic work and is now a professor teaching Mathematics in a university. This two-step process is targeted at how to score “perfect score” in the examinations.
- STEP 1: You have to understand everything in your lecture notes and tutorials very well to the point that you can reproduce them if they appear in the examination questions.
- STEP 2: You have to understand everything inclusive of the concepts in your lecture notes and tutorials very well to the point that even if the problems are modified, you can still solve them.
Right at the start, most students could not even fulfilled the first step. Many students do not read their lecture notes and do not complete all their tutorials, much less being able to reproduce them if they appear in the examination questions.
I am not advocating rote memory here. If you speak to any students who are good at Mathematics, they will tell you that they hate memory work. During my junior college days, students who took the subject, Further Mathematics, do not like to memorise formulae. Instead they will derive the formulae if they need to. By saying that you are able to reproduce them during examination merely meant that you can reproduce them because you have understood and mastered them. In fact, students who love Mathematics do not like subjects that require much memory work like Chemistry and Biology.
The second step definitely points out that this is not a memory work but one of understanding. It emphasizes that your objective is not to just solve problems the way you have solved them “just as they are” but understanding their concepts.
Simplistic as it may sound but this two-step process quickly locate many students where they are in their learning.
Most students have barely even read their lecture notes thoroughly and attempted their tutorials. This process can be used as an instrument to measure how well students have prepared themselves for examinations.
Compared to Mathematics taught to undergraduates, A-level Mathematics is still very simple in that besides examples of problem solving in lecture notes, all there is in the lecture notes are merely simple definitions, rules and formulae. There is hardly any theorem and proof to study and even proving of formulae has been de-emphasised and almost made obsolete.
Most students barely read and understand definitions given in the notes but merely focus on the examples. When solving problems, they merely look for examples that match the problems they are solving and hope to imitate the steps used in the examples in solving unseen problems. This is one of the main reasons why students find the topic “Functions” difficult to handle because they are not accustomed to learning and understanding definitions of mathematical terms which is the main emphasis in the study of the topic “Functions”.
Refining The Two-Step Process To Scoring Well In Mathematics
At this point, I will refine the two-step process by giving some checklists and guide to the journey to satisfactory attainment of the two-step process mentioned earlier.
Reading The Lecture Notes – How Should You Do It?
The bare essentials of your journey is to read your notes thoroughly and solve all tutorial problems given. This alone does not necessarily guarantee that you will do well in your examination for the simple fact that it does not guarantee that you have understood your concepts and acquire the necessary mathematical skills. Yet many students have not even done this bare minimal task.
A great percentage of students merely jumped into solving tutorial problems without reading their notes. A common practice is that they look for examples that are similar to the problems that they are solving and imitate the steps used in the examples’ solutions.
They do this without having fully understood the meaning of the notations, the questions, formulae and the logical connections of the working involved in the solution. At a low level in secondary schools where questions are simple and solutions are short, this approach may have worked but at A-level, it is inadequate towards mastery of the subject.
What should you do when you are reading your lecture notes?
While reading your lecture notes, you should always be holding a pen and having a stack of papers with you on the table. As you read each line from your notes, you should ponder about it and sort out your doubts and understanding in your head at the same time writing them down. Take for example the definition of a one-one function:
A function is said to be one-one if for every element
Definition of One-One Functionin the range of the function, there corresponds exactly one element
in the range of the function, there corresponds exactly one element 
).As I read each sentence, and in fact many a times, part of a sentence, I will write them down on my rough papers. As I write, I ponder on what those words mean. If the statement is stating a condition, I will try to list down several examples that satisfy that condition and other examples that do not satisfy that condition.
When I think that I have completed reading and pondering through all the words and sentences in the definition, I will try writing down the complete definition on a piece of paper while letting the meaning of the definition run through my mind as I write.
Usually when I have completed this process satisfactorily, I would have understood the definition completely. I would then be able to write down the definition on my own without having to refer to the notes. Take note that I did not memorise the definition blindly through recitation. There is no need to memorise the definition when I have understood the definition completely. That goes the same for any theorem or formula.
How Do You Study Examples And Solve Problems
Point 1 – Understand and retry until perfection is reached
When you are reading the examples given in your lecture notes, it goes without saying that the process of writing, pondering and understanding the logical connections which I have just mentioned above still hold. After you have understood an example, you should attempt to solve the problem stated in the example on your own without referring to the solution given in the example.
If you have never done this before, you would probably be surprised that you are unable to do it because you thought that you have understood the example. When you do this, you will realise that there is much that you still do not understand. This probably answers the question why you have done badly in your test or why you were unable to solve your problems during your test when you thought that you have studied.
So when you get stuck in your attempt to solve the problem, go back to your notes again and try to understand the example at that point which you got stuck at. Repeat this process until you are able to solve the problem completely on your own. This applies also to your tutorial problems, revision exercises and your practice of examination questions.
Point 2 – Try alternative approach to solve the problems
Let’s say that your problem is to find 
. Instead of solving the problem by only one way, you may want to explore it in all possible ways that you can conceive, say the following 3 approaches to solving the problem.
Method 1:
Using Double Angle Formula 
Method 2: Using the trigonometric identity 
Method 3: Using the trigonometric identity 
This question was originally an examination question. Set as a question in the tutorial for years, I have observed that most students used either method 2 or 3 and not method 1 that is the simplest and most efficient approach.
If students were to explore more alternative methods of solving, they would have learned a more efficient method of solving using method 1. Nevertheless, even if a student would have attempted this question by method 1 initially, it would still benefit him/her by attempting to solve it by method 2 and 3. The benefits are as follow:
- Students will get to practice more on applying various trigonometric identities which may seem to be similar but with subtle differences, for example, and .
and 
.- Most students tend to focus on just getting the same answer that was printed for them in the tutorial and not focus on the process. After attempting all the 3 methods, students should put in time to analyse and compare the 3 methods, noticing how is simplified to by the 3 methods eventually. A careful analysis of the 3 methods will result in the students knowing and understanding their trigonometric identities much better.
is simplified to 
by the 3 methods eventually. A careful analysis of the 3 methods will result in the students knowing and understanding their trigonometric identities much better.In this example here, all 3 methods yielded the same result. However, integrating more complicated trigonometric functions with different approaches could very often yield very different results. Although the results are different, they could be proven to be identical but the task of proving them to be identical can be very challenging and time-consuming. This in itself is a journey of learning.
Point 3 – Figure out the similarities and differences between seemingly similar problems.
I will illustrate this point with a question from the topic of “Permutation and Combination”
There are 4 boys named A, B, C and D (for simplicity). Find the number of ways these 4 boys can be divided into 2 groups of 2 if they are playing (a) a game of police and thieves where there are 2 police and 2 thieves; (b) a game of tennis where a team of 2 boys compete against another team of 2 boys.
Question (a) and question (b) both required you to divide the 4 boys into 2 groups of 2. That seems to be very similar. However the subtle difference is that in question (a), there is a functional difference between police and thief.
One of the many possibilities is that boys A and B form a group {A, B} while boys C and D form another group {C, D}. However, the case where boys A and B playing as police and boys C and D playing as thieves is different from the case where boys A and B playing as thieves and boys C and D playing as police. These 2 cases are counted as 2 as there is a functional difference.
For question (b), if boys A and B formed a double and boys C and D formed the other double, it is counted as only 1 case because both teams are playing tennis (no functional difference). So the solutions for questions (a) and (b) are as follow:
Solution for (a):
Step 1: Choose 2 boys from 4 boys to form a team of police: 
Step 2: Choose 2 boys from the remaining 2 boys to form a team of thief: 
Answer: 
Solution for (b):
Step 1: Choose 2 boys from 4 boys to form a double:
Step 2: Choose 2 boys from the remaining 2 boys to form another double:
Notice that after these 2 steps the answer we obtained is also 6 just like the solution in (a). However, for every 2 doubles we formed in the 6 cases, there is an exact duplicate:
e.g.
{{A, B}, {C, D}} = {{C, D}, {A, B}},
{{A, C}, {B, D}} = {{B, D}, {A, C}},
{{A, D}, {B, C}} = {{B, C}, {A, D}}
Thus, we need to divide the number of cases obtained step1 and step 2 by 2 to obtain our final answer.
Answer: 
Point 4 – Modify the problem you have just solved and see if you can still solve it. Ask yourself what are the similarities and differences in your solutions between the original problems and the modified problems.
Let’s say that a student was initially required to find the term in 
in the series expansion of 
.
By using the general term in the series expansion of , he will have the following working and result:
I would urge students not to stop here after getting the right answer but to explore further by modifying the question for themselves into finding the term in in the series expansion of 
.
Similarly, by using the general term in the series expansion of , he will have the following working and result:
At this point, students may or may not see the trend and pattern of such a question. However if students are willing to push themselves further to do more by modifying the question even further into finding the term in in the series expansion of 
, their understanding and competency in questions like this will improve tremendously.
The working and result of this question using the general term in the series expansion of is as follow:
If a student have done thus far, he/she should have noticed a pattern in the working and the result when the power of 
is reduced by 
each time. He/she would probably even be able to write the result in a general form if the power of is 
where 
is a positive integer.
There is much room for exploration in this question. A coefficient can be given to the term 
changing the question to either 
or even 
.
Instead of finding the term in , we can find the term in 
as well. The more you modify the question and the more you explore, the more you will learn and understand. In fact that is how questions are modified from other questions when test and examination questions are set.
Conclusion
I believe that many students upon reading this article will feel that this is an impossibility because the requirements spelt out here far exceed what they normally do and what they are willing to do. This is a good gauge to measure themselves on how far and how much effort they have put in towards learning and mastering their Mathematics. It also answers their questions as to why they have not achieve the desired outcome and grades that they have desired for.
A simple answer is, “You have not understood your work adequately.”
I would like to encourage any student who is reading this article that if you choose to employ this strategy into studying, you will find yourself improving and maturing in your studying habits and understanding like never before. Even if you have not been able to do all of the above recommendations but only partially, you will definitely experience a tremendous change.
I believe that any students who are willing to put this into practice, you will see the fruits of your labour and I will see you right at the top.
