Learning Mathematics is progressive.
Learning Mathematics is the same as learning anything else.
Like anything else, learning is progressive.
Are you struggling with learning your Mathematics? Perhaps you are struggling with a new concept and you do not even know how to solve a simple, basic problem in that new topic taught to you. May be you could handle the simple, basic techniques but when the problem is compounded by a series of smaller problems, you get overwhelmed. Do not lose heart!
Such obstacles are common to learning any skills because learning is progressive. People who have mastered amazing skills succeeded through making small achievements step by step, stage by stage. Let me bring your thoughts to how we have achieved skills which we are already proficient with in small steps and stages because we did not give up but persisted in mastering them.
Table of Contents
Mastering Pen/Pencil Spinning As An Analogy
Back during my generation and earlier generations, students were fond of a simple pen/pencil spinning skill. When I got a little older during my junior college days, I was fascinated with some “Card Flourishes” I saw on TV programmes. Such basic playing card manipulative skills like the “Ribbon Spread” and “Card Fans” are considered basic in today’s context. I began putting in some effort and I managed to master the “Ribbon Spread”.
In the case of spinning a pen/pencil, it is simple enough that most people eventually master them. First we learn to spin them forward (after ending up with several leaking ball point pens) although sadly some people never really could master them even though it was the most basic move in pen spinning.
The key lies in holding the pen/pencil with its centre of gravity slightly below the tip of your middle finger. You then push the centre of gravity of your pen/pencil with just the right amount of force and let it spin about your thumb. Using too little or too much force and/or positioning the pen/pencil wrongly will result in a failure.
By the way, I have noticed that pen/pencil spinning is a lost art among students for a few decades since the emergence of portable electronic game devices such as Nintendo and now students would rather play with their hand-phones instead of spinning pen/pencil.
Learning is progressive and achieved from stage to stage – from basic to intermediate and then to advanced, although we don’t always demarcate them that clearly. As I was using pen spinning as an analogy to learning here, I said that the first stage was the forward spin. After mastering that and I got bored, I learned to spin with my left hand and eventually both hands simultaneously. However, spinning backwards was still a challenge to me. Naturally, I moved on to learn the backward spin and eventually mastered it. Eventually, I learned to combine the forward and backward spin in one movement.
Combining the forward spin move with the backward spin move together is many times harder than mastering these two moves individually. Both the forward spin and the backward spin have to be mastered to a point that executing them is so easy and instinctive that they can be performed subconsciously.
Learning Stages In Mastering Psychomotor Skills
In learning these psychomotor skills, you have to be extremely comfortable with skills at the previous stage to a point that it becomes automatically a part of your reflexes. That is, you can do it with ease subconsciously without much concentration. In fact it becomes part of your reflex action. It is only then that you can move on to the skill at the next stage as it requires the move and the touch from the earlier stage. It is like you trying to attempt some Jackie Chan’s or Tony Jaa’s stunts when you could not even stretch your legs or do a somersault. You will be courting suicide. How could you run 2.4 km when you will faint after running just 400 metres?
Learning Stages In Mastering Mathematical Skills
Similarly in Mathematics, you have to master your basics and be very confident with them before you can move on to more advanced skills and concepts and retain them in your mind meaningfully and efficiently. Let me use the topic “Inequalities” as an illustration here. Solving an inequality question such as 
involves the following:
- factorising the quadratic expression on the LHS to
- using the ‘sign test’ to obtain the solution or
or 
This is considered as entry level skill. Solving the inequality problem
is the same but many students failed to solve it because they do not have a good cognitive understanding of the skill required in solving the previous problem
.
Reasons Why Students Failed To Handle New Problems Correctly
Even though they may be able to solve previously but they did it merely based on repetitive algorithm without a good understanding of the concept and the finer point of the technique involved. I regard students in such circumstances as having only learned the techniques or algorithm for solving basic inequality questions “behaviouristically” without a “cognitive” grasp of the concept. To them, the algorithm is merely a programmed mechanical behaviour. However, the logic behind the algorithm was not assimilated cognitively.
In learning Mathematics, students generally tend to relinquish a cognitive exercise to a repetitive behaviouristic exercise. This has been cultivated in their earlier years of learning when mathematics was much more easily mastered through repetitive exercises without the need for deeper thinking. Bad habits and the lack of understanding went unchecked and uncorrected and are subsequently carried forward habitually with increasing resistance to change to what is mathematically correct.
This eventually becomes a great emotional and habitual hindrance to learning in the advanced stages. This is clearly manifested when students often objected and argued with such statements,
“I have always done it this way. Why can’t I continue to do it the same way.”
The answer to such objections is one of the following,
“It is because what you have been doing is wrong.”
“This is a different scenario and what you did in the past cannot be applied here.”
For years, batches after batches, I found that many students could not master an elementary pure mathematics topic like Inequalities merely because they simply would not
- pay attention to what is being taught;
- put in the necessary effort to understand and apply the essential concepts but instead rely on some shallowly understood method learned in a previous problem which is technically a different kind of problem, and apply the techniques blindly to a new problem.
A third reason was that they hung on to some wrong concepts and misunderstood techniques acquired at O-level and thought to themselves that “they’ve got it” and make no room and no effort for new concepts and understandings. Have you seen or recall your own experience in failing to accomplish a new task using an old acquired method repeatedly? I am sure we have heard the following remarks thrown at us sarcastically:
“Insanity is doing the same thing over and over and expecting different results.”
– Einstein
Understanding And Applying The Concept Of Absolute Functions
Like I’ve mentioned that solving was basic, the next and more difficult stage was to learn how to handle absolute functions in solving . This involves knowing the definition of the absolute functions and how it is applied to problems. In addition, there are three other properties involving absolute functions listed below:
- or where
- and where
or 
where 
and 
where Applying Logical Operators To Inequalities Problems
In applying the second and third properties of absolute function above, students have to learn that the two words “OR” and “AND” are logical operators. “OR” is equivalent to the union of sets and “AND” is equivalent to the intersection of sets.
Solving the problem requires cognitive understanding and involve more than just basic skills in solving merely 
.
The skill sets required to be mastered in the topic “Inequalities” are much more than what has been discussed. Nevertheless, this is just an illustration of the process of learning and so I will not discuss at greater length the essential skill sets required in the topic “Inequalities” here further.
Obstacles and Breakthroughs Encountered during Learning
During each stage of learning, it is part and parcel that we encounter obstacles and failures. For example, in learning to spin a pen forward, your pen will drop or totally spin away from your hand as you learn to adjust the amount of force you exert on the pen. It can be very frustrating but it is your interest and determination at mastering the skill that keeps you going.
My Experience In Learning LATEX
In typing out this article, it is the beginning stage that I am learning a mathematical coding language called LATEX to input the mathematical symbols you have seen earlier. It required constant searching for codes from the manual, referring over and over again just for a few simple basic symbol.
After referring to the manual, I had to test them out just to encounter failure after failure which brought me back to looking up the manual again. Sometimes it is just a mere negligence, carelessness or a simple typo error that is causing the error. The more time I spent, I begin to notice subtleties I have not noticed or have taken for granted such as a mere difference in capital or small letter in the coding. This is just like a subtle change in force I need to get my pen spinning or flying off from my hand.
The process itself unknowingly have actually taken easily one to two hours and perhaps more while I was engrossed with getting my coding right.
Recall Your Experience In Advancing To Higher Levels In Computer Games
I am sure boys, perhaps girls too, are familiar with the process of facing defeat at different stages of computer game. Your obsession with getting through the stages you were stuck at led you to try and try again until you clear them (and realise that it is already morning and it is time to go to school). Sometime it got so difficult that it seemed impossible. You went for a nap, woke up and tried some more. Sometime you search for a solution from the internet or ask some friends to attempt new strategies until you finally clear the stage.
Back To Mathematics And Inequalities
Using the topic of “Inequalities” above for reference again, you probably misunderstood what was taught in class or may be you were not attentive enough or more often than not, your basic understanding was insufficient for you to grasp what was taught in class. You simply have to go back to your notes, friends or teachers to clarify it again and again until you figure out what were your mistakes. It could be just a careless mistake you made or a wrong idea you were applying.
Common Sense Is Uncommon
Eventually, you must master it to a point where it becomes so-called “Common Sense” to you which in reality is uncommon. For the longest time, I have been telling everyone,
“Common sense is uncommon! Common sense is acquired!”
since my days as a student. Yes, I originated that statement. I said it so often in class and I had a student who diligently wrote down those unique Daniel Sun’s quotes in his notebook. After a few decades, that statement has become a frequently heard statement. It does not matter if you disagree and do not believe that I originated that statement. That is immaterial. My early batches of students will be able to testify to that.
It is extremely frustrating for many people to tell you something is “common sense”, especially your teachers, when you do not understand something.
It is an insult to another person when you say, “It is common sense!” because that person meant 2 things:
- It is so easy that I need not explain to you how and why.
- You are so stupid because it is so easy and you do not understand.
All simple little things which we take for granted in life as common sense had to be learned earlier in life except that they have become so natural to us that we thought that it is so common that everyone knows them or think that everyone should know them.
Summary
In summary let me reiterate my points here.
- Learning is a process and it occurs in stages starting from the elementaries.
- The earlier stages of the process have to be mastered to a point that they become part of our reflex action and common sense where psychomotor and cognitive skills are concerned respectively. Failing which, the next stage of the process could not truly be mastered. Sometime you may have accidental success (we called it fluke!) but never complete mastery and the knowledge or skill set will be forgotten easily.
- A fully mastered skill or knowledge can always be picked up easily even if it has been forgotten over time.
- Encountering obstacles is unavoidable and it is necessary. It is through repeated attempts in overcoming the obstacles that cement the skill and knowledge into something that is second nature and intuitive to us which become eventually effortless for us to recall and perform.
- In learning mathematical skills to solving new mathematical problems, do not assume that new problems are similar to old problems and attempt to apply old approaches to new mathematical problems. Understand the concepts and rationales of how both old and new methods are applied to their respective problems. Do not use a hammer to do the job of a screw driver.
Remark:
This article was written and published on 14 July 2011 in my expired and defunct blog I created for my Mathematics Department. It has been edited and updated to be republished in this new website.
