1.2 What s needed to ensure good grades?
1) Knowledge of the material2) Practice in applying said knowledge to exam questions3) a) Problem solving knowledge and practiceb) Confidence in your ability to solve problems
I ve called the third and fourth points above 3 a) and b) because they are both so strongly linked that they are almost the same thing. There are two main reasons why lack of confidence is so detrimental to maths performance:If you are unsure of your ability to solve a problem, you are far more likely to give up easily, not try everything you could and will tend to abandon prematurely paths that would have led to the answer. If, however, you believe you are capable of answering any problem that is likely to come up, then when a question comes up that seems weird or unfamiliar, instead of defaulting to the idea that you can t do it, you re far more likely to assume the method to solve it is in your toolkit and so you will persistently try different things until one works
Even more important, unfortunately, is the fact that even basic maths is almost impossible to do while in a state of panic and anxiety. When panicking and anxious it is very difficult to think logically or coherently at all. What this means is that, in an exam setting, often students are thrown off by a question early on, start panicking and then find they can t think straight, start making silly mistakes and are unable to do later questions which they could easily have done had they been calm. Unfortunately this is a bit of a downward spiral and is often the cause of drastic exam underperformance.
Confidence is also one of the main causes of the good at maths , bad at maths false dichotomy that tends to emerge throughout the course of school. What tends to happen is that students branch into two camps quite early on (though unfortunately it is very common to be knocked into the lower camp at any stage and much less common to be knocked back into the higher branch).
The good at maths camp is usually someone who was encouraged at an early age to pursue maths and had reinforcement from somewhere that it was something they could do and were good at. This leads to more enjoyment of the subject and hence more time investment. It also leads to less fear and the belief that new concepts will be easy, or at least possible, to understand. Therefore, when a concept comes up that seems confusing, the student expects that they will be able to figure it out and, perhaps more importantly, feels that they should figure it out because, after all, they are good at maths and so not figuring it out conflicts with their identity and beliefs about themselves. Naturally, these confusing concepts unravel after some time is devoted to them and thus the belief is reaffirmed even more of course it wasn t actually confusing or difficult - they knew they must be able to figure it out since they re good at maths . Then they did figure it out, proving to themselves further that they are good at maths . This extra time investment and, more importantly, belief in the ability to grasp concepts and hence persistence in understanding new topics builds on itself over time. Whenever a gap in knowledge comes up it is quickly filled as it doesn t make sense that they can t grasp something - they re good at maths - it must just not have been explained well! Just a bit of extra thought or time on the topic will make it make sense. Additionally, all the way through school, when a question comes up that they re not sure how to solve, they know they can solve it so they get used to trying things until something works, and hence also learn how to problem solve. Maths is something that they re often praised for and are good at, so there s incentive to spend time doing it, or at the very least no major reason to avoid it.
However, that camp requires good teaching and encouragement from early on and throughout. Humans unfortunately have a massive negative bias (it s what kept us alive as we evolved - quite annoying in the 21st century though...), which means we are much more likely to focus on negative feedback than positive feedback. Additionally, maths is something which is far more easily made to seem impossible than approachable. Then there s the fact that maths tends to build on itself such that one small gap in knowledge often makes the next topic impossible to grasp, which makes the next topic impossible and so on. Insidiously, after a few topic changes, the original cause of the issue - a gap in knowledge that was just never taught - is forgotten and, due to the resulting inability to solve problems for several topics now, the student s confidence is shattered and the belief that they are simply bad at maths begins to set in. At this point, if a new topic seems confusing, the student s default is to believe it is truly extremely difficult and that they can t get it because they are bad at maths. Why should they bother trying to figure it out? They probably can t anyway it would take ages and might not even pay off. Besides, they ll never be good at maths so what s the point? When a question comes up that they re not sure how to solve, they think that odds are they don t know how to solve it because they don t understand the topic. They re probably missing the required knowledge anyway and you probably would have to be good at maths to solve it too. The default response is to not even try because it probably wouldn t be successful, and this doesn t conflict with their beliefs about themselves so they have no problem leaving it unsolved. This spiral builds over time they fall further behind in knowledge due to new topics being incomprehensible because of old gaps, they never practice problem solving and they avoid doing maths as much as possible because the experience is so unpleasant.
Up to this point, I haven t mentioned how natural talent or ability comes into play. It does influence things, but not nearly as much as people think. Slightly more able individuals are far more likely to be sorted into the good camp early on and so by the time A-levels are reached they have 12 odd years of positive feedback loops working for them to increase this effect. Natural ability does start to become more important at the very highest levels, but even then hard work and practice is far more of a deciding factor than you might think. Yes, not everyone can do a Maths PhD at Oxford, but a friend of mine is currently doing just that and he was in maths set 2 until year 10 and never got even a bronze in the maths challenge before year 11 (yes, that story is completely true). How did it happen? He worked. Hard. In year 10 would anyone have said he was gifted or had natural talent in maths? No. Ask anyone, does someone doing a PhD at Oxford in Maths have a gift or a natural talent in maths? Yes. There s no weird explanation or special factors here he wasn t that good at maths but found it interesting and so started working hard at it from year 11. As a result, he slowly improved over the years and, well, the situation speaks for itself.
My own experience is slightly less dramatic but has similar themes. In order to get into Cambridge for Maths you have to sit two entrance exams at the end of sixth form STEP II and III. When I first started practicing for these in January, I freaked out completely at the fact that I simply could not do the questions. I knew the material, I understood the solutions when shown them, I just could not seem to do the questions myself. I also experienced, for the first time, what panicking and anxiety mixed with lack of confidence does in a Maths exam. When panicking due to a mock exam paper going badly I discovered it was actually impossible to think straight. This led to second guessing myself, not following through on ideas that were correct because I didn t believe they would be right and just generally caused my exam performance to fall apart entirely.
What does this story mean? Perhaps I was just not cut out to go to Cambridge and wasn t good enough? As it turns out, no, that was not the explanation. By obsessing and practicing a lot, by the time June came around I was prepared for the papers and scored in the 96th percentile. However, had I sat the papers in January I would have failed, badly. The difference was entirely down to the fact that I was learning how to properly approach difficult problems for the first time, exposure to which I had been lacking up to this point. I didn t know it at the time, but in January I was effectively way behind in problem solving practice compared to my competition, due to my having been lazy and coasting up to this point. In January I have no doubt many more sensible students would (and probably did) decide they weren t cut out for Cambridge due to lack of natural ability/talent and give up. However, that would have been a lie the truth is actually that the playing field was incredibly uneven and they were behind in teaching and practice, not talent.
While not everyone can be the next Einstein, and some people do have to work harder than others to achieve the same result, the important truth is that anyone who hasn t put huge amounts of time and effort into it already, especially if they have a long history of fear and/or hatred of maths, has the potential to improve their maths ability hugely. It is not an exaggeration to say that, if they are willing to put in enough of the right kind of preparation and practice, practically everyone taking A-level Maths is capable of getting an A* or an A. Getting into the international maths olympiad team might be an impossible task without a serious natural talent, but up to and including A-level, the questions are at a level such that they can be made approachable to almost anyone. That is not to say they are easy by default - anything is borderline impossible if you aren t taught it well and haven t practiced it. It s just saying that you can get to a point where you find most of them reasonably easy. Whether or not you do get to that point depends on whether you put in the time and work smart. It s just important to know that it is possible.
While everything written so far is true, it is so contradictory to such widespread publicly held beliefs that you may want to dismiss it as wishful thinking or a motivational lie. I used to be firmly in the natural talent is set and dictates results camp it is only through both seeing this disproven so many times and doing a large amount of research into the findings of studies in the real world on the topic that I was forced to change my view. If you aren t swayed by the arguments here and don t believe my claims then I recommend you look up what the science says on the natural talent vs. hard work debate unsurprisingly it is something into which huge amounts of research has been conducted.
