“Practice makes perfect”
This is a phrase that is usually followed by eye-rolls, sideward glances and scoffs from unimpressed students who have heard it all far too many times before. But when it comes to success in maths, a more apt description could not be made; the reason? Maths is an obstacle course.
To explain the reasoning behind this, we’ll have to take a look at the structure of maths and how it compares with other subjects. Most academic subjects are usually knowledge based (such as the sciences and languages) or research/essay based (such as English and History) but maths seems to place itself in a whole other category entirely which, for the purposes of this article, we shall call ‘challenge based’.
Knowledge based subjects require the understanding of various facts, words, systems etc. and how these can be correctly applied to real-life situations. Essay based subjects require a thorough understanding of texts, history etc. and the ability to compile these into compelling and coherent arguments. Maths requires the ability and skillset needed to solve various problems (or overcome various obstacles). If we were to compare these to game shows then knowledge based subjects would be University Challenge, essay based subjects would be Strictly Come Dancing and maths would be…Total Wipeout.
The need to use practiced skills to solve various problems is recurrent in maths papers but also within the individual questions themselves. In the same way an athlete must train themselves in various fields for a heptathlon, students must develop and hone their skillset to be able to tackle any question that comes their way. Let’s look at an example problem:
Solve the simultaneous equations
y – 2x – 4 = 0 4x^{2} + y^{2} + 20x = 0
Skills needed to solve this problem:
- Rearranging and substitution
- Factorising quadratics
- Possible use of the quadratic formula
As we can see, even individual questions (whilst not to the degree of the entire exam paper) require students to be experienced with a varying skillset. This is where the term ‘practice makes perfect’ comes into play; pupils will need to have a sound understanding of each of these skills if they wish to solve the problem and this understanding will only come about via practice and more practice! The big advantage of an obstacle course is that you know what you will be facing. Whilst students may not know the specific numbers and wording of all the questions that will appear in their exams, they can learn the exact style of the obstacles and the skills that they will need to put into action to overcome them. If they have sufficiently practiced similar problems before, tackling maths questions and maths papers will become second nature.
Finally, let’s look at an outline for a practice obstacle course and how these may be tackled:
Obstacle 1: Quadratic equation with unknown constants
Possible skills needed:
- Factorising quadratics
- Knowledge and application of b^{2}- 4ac
- Solving inequalities
Obstacle 2: Fractional equation of a curve with x terms in numerator and denominator
Possible skills needed:
- Simplification of fractions
- Expanding brackets
- Finding the differential
- Finding the equation of a tangent
Obstacle 3: Equation with x and y sharing a logarithmic relationship
Possible skills needed:
- Knowledge and application of log function
- Rearranging and substitution
The three obstacles above were actually three questions in the A-level Mathematics C1 Paper for June 2015. While the questions and answers may not be comprehensive in this article, the idea of skills solving problems is prevalent. I believe that with enough practice, students can fill their mathematical toolbox with the necessary skills to make solving mathematical problems no longer an obstacle.