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# The Secret to Success in Mathematics

There are so many people who struggle with mathematics and yet some people seem to have no trouble whatsoever. Is there something different in their DNA or some super advanced part of their brains, or do they simply know some secret trick that helps them in their understanding?

The answer is a little of the second, but mostly the last. Scientific research suggests that those who are good at mathematics do have some areas of their brain more active during mathematical activities, however, this increased brain activity could be a result of increased mathematical training. Much like how an athlete’s muscles improve with training, regular mathematical exercises help to improve the performance of those brain areas associated with mathematics. Again, like an athlete in training, there needs to be some underlying skills and a good coach. All of us possess the minimum underlying skills, but not all of us are fortunate enough to have a good coach at the critical time when we are ready to start serious mathematical training.

That’s not to say that you must have a brilliant mathematics teacher in order to succeed, but rather that when you are ready to start learning about mathematics beyond simple addition and subtraction, you need someone who can help you identify that mathematics is not a series of boring repetitive exercises from a textbook. Nor is mathematics made up of a set of discrete topics that package neatly into textbook chapters. In short, you need someone to tell you the secret to success in mathematics.

The secret to success in mathematics is understanding and accepting the following:

1. Mathematics is a language
Mathematics is not a set of confusing hieroglyphs designed by some evil conspiracy to torture school children and university students, but rather a language built on rules. In the same way we learn any language, we must learn the alphabet and the rules used if we are to have any success in expressing our ideas or understanding others using this language. The primary rule of the language of mathematics is the “Order of Operations”.
2. Mathematics is based in logic
Mathematical rules are firmly based in logic. For example, if we accept that  $1 \textless 2$  (i.e. one is less than two) and  $2 \textless 3$ , then it follows that  $1 \textless 3$ . Whilst this is a simple example, complex mathematical rules are built by joining together many more simple rules. When the logic is violated, the result can be a set of mathematics that seems to prove the ridiculous, such as, that  $1 + 1 \neq 2$  (see post). If you are careful about how you apply the rules and ensure you don’t violate the logic behind them, you will be unlikely to go wrong.
3. Mathematics is interconnected
Like many things in life, mathematical ideas are often connected to more than one other idea. Techniques, such as those that apply to linear functions, often reappear in other areas of mathematics, for example, in the statistical technique of Simple Linear Regression. As such, it is dangerous to treat mathematics as a set of discrete skills, learned for one chapter of a text and forgotten shortly after the topic test. Instead, one should consider each newly acquired skill or technique as part of an arsenal or toolkit to be drawn upon for future problems.
4. Mathematics is everywhere
Much of mathematics taught in schools and universities suffers from an inability to answer the highly intelligent question, “Where will I use this in real life?”. Unfortunately, it is often difficult to point to a real world situation and make a plausible justification for the direct use of much of the mathematics taught in high school. Pythagora’s theorem ( $a^{2} + b^{2} = c^{2}$ ) is a classic case. The ancients knew that if you took a rope, placed knots at equal distances and then used this to construct a triangle that had sides of 3 knots, 4 knots and 5 knots, then the triangle contained a right angle (very useful for constructing buildings that won’t easily fall down). Despite this, it is unlikely that many modern students will need to build a pyramid for their pharaoh, so why should they learn it? Well, Pythagora’s theorem is about more than right-angled triangles, it also tells us about the relationship between many numbers and is a theorem that is the basis of much of the mathematics used in modern computing, communications technology and cryptography.
5. Mathematicians are lazy
Like many people, mathematicians don’t want to do more work than they have to. As such, many of the techniques employed in mathematics are about reducing a problem down to a set of previously solved sub-problems. In other words, we mathematicians want to re-use our previous work (or someone else’s work) wherever possible. What this often means is that students are often presented with almost identical problems as they learn different techniques. Unfortunately, this can lead to the perception that mathematics is nothing more than a set of boring repetitive exercises from a textbook, which is completely not true. Instead, perhaps it would be better to think that mathematics is very eco-friendly, recycling previous problems for reuse in more complex learning situations.
6. Mathematicians like simplicity
The axiom of “The simplest solution is often the best” is truly at home in mathematics. As such, we prefer solutions that are given in the simplest form, whether that is as a surd, a fraction, a decimal or a whole number. Wherever possible, you should try to express your solutions in the simplest, but most correct, manner. For example, one should write  $\sqrt{2}$  rather than  $1.414\ldots$ .
7. Mathematicians like order
Some might suggest that mathematicians suffer from some kind of obsessive compulsive disorder, but the truth is that mathematics is based upon order, be that the counting order of numbers or the “Order of Operations”. The same is true about how we like to write equations in descending order of powers, e.g.  $y = x^{2} - 3x + 5$  rather than  $y = 5 + x^{2} - 3x$ .
8. Statistics is a special kind of mathematics
Statistics uses all the techniques of mathematics, from simple algebra through to calculus and beyond, but it also uses a special set of skills called “Reasoning with Uncertainty”. Statistics is a branch of mathematics devoted to the science of uncertainty (risk, odds, chance, probability, likelihood). Unfortunately, it has gained a fairly bad reputation, mostly because, like weather forecasts, it does not provide 100% guarantees about the results, only the methods used to obtain them. Instead, statistics provides insight into the possibilities, which, when combined with other facts, should lead one to a reasonable (but still possibly wrong) conclusion based upon the sample data. Mark Twain once quipped, “There are three kinds of lies: lies, damned lies and statistics”, hinting at this possibility of an incorrect conclusion. Whilst statistics is the “Science of Uncertainty”, there is no uncertainty in the science of statistics. If you perform the same statistical technique many times on the same data, it will always lead to the same conclusion. The uncertainty comes from the data and whether or not it is an accurate and representative sample of the population of interest.
9. Statisticians are “frightened” of negative numbers
Most statistical techniques are based upon differences and as such, many of these differences are negative. Unfortunately, if you add together a bunch of positive and negative differences, many of them will cancel each other out. To avoid this, a number of statistical techniques use the trick of squaring the differences first, before the summation, and then square-rooting the result. A classic example occurs in the calculation of the standard deviation. This often leads people to comment that statisticians seem “frightened” of negative numbers.

Once you have understood and accepted the above truths about mathematics (and mathematicians), it should hopefully become clearer to you why it is that we mathematicians do certain things. Equally, these truths should help you to unlock the secret to your success in learning the techniques of mathematics and statistics.

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