Maths Study Guide, Study Tips and Techniques, Maths Strategies

For a discussion of the strategies that Master Coaching uses to ensure students success in maths studies, see Maths Tuition. In this article we look beyond that, to the logic of learning maths.

Once the most important part of mathematics was being numerate, able to do simple calculations such as multiplication, division, and having a working knowledge of decimals, fractions and percentages. The introcduction of the calculator completely changed this. Now calculators are so common most mobile phones have a simple scientific calculator.

The problem with reliance on calculators is many people today have little or no number concept. They can be over-charged massively without even realising it. Prior to calculators most people had a feel for numbers and were more easily able to recognise when numbers quoted were inflated or wrong.

The shift in maths has been away from a thorough understanding of, and skill level in calculation to other branches of the subject.

After calculation, the next feature of maths is the expounding of logic, ie looking at logic of a formula or equation such as A + B =C or using logic to show that : If A = B and B = C then A =C.

This is called transitive logic. Genuine mathematicians enjoy the challenge of finding such relationships and the really gifted mathematicians have the wonderful ability to see relationships where other mortals continue to over look or ignore.

For example, consider the following that I have many times used with gifted students. First give the student the following two pieces of information which he/she is very familiar with

Very few students see the connection which leads to the answer as set out in full here

These three examples encapsulate the true meaning of mathematics, the ability to apply logic and to find relationships. Learning maths requires the patience to test relationships, to write possibilities and to look at the results.

The best way to learn maths is to be continually writing. The same applies in a mathematics examination: let your pen do your thinking, don’t just stare at the question.

At Master Coaching we tutor the students to think and explore in their problem solving. As well, we have a number of unique ways to solve questions which eradicate some of the mathematical language that stands between the student and understanding.

Consider Pythagoras’ Theorem. Most students know that
a² + b² = c²
but very few students know that ab is divisible by 12 and that abc is divisible by 60 (IF a, b, and c are integers).

You can check by considering some of the more well known Pythagorean Triads such as 3,4,5, 5,12,13, 8,15,17, 7 24, 25, 20,21,29, etc. The proof of this is fairly difficult in our base ten number system, but if we use a base three number system, then every square number ends in a 0 or 1.

Consider the table of all four outcomes for a,b,c. Last digit in base 3 for each number

To prove that ab is divisible by four you just have to use the same table because in base 4 every square number ends in a 1 or 0.

The preceding pages have shown some of the relationships that mathematicians have discovered over time. The exciting thing is that there are still an unlimited number of discoveries in maths just waiting for that spark of genius to reveal.

In our mathematics workshop there is a free down-load of a very simple geometry problem that can be solved using year 7 geometry. The only assumed knowledge needed is that the angles of a triangle add up to 180 degrees, and an understanding of congruent triangles. This is the knowledge of the ancient Greeks, the pioneers of geometry and logic. The problem is very simple to solve using trigonometry and/or similar triangles, but try solving the problem using just geometry!

Students who study maths are well advised to make a short study of the history of some of the most famous mathematicians as it makes fascinating reading, and can only inspire you to greater results.