Elements of mathematical skill and knowledge are like bricks in a wall that is being built upwards. The more advanced elements are dependent on the support of properly established lower elements.
For study at any level of mathematical attainment, such as GCSE, there will be the range of these elements that need to be already established. If this is not so, progress and achievement will be difficult.
My first tuition session with a GCSE pupil is used to identify the gaps and weaknesses in the elements of mathematical skills and knowledge already expected to be held at this level.
To be precise, I will ask the student to demonstrate the following techniques, in this order. A calculator will only be used to check manual calculations on paper.
1) Addition and subtraction of large numbers on paper, with clear working.
2) Long multiplication and division, using a recognised method, and clearly presented.
3) Fractions - Finding a fraction of a number, using division and multiplication skills.
4) Fractions - Arithmetic using fractions as numbers.
5) Percentages - Finding 10%, 20%, 5%, 15% of simple numbers.
6) Decimals - Arithmetic using simple decimals as numbers. NOTE: A reliable method for multiplication of decimals is a gap in the knowledge of most students at this level, but is easily introduced to them.
7) Basic Algebra - Solving simple equations such as 2x + 1 = 11, showing clear working of how the solution has been found. If possible, progress to equations such as 3x + 1 = 2x + 7.
I do not push my pupils to work at anything other than their own pace. I make it clear that it makes no difference how long it takes to get a technique working effectively. If a student makes a mistake, the important thing is for THEM to see what went wrong, and to have another go.
Whatever errors and repetitions are necessary in that first hour, in my experience ALL these techniques can be addressed in that session. Of course, most of this will need to be revisited in subsequent sessions, but this is to be expected for establishment of skills and knowledge.
I'm sure that many parents watching this initial session will be thinking that a GCSE pupil shouldn't be having any problems with what they might see as very basic maths, for example, struggling to do subtractions without a calculator.
Well, just because a student SHOULDN'T be having a problem, that isn't a reason for a tutor not identifying such gaps of skills and knowledge and addressing them. When I start with any pupil, we have to start from where they are, not where we would like them to be.
