Posted by: Alexandre Borovik | June 20, 2011

## GCSE and university level mathematics

Why should university mathematicians care about the National Curriculum and GCSE?

1. To take one example, the levels of geometric intuition and spatial thinking in our (university!) students are pre-determined by their exposure to geometry at KS1-KS4 level.
2. Prerequisites for many university mathematics courses in Year 1 are rooted in GCSE mathematics. Linear algebra is a primary example. The way algebra is introduced and taught at KS3-KS4 directly affects our students’ understanding of university algebra.
3. We also teach service and foundation courses and directly deal with students whose mathematics background is restricted to GCSE.

This list can be easily continued.

Point 2 becomes especially obvious in matrix algebra the way it is needed, and is taught, to students in various economics and finance-related degree programmes in universities: it does not require school level calculus, trigonometry, vectors, complex numbers. What is needed is a deep interiorisation of basic principles of algebra at the level of confident “structural arithmetics” and fluid symbolic manipulation.

As an university teacher of linear algebra and Boolean algebra (essential components of mathematics in 90 percent of mathematically intensive university degree programmes) I have vested interest in the way GCSE mathematics is taught. I need students who have deeply interiorised concepts of associativity and distributivity. Distributive law of arithimetic is a germ of the future concept of linearity in linear algebra. It also appears in Boolean algebra (which is the foundation of digital circuits in electronics and programming languages in computating) as a mutual distributivity of conjunction and disjunction.

In mathematics, to learn something means to learn it at least twice at different levels. Therefore I would rather teach students who have learnt basics of their future linear algebra in KS2–KS4 first doing structured arithmetic problems of the kind 17×13 + 17×7 =?, and then re-enforcing their intuitive understanding of linearity by learning to express it in a symbolic form.

## Responses

1. But surely, having to solve little systems of linear equations, manipulating the basic properties of modular arithmetic and beginning to use complex numbers give high-school students at least 3 further occasion to see commutativity, associativity and distributivity in action and as useful theorems that one can demonstrate in class and apply as broadly as one’s imagination.

2. This one is really wonderful information about linear equations, manipulating the basic properties of modular arithmetic. Matrix algebra is one of the most important factor in calculating structural arthmetics