# The Basics

This post was provoked by the timing of two recent things: firstly, I have been investing a lot of time, reading and general thought into how a scheme of work could best support outstanding teaching in A level Maths and, secondly, a snippet of conversation where teachers bemoaned their students abilities with the basics and how there isn’t time to deal with these issues.

As A level teachers we often talk about the basics but perhaps never really think explicitly about what we are putting into that category: general fluency with algebra? ability to manipulate negative numbers? indices? surds? linear simultaneous equations? quadratics? The list quickly grows and eventually covers a substantial slab of AS Core. The most common complaint is that these are GCSE topics and thus, it is believed, we should be able to take them for granted and make progress with the ‘real’ A level stuff. There’s not enough time to waste covering things they should already know. Students who are weak on the basics are either not welcome to study A level or are perhaps expected to independently improve their understanding of the basics in their own time.

Two things occur to me: one is to consider the reason why there is such an apparent overlap in GCSE and AS content, and the other is whether the basics is the most appropriate label.

## The Overlap

My own inference is that AS core revisits GCSE topics to ensure that there is a sound and thorough understanding of these topics, fully preparing students for all the future material which heavily depends on the basics. There are also the apocryphal tales of GCSE classes skipping topics and focussing on the ones that have the highest mark:ease ratio. The AS core thus helps ensure equality of access to the A level curriculum.

Perhaps there are other reasons? Please let me know your thoughts. Perhaps the GCSE reforms and proposed A level reforms will change this, but I’d be surprised.

## The Fundamentals

I think a much more appropriate term for the basics might be the fundamentals. One thing I would like an A level SoW to consider is the future implications and connections that develop from whatever is studied.

Are students learning the current material in the most appropriate way to enable them to develop their understanding and relate it to relevant material in the future?

To test out the idea, I brainstormed the topic of straight line graphs. In Core 1, students are expected to be able to form and manipulate equations of straight lines, identify parallel and perpendicular lines and be able to find the point of intersection of two lines and the midpoint of two specified coordinates. All straightforward techniques, all on the GCSE syllabus (I believe), and all included in most teachers’ idea of the basics.

Initially, the obvious connection is with tangent lines and normals which pervade the calculus of Core 1-4 (not forgetting curves defined implicitly or parametrically at A2). With Edexcel this also crosses the border into the study of conics in FP1 and later. But then there is regression in Statistics where students are not only required to find the equation of a line but also to interpret its gradient. Do we emphasise the meaning of gradient so carefully in C1? And the topic of linear coding of data? For Edexcel this is a key theme running through S1. In Core 4, students meet the vector equation of a line in 3 dimensions. Did we ever consider showing them the vector equation of a line in 2 dimensions back in C1? (As a side note, there is an excellent resource in the CMEP pilot materials – precisely designed for students to think deeply about different representations of lines in 2D.) If students are studying D1 they need to be adept in plotting lines, shading for inequalities, and solving for points of intersection. Even in Mechanics, linear simultaneous equations are everywhere! Moreover, the concept of a 2D vector description of a line comes in when they consider motion of a particle moving with a constant 2D vector velocity.

So, rather than referring to straight line coordinate geometry as part of the basics, can we instead describe it as one of the fundamentals? Fundamental as in the theorems of Arithmetic, Algebra and Calculus.

## The Time

But we don’t have time to waste covering the basics” is a short-sighted viewpoint. Time invested in the fundamentals must surely pay off throughout the rest of the teaching and learning.

## Think, Pair, Share

• What is your opinion on the amount of overlap that exists between GCSE and AS specifications?
• When planning the teaching and learning of a particular topic, does your department consider the future needs and uses of that topic?
• Take another AS topic (for example, inequalities initially appears to be a brief and relatively self-contained topic in C1) – can you find examples of where it crops up in every other module that your students will study?
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