The Further Mathematics Support ProgrammeOur aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.

To find out more please visit www.furthermaths.org.uk

The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.

New specifications for A level Mathematics and Further Mathematics

� All A levels now linear

� AS can be co-taught with A level, but will have a separate exam which will not count towards the A level

� AS is the first half of A level

� AS content will be examined at ‘A2’ level in A level, but at a lower level for the AS exam

� AS can be taught over 1 year or 2 years

General points

� ALCAB recommended that first teaching should be from 2017, so that the first students have also done the new GCSE and will thus be better prepared. This has been agreed by the government.

� Only 24% of respondents said that the content for A level maths was appropriate. It doesn’t give a breakdown of whether respondents were from schools or higher education. But very little has been changed.

Consultation Response

� ALCAB noted that most decisions about which modules are made on the basis of teacher expertise/preference, not student plans, so using this as a reason for retaining flexibility is considered spurious. That seems reasonable to me!

� Universities are unanimous in wanting 100% prescribed content for A level Mathematics, with good reason. It is not possible to please everyone, and there are many advantages in this prescription. We have to accept this.

Consultation Response

� Some content has been removed, and some moved from AS to A2 to respond to concerns about too much content in AS.

� ‘Sections of proof (have been) strengthened’

� Decision Maths is out. This has been controversial, but again, we have to accept this. There are good arguments for removing it as well as for retaining it. It does at least give something to teach y12 further mathematicians which is not dependent on A level maths!

Consultation Response

� Remember this is building on the new GCSE

� Ensuring your students are ‘A level ready’ will now match preparing them for GCSE more effectively

� Content is 100% specified for Mathematics and 50% specified for Further Mathematics

Content - General

Specifications must encourage students to:

� reason logically and recognise incorrect reasoning

� generalise mathematically

� construct mathematical proofs

� …decide on the solution strategy

� …solve a problem in context

� …understand the relationship between problems in context and mathematical models…to solve them

Content – General (Maths and FM)

Specifications must encourage students to:

� read and comprehend mathematical arguments…

� read and comprehend mathematical articles…

� use technology such as calculators and computers effectively…

Content – General (Maths and FM)

� Specifications must require students to demonstrate the following overarching knowledge and skills.

� These must be applied, along with associated mathematical thinking and understanding, across the whole of the detailed content.

Overarching themes (Maths and FM)

� Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics

� Construct extended arguments to solve problems presented in an unstructured form, including problems in context

� Understand the concept of a mathematical problem solving cycle

� Translate a situation in context into a mathematical model, making simplifying assumptions

Overarching themes (Maths and FM)

� The use of technology, in particular mathematical and statistical graphing tools and spreadsheets, must permeate the study of AS and A level mathematics

Use of technology

� Calculators used must include the following features:

– an iterative function

– the ability to compute summary statistics and access probabilities from standard statistical distributions

– the ability to perform calculations with matrices up to at least order 3 x 3 (FM only)

GRAPHICAL CALCULATORS ARE THEREFORE GOING TO BE COMPULSORY

Use of technology

� Must use y – y1 = m(x – x1)

� Some circle geometry:

– Angle in a semicircle is a right angle

– Perpendicular from centre to chord bisects the chord

– Radius perpendicular to tangent

� Trigonometrical functions for all angles

Content – AS pure: In!

� y = ex including (informally) differentiating y = ekx; y = ln x

� Use of exponential growth and decay in modelling

� Differentiation from first principles

� Vectors in 2D, not including scalar product

Content – AS pure: In!

� No series work except binomial expansion (all in A2)

� No radian measure

� Trapezium rule

Content – AS pure: Out!

� Sampling techniques

� Interpreting diagrams

� Interpreting (but not calculating) regression lines

� Measures of central tendency and variation, including calculating standard deviation

� Binomial distribution as a model

� Hypothesis testing using binomial model

Content – AS: statistics

� Vectors in 2D

� suvat equations

� Displacement/time and Velocity/time graphs, gradients and areas

� Use calculus in kinematics

� Newton’s laws

� Weight and motion in a straight line under gravity

� Equilibrium

Content – AS: mechanics

� Proof by contradiction (including irrationality of √2 and infinity of primes, and application to unfamiliar proofs)

� Use of functions in modelling, including consideration of limitations and refinements of models

� All work on series now in A2

� Small angle approximations for sin, cos and tan

� Exact values for sin, cos and tan

� Arcsin, arccos, arctan

Content – A2 pure: In!

� Proofs of addition formulae

� Find your own substitutions for integration (simple)

� Newton-Raphson method

� Trapezium rule now in A2

Content – A2 pure: In!

� Volumes of revolution

Content – A2 pure: Out!

� Conditional probability, Venn diagrams

� Modelling with probability

� Normal distribution

� Hypothesis testing with correlation coefficients (not including calculating them)

� Hypothesis tests for the mean of a Normal distribution with known, given or assumed variance

� Interpret in context

Content – A2 statistics

� Projectile motion

� Resolving forces

� Dynamics for motion in a plane

� Friction

� Moments in simple static contexts

Content – A2 mechanics

� 50% content specified, all pure

� This is where you’ll want to look carefully at the specifications from each exam board

� They should all be producing specifications that can run alongside Mathematics

� Probably they will, after noting the experience of Edexcel in failing to do that initially and having to rewrite their specifications when they found everyone was doing other exam boards for FM!

Further Mathematics

� Note that this is designed to be a separate, valuable qualification

� But that it must be able to be co-taught with the full A level in Further Mathematics

� 20% of AS FM is specified

� A further 10% must be chosen from the compulsory material for A level FM

AS FM

� Complex Numbers

– Solving any quadratic

– Four rules

– Polynomial equations – roots in conjugate pairs

– Argand diagrams

– Modulus-argument form and conversion

– x and ÷ in mod-arg form, including use of radians and compund angle formulae

– Simple loci

Content – AS Further Mathematics

� Matrices

– +, -, x

– Linear transformations in 2D and some 3D (3D vectors assumed)

– Successive transformations in 2D

– Invariant points and lines

– Determinants and inverses of 2x2 matrices

� Roots and coefficients of polynomial equations up to quartics

Content – AS Further Mathematics

� Plus a small selection of topics from the remaining compulsory material for A level Further Mathematics

– This will be chosen by each exam board, but in practice there will be little choice given dependency on A level mathematics

– Suitable topics for AS highlighted in bold in the following slides

Content – AS Further Mathematics

� Proof by induction (series, divisibility tests, mat rices)

� Complex Numbers

– De Moivre

– eiθ

– Complex roots

– Geometrical problems

� Matrices

– Inverses of 3 x 3 matrices

– 3 linear simultaneous equations and geometrical interpretations

Content – A2 Further Mathematics

� Standard formulae for sums of series

� Method of differences including partial fractions

� Maclaurin series including awareness of validity

� Vectors

– Equation of line in 3D and plane in vector and Cartesian forms

– Scalar product and applications

– Intersection of line and plane

– Perpendicular distance between 2 lines, point to line, point to plane

Content – A2 Further Mathematics

� Calculus

– Integration where integrand extends to infinity

– Volumes of revolution

– Mean value of a function

– Integration using partial fractions with quadratic factors

– Integration inverse trig functions

– Integration using trigonometric substitutions

Content – A2 Further Mathematics

� Polar coordinates including area enclosed by polar curve

� Hyperbolic functions

– Definitions

– Graphs

– Differentiation and Integration

– Inverse hyperbolic functions including logarithm forms

– Integration with hyperbolic substitutions

Content – A2 Further Mathematics

� Differential equations

– Integrating factor

– General and particular solutions

– Modelling

– Second order equations (auxiliary equation for homogeneous equations; non-homogeneous equations using complementary function and particular integral)

– Simple harmonic motion equation

– Model damped oscillations

– Model situations with 1 indpt and 2 dpt variables as a pair of coupled 1st order simultaneous equations eg predator/prey

Content – A2 Further Mathematics

� You could think of the compulsory AS FM as being MEI FP1 without the graphs and inequalities and with the extra contexts for proof by induction from Edexcel FP1

– Assuming they put proof by induction and standard summations in AS FM

– It’s difficult to see what else from the compulsory material could go into AS FM, so the apparent choice is somewhat misleading

– It represents about 1/3 of the content

Comparison with current specs

� Remaining content for AS FM?

– There are likely to be options even within exam boards

– Choice of options will depend on whether you are running AS FM alongside or after AS Maths

– Decision Maths is likely to be a part of it – what else can you do that doesn’t depend on other things?

Comparison with current specs

� Compulsory material for the full A level FM is basically the union of Edexcel and MEI FP2 specifications but without the complex transformations

� With the AS material, this makes up half the A level:

– MEI: FP1 + FP2 + Differential Equations + Volumes of revolution

– Edexcel: FP1 (part + some extra) + FP2 + Matrices + Volumes of revolution

Comparison with current specs

� Remember content is 100% specified for Mathematics, so all boards will cover the same content

� Choose the exam board which looks most fun to teach, not the one that looks ‘easiest’ – learn from our experience with the current setup: easy leads to boring, and unmotivated students don’t achieve high grades

� At the moment I can only see small reasons for using the same exam board for Mathematics and Further Mathematics (eg familiarity with the formula book) – there are no modules to swap around, so be open-minded

Which exam board?

� Don’t make rushed decisions

� We’ll set up a network meeting for examining the different specifications and discussing the pros and cons of each, as soon as all the specifications are published

� Keeping the results of this, we can have a second meeting looking at resources

� Consider the advantages of neighbouring schools doing the same exam board - but only if you would be happy with the choice

Which exam board?

� Mathematics

– Work out your CPD needs: you can do this from the DfE documents – there is little room for manoeuvre for the exam boards

– Let Alexandra know what support you need

– Wait for the exam boards to publish their specifications

– Wait for the publishers to produce resources

– Set aside time to research the options and make an informed decision about the best exam board for you and your students

What now?

� Further Mathematics

– Work out some of your CPD needs: you can do this from the DfE documents

– Let Alexandra know what support you need

– Wait for the exam boards to publish their specifications

– Wait for the publishers to produce resources

– Set aside time to research the options and make an informed decision about the best exam board for you and your students

– Confirm any additional CPD needs

What now?

� Thursday 30 th April – post-16

– Venue and theme tbc – suggestions please!

� Wednesday February 25th: Subject Leaders' Coffee and Pi for HoDs or their representatives2pm to 6pm Venue TBAPreparing to teach the new National CurriculumWe hope to collaboratively produce differentiated resources to support teaching of the 'pockets' of understanding our current KS3 students may have missed out on but will need for the new GCSEs.Please sign up by Friday February 20th

Next Network meetings…