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Using further mathematics

Introduction

The purpose of this material is to share information on how schools and colleges are using further mathematics. It is based on visits to 10 further education colleges, three sixth form colleges and two schools. It describes common approaches to:

• how further mathematics is used within the curriculum
• which students are offered further mathematics
• how further mathematics is marketed
• teaching and supporting students.

Further mathematics is available at both AS and A level and the course is offered by the four main awarding bodies. The number of students studying further mathematics in the schools and colleges ranges from about five to 50.

Why offer further mathematics?

In most schools and colleges, further mathematics is introduced to challenge able students and prepare them for a degree in mathematics or a mathematics-related subject such as statistics, physics, engineering, computer science or economics.

How is further mathematics used within the curriculum and taught?

There are a variety of ways in which schools and colleges offer further mathematics. These are:

• completing six modules in year 12 (equivalent to a single A level mathematics). Some students then drop further mathematics to concentrate on their other A level subjects. This, although discouraged by some centres, enables students to gain an extra A level
• completing nine modules and dropping the subject part way through year 13, with the outcome of an A level in mathematics and an AS level in further mathematics
• taking AS level further mathematics in year 13, following AS level mathematics in year 12
• completing the full 12 modules, by continuing study in year 13. This option is chosen by most students and is favoured by most school and colleges.

The stated aim of teaching further mathematics is to stretch students. For this reason topics are covered quickly and in depth and students are given a lot of homework (up to five hours a week). There are three main approaches used to teach further mathematics:

1. Taught separately throughout the course
   
   Students are taught in separate groups from the A level mathematics groups throughout the course from the start of year 12. Some centres offer double time to their further mathematics group, allocating teaching time for a full subject, while others manage to deliver the course with half to two-thirds more teaching time than required for a single mathematics group.
   
   
2. Taught partly with single A level mathematics group and partly separately
   
   Students are taught in a single A level mathematics group throughout their course with additional separate sessions for further mathematics. This is less demanding of teaching time than if further mathematics is taught separately but requires extra slots in the timetable. These lessons have to be fitted in when students are not being taught other subjects, such as in non-contact periods, lunchtime or after school.
   
   It also requires careful management of the order in which topics are taught so that single mathematics topics are taught before they are required for further mathematics. Some centres teach further mathematics students in year 12 and 13 together for some topics, with a consequent reduction in time required.
   
   
3. Further mathematics only taught in year 13
   
   Students taking a mathematics A level course (three units or six units) in year 12, defer their choice to study further mathematics (three units or six units) until year 13. The course is covered in one year and students can gain either an AS (if three units are completed) or A level (if six units are completed) in further mathematics at the end of year 13. This approach can reduce the amount of teaching time by running separate further mathematics lessons only in year 13, however, it does put extra pressure on students because they only have two terms.

Further mathematics can also be taught through distance learning schemes that enable schools and colleges with few candidates or no previous experience of further mathematics to offer it to students.

In most centres the students decide at the end of year 12 whether to continue in year 13. However, in some centres it is the teacher who decides depending on the individual student’s achievements.

The order in which modules are taught in schools and colleges varies. Some complete C1–C4 and two applied modules in year 12, which allows successful candidates to complete A level mathematics in one year. Those continuing with further mathematics in year 13 would then study FP1, FP2, FP3 and three other modules. Other centres start teaching FP1 in year 12 to allow students to gain the modules necessary for an AS in mathematics and an AS in further mathematics at the end of year 12.

Which students are offered further mathematics?

Further mathematics is usually offered to students who gain an A* or A at GSCE. Students who gain a grade B at higher tier GCSE are sometimes able to study further mathematics if recommended by their teacher. However, it can be offered to all A level mathematics students, regardless of prior attainment as shown in the High Pavement Sixth Form College case study.

How is further mathematics marketed?

Further mathematics is marketed to students at open days, options choices events and during visits from potential school or college teachers. In schools, suitable students are encouraged from year 10 and 11 to consider further mathematics through looking at course materials or inspired by the enthusiasm from staff and students on the course.

Further mathematics is presented by schools and colleges as:

• a highly regarded qualification
• a fascinating subject and a tool for other subjects
• a means of showing ability in a competitive market
• a way of increasing students’ confidence (and their grade) in single mathematics
• a way to impress universities and a positive entry on a UCAS form. It is vital for studying some mathematics or mathematics-related subjects. Some universities include further mathematics in their offer to students and even give a reduced offer when further mathematics is included
• the right choice for able mathematicians, providing an opportunity to do more mathematics, stretching students and providing excitement
• a way for able students to gain an A level in one year (year 12) keeping options open in year 13 to continue with further mathematics, take up another subject at AS level or concentrate on their other A level subjects.

Here is a flowchart used by Aquinas College to market further mathematics to potential students.

    

Approaches taken to support the students

At the beginning of the course students are usually provided with an introductory booklet or access to a subject guide on the school or college’s intranet. These can include information on:

• the structure of the course
• schemes of work
• the content of the modules
• how the course is assessed
• how the subject is taught and teaching time allocated
• the expectations on students
• resources provided and needed
• revision exercises in algebra and numerical fractions, with worked examples and answers
• additional support, such as details of workshops.

Throughout the course, support materials are available on school or college intranets or in workshops, and include revision sheets, profile sheets which test each topic, model answers to some homework questions, enrichment material, textbooks for each module, STEP papers, past papers and solutions.

Workshops are provided by most schools and colleges and are mainly run in the teachers’ and students’ own time. Workshops available to all students are drop-in help clinics or revision workshops run in the lead up to examinations. Workshops for particular students are held:

• to help students scoring low marks in a diagnostic test given at the beginning of the course
• to help students retaking modules
• to provide more teaching time for the C1 module
• to practise for the Senior Mathematics Challenge that most further mathematics students volunteer to enter
• to provide extra support for students aiming for Oxford or Cambridge in preparation for the entrance exams
• to support students who join the course in year 13
• to study optional additional modules.

Many schools and colleges develop students’ self-reliance and independent learning, as essential skills for the course and future studies in university. In some, students are encouraged to mark their own work and learn to find their mistakes (the work is also checked by teachers). Others encourage students to explain problems they understand to students that do not understand. One college teaches students to follow five steps in mathematical thinking when tackling questions.

1. List the key words, drawing diagrams if appropriate and rewriting the question in their own words, to help understand the questions.
2. Plan how to do the problem by specifying the steps.
3. Do the problem.
4. Review the problem by devising a set of questions to check that the answer is correct and analysing what has been learnt.
5. Present the review to other students.

Students are also encouraged to do speed reading where they flick through the textbook ,recognising familiar words in preparation for the examination.

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Page updated: 12 June 2008
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