Foundations of Advanced
Mathematics
MEI Conference
June 2009
Presenter: Michael R Ling
“God’s Providence”
12 Trowell Grove
Long Eaton
Nottingham
NG10 4AZ
Michael_ling@btinternet.com
Foundations of Advanced Mathematics is a free standing Mathematics Qualification at
Intermediate level.
It was first examined in 1993, though it has undergone changes and evolution since then.
FAM is a broad-based course covering arithmetic, algebra, graphs, trigonometry and statistics.
The only other Intermediate FSMQs are offered by AQA - 'Calculation Finances,' 'Handling and
Interpreting Data' and 'Using Algebra, Functions and Graphs' which are limited in scope.
The original intention of FAM was as an access course for those wishing to study Mathematics in
Further Education but without the ground base of a certificate at GCSE Higher Level.
The objectives are:
• To provide access to AS and A level Mathematics for those not yet confident in being ready
to undertake them.
• To provide mathematics for students taking advanced courses in other subjects and those
preparing for higher education whose studies include a numerate element.
• To support the Key Skill Application of Number qualification.
A website on FSMQs suggest that the purpose of Intermediate awards is to provide a qualification
for students in yr 11 who have taken GCSE a year early. It is indeed being taken by students in this
category
Routes of progression
Students wishing to access AS and A level courses but whose mathematics at present is not
sufficiently strong for them to have reasonable prospects of success.
Many of these students will typically have a grade C, either by doing well on the limited syllabus
content of Foundation Tier or not having gained a very good grasp of the Higher Tier material.
There will be students who are returning to education after a break and this provides a good access
course.
There are students who learn their mathematics overseas. Their base of knowledge may be different
from typical students who have come through the British National Curriculum.
Students taking courses in subjects other than mathematics
Many GCE students taking sciences, business studies, etc are unable to fulfil their potential because
of lack of confidence in the use of Mathematical techniques.
Students who are undertaking or preparing to undertake a course in Higher Education which
includes a numerate element but who are not studying any other mathematics.
The majority of such students will be taking AS or A level course and some VCE courses or access
to Higher Education courses.
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FAM
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Students who take GCSE a year early but who demonstrate only a limited grasp of the Higher
Tier material and for whom “Additional Mathematics” might be too much
.
The broad base of the course will enable all these students to develop and consolidate the
techniques that they are most likely to use in future studies.
Guided learning hours.
It is anticipated that about 60 guided learning hours are needed to cover the course.
Overlap with GCSE Higher Tier.
There is a considerable amount of material common to both courses. However, a different approach
to the material is expected through a greater use of both vocational contexts and examples that are
appropriate to post- 16 candidates.
Specification aims
The aims of the specification are to encourage students to:
• Develop their understanding of mathematics and mathematical processes in a way that
promotes confidence and fosters enjoyment;
• Apply mathematical skills and techniques to solve problems in other subjects:
• Recognise how a situation may be represented mathematically and understand the relationship
between “real world” problems and standard and other mathematical models, and how these
can be refined and improved;
• Use mathematics as an effective means of communication;
• Acquire the skills needed to use technology such as calculators and computers effectively,
recognise when such use may be inappropriate and be aware of limitations;
• Develop an awareness of the relevance of mathematics to other fields of study, to the world of
work and society in general;
• Build a firm foundation for further study;
• Develop the skills required for the Application of Number Key Skill.
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Assessment Objectives
A course based on this specification requires students to demonstrate their knowledge,
understanding and skills in the following assessment objectives.
AO1 Recall, select and use their knowledge of appropriate mathematical facts, concepts and
techniques in a variety of contexts.
30 – 40%
AO2 Recognise and understand precise statements, logical deductions and inference; manipulate
expressions.
40 – 50%
AO3 Recognise and understand given mathematical representations of situations; interpret results
from such representations.
5 – 15%
AO4 Comprehend translations of common realistic contexts into mathematics.
5 – 15%
AO5 Appreciate whether or not given information is reasonable and is given to an appropriate level
of accuracy.
5 – 15%
In the context of this specification with 40 questions:
30 – 40% means 12 – 16 questions
40 – 50% means 16 – 20 questions
5 – 15% means 2 – 6 questions
Assessment
One examination of two hours is set in January and June each year. It is perhaps the only
mathematics examination which is multiple choice. Four responses are offered and candidates have
to choose one of the responses.
Calculating aids
The use of a scientific calculator is permitted. Computers, graphical calculators and those with
computer algebra facilities are not permitted.
Design Thresholds
The result is reported as a grade A, B, C, D, E or U.
Grade
Design Threshold/40
A
32
B
28
C
24
D
20
E
16
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FAM Statistics
June 2003
Jan 2004
June 2004
Jan 2005
June 2005
Jan 2006
June 2006
Jan 2007
June 2007
Jan 2008
June 2008
Jan 2009
Entry
516
764
712
837
662
788
672
768
1018
825
1265
702
Mean
23.2
24.28
22.01
24.7
22.1
23.4
21.1
23.1
21.5
23.4
22.3
24
SD
5.6
6.68
6.52
6.1
5.5
6.8
6.4
7.2
6.4
7.1
6.4
6.4
A
31
32
31
32
31
31
30
30
31
31
32
31
11.6%
14.8%
11.6%
15.7%
7.4%
16.9%
11.5%
18.1%
8.1%
18.5%
8.8%
17.1%
B
27
28
27
28
27
27
26
26
27
27
28
27
27.1%
32.6%
27.1%
32.1%
18.7%
32.7%
23.4%
37.0%
22.5%
34.1%
21.1%
35.6%
C
23 53.4%
24 53.7%
23 53.4%
24 55.9%
23 45.4%
23 53.7%
22 46.6%
22 58.0%
23 44%
23 53.7%
24 42.1%
23 54.7%
D
19
20
19
20
19
19
18
18
19
19
20
19
79.9%
73.8%
79.9%
77.7%
72.0%
72.3%
70.3%
76.9%
66.5%
75.5%
65.1%
80.2%
E
16
16
16
16
16
15
14
15
15
15
16
16
Number of questions in each strand
Number
Algebra
Graphs
Trigonometry
Vectors
Sampling
Display
Probability
Sampling/Display
Probability/display
Algebra/Trig
Number/Algebra
Number/Display
Trig/Graph
Jan
2007
11
12
4
4
2
June
2007
12
12
4
4
2
3
2
2
1
1
Jan
2008
10
12
5
4
2
1
2
2
June
2008
12
12
4
4
2
1
1
1
1
1
Jan
2009
12
13
5
3
2
1
2
2
June
2009
12
13
4
4
2
4
1
1
1
1
1
1
1
1
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92.3%
90.2%
92.3%
93.1%
90.0%
90%
87.9%
88.4%
85.5%
88.2%
85.5%
91.7%
Coverage of specification
There are syllabus statements which are difficult to test. E.g. FD2. “Be able to construct cumulative
frequency diagrams”.
This can be solved in part by asking candidates to complete diagrams and then interpret them. E.g.
June 2008, Q24.
E. g. “solve the following pair of simultaneous equations.”
In GCSE the instruction would be “solve algebraically….”.
E.g. B293, June 2009 Q6. There were 4 marks allocated, but the correct answer given with no
working gained only 1 mark.
What is to stop candidates substituting by trial and error? In which case the syllabus statement Fa14
“solve simultaneous equations” has not been met.
One way of getting round this is to set out a typical algebraic solution which is wrong and ask
candidates to identify where the first error is. E.g. June 2008, Q33.
Questions and distracters
We actively try to ensure a range of difficulty of questions. So there should be some with high
success rates and ideally these will be near the start of the paper to give candidates confidence.
The critical success rate is 25% that would result from guessing. Any figure significantly below this
means that candidates are actively getting it wrong and that one (or more) of the distracters is
picking up a common mistake or misconception or is confusing or unclear. The former is no bad
thing, the latter is unfortunate and a situation we would not choose to have.
Required responses
“Three of the following statements are true and one is false. Which one is false?”
This is the norm demand. It means that common errors have been identified and built into a
response that would lead to an incorrect answer.
Questions that include such a required response may take 4 different topics (e.g. June 2008, Q1) or
on related topics (e.g. June 2008, Q2).
“Which one of the following is …….”
This involves one question only, with (usually) one of 4 different responses. Candidates need to be
aware that the three incorrect ones come from typical errors.
e.g. June 2008, Q5, 10
This also includes options such as in June 2008, Q18.
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Strategies for candidates
• An “obvious” answer may not be obvious!
• The examiner is “guessing” what sorts of typical errors might be made. Have I made such an
error?
• Am I sure that the other three are correct/false?
• Is it ok to guess?
There may be others!
Some examples
June 2008 Question 34. Algebra
34
In the four statements below, n stands for an integer.
Three of the following statements are true and one is false. Which one is false?
A
n – 2 > 3 for the integers 6, 7, 8,… .
B
0, 1, 2 and 3 are the only integers for which n2 ≤ 9.
C
3 – 2n > 1 for the integers 1, 0, –1, –2, … .
D
2 < n + 6 < 10 can be rewritten as –4 < n < 4.
The response for this was A 11%, B 26%, C 26% D 37%
Compare this to a question on B293 this summer:
Solve the inequality 6x – 5 < 7
[2]
June 2008 Question 1 Arithmetic
1
Three of the following statements are true and one is false. Which one is false?
A
The highest common factor (HCF) of 42 and 70 is 14.
B
97 is a prime number.
C
1 1 1
+ =
4 12 3
D
15% of £80 is £10.00.
The response for this was A 5% B 3% C 4% D 89%
© MEI MEI Conference, 2009
FAM
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