Mathematics and the transition from Alevels to physics and engineering degrees
Professor Peter Main
Mathematics in STEM: Transition Issues
RAEng 16 th January 2012 peter.main@iop.org

Two Studies
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Mind the Gap
 A study based on the perceptions of students and academics
 Students include those who did choose physics or engineering as well as those who could have done but chose not to.
SCORE Maths in Science A-levels
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Investigations and characterisation of the mathematical content of science A-levels in absolute terms and relative to what is in the curriculum
Also feedback from academics and employers

Mind the Gap : Specific objectives


To understand the extent to which students are prepared to deal with the maths aspects of physics and engineering undergraduate courses




Transition from A-Level to degree
Reactions to mathematical aspects of degree courses
Most and least challenging mathematical aspects
Gaps in preparation
To understand reasons for not pursuing physics to degree level

A-Level subjects studied (%)
A Level Physics
AS Level Physics 1
0
2
0
A Level Maths (pure + mechanics)
Subject(s) studied at A/ AS Level
90
96
88
70
AS Level Maths (pure + mechanics)
A Level Maths (pure + statistics)
AS Level Maths (pure + statistics)
A Level Further maths
AS Level Further maths
0
5
6
4
3
43
47
40
40
14
16
13
10
19
23
16
13
26
35
42
50
20 40 60
84
85
83
83
Total (393)
Physics (180)
Engineering (183)
Computer Science (30)
80 100

A-Level grades high overall, but especially amongst those studying Further Maths
Percentage of respondents achieving A or A* grade
Maths (mechanics) (349)
Maths (statistics) (225)
97
75
85
97
Further maths
No further maths
Physics (357)
64
78
0 50 100
“Starting my A-levels, I was thinking about what degrees I might want to do.
My teachers said if you want to do Maths or Physics, you should try Further Maths, so that you’re prepared for degree level.
It made the A-level Maths seem easier in comparison” Physics student

A number of factors steered potential students away from
Physics degrees
Aptitude for
Maths at
GCSE = study at A-
Level
Maths &
Physics are a logical fit
Physics perceived more as
‘learning’ than
‘problem solving’
Compared to Further
Maths,
Physics not as challenging
Lack of awareness of potential career options
“I guess it’s the only thing that truly made sense to me I was good at it and the problem-solving and satisfaction that you get from that made me more enthusiastic about it. It just seemed like the natural thing to want to study it further” Maths student
“What you could do with a Physics degree was never mentioned to me or gone over in my classes. I don’t think it was ever mentioned. A lot of people who were interested in Physics did Engineering”
Maths & Physics joint honours student

Variation in extent to which expectations around mathematical content were met, both in terms of amount…
Further Maths: 38%; No Further Maths: 57%
Total (393) 14 33 47 5 1
Physics (180) 11 38 49 2
Engineering (183) 15 30 45 9 1
Computer Science (30) 23 27 47 3
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
There is a great deal more mathematical content than I expected
There is somewhat more mathematical content than I expected
The degree of mathematical content meets my expectations
There is somewhat less mathematical content than I expected
There is a great deal less mathematical content than I expected

…and difficulty
Further Maths: 36%; No Further Maths: 56%
Total (393) 9
Physics (180) 4
Engineering (183) 14
36
42
42
42
39
12
11
1
1
32
Computer Science (30) 7
0%
30 53
20% 40% 60%
It is a great deal more difficult than I expected
It is somewhat more difficult than I expected
It is at the level I expected
It is somewhat easier than I expected
It is a great deal easier than I expected
80%
13 2
10
100%

Students and academics differed in their perceptions of students ability to cope with mathematical content
Further Maths: 88%; No Further Maths: 66%
60 19 Total (393) 5
Physics (180) 5
17
16 58
Engineering (183) 5
Computer Science (30)
16 61
30 60
Not at all well Not very well Quite well Very well
21
17
10
Academics (40)
2 53 43 2
Not at all well prepared Not very well prepared Quite well prepared Very well prepared

Vast majority of academics also agreed that students joining their course lacked fluency in Maths
Level of agreement that incoming students lack fluency in Maths
Academics (40)
2 2 5 63 28
0% 20% 40% 60% 80% 100%
Strongly disagree Somewhat disagree Neither agree nor disagree Somewhat agree Strongly agree
*Base: 36 academics
“They don’t usually admit that they’ve got a problem. They don’t quite understand what problem they’ve got. They know they are not quite understanding it but they can’t pin point where the problem lies”
Engineering academic

Specific areas of difficulty according to students and academics
Elements that academics perceived students to find particularly challenging
Integration
Being able to identify the…
Vectors and scalars
Complex numbers
Calculus
Algebra
Matrices
Logs
Differentiation
Exponentials
Transposing equations
Geometry
Other
None
Elements that students found particularly challenging
Integration
Being able to identify the appropriate…
Vectors and scalars
Matrices
Calculus
Complex numbers
Differentiation
Transposing equations
Geometry
Exponentials
Logs
None
Total (393)
Physics (180)
Engineering (183)
Computer Science (30)
0 50 100
0 50 100

Many academics believed there could be long-term consequences
92% academics felt a lack of mathematical fluency could be an obstacle to achieving full potential
3 5 61 31
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Strongly disagree Somewhat disagree Neither agree nor disagree Somewhat agree Strongly agree
85% academics felt this affected their departments’ ability to deliver an optimal programme of study
2 5 8 54 31
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Strongly disagree Somewhat disagree Neither agree nor disagree Somewhat agree Strongly agree

... long-term consequences
“The lack of any proper maths at A-Level
Physics meant that I felt quite overwhelmed and had to learn the skill of deriving physical meaning from maths, something I'm still having to pick up on”
Physics student


This can prove to be an obstacle throughout the course
A small number (ranging from 5-15%) drop out each year, but this cannot be directly linked to a struggle with mathematical content
“Before I came to university I was really confident with my mathematical ability. The first year I was good, but now the problems are getting more and more vast, hard and complicated. Sometimes I look at a problem and barely even attempt it because I just get it in mind that it's really difficult. Two years ago I would never have done that, so it’s starting to get a bit more daunting” Physics student
“Mostly they’ve covered a wide and appropriate range of mathematical concepts and techniques.
I do think they lack practice and fluency. I think many of them have a rather poor feel for numbers and find it very difficult to do estimates, quick checks, if there is a reasonable answer […] I think they’re quite poor at unstructured problem solving”
Physics academic

A number of reasons offered for perceived lack of fluency
TEACHING TO EXAMS
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
Many academics believed that A-Level Maths is taught too heavily to exam
Some agreement from students
 Equations/ techniques taught by rote
 Leads to a lack of understanding of the context in which techniques should be applied
“I always say you often see equation democracy. If you look at A-Level Maths text books nowadays they look like every equation is as important as every other equation. The only way you could possibly do maths is to memorise every equation in the world, whereas there are relatively few equations that are important….” Engineering academic
“I was startled by how different university level maths is, lots more rigour. I wasn't very well prepared for this by my school's teaching to pass exams” Physics student

A number of reasons offered for perceived lack of fluency
COMPARTMENTALISATION
 Too little connection between Maths & Physics at A-Level
 Too little connection between modules
 Further compounds inability to see the link between mathematical techniques and their place within Physics/ Engineering
“If you learn technique a in a unit, let’s say, in a module on technique a, you are never ever expected to use it in a course or module in something different. So there’s absolutely no transferable skill that they are learning…..
They’ve never seen maths as toolbox for physics before. It is purely down to compartmentalisation” Physics academic

Possible solutions suggested from academic perspective
“Introducing more mathematical, problemsolving type questions into the Physics A-
Level seems more appropriate. Getting across the importance of Maths and practising how it is used in Physics would be very valuable”
Academic
“Change the way Maths is taught at A-
Level; provide the fundamentals, not just recipes to apply in specific cases”
Academic
“The content [of Further Maths A-Level] is clearly helpful. In their first semester here, a lot of the
Maths that we’re teaching them is essentially the content of A Level Further Maths, because we take students who haven’t had maths. So they have a gentler lead in and more time to get used to the differences in style and shape of problems that we’re asking them to tackle” Physics academic
“If they were music students they would need to practice their scales.
They just need more practice, until algebraic manipulations, powers, logs etc all come naturally and quickly to them. Then they can start to play the sonatas, concertos and symphonies that are Physics”
Academic
“Those who have taken Further Maths at A-level are fairly well-prepared. It appears necessary to give those without this background a different introduction. It is not so much a question of mathematical ability but more using maths in a Physics problem
– reducing a Physics problem to a Maths problem that they can solve.”
Academic

Key points
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Lack of awareness of what a physics degree involves or can lead to
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A level should give a better foretaste of degree work
Stronger careers promotion needed
Academics and students recognise problems with maths preparation
 Academics perceive a greater problem than students
Further Maths is regarded as helpful for degree preparation
Practice makes perfect
Teaching to the Test and Compartmentalisation at
A level
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Does not encourage use of maths within physics
Does not give a realistic foretaste of degree work

Physics
1e. Trig. F’ns in calculators
2c. Means
2d. Percentages 5d. Log graphs
3f. Solve e.g. y=k/x 5f. y = mx + c
5g. Rate of change
5h. Tangent

1. Influence on what is taught
• Authenticity of experience of the sciences
• Providing tools to fully understand the science
• Rewarding explanations
• Preparation for further study
2. Confidence in standards
3. Confidence in the system