McCabe, Michael

English

Portsmouth University Research Portal (Pure)

The exponential growth of mathematics
and technology at the University of
Portsmouth
MICHAELMCCABE
Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace,
Portsmouth, Hampshire PO1 3HF, UK
[Submitted June 2009; accepted October 2009]
The number of students studying university mathematics in the UK has been increasing
gradually and linearly since 2002. At the University of Portsmouth, number of students
studying mathematics doubled from 30 to 60 between 2002 and 2007, then increased by
240% in just 1 year to over 140 in 2008. This article explains how learning technology has
supported such a dramatic growth and continues to provides inspiring resources for first
year students (see McCabe, ICTMT-x, x=3 . . .8). Maple 12 and Matlab applications are
provided inWebCT Vista with e-assessment questions delivered in QM Perception and
MapleTA. This combination has been extremely effective in scaling up course delivery
to ‘exponentially’ larger classes. In particular, the assessment of mathematical problem
solving with reusable assessment objects has encouraged students to learn.
1. Background
Past papers by the author at ICTMT conferences have focused on different aspects of technology use in
teaching mathematics at the University of Portsmouth: computer-based learning with computer algebra
(McCabe & Watson, 1997); online e-assessment (McCabe, 1999); computer-assisted assessment for
higher level learning (McCabe, 2001); interactive classrooms (McCabe & Lucas, 2003); blended
learning in assessment (McCabe, 2005); and reusable assessment objects in mathematics teaching
(McCabe, 2007). During this period, number of students starting courses in mathematics was rarely
more than 60 (Fig. 1). In 2008–2009, this figure rose dramatically to over 140 and is expected to be
comparable in 2009–2010.
While some uses of technology are difficult to scale up, others are more straightforward to use
without compromising teaching standards. Our growth in student numbers brings this issue into
sharp focus.
2. Scaling of mathematics technology
The scalability of human, traditional and technology resources for teaching and learning varies
(Table 1). Lectures can be fitted into larger rooms without an increase in staffing, whereas small

Email: michael.mccabe@port.ac.uk
Teaching Mathematics and Its Applications (2009) 28, 222^227
doi:10.1093/teamat/hrp021 Advance Access publication 24 November 2009
 The Author 2009. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications.
All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org
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group activities or individual support require more staff. Our university provides distributed mathe-
matics support through personal consultations at a large number of Maths Cafe´ locations around the
campus, a labour-intensive and time-consuming activity, where the benefits outweigh the scalability.
The printing of extra copies of lecture notes carries few overheads, while the purchase of extra books
is possible—at a price. Traditional mathematics assessment, whether exam papers or coursework,
is generally time-consuming to produce and even more time-consuming to mark. Scalability has to
be judged by many different criteria, e.g. cost, time, efficiency and effectiveness.
The introduction of technology does not always imply greater scalability. For example, handsets and
other tools for supporting interaction in the classroom (McCabe & Lucas, 2003) can be straightforward
to use and manage for moderate class sizes, but can become difficult to manage for larger (>100)
classes. In contrast, online resources are readily made available to large numbers of students and
a WebCT Vista (Victory) ‘presence’ is a requirement for all first year units at Portsmouth (Fig. 2).
In one mathematics course unit, Victory has been used for the submission of weekly student learning
diaries, including a portfolio of worked examples and Maple worksheets.
FIG. 1. The growth of mathematics at the University of Portsmouth. Student entries to year 1 (solid line) and
UK university entry exams (dotted line). (This figure appears in colour in the online version of Teaching
Mathematics and its Applications.)
TABLE 1. The scalability of teaching resources
Resources Human Traditional Technology
Scalability
High Lectures Printed lecture notes Large/dynamic question banks, basic use
of virtual learning environment (VLE),
computer based learning (CBL).
Medium Small-group tutorials, computer
practicals
Books Small/static question banks, computer-
generated coursework, mathematical
software
Low Individual support (Maths Cafe´) Written exam papers and
coursework
Classroom handsets e-learning diaries/
portfolios
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This is easy enough, but the assessment of these artefacts is definitely not scalable, if they are to be
examined in their entirety. It is straightforward to distribute Maple worksheets for students to try out
and modify, but nearly impossible to check through the detailed workings generated by a large class.
3. Computers and assessment
Question Mark Perception online tests have been the workhorse for assessing first year algebra and
calculus at Portsmouth for >10 years. This software has been used to deliver computer-marked tests
with apparent scalability as number of students increase. The main problem is that the question banks
used are relatively small and static, requiring strict security measures and safeguards to ensure that
their integrity is not breached.
MapleTA has the ability to generate ‘reusable assessment objects’ (McCabe, 2007), which provide
genuine scalability by delivering questions and providing feedback dynamically. Since its introduction
in 2006, an increasing variety of MapleTA questions have been developed (Table 2) to create ‘almost
infinite’ banks of reusable questions. These questions are freely available for non-commercial use from
the author.
The question banks are in three main categories:
(1) support for HELM (Helping Engineers Learn Mathematics) worksheets, including some topics
covered in first- and second-year mathematics courses;
(2) problems which are difficult to solve by hand, specifically designed for solution using mathe-
matical software, Maple or Matlab;
(3) problems in graph theory, including networks.
Exhaustive testing of dynamically programmed questions is always time-consuming, because many of
the question components change. A large proportion of these questions have been tested through ‘peer
moderation’, by releasing them to students in advance of formal assessments. Any faults are quickly
spotted by the stronger students and are corrected long before the weaker students attempt them! The
variability of the questions reduces the need for tight security and online tests can even be made
available over an extended period as coursework rather than as invigilated exams.
The incentive for students to attempt practice questions is high and almost 100% try out tests which
they know to be closely comparable with their summative assessments.
FIG. 2. Victory, the University VLE, used for learning diary submissions. (This figure appears in colour in the
online version of Teaching Mathematics and its Applications.)
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A significant concern is therefore that students learn to solve these reusable questions ‘by rote’,
without an adequate understanding of the mathematics involved. Regular computer-marked tests
generally lead up to a more traditional written exam and detailed analysis of the latter results strongly
supports the effectiveness of this approach. More subjective feedback from students is also encourag-
ing, routinely quoting the availability of practice tests as one of the most useful study resources which
support their learning.
4. Engaging applications
Mathematical problem solving needs a wide variety of interesting applications to maintain student
interest. For example, at ICTMT-7, Challis & Gretton (2005) described how rollercoasters can ‘give
mathematics a lift’. Students can use a computer algebra system, such as Maple, to experiment with
different fairground designs, e.g. to determine the jerk and jounce (third and fourth derivatives) of a
track or find any track with a specified jerk and jounce (Fig. 3). MapleTA has been used to generate
a bank of questions covering applications like this, which require a computer algebra system such
TABLE 2. Question banks of reusable assessment objects for MapleTA
Basic Maple arithmetic (21 questions)
Basic Maple algebra (15 questions)
Basic Maple plotting (16 questions)
Further Maple algebra (21 questions)
Maple equations, inequalities, functions and limits (11 questions)
Maple differentiation (10 questions)
Maple integration (10 questions)
Maple applied max/min (10 questions)
Matlab and matrices (33 questions)
HELM 02: Basic Functions (21 questions)
HELM 03: Equations and Inequalities (11y questions)
HELM 07: Matrices (20 questions)
HELM 08: Matrix Solution of Equations (7 questions)
HELM 09: Vectors (6 questions)
HELM 10: Complex Numbers (2y questions)
HELM 18: Functions of Several Variables (16y questions)
HELM 19: Differential Equations (11 questions)
HELM 22: Eigenvalues and Eigenvectors (12y questions)
HELM 28: Differential Vector Calculus (66 questions)
HELM 29: Integral Vector Calculus (26 questions)
HELM 35: Sets and Probability (11 questions)
HELM 36: Descriptive Statistics (5 questions)
HELM 37: Discrete Probability Distributions (8 questions)
HELM 38: Continuous Probability Distributions (4 questions)
HELM 39: The Normal Distribution (6 questions)
HELM 40: Sampling Distributions and Estimation (5 questions)
HELM 41: Hypothesis Testing (4 questions)
HELM 43: Regression and Correlation (6 questions)
HELM 46: Reliability and Quality Control (43y questions)
Graph Theory—Graphs (20 questions)
Graph Theory—Networks (17 questions)
Astrobiology—ABAQUS question bank (300 questions)
Helping Engineers Learn Mathematics (HELM), http://www.lboro.ac.uk/research/helm
yQuestions generated by final year student projects.
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as Maple to solve them. Questions are both closed- and open-ended. Above all, they are scalable,
because their random components allow them to be reused.
5. Conclusion
The ‘exponential’ growth of number of students studying mathematics at the University of Portsmouth
is encouraging. A proportionate increase in staff numbers helps, but does not deal with all the issues,
especially assessment. Scalable technology is important, but it means more than simply making paper
resources available electronically. ‘Reusable assessment objects’ are valuable tools for maintaining
scalability without reducing student engagement.
REFERENCES
CHALLIS, N. & GRETTON, H. (2005) Giving Mathematics a Lift, ICTMT-7, Bristol (see http://maths.sci.shu.ac.uk/
mathsstaffpages/hwg/Papers/bristol2005.pdf) [accessed 12 November 2009].
MCCABE, E. M. (1999) WHAM! Web Hosted Assessment of Mathematics, ICTMT-4, Plymouth, ISBN 1-84102-
056-7 (see http://userweb.port.ac.uk/mccabeem/ictmt4paper.doc) [accessed 12 November 2009].
MCCABE, E. M. (2001) Computer Assisted Assessment (CAA) of Proof=Proof of CAA New Approaches to
Computer Assessment for Higher Level Learning, ICTMT-5, Klagenfurt (see http://userweb.port.ac.uk/
mccabeem/ictmt5paper.doc) [accessed 12 November 2009].
MCCABE, E. M. (2005) Assessment, Blending and Creativity: The ABC of Technology in Mathematics Teaching,
ICTMT-7, Bristol (seehttp://userweb.port.ac.uk/mccabeem/ictmt7paper.doc) [accessed 12 November
2009].
MCCABE, E. M. (2007) Reusable Assessment Objects in Mathematics Teaching, ICTMT-8, Hradec Kralove
(see http://userweb.port.ac.uk/mccabeem/ictmt8paper.doc) [accessed 12 November 2009].
MCCABE, E. M. and LUCAS, I. (2003) Engagement with Mathematics in an Interactive Classroom, ICTMT-6,
Volos (see http://userweb.port.ac.uk/mccabeem/ictmt6paper.doc) [accessed 12 November 2009].
FIG. 3. Maple Rollercoaster design and analysis with MapleTA assessment. (This figure appears in colour in the
online version of Teaching Mathematics and its Applications.)
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MCCABE, E. M. and WATSON, J. (1997) From MathEdge to Mathwise: The Cutting Edge of Interactive Learning
and Assessment in Mathematics, ICTMT-3, Koblenz, ISBN 3-00-002330-5.
Michael McCabe is a Principal Lecturer and National Teaching Fellow in the Department of Mathematics
at the University of Portsmouth. He has actively promoted the use of technology in the teaching and
assessment of both mathematics and astronomy for the past 20 years. Previously he worked for 10 years as a
research officer in the UK electricity industry. His research degree was in astronomy and chemistry, having
Nobel laureate Sir Harry Kroto as an internal examiner.
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227 at University of Portsm outh Library on

Assessment, Blending and Creativity:

Computer Assisted Assessment (CAA) of Proof=Proof of CAA New Approaches to Computer Assessment for

Giving Mathematics a Lift, ICTMT-7,

Maple Rollercoaster design and analysis with MapleTA assessment. (This figure appears in colour in the online version of Teaching Mathematics and its Applications.) EXPONENTIALGROWTH OF MATHEMATICS AND TECHNOLOGY226 at University of Portsm outh Library on

Reusable Assessment Objects

WHAM! Web Hosted Assessment of Mathematics,

The exponential growth of mathematics and technology at the University of Portsmouth

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The exponential growth of mathematics and technology at the University of Portsmouth

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