RBrown CAME Reims 2003
Comparing system wide approaches to the introduction of
Computer Algebra Systems into examinations.
Roger Brown
International Baccalaureate Research Unit
University of Bath, UK.
There is a move towards the adoption of Computer Algebra Systems (CAS) assumed
examinations within many examination systems, this paper will consider two such
systems, the Danish Ministry of Education (DME) and the Victorian Curriculum and
Assessment Authority (VCAA) and will be guided by two questions


How are the examination writers responding to the introduction of CAS?
What is the role of the CAS within the examinations?
The paper will use a framework developed by the author Brown, (in Press-b) to
investigate the role that the CAS plays within the CAS assumed examinations in
comparison with similar non-CAS graphics calculator assumed examinations. The results
reveal surprisingly little difference in the types of questions used; however, the CAS
greatly enhances the range of solution strategies available to students.
Computer algebra systems in examinations
Many authors have described the impact of the introduction of CAS into mathematics
education and examinations, for example Meagher (2002) provided a review of the
implications of the use of computer algebra system (CAS) in curriculum and assessment.
Monaghan (2000), on the other hand, described a study in the United Kingdom, which
reviewed the implications of the introduction of CAS into A-level examinations. Whilst many
other authors have written about the possible effects of the introduction of the CAS into
mathematics examinations (e.g. Leigh-Lancaster & Stephens, 1996; Stacey, 1997; Drijvers,
1998; Hong, Thomas, & Kiernan, 2000; Kissane, 2000; Stacey, McRae, Chick, Asp, & LeighLancaster, 2000) only a small number of systems have moved to develop examinations in
response to these possible effects. Brown (2001) has provided an overview of CAS
examinations in Europe and the USA and detailed alternative styles of examination questions.
Additionally, a number of authors have developed schemes for classifying examination
questions, in terms of the impact that a CAS will have on the solution of traditional
mathematics examination questions (Kokol-Voljc, 1999; MacAogáin, 2000; Flynn &
McCrae, 2001). More recently Flynn & Asp (2002) and Ball & Stacey (2003) have
investigated the use of CAS in examinations from a students’ perspective.
Two examination boards
At the conclusion of a three-year pilot project, the Danish Ministry of Education (DME)
sanctioned the use of CAS assumed examinations from 2001. The policy enabled schools to
either choose to use examinations for which the graphics calculator (non-CAS) was assumed,
or examinations in which the CAS was assumed. The Victorian Curriculum and Assessment
Authority (VCAA) commenced a pilot project in 2001 in which the use of a CAS was to be
assumed in examinations, with the first cohort of students sitting these examinations in 2002.
These two examining boards therefore, are indicative of the early stages of the development
CAS assumed examinations and also provide examples of different forms of examination, see
Table 1.
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RBrown CAME Reims 2003
Table 1 Examining boards, the mathematics subjects offered and the forms of examinations.
Examining
Board
Title
Danish
Ministry of
Education
(DME)
B-level
Mathematics
(CAS Examination)
A-level
mathematics
(CAS Examination)
Victorian
Curriculum
and
Assessment
Authority
(VCAA)
Mathematical
Methods
(Pilot CAS
Examination)
Forms of Examination
1.
2.
1.
2.
1.
2.
CAS free short response.
CAS active extended answer
examination.
CAS free short response.
CAS active extended answer
examination
i) CAS active multiple-choice
ii) CAS short response
examination
CAS active extended answer
examination
For this paper the focus will be on the DME A-level and the VCAA Mathematical Methods
examination papers. The VCAA multiple-choice question examinations will not be included
in this study but form part of a further study by (Brown, in Press-a)
Analysing CAS examinations in Denmark and Victoria
In order to investigate the impact of the role that the CAS plays within examinations, it is
useful to firstly consider what role the graphics calculator has played in comparable
examinations to those in which a CAS is assumed.
The role of the computer algebra system and graphics calculator in end of
high school examinations
In a previous study of the transition from scientific calculators to CAS, Brown (2001) it was
reported that, as a result of the introduction of the CAS into the Danish A-level examinations
the following conclusions could be drawn.
 There was minimal change to many questions.
 Students do need to be very familiar with the functionality of their CAS and need
to be fully aware of the steps required to complete complex problems.
 The method of solution may be different for a CAS than for traditional pencil and
paper techniques.
 The form of the answer provided by the CAS will likely be different when using
the CAS
In an analysis of the 2001 DME CAS assumed examination paper DME (2001b) there were 4
questions (out of a possible 7) that were identical to the graphics calculator (non-CAS)
assumed paper DME (2001c) (amounting to 85% of the marks) whilst two questions (out of a
possible 7) included only minor changes to the wording of the question. This approach was
repeated in 2002 with 3(out of a possible 7) questions being identical for both the CAS, DME
(2002a), and the graphics calculator (non-CAS) assumed paper DME (2002b) (amounting to
75% of the marks), with a further 3(out of a possible 7) questions having minor word changes
as exemplified in bold in Figure 1.
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RBrown CAME Reims 2003
Let O(x) be the sum of costs, in million kroner, for a production of x tonnes of a commodity. The
function O(x) is given by
O( x) 17 x  100 cos(0.16 x)  150 , x 0 ; 30 
Find O’(x) and show that O(x) is an increasing function.
O( x)
There exists exactly one value of x, for which the unit cost
is equal to the marginal cost O’(x).
x
[Use your graphics calculator (in graphics calculator paper) to] Find this value of x.
Every tonne produced can be sold for 30.8 million kroner. Let F(x) be the profit, in million kroner, for
a production of x tonnes of the commodity.
Show that F ( x) 13.8 x 100 cos (0.16 x)  150 , x 0 ; 30 
And find x, so that F(x) is the greatest possible.
Figure 1 Question 3 taken from the 2002 DME A-level CAS and graphics calculator papers (DME, 2002a,
2002b)
An analysis of the 2002 VCAA graphics calculator assumed, and the CAS assumed short
response examination (Examination 1 part ii), VCAA (2002a, 2002c), along with the
extended response examination (Examination Two), VCAA (2002b, 2002d), provided a
similar result to the DME, with 34(out of 42) parts of the questions being identical to the
equivalent non CAS graphics calculator assumed paper, amounting to a total of 68% of the
marks, with only one of the short answer questions remaining unchanged for the CAS
examination. However, around 80% of the extended response questions remained unchanged
for the CAS examination when compared with the graphics calculator (non-CAS)
examination.
The results for the first year of implementation of the VCE, when compared with the first two
years of the DME are similar, indicating that questions are being written which are not
specific to either a graphics calculator or a computer algebra system. This is an interesting
result, which I will return to later.
The role of the computer algebra system in examination questions
To investigate further the style of questions that are being used in the graphics calculator
assumed and computer algebra system assumed examinations it has been necessary to
consider the role that the graphics calculator or computer algebra system is playing in the
solution process. A modified form of categorisation scheme developed by Brown (in Press-b)
for the analysis of the role of the graphics calculator within system wide examinations has
been used for this purpose and is described in Table 2.
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RBrown CAME Reims 2003
Table 2 Categorisation scheme used for the analysis of the potential role of the graphics calculator or
computer algebra system plays in the solution process of examination questions
Potential Role of graphics calculator and or computer algebra system in
examination questions
The question cannot be reasonably solved without the use of a
 graphics calculator (on a graphics calculator assumed
examination)
Required
 computer algebra system (on a computer algebra system
assumed examination).
A graphics calculator or computer algebra system could contribute to
Optional
the solution of the question but its use is not necessary.
The graphics calculator or computer algebra system has no potential
to contribute to the solution of the question over and above
 a scientific calculator for a graphics calculator assumed
Neutral
examination
 a graphics calculator for a computer algebra system assumed
examination
The question is deliberately structured (or worded) in such a way
that a graphics calculator or computer algebra system cannot be used
directly in its solution; although an alternative structure (wording)
Excluded
would allow a graphics calculator contribute significantly to the
solution of the question.
The use of the technology excluded category for CAS is problematic. In the graphics
calculator assumed examinations there is continuing evidence Brown (in Press-b) of questions
which explicitly exclude the use of the graphics calculator. The exclusion is achieved by the
use of code words such as “Use calculus” in the VCE mathematics examinations and
“Beregn” (calculate [by hand]) in the DME examinations. Whereas in the DME CAS
examinations, the official instructions indicate that words such as “Beregn”, when used in the
CAS examinations, do not imply the requirement for pencil and paper techniques, which is
not the case for the graphics calculator assumed examinations (DME, 2001a). Whilst for the
VCE, the implicit instructions in the graphics calculator examinations, such as “Use
Calculus”, do not appear. However, the VCAA indicate that there will be an increased
requirement for exact answers along with the use of graphical, numerical and analytical
techniques (VCAA, 2001). The VCAA have also recognised that students are unlikely to
complete the papers in the allocated time if they rely exclusively on by hand or by CAS
approaches (Leigh-Lancaster, 2003).
The results of the analysis of the 2001 and 2002 DME graphics calculator and CAS assumed
examinations is summarised in the form of parallel-segmented bar charts in Figure 2.
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RBrown CAME Reims 2003
100%
80%
Technology Neutral
60%
Technology Excluded
Technology Optional
40%
Technology Required
20%
0%
DME A-level DME A-level DME A-level DME A-level
Mathematics Mathematics Mathematics Mathematics
GC 2001
GC 2002
CAS 2001
CAS 2002
Figure 2 The overall profile for technology usage in the DME A-level examinations for 2001 and 2002.
The results of the analysis of the 2001 and 2002 VCAA graphics calculator and CAS assumed
examinations is summarised in the form of parallel-segmented bar charts in Figure 3.
100%
80%
Technology Neutral
60%
Technology Excluded
Technology Optional
40%
Technology Required
20%
0%
Mathematical Mathematical Mathematical
Methods GC Methods GC Methods CAS
2001
2002
2002
Figure 3 The overall profile for technology usage in the VCAA Mathematical Methods examinations for
2001 and 2002.
It is evident from the analysis of the graphics calculator assumed examinations, in Figure 2
and Figure 3 that there is reasonable consistency between the two years, for each examination
board, in terms of the pattern of use of the graphics calculator. It is also evident for both the
DME and the VCAA, about half of the questions are technology active (that is technology
required or technology optional) but the DME A-level contains a far greater percentage, 48%
in 2002, of technology excluded questions.
When considering the role of the CAS in examinations questions the results in Figure 2 and
Figure 3 indicate that a significant proportion of questions on the CAS papers are CAS
optional, 78% for the DME and 55% for the VCAA. An example of a CAS optional question
is shown in Figure 4. The content of style of the question, in Figure 4, is consistent with the
form of question that has appeared in the extended response examination over the past few
years, VBOS (1996) and VCAA (2002d), throughout this time students have been required to
find solutions to a number of aspects relating functions from both a graphical and symbolic
perspective. In many ways this question is no different though the need for symbolic
manipulation appears to have been reduced.
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RBrown CAME Reims 2003
Question 4. On an adventure park ride, riders are strapped into seats on a platform which starts 15
metres above the ground and goes up and down. The distance, x metres, of the platform above the
ground, t seconds after the ride starts, can be modelled by the formula
 t 
x(t )  15  6 sin  
3
a. i. According to this model, find the maximum height above the ground reached by the platform.
ii. According to this model, how many seconds after the ride starts is the platform first exactly 9 metres
from the ground?
Tasmania Jones is redesigning the ride so that the platform moves further up and down each cycle.
During the first 60 seconds of the redesigned ride, the distance, y metres, of the platform above the
ground, t seconds after the ride starts, can be modelled by the formula
 t 
y(t ) 15  e 0.04t sin  , 0  t  60
3
b. According to this model, the platform is exactly 6 metres above the ground for the first time about
58 seconds into the ride. Find this time correct to two decimal places of a second.
c. According to this model, how many times is the platform exactly 15 metres above the ground from
t = 40 to t = 59?
d. According to this model, find the time which passes from when the ride starts until the platform first
reaches 24 metres above the ground. Give your answer correct to the nearest second.
Figure 4 Question taken from the VCAA 2002 Mathematical Methods CAS Pilot project Examination. 2 as
an example of a identical graphics calculator and computer algebra system technology optional question.
(VCAA, 2002b)
It can be clearly seen from Table 3 that this question could have been solved using pencil and
paper techniques, graphics calculator or a computer algebra system equally well. This result is
understandable given the “newness” in the use of CAS in examinations and the careful
approach of examiners when introducing a new technology into examinations.
Table 3 Solution of part a. of question provided in Figure 4 showing the alternative solution
strategies available.
 t 
i. According to this model, x(t )  15  6 sin   , find the maximum height above the ground reached
3
by the platform.
ii. According to this model, how many seconds after the ride starts is the platform first exactly 9 metres
from the ground?
Pencil and paper
Graphics calculator
Computer algebra system
i. By inspection Whilst it the use of a graphics calculator and or CSA may be not necessary they
maximum occurs are included for comparative purposes.
when
 t 
sin    1 thus
3
maximum height
is 21 metres.
Window: x  [-0.5, 2π], y [-1,
22].
The solution found using the CAS will
require some interpretation by the student.
Whilst it was expected by the examiners that most students would be able to solve part i of
the question by inspection, the use of a graphical calculator solution, or a CAS, would be
possible. However, in the case of the CAS solution, it is clear that the form of the answer is
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RBrown CAME Reims 2003
likely to confuse, not help, unless the student has a good understanding of the syntax of the
CAS.
For part b of the question, given in Table 4, it is evident that in the graphics calculator only
examination the use of the graphics calculator is necessary to find a solution. Whereas for the
CAS examination the students could use the solve command, as indicated in the diagram, or
else they could solve the question graphically and or use tables, thus the problem can be
solved in a number of ways when a CAS is available.
Table 4 Solution of part b of question provided in Figure 4 showing the alternative solution strategies
available.
 t 
b. According to this model, y(t ) 15  e 0.04t sin  , 0  t  60 , the platform is exactly 6 metres above
3
the ground for the first time about 58 seconds into the ride. Find this time correct to two decimal places
of a second.
Pencil and
Graphics calculator
Computer algebra system
paper
Not available
Window: x  [57, 60], y [-1, 28].
Or using the numerical solver
There is evidence that the impact of the introduction of the CAS into mathematics
examinations has been to increase the range of solution strategies available to the student,
enabling them to select their preferred method of solution. This is less apparent prior to the
introduction of the graphics calculator and the computer algebra system into examinations. As
a consequence, students with access to a CAS need to become proficient at selecting the
appropriate methods to solve the problem when they have access to such technology.
Evidence presented by Boers & Jones (1993) has already indicated the difficulty that students
can have in reconciling alternative approaches to solving a problems, such as graphical
solution versus an algebraic solution, which could be further compounded with the
introduction of the computer algebra system in to examinations.
Conclusion
The introduction of CAS into examinations has the potential to allow the student to move
away from an examination where the examiner controls the solution strategy, to one in which
the student controls the solution strategy and consequent form of the solution. That is, the
students choose the way in which they undertake to solve the problem and present their
solution. Examiners, therefore, when setting questions will need to be aware of these different
solution strategies and as such be prepared to reward them equally, or alternatively, constrain
the question to one solution method. The examiners will also need to decide on how they will
apportion marks within the question, and as a consequence determine whether the focus of the
question will be on the solution or the process in gaining that solution.
These decision will become increasingly important as we move towards a situation where all
forms of technology are allowed and assumed within examinations and these technologies are
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RBrown CAME Reims 2003
considered to present solutions of equal value. The DME and the International Baccalaureate
Organization (IBO) are already beginning to move down this path of allowing students to
choose which technology they use and when they use it in an examination environment. It is
apparent therefore that examination question writers will need to respond to such questions as
how do they achieve equality of method when using any technology, are all methods equally
valid, and are all responses equally valid e.g. exact solutions versus approximate solutions?
Alternatively, if this approach is adopted will it lead to the writing of technology neutral
questions, which are easier to write, and do not reward the use of technology (Drijvers,
1998)? These developments will be monitored by many examining boards in the light of the
changes already taking place in examination question writing in a CAS environment.
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