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Leigh-Lancaster, David
The Victorian Curriculum and Assessment Authority
Mathematical Methods Computer Algebra Pilot Study and
Examinations.
2003-06-23
34p.; Produced by the Victorian Curriculum and Assessment
Authority. Paper presented at the CAME Conference (3rd,
Rheims, France, June 23-24, 2003).
Reports
Research (143)
EDRS Price MF01/PCC)2 Plus Postage.
*Academic Achievement; *Computer Assisted Instruction;
Computer Uses in Education; Evaluation; Mathematical
Applications; Mathematical Concepts; Mathematics Curriculum;
*Mathematics Instruction; Secondary Education
Computer Algebra

ABSTRACT

Computer algebra systems (CAS) have now become much more
readily accessible for use in secondary school mathematics on both hand-held
and computer platforms. While the initial focus of work with CAS from the
early 1980's hars generally been with respect to pedagogical and curriculum
issues, as familiarity with CAS in senior secondary mathematics contexts has
evolved around the world, systems and organizations have responded in various
ways to the increasing availability of CAS and its impact on assessment, in
particular end of secondary schooling formal examinations. This paper
discusses key design and development aspects of the first examinations for
the VCAA Mathematical Methods (CAS) pilot study in 2002, and provides some
preliminary analysis and commentary with respect to student performance on
these examinations. (Author)

Reproductions supplied by EDRS are the best that can be made
from the original document.

The Victorian Curriculum and Assessment Authority
Mathematical Methods Computer Algebra Pilot Study and
Examinations

David Leigh-Lancaster

U.S. DEPARTMENT OF EDUCATION
Office of Educational Research and Improvement
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originating it.

PERMISSION TO REPRODUCE AND
DISSEMINATE THIS MATERIAL HAS
BEEN GRANTED BY

D. Leigh-Lancaster

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TO THE EDUCATIONAL RESOURCES
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Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

The Victorian Curriculum and Assessment Authority
Mathematical Methods computer algebra pilot study
and examinations
David Leigh- Lancaster

Victorian Curriculum and Assessment Authority
Abstract
Computer algebra systems (CAS) have now become much more readily accessible for use in
secondary school mathematics on both hand-held and computer platforms. While the initial
focus of work with CAS from the early 1980's has generally been with respect to pedagogical
and curriculum issues, as familiarity with CAS in senior secondary mathematics contexts has
evolved around the world, systems and organisations have responded in various ways to the
increasing availability of CAS and its impact on assessment, in particular end of secondary
schooling formal examinations. This paper discusses key design and development aspects of
the first examinations for the VCAA Mathematical Methods (CAS) pilot study in 2002, and
provides some preliminary analysis and commentary with respect to student performance
on these examinations.

Background
While earlier considerations on the use of CAS in mathematics education focussed on
pedagogical and curriculum issues, these issues do not arise in isolation from assessment
(see Leigh-Lancaster and Stephens, 1997, 2001). The notion of congruence between pedagogy,

curriculum and assessment is a central part of the discourse on the use of such technology
(see Leigh-Lancaster, 2000, HREF1). Here congruence refers to the alignment between
curriculum structure and aims, approaches to working mathematically, and the nature and
purpose of assessments, in particular examinations. The use of technology in the senior
mathematics curriculum, and end of secondary schooling mathematics examinations in
Victoria, has evolved over the last several decades as different technologies have become
more widely available and integrated into mainstream teaching and learning practice:
1970

slide rule and four figure mathematical tables;

1978

scientific calculators;

approved graphics calculators permitted (examinations graphics calculator

1997/ 8

'neutral');
1999

'assumed access' for graphics calculators in Mathematical Methods and

Specialist Mathematics examinations, permitted for Further Mathematics
examinations;

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

2000

'assumed access' for graphics calculators in all mathematics examinations,

examinations for revised Victorian Certificate of Education (VCE) Mathematics study
2000 - 5 incorporating some graphics calculator 'active' questions;

November 2002 Mathematical Methods (CAS) pilot study, 'assumed access' for
approved CAS in pilot examinations.

Government policies, directions and resources for various sectors related to ICT, reflect
cognisance of the role that such technologies play in the economy, education and society in
general (see, for example, it reality bytes, 2001, HREF2, and the Ministerial Statement on
Knowledge and Skills for the Innovation Society, HREF3). Particular curriculum and assessment
projects, such as the VCAA Mathematical Methods (CAS) pilot, necessarily occur within the

context of the corresponding policy framework, for example, the VCAA is explicit in its
Strategic Plan 2002 -2004 that it will ensure ICT and innovative thinking are embedded
throughout the curriculum (HREF4). Important ICT related issues in this context are the
potential for flexible delivery of curriculum and assessment in senior certificates, support for
student engagement, and social justice considerations of equity, access and retention.

Mathematical Methods (CAS) Units 1 4 is an accredited pilot study of the Victorian
Curriculum and Assessment Authority for the period from January 2001 December 2005.
The pilot study is monitored and evaluated as part of the ongoing review and accreditation
of VCE studies, and, following on from a successful conclusion to the pilot, there would
likely be a subsequent period of overlapping accreditation for the revised Mathematical
Methods and Mathematical Methods (CAS) courses. Details of the pilot, including the study
design for Units 1 4, sample and 2002 examinations, assessment reports and other teacher
resources, can be accessed from the VCAA website (HREF5).

The first phase of the VCAA pilot study 2000 2002, involved students from three volunteer
Stage 1 schools, and was implemented in conjunction with the CAS CAT project, a research

partnership between the VCAA, the Department of Science and Mathematics Education at
the University of Melbourne, and three calculator companies (CASIO, Hewlett-Packard, and
Texas Instruments). The CAS CAT project has been funded by a major Commonwealth
Australian Research Council (ARC) Strategic Partnership with Industry Research and
Training (SPIRT) grant (HREF6). In November 2002, 78 students from the three Stage 1
schools sat end of year final Mathematical Methods (CAS) Unit 3 and 4 examinations, for
which student access to an approved CAS calculator (TI-89, CASIO ALGEBRA FX 2.0 or HP
40G) was assumed.

The second and third stages of the VCAA expanded pilot study 2001 - 2005, incorporate the
original three schools (implementing Mathematical Methods (CAS) Units 1 and 2 from 2001
and Units 3 and 4 from 2002) and include two additional groups: nine volunteer Stage 2
schools implementing Units 1 and 2 from 2002 and Units 3 and 4 from 2003, and a further
seven volunteer Stage 3 schools implementing Units 1 and 2 from 2002 and Units 3 and 4

from 2004. The schools in the expanded pilot include co-educational and single sex,
metropolitan and regional schools from government, catholic and independent sectors, using
a range of different CAS. Thus, there will be slightly over 250 students enrolled in Units 3
and 4 from 11 schools of the expanded pilot in 2003. This will include students using the
CAS TI Voyage 200, Derive and Mathematica in one school for each of these CAS.

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

Early use of CAS by teachers and students in Victorian senior secondary mathematics in the
early to mid 1990's focussed on its use as a pedagogical tool for improving student learning
within existing courses, and to support student work in responding to the complexity and
generality of mathematics in investigations, modelling and problem-solving tasks such as
the centrally set, but school assessed, extended VCE mathematics common assessment tasks
(see, for example, Tynan, 1991; Woods, 1994; Delbosc and Leigh-Lancaster, 1995). The
revised VCE mathematics study, implemented from 2000, is explicit about the effective and
appropriate use of technology to produce results which support learning mathematics and
its application in different contexts:

The appropriate use of technology to support and develop the teaching and
learning of mathematics is to be incorporated throughout each unit and course.
This will include the use of some of the following technologies for various areas
of study or topics: graphics calculators, spreadsheets, graphing packages,
dynamic geometry systems, statistical analysis systems, and computer algebra

systems. In particular, students are encouraged to use graphics calculators,
spreadsheets or statistical software for probability and statistics related areas of
study, and graphics calculators, dynamic geometry systems, graphing packages
or computer algebra systems in the remaining areas of study systems both in the

learning of new material and the application of this material in a variety of
different contexts. (Board of Studies, p 12, 1999).

The Mathematical Methods (CAS) pilot "study develops these considerations with respect to

congruence between pedagogy, curriculum and assessment for computer algebra system
technology. Consultation with universities and the Victorian Tertiary Admissions Centre
(VTAC) took place throughout the development and accreditation of Mathematical Methods
(CAS) Units 1-4 for the pilot study and in March 2001, VTAC informed the VCAA that the

pilot study design had been approved by all universities for prerequisite purposes from
2003. Further details about the VCAA pilot program and its progress can be found in LeighLancaster (2002) and Leigh-Lancaster (2003).

Benefits and concerns
A range of potential benefits for the use of CAS are typically articulated, including the
following:

the possibility for improved teaching of traditional mathematical topics;
opportunities for new selection and organisation of mathematical topics;

access to important mathematical ideas that have previously been too difficult to
teach effectively;
as a vehicle for mathematical discovery;
extending the range of examples that can be studied;
as a programming environment ideally suited to mathematics;

emphasising the inter-relationships between different mathematical representations

(the technology allows students to explore mathematics using different
representations simultaneously);
as an aid to preparation and checking of instructional examples;
promoting a hierarchical approach to the development of concepts and algorithms;

long and complex calculations can be carried out by the technology, enabling
students to concentrate on the conceptual aspects of mathematics;

the technology provides immediate feedback so that students can independently
monitor and verify their ideas;

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

the need to express mathematical ideas in a form understood by the technology helps
students to clarify their mathematical thinking;
situations and problems can be modelled in more complex and realistic ways.
(Conference proceedings, ICME 5, 1984 pp 162
165 and the Mathematical
Association, 1997,

pp 43

46).

For systems these potential benefits need to be considered along with the various concerns
about potential negative effects that are expressed at times by academics, teachers, parents
and students, including those who are nonetheless positive about the overall benefits of
CAS:

the extent to which the use of CAS may reduce students knowledge and skills with
important and valued conventional by hand or mental techniques;
how students, including those who may be less mathematically inclined, will cope
with a more conceptually demanding curriculum;
a diminished role for teachers in terms of traditional (and valued) pedagogy;
whether appropriate cognisance has been given to the role of by hand approaches in
the development of important mathematical concepts, skills and processes.
A principled and coherent response to the natural questions of what mathematics? (selection
from discipline and domain knowledge, theory and application); for whom? (subsets of the
cohort); how? (curriculum and assessment study requirements and related advice on possible
pedagogies); and why? (rationale and purpose), is central to the responsibilities and work of
curriculum and assessment authorities.

Systems and the nature of CAS use in examinations
The use of CAS in final senior secondary mathematics examinations can be precluded,
permitted or assumed for part or all of these examinations (an interesting and related
question for curriculum and assessment authorities is whether / how CAS might be
profitably used in teaching, learning and assessment of other studies, for example, physics).

Systems use a variety of approaches, with underpinning value emphases and process,
knowledge and skill aims, to ensure congruence between their curriculum goals and
corresponding examination structures and question design. Thus various notions such as
CAS 'free', 'independent', 'advantaged', 'privileged', 'trivialised', 'neutral', 'active'(or not)
and the like, can be found in the corresponding discourse, with at least as many nuances of
meaning as there are notions. At this stage it appears to be the case that each system where
CAS can be used, at least in part for some examinations, has a distinct structure for these
examinations. These are summarised in Table 1 below:
Table 1: com arison of s stems and examination structure
System
France

Victoria, Australia
program

Examination structure

CAS neutral questions, unrestricted access to approved CAS for all parts
of examinations, pure mathematical emphasis.
pilot Assumed access to approved CAS for all parts of examinations (multiple
choice, short answer and extended response), application emphasis in

extended response questions. Common questions on corresponding
papers for CAS and non-CAS versions of the same paper, for parallel
courses with common and distinctive content.
College Board (USA)

Advanced Placement

CAS 'not an advantage' questions, access to a broad range of approved

graphics calculators or hand-ild CAS permitted in parts, other parts

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.
Calculus

technology free.

Denmark

Access to broad range of hand held and computer based CAS, technology
assumed access parts and technology free parts, common and distinctive
questions for CAS and non-CAS versions of the same paper and course.

International

Assumed access to a single approved CAS, technology assumed access

Baccalaureate*

parts and technology free parts.

proposed pilot program for Hig er Level course in conjunction with current review process.

Mathematical Methods (CAS) Units 3 and 4 examinations: structure and
design
Mathematical Methods Units 1 - 4 and the pilot Mathematical Methods (CAS) Units 1

4 are

parallel and like subjects with function, algebra, calculus and probability areas of study.
They are considered by VTAC and the universities to be alternative but equivalent prerequisites for the same range of university courses. The examinations for Mathematical
Methods Units 3 and 4, with assumed access to a graphics calculator, and the examinations
for the pilot Mathematical Methods Units (CAS) Units 3 and 4, with assumed access to CAS,

are intended to make comparable demands on students. Both courses are designed to
provide students with a suitable basis from which to undertake post secondary mathematics

and mathematics related subjects that develop these areas of study further as well as
introduce new material. One examination (Facts, skills and standard applications task)
consists of multiple choice and short answer questions, while the other examination
(Analysis task) consists of four extended response questions each with several parts of
increasing complexity. Three of the fours analysis task questions are based on an application
context, and one of these questions covers material from the probability area of study.
The examinations for both subjects are time-tabled concurrently, and have the same structure
and format, with substantial common material (HREF7). They include questions for which
access to either CAS or a graphics calculator are not likely to be of assistance, as well as

questions for which both may be of assistance, using comparable functionalities such as
numerical equation solving or drawing graphs, and for which neither technology is likely to
confer an advantage with respect to the other. For the 2002 examinations, together these
types corresponded to about 80% of the multiple choice and extended response questions,
but only around 20% of the short answer questions. These common questions provide a
basis for comparison of the performance of the two cohorts.

A small number of common questions which some might consider to be 'trivialised' by
access to CAS are also included. Researchers such as Herget, Heugl, Kutzler and Lehmann
(2000) and Stacey (2000) have considered the issue of access to CAS 'trivialising' certain types

of symbolic manipulations, and argue that these could (or should) be 'given over' to the
technology, with an emphasis placed upon the use of CAS in application contexts, supported
by a sense of 'algebraic expectation' or 'algebraic insight'. Other researchers, such as Drijvers
and Van Herwaarden (2000) and Monaghan (2001), have discussed the complexities of such
considerations, in particular, that there is a significant issue of by hand-technology interplay

in the dimension of mathematical understanding and competency. Gardner (2001) in

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

particular cautions that cognisance of these connections is essential. Access to a given
functionality or repertoire of functionalities, does not necessarily confer their effective use in
practice, or understanding of important ideas and principles underpinning such use.

On the other hand, effective use of CAS can support student engagement towards
completions of questions with which they might otherwise falter, either through error at an

early stage or continuing reliably and accurately with the complexity of manipulations
involved. Survey feedback from teachers in the pilot is supportive of this view. This
assistance would have a necessary rather than sufficient impact, since CAS use in itself does

not confer the insight required to apply appropriate solution processes, although it may
facilitate the development of such insight through the teaching and learning process. Thus it
is reasonable to anticipate that access to CAS would likely have some benefit in this regards.

Questions included in the CAS examination papers only, also cover content that has been
included in the Mathematical Methods (CAS) subject, but had not previously been generally
or readily available as study content for Mathematical Methods. This has typically been the
case due to a combination of technical difficulty, conceptual complexity and time available to
cover this material suitably. Examples of such material include continuous random variables

and functional equations. Corresponding examination questions are likely to involve
functions without rules being specified (such as a function f which is differentiable over a
given domain); general forms involving the use of parameters, and more complex symbolic
expressions, including those arising in application situations.

For other questions, the use of mental or by hands approaches will be simpler and more
efficient. In many cases use of CAS will also provide students with a means of obtaining an
answer, although this may not be in a convenient form, or obtainable through a simple
process. A characterisation of this aspect of examination design is that students are unlikely
to complete the papers in the allocated time if they rely solely on the use of either CAS or a
combination of mental and by hand methods (indeed they will not be able to readily answer
some questions, or parts of questions in the latter case). The examiners also expect students
to be able to identify equivalent algebraic forms as different CAS use a variety of routines to

'simplify' expressions and carry out other computations, for example, integration of
combined functions involving circular functions. The VCAA Assessment Reports for 2002
examinations provides commentary on student performance by the chief assessors for the
2002 examinations (HREF8).

Some comments and observations
The following comments and observations are based on preliminary analysis of data (see
Appendix 1) related to cohort proportions of correct responses (multiple-choice questions)
and cohort mean and available scores (short answer and extended response questions),
discussions with panel setting chairs and chief assessors, and discussions with pilot study
teachers. It should be noted that the data for Mathematical Methods is for the state-wide
cohort of around 20 000 students, and that the data for the Mathematical Methods (CAS)
pilot is for the 78 students from three volunteer pilot schools. These schools comprise two co-

educational schools (one government sector, metropolitan school and one independent
sector, regional school) and one single-sex girl's school(catholic sector, metropolitan school).
Thus it is only possible to make tentative comments and observations of interest at this stage

Paper presented at the third CAME Conference, 23 - 24 June, Rheims, France.

of the pilot. For students in each VCE study, the VCAA computes a study score in the range

0 to 50, which comes from a truncated normal distribution with mean 30 and standard
deviation 7. For VCE mathematics, this study score is based on two examinations, each
worth 33% of the final weighting, and a school based coursework assessment score, worth
34% of the final weighting, and statistically moderated with respect to the examinations.
VTAC re-scales these study scores to take into account differences in relative difficulties of
studies (based on analysis of how students perform across studies) and uses these re-scaled

scores in computation of a national tertiary entrance score, using a combination of best
subject scores, on a scale of 0 100 (HREF9). For 2002, the re-scaled means and standard
deviations for Mathematical Methods and Mathematical Methods (CAS) were 36.6 and 38.8
and 6.9 and 5.6 respectively. This accords with what the VCAA had anticipated, given the
nature of the three volunteer schools, a slightly higher (re-scaled) mean score with a slightly
smaller standard deviation. For the first two years of pilot examinations, VTAC has agreed to
scale the pilot study in the same manner as Mathematical Methods.
Table 2 summarises the difference in proportion of correct responses (as a percentage of each
cohort) to the 20 (out of a total of 27) common multiple choice items between Mathematical
Methods (CAS) and Mathematical Methods (thus a positive difference will indicate that a
higher proportion of the CAS cohort selected the correct response). The items have been
classified as technology independent (I), technology of assistance but neutral with respect to
graphics calculator or CAS (N) or use of CAS likely to be advantageous (C). Those items for
which technology is of assistance, but that are likely to be answered efficiently by conceptual

understanding, pattern recognition or mental and / or by hand approaches have been
indicated by a tick (/).
Examination 1- common questions

Table 2: summary of differences between proportions of correct responses to common Examination 1
multiple choice items given as number of items (question number/s)
Negative difference*
Item type
I

20%

Positive difference*

10 to 19 %

up to 9%

up to 9 %

10 to 19 %

(22)

(26)

(7, 17, 18)

(6, 11, 13)

2 st

4 sr

(4, 24)

(1, 3, 5, 25)

(27)

20%

1 ,/

(16)

(14)

(9, 15)

(20)

* there were no items with zero difference proportions.

The following comments are perhaps best considered as pointers to some areas of interest for
further analysis within the expanded pilot program. In broad terms, the summary data in

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

Table 2 indicates that, on balance, this pilot cohort was not disadvantaged on these common
questions when compared with the Mathematical Methods cohort. Questions 14 and 20 with

high proportion of correct responses (94, 90) and large positive difference (30, 40)
respectively, suggest that access to CAS can significantly improve accuracy and reliability of
some symbolic computations such as simple differentiation and anti-differentiation, or at
least the ability to recognise the correct form from a list of alternatives. On the other hand,

Question 22, formulation of a definite integral to represent area between two curves,
indicates that the anticipated benefits such as the view that use of CAS to carry out related
computations 'frees up' time to place more emphasis on formulation, do not necessarily
follow automatically. Similarly, Question 9 (C, 69, 55) which required students to identify the
linear factors of x4 + x3 3x2 3x over R, while having a better correct response rate than the
non-CAS cohort, was not as high as might have been expected if access to CAS were to have
practically trivialised such a question. Analysis of response rates for distractors indicates that
this was most likely due to students factorising over the rational field, Q, rather than the real
field, R (a 'popular' error for both cohorts).
With respect to the common Examination 1 short answer questions there were only 4 marks
of this kind (out of a total of 23 available marks). The type of question, maximum available,
mean Mathematical Methods (CAS) cohort and mean Mathematical Methods cohort scores
were respectively Question 5a (I, 1, 0.31, 0.24) specifying a sequence of transformations to
produce a given function rule; Question 5b (I, 2, 1.33, 1.11) stating the domain and range of
the transformed function; and question 6b (N, 1, 0.56, 0.49) -finding a numerical value for a
derivative.
Examination 1

CAS only questions

For the six CAS paper only multiple choice items, the correct response rates were Question 2:
sum of solutions to a simple circular functions equation over an interval (65); Question 8:

matching an explicit function rule to data (94); Question 10: functional relation (64);
Question 19: identifying an unknown function in a chain rule application (86); Question 21:

application of definite integrals to distance travelled for v(t) =

with exact value

answer form (58); and Question 23: evaluating probability for continuous random variable
with a linear function probability density function (60).

With respect to the CAS paper only short answer questions, it was noticeable that in
Question la (C,
2, 1.07): finding the value of the multiplicative constant, c, so that a
function with domain [0, 00) is a probability density function; and Question lb (C, 2, 0.99):
finding the median value of the random variable; while most students were able to correctly
formulate a suitable definite integral expression and related equation, this did not transfer to

corresponding correct calculations or evaluations to any where near the same extent.
Question 3 (I, 2, 1.55): drawing the graph of I f I given the graph of f (without a rule) was
done well, with the two main sources of error being 'rounding off' of the point of non differentiability, and incorrect concavity at the left and/ or right hand ends of the curve.

Question 4 required students to find the rule of a cubic polynomial function with
undetermined coefficients using a combination of conditions involving the values of the
function and its derivative. The three parts of the question involved formulation as a set of

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

simultaneous linear equations (2, 1.56), their representation in matrix form (2, 1.39) and
solution (by any method) to find the rule explicitly (2, 1.13).
Examination 2

common questions

With respect to Examination 2, 80 % of the material consisted of common extended response
questions, or common parts of extended response questions. Using the same classification as
previously, and consideration of the type of question, maximum available, and respective
mean Mathematical Methods (CAS) cohort and mean Mathematical Methods cohort scores,
the following observations can be made.

Where numerical computations and / or graph sketching is required, both cohorts performed
comparably, or the CAS cohort performed slightly better, for example: Question la.i (N,
3, 1.61, 1.53) drawing the graph of a transformed log function on a restricted domain in a
modelling context; Question la. iv (I, 1, 0.21, 0.20) stating the domain of the inverse
function; Question la.v (N, 2, 0.71, 0.72) sketching the graph of the inverse function labelling
key features.

Where algebraic manipulation was involved, access to CAS improved the reliability of
correct response: Question la. iii (C, 2, 1.77, 1.41) finding the rule of the inverse function;
Question lb (N, 2, 1.57, 1.15)

solving simultaneous equations to determine coefficients.

Similarly, where algebra and calculus were both involved in a theoretical context, such as in
Question 3, access to CAS improved the reliability of correct response: for example, Question
3 b.i (N, v, 3, 2.05, 1.53) showing a given linear function is normal to a quartic polynomial
function at a specified x coordinate value. However, the earlier observation with respect to
formulation involving definite integrals also receives some further support: Question 3c.i (I,
2, 0.97, 0.84) - writing down a definite integral expression for the area between the two curve
and their points of intersection.

On the common parts of the extended response probability question, involving binomial,
hypergeometric and normal distributions with mainly numerical calculations, (which
constituted most of the available marks) the CAS cohort performed slightly better. However
it was expected by examiners that there would be no appreciable difference between the two
cohorts here (despite the view of some that these functions were more easily accessible on
graphics calculators).

The final question, Question 4 involved a modelling context based on a transformed circular
function with multiples of .7r in the argument, leading into a more complex product function
model involving a transformed exponential function as well. While both cohorts performed
similarly on some parts of this question, for example: Question 4 b (N, 1, 0.58, 0.42) finding
the first time the product function attained a particular value in its range; for each part the

CAS cohort performed comparably, slightly better, or noticeably better, for example
Question 4e.i (C, 2, 1.75, 1.07) finding a symbolic expression for the product function
model.

On the other hand, for a question where a simple parameter was incorporated into the model
as a multiplied coefficient, and students were asked to determine the greatest ( decimal)
value of the parameter such that the derivative satisfied certain constraints, while the CAS
cohort did perform better, neither cohort did particularly well (C, 3, 0.53, 0.35).

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.
Examination 2

CAS only questions

The main areas of interest here are where alternative CAS and non-CAS formulations for
parts of questions have been used, or where distinctive questions have been used.

The panel setting chairs and chief assessors for Mathematical Methods (CAS) who also have
considerable experience with setting and marking Mathematical Methods papers, have also
commented favourably on the notable lack of 'blank spaces' where student responses would
be expected to short answer or extended response questions, in examination scripts for the

pilot CAS cohort. Question lc was designed to require general symbolic calculation
involving the use of a parameter, k, and a fixed real value T, but also to be amenable in part
to conceptual graphical analysis. Question lc.i (C, 1, 0.69) involved finding the solution to an
equation, in the form P' 71. Question lc.i (C,
2, 0.21) then required students to find, for a
given (but unknown) value of T, the largest value of the parameter, k, such that an equation
has a solution over the domain of the underlying modelling function. The modest success
rate reinforces advice from the chief examiner of the Danish pilot study to the author that,
based on their experience, such questions are challenging to students, and what appears to
be a small increase in complexity of question design can be a substantial increase in difficulty
for students, especially where a parameter is involved. This also accords with the experience
of the author as a setting panel member and assessor for Mathematical Methods Units 3 and
4 graphics calculator not permitted, graphics calculator 'neutral' examinations, and graphics

calculator 'assumed access' examinations where a parameter has been used in question
formulation.

In Question 3a.i (C, 1, 0.81) and Question 3 a.ii (C, 2, 1.76) which involve factorisation of a

quartic polynomial function into a particular form, and finding the exact values of the
zeroes, access to CAS was clearly beneficial in enabling students to obtain the required
results.
Other related data

In Term 3 of 2001 (for Units 1 and 2, Year 11) and Term 3 of 2002 (for Units 3 and 4, year 12),

the VCAA gathered data where students of both Mathematical Methods and Mathematical
Methods (CAS) cohorts across the three pilot schools undertook an algebra and calculus
skills test, related to content covered to that stage, without access to either graphics calculator
or CAS technology. The results clearly show that students from the CAS cohort were able to
perform as well or better on this material as the non-CAS cohort.

The VCAA has also collected qualitative data from teachers on their implementation of the
pilot study in 2001 and 2002. VCE Mathematics units have a two-tiered assessment structure,
demonstration of achievement of a set of outcomes (with specified key knowledge and key
skills) for satisfactory completion of the unit, and level of performance assessment. Award of
the Victoria Certificate of Education is based satisfactory completion of a minimum number
of units (typically across two years) including a specified number of Unit 3 and 4 sequences

(four sequences as well as a compulsory English language sequence) For example,
Mathematical Methods (CAS) Units 3 and 4 form a sequence. At the Unit 3 and 4 level,
students will not receive a study score for a sequence unless they have satisfactorily
completed both of its units. Coursework assessment is also based on the outcomes, so their

associated key knowledge and key skills are also assessed through the coursework
assessment tasks for Units 3 and 4.

4 clearly specify expected mental,
by hand and CAS concepts, skills and processes, and teachers, parents and students from
pilot schools have clearly affirmed their valuing for the effective development of important
The outcomes for Mathematical Methods (CAS) Units 1

Paper presented at the third CAME Conference, 23 24 June, Rheims, FranCe.

mental and by hand algebra and calculus skills. Thus, in discussion of the role of teaching by
hand algebraic manipulations and differentiation rules for combined functions, pilot teachers
felt that this was still important (yet were comfortable about students choosing when to use
CAS or by hand approaches when tackling problems), for the following sorts of reasons, as
summarised by one of the original pilot teachers who is a highly regarded and expert teacher

with considerable experience over many years in the use of technology in school
mathematics:

so that students can see a process first (or else they think it is magic);
so students can recognise equivalent forms of a solution;
so that students can access a range of methods (for versatility);

so that students can make an informed choice about the tool they use
(head/ hand /CAS or combination thereof);
so that students can develop a sense of whether a result is plausible.

Teachers and students from the pilot study also reported affirmation of a range of the
potential benefits outlined earlier, in particular a more in depth treatment of existing
material, access to new and interesting content, enhanced engagement and persistence of
students in mathematical work, increased accuracy and reliability of student work, including

in application contexts, and improved understanding of the concepts of function and
variable, and facility with symbolic notation and exact values. A greater confidence and
independence of student work in mathematical activity has also been noted. Certainly,
teachers have commented that their own mathematical horizons have been extended, and
that they have developed an expanded pedagogical repertoire (see Garner, 2003) as a
consequence of their involvement in the pilot study.

Biographical details
David is the Mathematics Manager at the Victorian Curriculum and Assessment Authority,
and the project manager of the VCAA Mathematical Methods (CAS) pilot study 2001 2005.
He has taught secondary mathematics for around 20 years, including 12 years as a head of

faculty. During this time he has been extensively involved in curriculum and teacher
professional development, examinations and school based tasks. He has a long-standing
interest in the nature of mathematical inquiry, related teaching and learning approaches and
assessment tasks.

David's mathematical background is in pure mathematics, in particular mathematical logic
and the history and philosophy of mathematics. He has worked with the application of
technology in mathematics throughout his career, and has used the CAS Mathematica with
students from Years 9

12 at Kingswood College, Victoria from 1993

1998.

References
Board of Studies. (1999). VCE Study design

Mathematics. Carlton: Author.

Del Bosc, M. & Leigh-Lancaster, D. (1995). Specialist Mathematica. In Celebrating Mathematics
Learning. Mathematical Association of Victoria. Melbourne.

Drijvers, P. & Van Herwaarden, 0. (2000). Instrumentation of ICT-tools: the case of algebra
in a computer algebra environment. International Journal of Computer algebra in
Mathematics Education 7(4), pp 255 - 275.

Gardner, T. (2001). Education or CAStration?. Micromath, Spring 17 (1), pp 6

Paper presented at the third CAME Conference, 23

24 June, Rheims, France.

Garner, S. (2003). How has the use of the CAS calculator affected the teaching and learning of

Maths in the Year 12 classroom? In Mathematics-Making Waves. Ed. M. Goos.
Australian Association of Mathematics Teachers. Adelaide.

Herget, W., Heugl, H.,Kutzler, B.&Lehmann, E. (2000). Indispensable manual calculation
skills in a CAS environment. Micro math, autumn 16 (3) pp 8 -17.
ICME 5. (1984). Proceedings of the Fifth International Congress of Mathematics Education (ICME

5). Ed M.Carss. Birkhauser. Boston.

Leigh-Lancaster, D. (2002). The Victorian Curriculum and Assessment Authority (VCAA)
Mathematical Methods (CAS) pilot study 2001 2005. Australian Senior Mathematics
Journal, Volume 16, Number 2, pp 20

28.

Leigh-Lancaster, D. (2003). The Victorian Curriculum and Assessment Authority (VCAA)
Mathematical Methods (CAS) pilot study 2001 2005. In Mathematics Making Waves,
Ed. M Goos. Proceedings of the 19th Biennial Conference. Australian Association of
Mathematics Teachers, Adelaide.
Leigh-Lancaster, D. & Stephens, M. (1997). Anticipating the impact of widely available access
to Computer Algebra Systems on Teaching and Assessment. In Mathematics: the next

millennium. Ed. K. Baldwin and J. Roberts. Australian Association of Mathematics
Teachers, Adelaide.

Leigh-Lancaster, D. & Stephens, M. (2001). Responding to the impact of increasing
accessibility of Computer Algebra Systems (CAS) on national and other school
systems at the senior secondary level. In Proceedings of the 12th ICMI Study
Conference: The Future of the Teaching and Learning of Algebra. Ed H. Chick.
Melbourne. The University of Melbourne.

Monaghan, J. (2001). Reaction to indispensable manual calculation skills in a CAS
environment. Micromath, Spring 17 (1), pp 9

11.

Stacey, K., McCrae, B., Chick, H., Asp, G., & Leigh-Lancaster, D. (2000). Research-led policy

change for technologically-active senior mathematics assessment. In J. Bana and A.
Chapman (Eds.), Mathematics Education Beyond 2000 (Proceedings of the 23rd annual
conference of the Mathematics Education Research Group of Australasia, pp. 572579). Freemantle. MERGA.

The Mathematical Association. (1996). Symbolic Manipulation by Computers and calculators. Ed.
A. Oldknow and J. Flower. The Mathematical Association. London.

Tynan, D. (1991). Symbolic Algebra Processors - A Manipulative Revolution. In Mathematics:

Ideas. Eds. J. O'Reilly & S. Wettenhall. Melbourne. Mathematical Association of
Victoria.

Woods, N. (1994). Mathematics software and the effect on curriculum development. In

Mathematics Without Limits. Eds. C. Beesey & D. Rasmussen. Melbourne.
Mathematical Association of Victoria.
HREF1:

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

http:/ /www.math.ohio-state.edu/waitsb/ papers/ t3 posticrne2000/

Teachers Teaching

with Technology, Post ICME 2000 Conference, Japan, Mathematics
Department, Ohio State University.
HREF2:
http:/ / www.rnrnv.vic.gov.au Reality Bytes, State Government of Victoria.
HREF3:

http: / / www.otte.vic.gov.au / publications / KnowledgeandSkills Ministerial Statement on
Knowledge and Skills for the Innovation Society, State Government of Victoria.
HREF4:

http:/ / www.vcaa.vic.edu.au /board /pdfs/stratpla.pdf Strategic Plan 2002 4, VCAA.
HREF5:

http: / / www.vcaa.vic.edu.au / vce / studies/ MATHS/ caspilot.htm -the Mathematical
Methods (CAS) pilot study website, VCAA.
HREF6:

h ttp: / / w w w.edfac.0 n imelb.edu.a u / DSMEI CAS-CAT
University of Melbourne.

the CAS CAT project website,

HREF7:

http: / / www.vcaa.vic.edu.au / VCE /Exams/ subject.htrn#maths02 mathematics
examinations, VCAA.
HREF8:

http:/ /www.vcaa.vic.edu.au/VCE/Assessrnent/ AssessReports/subject.htm#maths
mathematics assessment reports, VCAA.
HREF9:

http:/ / www.vtac.edu.au /articles/ scaling, report.pdf Victorian Tertiary Admissions Centre.
Each of these sites was accessed at 28103103.

EST COPY AVAITABLIE

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

Mathematical Methods and Mathematical Methods (CAS)
examinations 2002 common questions
Table 1 summarises the totals of marks from common questions across each component of
the examinations, and the aggregate total across all components.
Table 1

Component

Nature

Proportion of marks

Examination 1 - Part I

Multiple choice

21 out of 27 (.., 78%)

Examination 1 - Part II

Short answer

4 out of 23 (.. 17 %)

Examination 2

Extended response

44 out of 55 (80 %)

Aggregate

69 out of 105 (... 66 %)

Paper presented at the third CAME Conference, 23

24 June, Rheims, France.

Table 2 lists the actual common questions for each part, and associated marks.

Component

Examination 1

Part I

Questions

Marks

1, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18,

1 mark each

20, 22, 24, 25, 26, 27

(Total

marks)

Examination 1 Part II

Question 5 parts a and b

1 + 2 marks

Question 6 part b

1 mark
(Total 4 marks)

Examination 2

Question 1 parts ai, aii, aiii, aiv and av

3+1+2+1+
2 marks
2 marks

Question 1 part b

Question 2 MM part a = MM(CAS) part b
Question 2 MM part b = MM(CAS) part c

Question 2 MM part d = MM(CAS) part d

2 marks

4 marks
4 marks

3 + 4 marks

Question 3 parts bi, bii

2 + 1 marks

Question 3 parts ci, cii

1 mark

Question 4 part ai
Question 4 MM part bi = MM(CAS) part b
Question 4 MM part bii = MM(CAS) part c

Question 4 MM part biii = MM(CAS) part d
Question 4 MM part ci = MM(CAS) part ei
Question 4 MM part cii = MM(CAS) part eii
if

1 mark
1 mark
2 marks
2 marks
3 marks

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.
Question 4 MM part e = MM(CAS) part f

3 marks

(Total
marks)

for example, sketch of a graph or

Description

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Question

Type

Examination 1 Part I: Multiple choice items (78% common material)

% correct
responses

straightforward mental or by hand approaches have been indicated by a tick V) in the type column following.

responses

% correct

Questions where technology is of assistance, but that are likely to be answered efficiently by conceptual understanding, pattern recognition,

for example, symbolic manipulation for finding exact value roots of an equations,

factors of an expression, symbolic expressions for a derivative or anti-derivative.

Use of a CAS calculator likely to be an advantage

numerical solution to an equation or numerical evaluation of a derivative or definite integral

Technology of assistance, but neutral with respect to graphics calculator or CAS calculator use

Technology independent for example a conceptual question or analysis of an non-scaled graph without rules for functions

The following classifications for type of item have been used:

Mathematical Methods and Mathematical Methods (CAS) examinations 2002 cohort comparative
performance on common questions

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

possible rule for the function.

Given graph of a quartic polynomial function identify

specified domain.

Given rule of a transformed circular function in a
modelling context, identify maximum value on a

graph.

Identify rule of circular function that matches a given

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

MM (CAS) cohort

MM cohort

-5

MM (CAS) - MM

1 function on 1:1

axes scales, identify the graph of the inverse function.

Given the graph of undetermined 1

identify the corresponding graph forflx).

Given the graph of a function with unspecified rule f(x),

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Given graph of a transformed basic hyperbola, find the
values of the parameters that define the corresponding

fLmction rule.

Description

Question

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

Type

+12

MM (CAS) - MM

MM cohort

MM (CAS) cohort

responses

% correct

function.

Identify the derivative of a log composition circular

corresponding part if its derivative function.

horizontal asymptote) identify the graph for the

Given part of the graph of art unspecified function (with

which it will have an inverse function.

Given the rule of a quadratic polynomial function in
completed square form, identify the domain set for

field.

Given the rule of a quartic polynomial function with no
constant term, identify the linear factors over the real

24 June, Rheims, France.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Paper presented at the third CAME Conference, 23

+30

+11

+10

+14

corresponding graph for f

information about the function at points and the
derivative at points and over intervals, find the

For an unspecified function f, given a combinations of

of linear approximation formula.

Identify substitutions necessary for correct application

exponential function base e at a given exact point.

Find the exact rate of change for a simple transformed

product function at a given exact value point.

C or N

Find the normal to a curve whose rule is a simple C or N

24 June, Rheims, France.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Paper presented at the third CAME Conference, 23

+10

+12

of success, p.

deviation in a particular context, identify the probability

Binomial distribution, given mean and standard

replacement application context - 'at least one'.

Hypergeometric distribution, sampling without

related intercepts.

for unsigned area between curve, horizontal axis and

correct symbolic expression involving definite integrals

Given the graph of an unspecified function, with
portions above and below horizontal axis, identify

derivative function.

June, Rheims, France.

Given derivative function that is a simple dilated
circular function, identify a corresponding anti-

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Paper presented at the third CAME Conference, 23

-6

-10

+40

closest value to required actual mean to ensure that only
1% of packets are under weight.

Normal distribution in context (coffee packets), with
'labelled' mean and standard deviation given. Identify

transformation to standard normal distribution.

probability is required, identify the corresponding

variable, and a value for which the 'less than'

Given parameters for a normally distributed random

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

1
50

+12

-7

Apply a sequence of transformation to the rule of the

For a transformed sin function, find the numerical value
of the derivative at a specified x value.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

State the domain and range of the transformed function.

basic hyperbola to determine the rule of the transformed
hyperbola.

Description

Question
Type

Maximum

available marks

Examination 1 Part II: Short answer items (17% common material)

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

0.56

1.33

0.31

cohort

MM (CAS)

Mean score

0.49

1.11

0.24

MM cohort

Mean score

Explain why the function has an inverse function.

with their coordinates.

context, labelling any asymptotes and any endpoints

Sketch the graph of a transformed log function (decimal
value coefficients) on a restricted domain in a modelling

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

la.ii

la.i

part

Question

Description

Type

available marks

Maximum

0.81

1.61

cohort

MM (CAS)

Mean score

0.63

1.53

MM cohort

Mean score

b loge(x) on a restricted

extended response questions with several parts of increasing complexity (80% common

Question 1 is a modelling context based around a transformed natural logarithm function with rule of the form f(x) = a
domain.

Examination 2
material)

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

Sketch the graph of the inverse function, labelling any

la.v

coefficient values for different modelling data.

Solve simultaneous equations to determine new

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

State the domain of the inverse function.

la.iv

asymptotes and any endpoints with their coordinates.

Find the rule of the inverse function.

la.iii

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

1.57

0.71

0.21

1.77

1.15

0.72

0.20

1.41

Conditional and total probability.

deviation.

random variable, determine the mean and standard

Given probabilities associated with normally distributed

distributed random variable.

Basic probability calculation Pr (X < a) for normally

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

(MM 2b)

(MM 2a)

part

Question

Description

Type

available marks

Maximum

0.58

1.62

1.77

cohort

MM (CAS)

Mean score

0.55

1.61

1.57

MM cohort

Mean score

Question 2 is a probability modelling context based around various distributions binomial, normal) related to characteristics of two fictitious
species of butterflies.

Paper presented at the third CAME Conference, 23 24 June, Rheims, France.

functions defined by their points of intersection.

Write down a definite integral expression for the area
between the quartic and the linear (normal/tangent)

value coordinates of that point.

polynomial at another point, and determine the exact

Show that this normal is tangent to the quartic

quartic poynomial at a specified x coordinate.

Show that a given linear function is normal to the

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

3 c.i

3 b.ii

3 b.i

part

Question

Description

Type

available marks

Maximum

0.97

2.19

2.05

cohort

MM (CAS)

Mean score

0.84

1.46

1.53

MM cohort

Mean score

Question 3 is a theoretical context involving a quartic polynomial function, differentiation and integration, tangents and normals, points of
intersection and area between two curves.

Paper presented at the third CAME Conference, 23 - 24 June, Rheims, France.

Evaluate the definite integral to a specified accuracy.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

3 c. ii

Paper presented at the third CAME Conference, 23 - 24 June, Rheims, France.

0.41

0.26

24 June, Rheims, France.

exponential and transformed sin function.

during a specified time interval, for the product

Find how many times a particular height is achieved

Find the first time a particular height is reached for the
product exponential and transformed sin function.

in a modelling context.

Find the maximum value of a transformed sin function

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

b.ii)

(MM 4

b.i)

(MM 4

4 a.i

part

Question

Description

Type

available marks

Maximum

0.73

0.58

0.95

cohort

MM (CAS)

Mean score

0.60

0.42

0.84

MM cohort

Mean score

secondly a product function involving an exponential function and a transformed circular function. The problem involves solving equations, rates
of change and differentiation.

Question 4 is a modelling context involving firstly a height-time transformed sin function with multiple of g in the argument of the function, and

Paper presented at the third CAME Conference, 23

answers required).

the value of the minimum height (decimal value

Use dy/ dt to write down an equation, one solution of
which gives the value of t for the minimum height, and

exponential and transformed sin function.

Find a symbolic expression for dy/ dt, the product

exponential and transformed sin function.

start until that height is reached, for the product

Given a specific height value find the time elapsed from

24 June, Rheims, France.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

c.ii)

(MM 4

4 e.ii

c.i)

(MM 4

4 e.i

b.lii)

(MM 4

Paper presented at the third CAME Conference, 23

1.66

1.75

1.60

0.95

1.07

1.06

required).

value over a given interval (decimal value answer

that the derivative of the function is less than a specified

Inclusion of a parameter as a multiplied coefficient of
product function, find greatest value of the parameter so
C

24 June, Rheims, France.

High Empress:Users:ericcsmee:Desktop:1-Reaction-LeighLancaster.doc

(MM 4 e)

Paper presented at the third CAME Conference, 23

0.53

0.35

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information requested, If you Want gikiic 10..prokiide only microfiche ordigitizettcoples,Of print
corium
met"tele cheek the leee ea Me right side 'of the page end previdt the *tested signaluMand
information. If you am submitting non-print items.er wish ERIC to onlY describe and =nowt= your
Matedials, without providing reproduedoru or any 'typo, please contact ERIC1CSMEE a.i Indicated bcfPwo
64 roi3oFst cIte ço rt1014te'reippdlicttOlt tett= form..

Submit the completed releaselorret along With two etspiesof theconknence paper or other &content
being subnithed. There must be niapozatc reIonto form for each item sUlamiated. Mall oil material°
the attentien o( hliqui Rechrthn Lathe address Indleased.

310r further Intone:ode:am contact:-

'Metal Ileckium
DoOkbast.COnrdiriatOr
1131LICCSMES;
1929. Kona), Ro.id

1400.2764462
(614)2924717,
ON) 292420 (P44.)
-.dlesiOStoio:edu (o.,ro411)

ColurnbO4VH 43210.1130

VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY

41 St Andrews Place
East Melbourne
Victoria 3002 Australia

David Leigh-Lancaster
Hey Learning Area Manager
Mathematics

TELEPHONE +61 3 9651 4537

FACSIMILE +61 3 9651 4324
EMAIL

leigh-laneasterdavid.d@edumaiLviegovau

Uietor4
The Ph. To at

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