ERIC Identifier: ED463947
Publication Date: 2000-12-00
Author: Meagher, Michael
Source: ERIC Clearinghouse for
Science Mathematics and Environmental Education Columbus OH.
Curriculum and Assessment in an Age of Computer Algebra
Systems. ERIC Digest.
The use of Computer Algebra System (CAS) technology in the teaching of
mathematics is entering a new and crucial stage. A considerable amount of work
to date has focused on support of existing curriculum topics and existing
curriculum models. Kutzler's seminal book on teaching and learning using DERIVE
(1996) includes the phrase "here is an example of how CAS can be used to support
an existing curriculum topic". Arguments for the use of CAS as an effective tool
in supporting teaching and learning of mathematics are well established. Now
comes the real work.
Curriculum models and assessment of curriculums are the structures that
schools and examining boards create to educate students. The two are deeply
connected and the real work of CAS advocates is now to develop curriculum models
in which the use of CAS is an integral part, not just of the practice of the
curriculum, but, of the conception of the curriculum. A thorough examination of
topics must be carried out to decide what really has to stay for conceptual
reasons and what is still extant only because of tradition.
A crucial part of this process is a demonstration of assessment using CAS.
Unless instruments can be found to measure the achievement, or lack of
achievement, of that learning which is possible through the use of CAS it is
destined to remain as a prop occasionally brought out to support and bolster
existing methods of teaching and learning.
We should take the current topics, the current models and using those topics
see how we can begin to assess students who are answering examination questions
with CAS as an aid. This exercise serves the very useful purpose of bringing to
light the relative triviality, in a CAS age, of some common current questions
and forces us to examine their real significance as mathematical concepts. The
close scrutiny of current examination questions also affords the possibility for
the incorporation of more complicated questions previously beyond the scope of a
time-limited written examination.
RESEARCH ON ASSESSMENT WITH CAS TECHNOLOGY
A vital issue is "How much is a question testing mathematics and how much is it testing ability
to use technology?" There is value in both, but there is a distinction.
Kokol-Voljc (1999a) chose to make this distinction by setting up a matrix with
four categories which answer two questions: (a) Is the technology immediately
put to use in answer to the question or does some work need to be done first
(i.e. Is the use of technology primary or secondary?), and (b) Once the
technology is being used are the operations straightforward or does the student
need good ability in using the technology (i.e. Is the use of the technology
elementary or advanced?). Kokol-Voljc has written more extensively about the
classification which has been refined and expanded in various ways (1999b).
Several have examined the impact of CAS on current national examinations with
questions posed about the future of such examinations in the CAS age. Aogain
(2000) looked at the Irish Leaving Certificate Examinations for 1999 and
classified questions as trivial, easy, difficult or CAS proof. He proposes a
calculation of an "index of suitability" for examinations that could be applied
to any examination.
A review of examinations in several European countries was undertaken by
Drijvers (1998) who noticed several approaches and classified them according to
the extent to which technology is allowed and, if allowed, the reward available
to those who use it.
Brown (1999) compared some examination questions from Australia, Denmark and
the USA and noted the clear difficulties that examining boards are having in
finding an appropriate approach to assessment. He offers the view that "There
will undoubtedly be an evolution in the development of setting questions that
encourage the use of technology while ensuring that the primary focus of the
mathematics assessment will remain on assessing mathematical concepts and
understanding." Others working in this area include Fazio (2000), Lokar &
Lokar (2000) & Ruthven (1997).
Kutzler (2000) makes the case for two-tier examinations, one part with no
technology allowed and a second part with any and every technology allowed. As
Kutzler says "Two-tiered exams would be a well-balanced compromise meeting both
the desires of technology supporters and the reservations of those who are
concerned about the use of technology in the classroom."
TEACHER ACCOUNTS OF ASSESSMENT WITH CAS TECHNOLOGY
discussion of assessment must be more far reaching than summative examinations,
and several articles address this wider discussion. Many of these are in the
form of teachers relating personal experiences of trying to incorporate CAS
technology into their teaching and assessment.
Torres-Skoumal (2000) found group assessment very effective both for
improving technical competencies and mathematics achievement. Other benefits she
observed were social integration, student awareness of assessment procedures,
and improvement in use of mathematical language.
Kempski (1988) uses DERIVE in the teaching and assessment of mathematics.
Based on several years of experience he suggested that "students who best use
the software are those who are more able and confident in Mathematics." Kempski
also considered the way that students used DERIVE in examinations and was
interested to observe that "more often than not, they demonstrated an inability
to make the best use of the software and did not even take full advantage of the
routine menu commands."
The proceedings of the 6th ACDCA Summer Academy in Portoroz, Slovenia,
published under the title "Exam Questions & Basic Skills in
Technology-Supported Mathematics Teaching (Proceedings Portoroz 2000)"
(Kokol-Voljc et al, 2000) contains many more articles related to teaching,
learning and assessment in the CAS age.
The question of curriculum itself in a
CAS age is at the early stages of discussion. Some researchers (Heid, 1988 and
Cabezas & Roanes-Lozano, 1998) are looking at the sequencing of certain
topics. McCrae and Stacey (2000) are investigating "the changes that regular
access to CAS calculators may have on senior secondary mathematics subjects and
to explore the feasibility of offering new subjects that use CAS extensively," Students in three volunteer schools have been participating in the study which
will likely result in formal assessment in a CAS-active environment only. The
project web site is www.edfac.unimelb.edu.au/DSME/CAS-CAT/.
Heugl (2000) has listed what he sees as seven important competencies in
algebra, such as the competence of recognizing structures, recognizing
equivalence of terms, and visualization. He discusses the seven competencies in
terms of the impact of CAS on them, and he suggests several different
examination models which may prove more responsive to the impact of CAS than
Finally, a very controversial addition to the debate is the paper arising
from the discussion between Herget, Heugl, Kutzler and Lehmann (2000) about what
they consider to be the essential skills in arithmetic and algebra in the CAS
age. They imagine an environment of a technology-free examination and classify
problems into essentially two types: those questions which a student would be
expected to answer in such a technology-free examination (i.e. without any
calculator or computer), and those questions which would not be asked in such an
examination. A third group in the classification are those questions about which
the authors have doubts as to where to classify them. The authors stress that
those questions that they would not ask in a technology-free examination they
find inappropriate for any examination, even one where powerful technology is
allowed. As they say "we would not ask these questions in a technology-supported
exam either, because these questions appear useless as such, their best use
might be to test how well a student can operate a calculator." The paper is
intentionally controversial and as the authors themselves say "we deliberately
wanted to be provocative and shake the mainstay of traditional mathematics
Brown, R. (1999). Graphical Calculator
Assessment in Australia: A comparison with other countries. "Proceedings of
ICTMT 4." Plymouth, UK.
Cabezas, J. & Roanes-Lozano, E. (1998). Do Computer Algebra Systems
Change the Order in Which we Should Teach Mathematics? Proceedings of the 3rd
International Derive and TI-92 Conference. Gettysburg, PA.
Drijvers, P. (1998). Assessment and the new Technologies. The international
"Journal of Computer Algebra in Mathematics Education", 5(2), 81-93.
Fazio, R. (2000). The Mathematics Final Exam Questions for the Italian
Experimental Scientific Liceo Discussed and Solved with Derive 5. In
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000).
Heid, M K. (1988). Resequencing Skills and Concepts in Applied Calculus using
the Computer as a Tool. "Journal of Research in Mathematics Education," 19(1),
Herget, W., Heugl, H. Kutzler, B., & Lehmann, E. (2000). Indispensable
Manual Calculation Skills in a CAS Environment. In Kokol-Voljc, V., Kutzler, B.,
Lokar, M. & Palcic, J. (2000). Also available at
Heugl, H. (2000). New emphasis of Fundamental Algebraic Competence and its
Influence in Exam Situation. In Kokol-Voljc, V., Kutzler, B., Lokar, M. &
Palcic, J. (2000).
Kempski, B. (1998). The Use of DERIVE in Assessed Student Work. "Proceedings
of the 3rd International Derive and TI-92 Conference." Gettysburg, PA.
Kokol-Voljc, V. (1999a). Exam Questions when using CAS for School Mathematics
Teaching. "International Journal of Computer Algebra in Mathematics Education."
Kokol-Voljc, V. (1999b). "Exam Questions for DERIVE &TI-92/89." bk
teachware, Hagenberg (in German). ISBN 3-901769-23-4.
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000). "Exam
Questions and Basic Skills in Technology-Supported Mathematics Teaching"
(Proceedings, July 2-5, 2000, Portoroz, Slovenia). bk-teachware, Hagenberg. ISBN
Kutzler, B. (1996). "Improving Mathematics Teaching with DERIVE."
Chartwell-Bratt, Bromley, UK. ISBN 0-86238-422-2.
Kutzler, B. (2000). Two-Tier Examinations as a Way to Let Technology. In
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000). Also available
Lokar, M. & Lokar, M. (2000). Slovene Final External Examination --
Matura in View of Computer Algebra Systems. In Kokol-Voljc, V., Kutzler, B.,
Lokar, M. & Palcic, J. (2000).
Mac Aogain, E. (2000). Assessment in the CAS age: an Irish perspective. In
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000).
McCrae, B. & Stacey, K. (2000). "Computer Algebra Systems in Schools --
Curriculum, Assessment and Teaching Project."
Ruthven, K. (1997). "Computer Algebra Systems in Advanced-Level Mathematics A
report to SCAA. Cambridge: University of Cambridge." Torres-Skoumal, M. (2000).
Alternative Assessment Model. In Kokol-Voljc, V., Kutzler, B., Lokar, M. &
Palcic, J. (2000).