ED463947 2000-12-00 Curriculum and

Assessment in an Age of Computer

Algebra Systems. ERIC Digest.

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Curriculum and Assessment in an Age of Computer Algebra Systems.

ERIC Digest................................................................... 1

RESEARCH ON ASSESSMENT WITH CAS TECHNOLOGY.......... 2

TEACHER ACCOUNTS OF ASSESSMENT WITH CAS

TECHNOLOGY............................................................ 3

CURRICULUM ISSUES....................................................... 4

REFERENCES.................................................................. 4

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.

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The use of Computer Algebra System (CAS) technology in the teaching of mathematics

ED463947 2000-12-00 Curriculum and Assessment in an Age of Computer Algebra

Systems. ERIC Digest.

Page 1 of 6
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).

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Page 2 of 6 ED463947 2000-12-00 Curriculum and Assessment in an Age of Computer Algebra

Systems. ERIC Digest.
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

The 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."

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ED463947 2000-12-00 Curriculum and Assessment in an Age of Computer Algebra

Systems. ERIC Digest.

Page 3 of 6
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.

CURRICULUM ISSUES

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 traditional examinations.

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 teaching."

REFERENCES

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

www.eric.ed.gov ERIC Custom Transformations Team

Page 4 of 6 ED463947 2000-12-00 Curriculum and Assessment in an Age of Computer Algebra

Systems. ERIC Digest.
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), 3-21.

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 http://www.kutzler.com/article/art_indi/indisp.htm

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." 7(1),

63-76.

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 3-901769-33-1.

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 at

http://www.kutzler.com/article/art_exam/exams.htm

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).

ERIC Resource Center www.eric.ed.gov

ED463947 2000-12-00 Curriculum and Assessment in an Age of Computer Algebra

Systems. ERIC Digest.

Page 5 of 6
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).

Michael Meagher is a Doctoral student in Mathematics Education at Ohio State

University and can be contacted at meagher.10@osu.edu.

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This digest was funded by the Office of Educational Research and Improvement, U.S.

Department of Education, under contract no. ED-99-CO-0024. Opinions expressed in

this digest do not necessarily reflect the positions or policies of OERI or the U.S.

Department of Education.

Title: Curriculum and Assessment in an Age of Computer Algebra Systems. ERIC

Digest.

Document Type: Information Analyses---ERIC Information Analysis Products (IAPs)

(071); Information Analyses---ERIC Digests (Selected) in Full Text (073);

Available From: ERIC/CSMEE, 1929 Kenny Road, Columbus, OH 43210-1080. Tel:

800-276-0462 (Toll Free); e-mail: ericse@osu.edu; Web site: http://www.ericse.org.

Descriptors: Computer Uses in Education, Evaluation, Mathematics Curriculum,

Mathematics Education, Secondary Education

Identifiers: Computer Algebra, ERIC Digests

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