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ICME11: 11TH International Congress on Mathematical Education
Discussion Group 27 Paper Proposal
Computer Algebra System (CAS) Usage and Sustainability in University Mathematics Instruction: New Opportunities and Consequences
Submitted to
Discussion Group 27: How is Technology Challenging Us To Rethink
the Fundamentals of Mathematics Education?
Lulu Healy, Topic Group Chair
HYPERLINK "mailto:c.hoyles@ioe.ac.uk" lulu@pucsp.br
Submitted by
Daniel Jarvis, PhD
Assistant Professor of Mathematics Education
Faculty of Education, Nipissing University
100 College Drive, North Bay, Ontario, Canada P1B 8L7
Tel: +01 (705) 474 3461 ext: 4445
Email: HYPERLINK "mailto:dhjarvis@sympatico.ca" dhjarvis@sympatico.ca
Zsolt Lavicza, PhD (ABD)
Graduate Student and Course Instructor
The University of Cambridge
184 Hills Rd., Cambridge, CB2 8PQ, United Kingdom
Tel: +44 7962 488 222
Email: HYPERLINK "mailto:zl221@cam.ac.uk" zl221@cam.ac.uk
Chantal Buteau, PhD
Assistant Professor of Mathematics
Department of Mathematics, Brock University
500 Glenridge Avenue, St.Catharines, Ontario, Canada
Tel: +01 (905) 688 5550 ext: 3167
Email: HYPERLINK "mailto:cbuteau@brocku.ca" cbuteau@brocku.ca
Computer Algebra System (CAS) Usage and Sustainability in University Mathematics Instruction: New Opportunities and Consequences
Daniel Jarvis, PhD
Assistant Professor, Faculty of Education, Nipissing University
Zsolt Lavicza, PhD (ABD)
Faculty of Education, The University of Cambridge
Chantal Buteau, PhD
Assistant Professor, Department of Mathematics, Brock University
Abstract
With a focus on new opportunities for teaching and consequences of Computer Algebra System (CAS) usage, this paper will report on the doctoral findings of Lavicza (2008), the ongoing pilot project work of Buteau (2008), and the preliminary research agenda and progress of the SSHRC International Opportunities Fund Development Grant project entitled, Computer Algebra Systems (CAS) usage and sustainability in university mathematics instruction (Jarvis, Lavicza, & Buteau, 2008). CAS usage and sustainability among individual professors and university departments, respectively, are timely topics within a technological era. Instructor beliefs regarding the nature of mathematics learning, cultural influences, required curriculum/assessment changes, and the availability of soft and hardware for technological explorations in the postsecondary classroom/lab are among the complex set of factors that affect the degree to which technology is adopted, adapted, and implemented within university programs.
Discussion Group 27 Aims and Focus (Excerpt from ICME11 Website)
Although technology has been a key aspect of mathematics education for several decades, its role and use continue to be subject to controversy and debate amongst mathematics educators and mathematicians. What is undeniable is that the introduction of technology into mathematics classrooms should bring transformations to both the educational processes and the mathematics itself. The central aims of this discussion group are to identify possible changes that are brought about by the use of technology within mathematics education, as well as the different factors which contribute to the debates surrounding the integration process and how (or if) it has impacted on the ways we do, learn and teach mathematics as well as on the mathematics that we teach, learn and do.
We hope that the discussions will both look to the future and reflect upon what we might learn from the past. Although we accept that the use of any technological tool brings potential changes to learning ecologies, in order to promote a focused discussion, we use technology as shorthand for digital technology and we welcome contributions associated with the use of technology tools as a means to express mathematical ideas (including handheld devices such as calculators as well as desktop machines). We also envisage that group discussion will consider the use of technology in the creation of new learning spaces and virtual mathematics education communities. We hope to receive contributions the reflect the diversity of both learning contexts and learners, since it is far from clear that technology impacts on our thinking about mathematics and its teaching in any uniform way. The following questions are suggested as starting points for the discussions:
What are the nature and goals of mathematics education and should we rethink the goals in an everevolving technological world?
What are the new opportunities that digital technologies offer to mathematics education?
What are the consequences (both positive and negative) that are likely to result from the use of technology in mathematics education and how do these vary in different countries and learning contexts?
Will the use of technology in mathematics education permit a more democratic and universal access to the development of certain sorts of mathematical insight and competencies or will it contribute to an undermining of the mathematical literacy of the student population?
Potential Contribution to DG 27
Of these four important questions that will be discussed during Discussion Group 27 sessions at ICME11, we believe that our ongoing research will enable us to contribute most meaningfully to questions B and C. As we continue to focus our attention on CASbased technologies and the pedagogical beliefs/attitudes of university mathematics instructors, we can provide insight into new opportunities within mathematics education, as well as into perceived/observed/reported consequences of CAS usage by individual professors and by mathematics departments using CAS over time.
Ongoing International Research Update
On April 30th, 2007, Jarvis, Lavicza and Buteau were awarded a Social Sciences and Humanities Research Council of Canada (SSHRC) International Opportunities Fund Development Grant (IOFDG) of $15 043. This funding has enabled the research team to familiarize themselves with each others institutions and research, to host two international technologyfocused conferences, and to consult a number of leading international colleagues with expertise in this particular area of research who have offered their input and support. They are now in the process of applying for the more substantial SSHRC International Opportunities Fund Project Grant (IOFPG) for 200809 with their application entitled, Computer Algebra Systems (CAS) in University Instruction: Implementing an International Research Study on CAS Usage and Sustainability. The research team consists of: Daniel Jarvis who is an Assistant Professor of Mathematics Education at Nipissing University located in North Bay, Ontario, Canada; Zsolt Lavicza who is a doctoral student preparing to defend his thesis at The University of Cambridge, Cambridge, UK; and Chantal Buteau who is an Assistant Professor of Mathematics at Brock University located in St. Catharines, Ontario, Canada.
All three were invited to participate in the International Commission on Mathematical Instruction (ICMI) Study # 17 entitled, Digital technologies and mathematics teaching and learning: Rethinking the terrain, which was held in Hanoi, Vietnam in December 2006. It was here that they first met, and in subsequent correspondence have applied for, and been granted, a SSHRC IOFDG funding package. This grant has capitalized on the strengths that all three had brought from their respective research experiences: Jarvis knowledge of professional learning models ADDIN EN.CITE Jarvis20062172217Jarvis, D. H.2006Tracking the TIPS mathematics document: Curriculum negotiation and professional development modelsJ. G. Althouse Faculty of EducationLondon, Ontario, CanadaThe University of Western OntarioDoctor of Philosophy (Education Studies) dissertationTracking TIPS: Multilevel participant perceptions of professional practice and developmentprofessional development, reform, perceptions, necessary distortion(Jarvis, 2006), Laviczas expertise with Computer Algebra Systems and his recent findings regarding how CAS have been used internationally ADDIN EN.CITE Lavicza2008149121491Lavicza, Zsolt2008The examination of Computer Algebra Systems (CAS) integration into universitylevel mathematics teachingCambridge, UKThe University of CambridgePhD Dissertationtechnology in education, CAS, comparative research in mathematics education(Lavicza, 2008), and Buteaus firsthand reports of the MICA success story at Brock ADDIN EN.CITE Buteau20061457311457Buteau, C.Muller, E.2006Evolving technologies integrated into undergraduate mathematics education.Le Hung SonNathalie SinclairJeanBaptiste LagrangeCelia Hoylesproceedings of ICMI Study # 17: Digital technologies in mathematics educationRethinking the terrainHanoi Institute of Technology, Hanoi, Vietnamtechnology in education, MICA, mathematics education(Buteau & Muller, 2006).
The research team has accomplished many of its key objectives in 200708 pertaining to the developmental grant (e.g., research planning, conference hosting, consulting with various colleagues). Jarvis, Lavicza, and Buteau now plan to implement a research framework for an international mathematics education study focusing specifically on the implementation and sustained usage of Computer Algebra System (CAS) technology within universitylevel instruction. The research methodology will feature a mixedmethods approach and will draw upon the research/literature review template developed by Lagrange, Artigue, Laborde and Trouche ADDIN EN.CITE Lagrange2003149271492Lagrange, J. B.Artigue, M.Laborde, C.Trouche, L.2003Technology and mathematics education: Multidimensional overview of recent research and innovation.Leung, F. K. S.Second International Handbook of Mathematics EducationDordrechtKluwer Academic Publishers1237270mathematics, technology, innovation, research, mathematics education(2003), and upon Laviczas online survey that he implemented with 1100 university mathematics professors from three countries (UK, US, and Hungary) in his doctoral study ADDIN EN.CITE Lavicza2008149121491Lavicza, Zsolt2008The examination of Computer Algebra Systems (CAS) integration into universitylevel mathematics teachingCambridge, UKThe University of CambridgePhD Dissertationtechnology in education, CAS, comparative research in mathematics education(Lavicza, 2008). By July 2008 when the ICME11 takes place in Monterrey, Mexico, the teams pilot study at Brock will be well underway, and the various research design components prepared to be used in the research proper.
Context/Background of the Research Project
A growing number of international studies have shown that Computer Algebra Systems (CASbased) instruction has the potential to positively affect the teaching and learning of mathematics at various levels of the education system, even though this has not been widely realized in schools and institutions ADDIN EN.CITE Kendal2002147201472Kendal, M.Stacey, K.2002Teachers in transition: Moving towards CASsupported classroomsZentralblatt für Didaktik der Mathematik (ZDM)345196203mathematics, technology, CAS, learning, reformSteinke2004126901269Steinke, TomSpry, KevinFerneyhough, Fred2004Computer Algebra Systems and the Ontario CurriculumOntario Mathematics Gazette4322730computer algebra systems, CAS, curriculum, calculators, technology, mathematics educationLavicza20061445231445Lavicza, Zsolt2006The examination of Computer Algebra Systems (CAS) integration into universitylevel mathematics teachingLe Hung SonNathalie SinclairJeanBaptiste LagrangeCelia Hoylesproceedings of ICMI Study # 17: Digital technologies in mathematics educationRethinking the terrainHanoi Institute of Technology, Hanoi, Vietnamtechnology in education, CAS, comparative research in mathematics educationBossé2004145801458Michael J. BosséN. R. Nandakumar2004Computer algebra systems, pedagogy, and epistemologyMathematics and Computer Education383298306mathematics education, CAS, university, technologyMeagher2001147301473Meagher, M.2001Curriculum and assessment in the age of computer algebra systemsThe International Journal of Computer Algebra in Mathematics Education818995mathematics, technology, CAS, curriculum, assessmentPierce2004147401474Pierce, R. L.Stacey, K.2004A framework for monitoring progress and planning teaching towards the effective use of Computer Algebra SystemsInternational Journal of Computers for Mathematical Learning95993mathematics, technology, CAS, planning, implementation, reformArtigue2002147601476Artigue, M.2002Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual workInternational Journal of Computers for Mathematical Learning73245274mathematics, technology, CAS, curriculum, technical work, reflections, reform(Artigue, 2002; Boss & Nandakumar, 2004; Kendal & Stacey, 2002; Lavicza, 2006; Meagher, 2001; Pierce & Stacey, 2004; Steinke, Spry, & Ferneyhough, 2004). In contrast to the large body of research focusing on technology usage that exists at the secondary school level, there is a definite lack of parallel research at the postsecondary level. Furthermore, although substantial research has been conducted surrounding the area of professional development for teachers (DarlingHammond & McLaughlin, 1995; Earl, Fullan, Leithwood, & Watson, 2000; Guskey, 2002), relatively little has focused specifically on professional development for mathematics educators (Ball, 2000; LoucksHorsley & Matsumoto, 1999; Snead, 1998), especially with regard to technology usage at the university level. As noted by Lawless and Pellegrino ADDIN EN.CITE Lawless2007149301493Lawless, Kimberly A.Pellegrino, James W.2007Professional development in integrating technology into teaching and learning: Knowns, unknowns, and ways to pursue better questions and answersReview of Educational Research774575614professional development, technology, questions, research, student outcomes(2007), large metaanalysis studies on professional development have indicated that there are several key components necessary for success, one of those being the access to new technologies:
A number of organizations and researchers have conducted elaborate reviews of the literature and evaluations in this area (e.g., Corcoran, Shields, & Zucker, 1998; LoucksHorsley, Stiles, & Hewson, 1996; National Foundation for the Improvement of Education [NFIE], 1996; National Staff Development Council, 2001; Porter, Garet, Desimone, Yoon, & Birman, 2000). This knowledge base has consistently indicated that highquality professional development activities are longer in duration (contact hours plus followup), provide access to new technologies for teaching and learning, actively engage teachers in meaningful and relevant activities for their individual contexts, promote peer collaboration and community building, and have a clearly articulated and a common vision for student achievement (Adelman et al., 2002; NFIE, 1996; Porter et al., 2000; Sparks, 2002). ADDIN EN.CITE Lawless20071493p. 57901493Lawless, Kimberly A.Pellegrino, James W.2007Professional development in integrating technology into teaching and learning: Knowns, unknowns, and ways to pursue better questions and answersReview of Educational Research774575614professional development, technology, questions, research, student outcomes(p. 579)
While our research team will not be using the term professional development with regard to university level mathematics instruction, as it is does not seem to be widely used, we will be focusing on instructor learning and technologybased change which will involve many/all of the same necessary components described in the PD literature. Since remarkably little research has been done relating to the support of mathematicians using CASbased instruction at the university level, this has become our primary focus.
Lavicza Doctoral Findings (2008)
Computer Algebra Systems (CAS) are increasingly being used in universitylevel mathematics education. However, little is known about the extent of CAS use and the factors influencing its integration into university curricula. Preuniversity level studies suggest that beyond the availability of technology, teachers conceptions and cultural elements are key factors in technology integration into mathematics teaching and learning. The Lavicza study featured an online survey of 1100 mathematicians as well as interviews with 22 mathematicians in three countries, namely, Hungary, United Kingdom, and United States, which examined mathematicians conceptions of CAS and its pedagogic uses. The findings showed some similarities, but numerous differences, between university and schoollevel research findings. Building on this international doctoral study, the team of Jarvis, Lavicza and Buteau have now planned a fouryear, mixedmethods research study to examine individual and systemic CAS usage.
Potential International Impact of the Research Project
Although the computer hardware and software options have been present for decades, we have still not seen a major shift in pedagogy within our education systems such as was widely predicted. . . . We need to dedicate perhaps 10% of our individual energy and working lives to the exploration of new ways of teachingof reconceptualizing how it is that we teach and students learn mathematics at all levels. It was with these powerful and challenging opening remarks that Dr. Seymour Papert addressed the international group of researchers attending the International Commission on Mathematical Instruction (ICMI) Study # 17 in Hanoi, Vietnam in December 2006. Dr. Papert is widely known for his research and innovations in the areas of Artificial Intelligence, the development of Logo for early childhood instruction, his work with Piaget in the 1960s around constructivism, his teaching and research at MIT, and his most recent initiativewhich he also presented to the larger group during the Hanoi Keynote Addressthe $100dollar laptop. The entire ICMI study conference, featuring working groups, lectures, and computer lab sessions, focused on technology usage within mathematics education at all levels.
Our SSHRC Development and Project Grant applications are, at least in part, a shared response of Jarvis, Lavicza, and Buteau to Paperts challenge to rethink the status quo mathematics pedagogy found in universities. We wish to facilitate, via the proposed international mixedmethods research study, a positive move forward with regard to digital technologies in the university mathematics curriculum. Our goal is to support the individual instructor and the university department as a whole, wherever pedagogical and program change is seriously sought, respectively. In this way, not only will Canadian research in mathematics instruction at the university level be impacted, but results and recommendations from this study may have the potential to have lasting and meaningful effects on institutions worldwide. ADDIN EN.REFLIST References
Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245274.
Boss, M. J., & Nandakumar, N. R. (2004). Computer algebra systems, pedagogy, and epistemology. Mathematics and Computer Education, 38(3), 298306.
Buteau, C., & Muller, E. (2006). Evolving technologies integrated into undergraduate mathematics education. In L. H. Son, N. Sinclair, J.B. Lagrange & C. Hoyles (Eds.), proceedings of ICMI Study # 17: Digital technologies in mathematics educationRethinking the terrain. Hanoi Institute of Technology, Hanoi, Vietnam.
Jarvis, D. H. (2006). Tracking the TIPS mathematics document: Curriculum negotiation and professional development models. Unpublished Doctor of Philosophy (Education Studies) dissertation, The University of Western Ontario, London, Ontario, Canada.
Kendal, M., & Stacey, K. (2002). Teachers in transition: Moving towards CASsupported classrooms. Zentralblatt fr Didaktik der Mathematik (ZDM), 34(5), 196203.
Lagrange, J. B., Artigue, M., Laborde, C., & Trouche, L. (2003). Technology and mathematics education: Multidimensional overview of recent research and innovation. In F. K. S. Leung (Ed.), Second International Handbook of Mathematics Education (Vol. 1, pp. 237270). Dordrecht: Kluwer Academic Publishers.
Lavicza, Z. (2006). The examination of Computer Algebra Systems (CAS) integration into universitylevel mathematics teaching. In L. H. Son, N. Sinclair, J.B. Lagrange & C. Hoyles (Eds.), proceedings of ICMI Study # 17: Digital technologies in mathematics educationRethinking the terrain. Hanoi Institute of Technology, Hanoi, Vietnam.
Lavicza, Z. (2008). The examination of Computer Algebra Systems (CAS) integration into universitylevel mathematics teaching. Unpublished PhD Dissertation, The University of Cambridge, Cambridge, UK.
Lawless, K. A., & Pellegrino, J. W. (2007). Professional development in integrating technology into teaching and learning: Knowns, unknowns, and ways to pursue better questions and answers. Review of Educational Research, 77(4), 575614.
Meagher, M. (2001). Curriculum and assessment in the age of computer algebra systems. The International Journal of Computer Algebra in Mathematics Education, 8(1), 8995.
Pierce, R. L., & Stacey, K. (2004). A framework for monitoring progress and planning teaching towards the effective use of Computer Algebra Systems. International Journal of Computers for Mathematical Learning, 9, 5993.
Steinke, T., Spry, K., & Ferneyhough, F. (2004). Computer Algebra Systems and the Ontario Curriculum. Ontario Mathematics Gazette, 43(2), 2730.
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