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The effect of Computer Algebra System (CAS) in the development of conceptual and procedural knowledge

The effect of Computer Algebra System (CAS) in the development of conceptual and procedural knowledge. Yılmaz Aksoy , Erciyes University, TR. aksoyyilmaz @ hotmail .com and Mehmet Bulut , Gazi University, TR. mbulut@gazi.edu.tr and

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The effect of Computer Algebra System (CAS) in the development of conceptual and procedural knowledge

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  1. The effect of Computer Algebra System (CAS) in the development ofconceptual and procedural knowledge YılmazAksoy, Erciyes University, TR. aksoyyilmaz@hotmail.com and MehmetBulut, Gazi University, TR. mbulut@gazi.edu.tr and ŞerefMirasyedioğlu, Başkent University, TR. serefm@baskent.edu.tr

  2. Outline • Introduction(Rationale of study) • A brief review of literature • Methodology • Results • Discussion

  3. Rationale of study • Thisreportexplorestheeffect of CAS in thedevelopment of proceduralandconceptualknowledge of firstyearundergraduatemathematicsandscienceeducationstudents. • MAPLE wasused as CAS in theteaching of Calculusconcepts.

  4. In this study, teaching of the derivative, the key concept of the Calculus has been studied. As derivative is usedmainlybymathematicsandscienceeducationlessons, wechoosethisconceptforcomparingthedevelopment of proceduralandconceptualknowledge of firstyearundergraduatemathematicsandscienceeducationstudents.

  5. A brief review of the literature Constructivisttheory: • According to constructivist learning theory, if an individual construct a concept through acting an active role while experimenting, conjecturing, proving and applying in learning environment, this learning can be called acquiring more than only receiving the information. By using CAS (Computer Algebra System) students have an active role in mathematics classrooms.

  6. Conceptualknowledge • is seen as theknowledge of thecoreconceptsandprinciplesandtheirinterrelations in a certain domain. • is assumedto be stored in some form of relationalrepresentation, likeschemas, semantic Networks orhierarchies (e.g. Byrnes & Wasik, 1991).

  7. Because of itsabstractnatureandthefactthat it can be consciouslyaccessed, it can be largelyverbalizedandflexiblytransformedthroughprocesses of inferenceandreflection. • It is, therefore, not boundupwithspecificproblems but can in principle be generalizedfor a variety of problem types in a domain (e.g. Baroody, 2003).

  8. Proceduralknowledge, • is seen as theknowledge of operatorsandtheconditionsunderwhichthese can be usedtoreachcertaingoals (e.g. Byrnes & Wasik, 1991). • Further, it allowspeopletosolveproblemsquicklyandefficientlybecause it is tosomedegreeautomated.

  9. Automatization is accomplishedthroughpracticeandallowsfor a quickactivationandexecution of proceduralknowledge, since itsapplication, as comparedtotheapplication of conceptualknowledge, involves minimal consciousattentionandfewcognitiveresources (see Johnson, 2003, for an overview).

  10. Itsautomatednature, however, impliesthatproceduralknowledge is not oronlypartlyopentoconsciousinspectionand can, thus, be hardlyverbalizedortransformedbyhighermentalprocesses. • As a consequence, it is tiedtospecific problem types (e.g. Baroody, 2003).

  11. Computer tools: • Since the early 1980s numerous general claims have been made about the likely benefits of using computer tools to improve understanding of calculus concepts • For example, Heid (1988, p.4), commenting on a body research conducted during the previous ten years, states “Computing devices are natural tools for the refocusing of the mathematics curriculum on concepts.”

  12. Ellison’s study indicated that using TI-81 graphing calculators and computer software assisted her 10 college students to mentally construct an appropriate concept image of the concept of derivative. However, not all the students developed a mature concept image.

  13. Using CAS in Calculus • Reporting informally on a remedial teaching program (for 22 college students) that integrated a CAS (Maple) into a course of calculus Hillel (1993, p.46) observed benefits to student learning: students coming out of it had acquired different types of insights and knowhows than the traditionally- prepared students - insights and knowhows which we felt were closer to the essence of calculus.

  14. Methodology • In this study quasi-experimental research design were used. • Sample of this study contains 49 first year students of mathematics education andscience education departments. • The students in both groups were encountered with the derivative concept for the first time. • In both of the groups, students have been studied as groups of 2 or 3 students. • The calculus potential test was administered to students in order to determine groups were taught in a computer based learning environment as a pretest.

  15. The lessons were taken in the laboratory and the students had the opportunity to use laboratory besides the lessons. In order to teach the derivative concept, student cantered activities have been designed. While designing these activities, guides were given to students to use the MAPLE. • Then students have been studied on certain problems which help to discover the mathematical concepts.

  16. Teaching the derivative concept has been designed as two consecutive steps: • At first step; students studied on the concept of derivative as rate of change. At this step real life problems about rate of change have been given to students. By solving these problems students have discovered the concept of the derivative. Students interpreted graphics of functions for developing conceptual understanding of rate of change. • At second step, activities have been designed as geometrical, numerical and symbolic (algebraic) representations of derivative concept.

  17. In student activities, which were designed by researchers before, used in computer learning environment for procedural and conceptual learning of the derivative concept. These activities administered with interactive worksheets prepared with maple, animated and non-animated graphics, plotted by maple, special applets in maple called maplet. • By using interactive maple worksheets and animated graphics, students have found the opportunity of numerous experiments that provide well understanding for them. • To provide conceptual and meaningful understanding for the student, a maplet has been designed to see, geometrical application of derivative as slope of the tangent line.

  18. At the end of the treatment, students’ understanding of derivative was elicited through written tasks administered to all students. • For this exam, students were given the opportunity, but not required to use the computer to solve the problems. These problems were considered to be “computer neutral”. Students were presented with tasks that assessed their conceptual understanding and representational methods of solution of derivative.

  19. The open-ended written tasks used in the examination instrument were mostly adapted from Girard (See [5]) common tasks used to assess student understanding of derivative, in Calculus I courses and found in most textbooks or adapted from other studies concerning student understanding of derivative concept. • The tasks were evaluated by a panel of mathematics instructors (two university level) for the reasonableness of the question for university Calculus I students. Recommendations from the expert panel were examined and changes were made to the instrument accordingly.

  20. Results • Scores for all questions were calculated by tworesearchers using rubrics designed for thisstudy. All of the questions were open-ended also required a written explanation. • To study the differences among students’ conceptual and procedural knowledge we performed two MANCOVAs using the The calculus potential test grades as covariate, proceduralandconceptualproblems’scoresas dependent variables. • For comparing means of the two groups’ scores on the questions, general linear model: MultivariateAnalysis of Covariance (MANCOVA) was used.

  21. Initially,we conducted an independent samples t test having student’s pretest attainment in the calculus potential test to examine whether there were statistically significant differences between the two groups.

  22. According to tables this independent-samples t-test analysis indicates thatstudents in mathematics group had a mean of 47,0227 total points and the students in science group had a mean of 46,4259 total points.So, there is not significant difference between groups’ pre-test scores at the p>.05 level(note: p=.814).

  23. Then, weconducted a multipleanalysis of covariance test (MANCOVA) havingstudent’s post-test attainment in thequestionsaboutconceptualandproceduralknowledge as dependentvariablesandthepre-test scores as covariates. • Theresultsshowedthattherewerestatisticallysignificantdifferences in students’ post-test attainmentbetweenthetwogroups, Pillai’s F(2,45)=16,959, p<0.05.

  24. Conceptualquestions • Table 3 presentstheresults of the MANCOVA test, showingthatthereweresignificantdifferencesbetweenthetwogroups. Table 3: DifferencesbetweenMathematicsandSciencegroups’ means in conceptualknowledge • We can concludethatthestudents of mathematicsgroupperformedsignificantlybetterthanthestudents of thesciencegroup in conceptualknowledgequestions.

  25. Then, weconducted a multipleanalysis of covariance test (MANCOVA) havingstudent’s post-test attainment in thefourquestionsaboutconceptualknowledge as dependentvariablesandthepre-test scores as covariates. • Theresultsshowedthattherewerestatisticallysignificantdifferences in students’ post-test attainmentbetweenthetwogroups, Pillai’s F(4,43)=7,625, p<0.05. • On theotherhand, in question 4 therewas not significantdifferences in students’ post-test attainmentbetweenthetwogroups, p>0.05.

  26. Proceduralquestions • Table 4 presentstheresults of the MANCOVA test, showingthatthereweresignificantdifferencesbetweenthetwogroups. Table 4: DifferencesbetweenMathematicsandSciencegroups’ means in proceduralknowledge • We can concludethatthestudents of mathematicsgroupperformedsignificantlybetterthanthestudents of thesciencegroup in proceduralknowledgequestions.

  27. Then, weconducted a multipleanalysis of covariance test (MANCOVA) havingstudent’s post-test attainment in thethreequestionsaboutproceduralknowledge as dependentvariablesandthepre-test scores as covariates. • Theresultsshowedthattherewerestatisticallysignificantdifferences in students’ post-test attainmentbetweenthetwogroups, Pillai’s F(3,44)=5,219, p<0.05. • On theotherhand, in question 5 therewas a significantdifferences in students’ post-test attainmentbetweenthetwogroups, p>0.05.

  28. Discussion • Most of thestudents in mathematicsgroupansweredthisquestioncorrectly. Students in sciencegroup can do thecomputations but theycouldn’tshowtheequality.

  29. Students in themathematicsgroupshowedbetterunderstanding of theconcept of thederivative (such as themeaning of thederivative) thanthesciencegroupandtherewasalso a significantdifference on proceduralskills.

  30. Specifically, theywereabletoexpressideas in theirownwordsandtheirconceptualizationswerebroader, clearer, moreflexibleandmoredetailedthanstudents in thecontrolgroup. Theseresults can be interpreted as evidencethatstudents can understandcalculusconceptsshowingthat it waspossibletoreorganizetheorder in whichcalculus is taughttostudents, tofocus on conceptspriortoteachingprocedures.

  31. The students reported feeling that the computer relieved them of some of the manipulative aspects of calculus work, that it gave them confidence on which they based their reasoning, and it helped them focus on more global aspects of problem solving. • During the instruction the students were involved in discussing ideas and were required to make sense of calculus related language, including terminology and symbols.

  32. References • [1] Artigue, M. Analysis. 1991. InTall, D. (Ed.), AdvancedMathematicalThinking. KluwerAcademicPub. pages 167-198. • [2] Baroody, A. J. (2003). The development of adaptiveexpertise and flexibility: The integration of conceptualand procedural knowledge. In A. J. Baroody & A.Dowker (Eds.), The development of arithmetic conceptsand skills: Constructing adaptive expertise (pp. 1-33).Mahwah, NJ: Erlbaum. • [3] Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptualknowledge in mathematical procedural learning.DevelopmentalPsychology, 27(5), 777-786.

  33. [4] Cooley, L. A. Evaluatingtheeffects on conceptualunderstandingandachievement of enhancing an introductorycalculuscoursewith a computeralgebrasystem. 1995. (New York University). DissertationAbstractsInternational 56: 3869. • [5] Ellison, M. J. Theeffect of computerandcalculatorgraphics on students’ abilitytomentallyconstructcalculusconcepts. 1993. (Volumes I and II). (University of Minnesota).DissertationAbstractsInternational, 54/11 4020.

  34. [6] Fey, J. T. Technologyandmathematicseducation: A survey of recentdevelopmentsandimportantproblems.1989. EducationalStudies in Mathematics, 20, pages 237-272. • [7] Girard, N.R. Students’ representationalapproachestosolvingcalculusproblems: • Examiningthe role of graphiccalculators. 2002. (University of Pittsburgh) • [8] Heid, M. K. Resequencingskillsandconcepts in appliedcalculususingthecomputer as atool. 1988. JournalforResearch in MathematicsEducation, 19(1), pages 3-25.

  35. [9] Tall, D.& West, B. Graphicinsightintocalculusanddifferentialequations. 1986. In A.G. Howson & J. P. Kahane (Eds.), Theinfluence of computersandinformation on mathematicsanditsteaching. Cambridge: Cambridge UniversityPress. pages 107-119. • [10] White, P. Is calculus in trouble? 1990. AustralianSeniorMathematicsJournal, 4(2), pages 105-110.

  36. Köszönöm szépen! Contact: Mehmet Bulut Gazi University Faculty of Gazi Education ANKARA-TURKEY mbulut@gazi.edu.tr

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