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References 1 (concept formation)

Critical look at distance learning and using of dynamic sketches LeTTET’05 , Savonlinna , Finland Martti E. Pesonen (Joensuu) Lenni Haapasalo (Joensuu) Timo Ehmke (Kiel).

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References 1 (concept formation)

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  1. Critical look at distance learning and using of dynamic sketchesLeTTET’05, Savonlinna, FinlandMartti E. Pesonen (Joensuu) Lenni Haapasalo (Joensuu)Timo Ehmke (Kiel)

  2. Based on the project From Visual Animations to Mental Models in Mathematics Concept FormationSponsored by DAAD and Academy of FinlandWeb pageMartti E. Pesonen (Joensuu) Lenni Haapasalo (Joensuu)Timo Ehmke (Kiel)

  3. References 1 (concept formation) • Haapasalo, L. 1993. Systematic constructivism in mathematical concept building. In P. Kupari & L. Haapasalo (eds.), Constructivist and Curriculular Issues in the Finnish School Mathematics Education. Mathematics Education Research in Finland. Yearbook 1992-1993. University of Jyväskylä, Institute for Educational Research. Publication Series B 82. • Haapasalo, L. 1997. Planning and assessment of construction processes in collaborative learning. In S. Järvelä & E. Kunelius (eds.), Learning & Technology - Dimensions to Learning Processes in Different Learning Environments. Electronic publications of the pedagogical faculty of the University of Oulu. Internet: http://herkules.oulu.fi/isbn9514248104 LeTTET'05

  4. References 2 (knowledge & techs) • Haapasalo, L. & Kadijevich, Dj. 2000. Two types of mathematical knowledge and their relation. Journal für Mathematikdidaktik 21 (2), 139-157. • Ehmke, T. 2001.Eine Klasse beweglicher Figuren für interaktive Lernbausteine zur Geometrie. Dissertation. University of Flensburg. • Pesonen, M. E. 2001.WWW Documents With Interactive Animations As Learning Material. In the Joint Meeting of AMS and MAA, New Orleans, January 2001. http://www.joensuu.fi/mathematics/MathDistEdu/MAA2001/index.html LeTTET'05

  5. References 3 • Haapasalo, L. 2003. The Conflict between Conceptual and Procedural Knowledge: Should We Need to Understand in Order to Be Able to Do, or vice versa?In L. Haapasalo & K. Sormunen (eds.) Towards Meaningful Mathematics and Science Education. • Pesonen, M., Haapasalo, L. & Lehtola, H. 2002. Looking at Function Concept through Interactive Animations. The Teaching of Mathematics 5 (1), 37-45. LeTTET'05

  6. References 4 • Pesonen, M. E. et al. 2003. Applying verbal, symbolical and graphical representations to studying basic mathematical concepts in interactive distance learning material (in Finnish). University of Joensuu, Finland. • Pesonen, M., Ehmke, T. & Haapasalo, L. 2005. Solving mathematical problems with dynamic sketches: a study on binary operations. To appear in the Proceedings of ProMath 2004 (Lahti, Finland). LeTTET'05

  7. References 5 • Tall, D. & Bakar, M. 1991. Students’ Mental Prototypes for Functions and Graphs. Downloadable on Internet at http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1991f-mdnor-function-pme.pdf • Tall, D. 1992. The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof. In D. Grouws (ed.), Handbook of research on mathematics teaching and learning. NY: MacMillan, 495-511. • Sierpinska, A., Dreyfus, T., & Hillel, J. (1999). Evaluation of a Teaching Design in Linear Algebra: the Case of Linear Transformations. Recherche en Didactique des Mathématiques, 19 (1), 7-40. LeTTET'05

  8. References 6 • Vinner, S. & Dreyfus, T. 1989. Images and definitions for the concept of function. Journal for Research in Mathematics Education 20 (4), pp. 356-366. • Vinner, S. 1991. The role of definitions in teaching and learning. In D. Tall (ed.): Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer. • Kadijevich, Dj., Haapasalo, L., Hvorecky, J. 2004.Using technology in applications and modellingICME 10. Copenhagen, Denmark. July, 2004. LeTTET'05

  9. Our study frame • scope: first year University maths course on Linear Algebra • concepts: function, binary operation • purpose: to evaluate students’ actions and understanding of different representations; verbal, symbolic and graphic • media: WebCT quizzes including interactive exercises/problems, with questions for feedback LeTTET'05

  10. Ingredients • mathematical: the concept definitions • pedagogical: concept formation • technical: dynamic Java applets, WebCT test tools Example of a dynamic applet LeTTET'05

  11. Theoretical background The 5 phases of concept formation1 • Orientation • Definition • Identification • Production • Reinforcement Emphasis on the first four steps. 1 see Haapasalo 1993, 1997, Pesonen et al. 2005 ) combined LeTTET'05

  12. Features of the interactive tasks • dragging points by mouse • automatic animation/movement dynamic change in the figure • tracing of depending points • hints and links (text) • hints as guiding objects in the figure • response analysis (in Geometria applet) LeTTET'05

  13. Aims • Student opinions of interactive applet questions? • Defects in metacognitive thinking? • Defects in the use of technology? • Advantages and disadvantages of interactive applets? • What kind of pedagogical tutoring is needed? • Problems caused by WebCT? LeTTET'05

  14. Methods • Second semester Linear Algebra students (N = 82) • 3 tests using WebCT quizzes (outside classroom)- Functions of 1 or 2 variables (11/46)- Internal binary operations (13/33)- External binary operations (16/24) • Most questions of typemc, matching or short text answer • An open-ended feedback questionexpressions interpreted and classified LeTTET'05

  15. Results 1aStudents’ generalopinions of the tests LeTTET'05

  16. Results 1b (detailed) LeTTET'05

  17. Results 1c (expressions) • The tasks were most suitable for testing one’s mastery of the function concept. (girl, 90 %) • Java figures are hard to understand, how to get information out of them and what they mean. However, they are nice to do, a bit different from ordinary exercises. (girl, 50 %) • Especially the figure-based taks difficult, because nothing alike was done before. (boy, 38 %) • Some problems easy, some not. Especially the problems concerning injection/surjections/bijections of two variable functions were not easy. (boy, 75 %) • The problems were difficult, since the concepts are not yet absorbed but sought. Training, training! (girl, 34 %) • Terrible tasks, even many of the questions are too difficult to understand. (boy, first trial, 40 %) • Well, it was moderately easy on the second try. Many problems were similar. (same boy, second trial, 95 %) LeTTET'05

  18. Results 2Defects in metacognitive thinking • experts’ vs. novices’ strategies in manipulating new tools • essential vs. irrelevant elements & actions • easily too many dimensions: mathematical, technical, observational • example: one variable ignoreddynamical picture LeTTET'05

  19. Results 3Problems in the use of technology • conflicts in using e.g. Javascript in the questions and orientation module in WebCT • browser problems with Java • browser problems with mathematical fonts LeTTET'05

  20. Results 4aAdvantages of interactive applets • students become engaged with the content and the problem setting • students get a ”feeling” for dependencies between the given parameters • dynamic pictures offer new possibilities to solve problems (e.g. draw a trace or use scaling) • automatic response analysis provides feedback and supports concept understanding and ”learning when doing” LeTTET'05

  21. Results 4bDisadvantages of interactive applets • new kind of representation form is unfamiliar for many students • computer activities are time consuming • problems in embedding to traditional curriculum • problems in measuring the results • students are conservative in new situations LeTTET'05

  22. Results 5The need of pedagogical tutoring • we start from two perspectives:1. appropriate pedagogical framework2. tutorial measurements from the teacher’s side • 1. In the ideal case the students’ social constructions lead to a viable definition for the concept. • 2. Metacognitive and technical tutoring: • a) face-to-face tutoring best for metacognitive defects, at least for less experienced students • b) for technical guidance also audio solutions should be taken into account LeTTET'05

  23. Results 6aAdvantages of the WebCT environment • questions can be authored using plain text style or html code (mathematics, pictures, applets) • easy to use for the students • after submitting the quiz the students can see the whole worksheet equipped with their own answers, together with the correct answers, and comments written by the teacher • quizzes can be corrected automatically, or at least by making minor revisions • data can be examined, manipulated and stored in many ways LeTTET'05

  24. Results 6bDisadvantages of the WebCT environment • becoming very expensive • the lack of support for (higher) mathematics • the system is not easy to use for the authors, e.g. navigation is complicated and running slow • although the problems of a test can be shown either one at a time or all at the same time as a long document, the evaluation and teacher comments cannot be seen before answering all the questions. Therefore the test system cannot be used efficiently for delivering learning material (“learning when doing”, “exam as a learning tool”) • it is not possible to correct all the answers to a certain problem manually in a row LeTTET'05

  25. Conclusions (1/3) • To shift from paper and pencil work towards technology-based interactive learning, an adequate pedagogical theory is needed. • While the focus of school teaching is on procedural knowledge, the university mathematics aims for conceptual understanding. • More or less systematic pedagogical models connected to an appropriate use of technology can help us to achieve both of these goals. LeTTET'05

  26. Conclusions (2/3) • Interactive applets can be used not only for learning but also for assessment and for increasing new kinds of complexity for the content. • Simultaneous activation allows the teacher to be freed from the worry about the order in which student’s mental models develop when interpreting, transforming and modelling mathematical objects. LeTTET'05

  27. Conclusions (3/3) • University mathematics can be learnt outside institutions by utilising web-based interactivities. • Most students’ difficulties appear in the steps of mathematising and interpreting. To validate this result, the correlation between test performance in IGR problems and in problems represented in symbolic form will be examined. • The on-going research in the DAAD project will focus on qualitative research of students’ thinking processes. LeTTET'05

  28. IBMT principle (Interaction Between Mathematics and Technology)by Kadijevich, Haapasalo & Hvorecky (2004): • “When using mathematics, don’t forget available tool(s); when utilising tools, don’t forget the underlying mathematics.” LeTTET'05

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