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Introduction

Solving mathematical problems based on dynamical sketches: an exploratory study ProMath 2004, Lahti, Finland Timo Ehmke (IPN, Kiel) Martti E. Pesonen (Joensuu). Introduction. the scope : first year University maths courses concepts : function, binary operation

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Introduction

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  1. Solving mathematical problems based on dynamical sketches: an exploratory studyProMath 2004, Lahti, FinlandTimo Ehmke (IPN, Kiel)Martti E. Pesonen (Joensuu)

  2. Introduction • the scope: first year University maths courses • concepts: function, binary operation • media: interactive exercises/problems • purpose: to evaluate students’ actions and understanding of different representations; verbal, symbolical and graphical ProMath 2004

  3. Ingredients • mathematical: the concept definitions • pedagogical: concept formation • technical: dynamical Java applets, WebCT test tools ProMath 2004

  4. Theoretical background The 5 phases of concept formation1 • Orientation • Definition • Identification • Production • Reinforcement Emphasis on the first four steps. 1 see Haapasalo (1993, 1997) and Pesonen (2001), Pesonen et al. (2004) ) combined ProMath 2004

  5. n ½ (n+m) m Theoretical background Verbal - symbolical - graphical representations (see Haapasalo 1993, 1997) verbal: mean value of two numbers n and m symbolical: 1/2 (n + m) graphical: Example with dynamical picture ProMath 2004

  6. 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) ProMath 2004

  7. General remarks Advantages • 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” ProMath 2004

  8. General remarks Disadvantages or weak points • these computer activities are time consuming • embedding to traditional curriculum problematic • measuring the results • students are conservative in new situations • most general students’ complaints: ”I don’t like computers.” ”I learn better with paper-and-pencil exercises.” ”I don’t understand what to do.” ”The time should not be limited.” ProMath 2004

  9. Study 1 • first semester Introductory Mathematics course in Joensuu (N = 42) • 2-hour exercise sessions in 2 groups • student actions recorded by screen capture program Camtasia (Techsmith) • material analyzed by Ehmke & students at IPN, Kiel The test material ProMath 2004

  10. Research Questions in Study 1 • What advantages are there in manual dragging, what in automatic animation? • What can be said about tracing? • What significance do the hints have, how much and what kind of guidance is ”optimal”? ProMath 2004

  11. Conclusions 1 What advantages are there in manual dragging, what in automatic animation? • dragging was very popular throughout the tests • in some problems it was crucial • dragging is advantageous when studying what happens in special places, and in controlling values • animation is useful in attracting students’ attention to special situations • most students used animations when it was helpful or necessary, but only 40% when not really needed ProMath 2004

  12. Dragging in special situations • differences caused probably by different levels of difficulty • the students had to find the special places themselves, not all managed in this • in Problem 5 varying the parameter a causes the whole function x ax change • dragging a around 1 was crucial in finding out the values for which the function is increasing ProMath 2004

  13. Example of tracing ProMath 2004

  14. Conclusions 2 What can be said about tracing? • about half of the students used tracing when it was available • tracing facility was not well guided, 67% did not clear the traces  problems with messy figure • faulty ideas or misconceptions: in Problem 2f five students gave the same wrong answer for the image of [0,1], none of them used tracing ProMath 2004

  15. Conclusions 3 What significance do the hints have, how much and what kind of guidance is ”optimal”? • applet hints must be offered only when crucial; the students stopped using hints as soon as they found them not useful (problems were easy) • the link to the formal definition was practically not used at all, this is perhaps caused by their weak understanding of it cf. the concept image vs. concept definition in Vinner (1991) ProMath 2004

  16. Research Questions in Study 2 a) Do different kinds of interactive graphical representations of the same operation lead to differences on the students’ performance? b) Are the student performances with interactive graphical problems in correlation with problems of other representation types, and with their overall grades? ProMath 2004

  17. Different graphical representations Do different kinds of interactive graphical representations of the same operation lead to differences on the students’ performance?Problems and tables: • Problem 21 (operation in R) • Problem 22 (operation in [-c, c]) • Problem 23 (operation in R2) • Problem 24 (operation in dics) • Problem 25 (operation in discrete set) ProMath 2004

  18. Definition Identification (internal BO) Student performance Descriptive statistics Correlations (also with ”pre-knowledge” Function Tests 1&2) ProMath 2004

  19. Definition Identification (int & ext BO) Student performance Descriptive statistics Correlations ProMath 2004

  20. VSG Identification (internal BO) Student performance Descriptive statistics Correlations (also with ”pre-knowledge” Function Tests 1&2) ProMath 2004

  21. Student performance VSG Production (internal BO) Descriptive statistics ProMath 2004

  22. Student performance VSG Production (internal BO) Correlations (also with ”pre-knowledge” Function Tests 1&2) ** Correlation is significant at the 0.01 level (2-tailed) * Correlation is significant at the 0.05 level (2-tailed) ProMath 2004

  23. Student performanceaccording to total achievement ProMath 2004

  24. Student performanceaccording to total achievement ProMath 2004

  25. Student performanceaccording to total achievement ProMath 2004

  26. References 1 • 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 ProMath 2004

  27. References 2 • Pesonen, M. E. 2001. WWW Documents With Interactive Animations As Learning Material. In the Joint Meeting of AMS and MAA, New Orleans, January 2001. URL: http://www.joensuu.fi/mathematics/MathDistEdu/MAA2001/index.html • Pesonen, M., Haapasalo, L. & Lehtola, H. 2002. Looking at Function Concept through Interactive Animations. The Teaching of Mathematics 5 (1), 37-45. • Pesonen, M. E. et al. 2004. Applying verbal, symbolical and graphical representations to studying basic mathematical concepts in interactive distance learning material (in Finnish). University of Joensuu, Finland. ProMath 2004

  28. References 3 • 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. ProMath 2004

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