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Building a bridge between school and university - critical issues concerning interactive applets Timo Ehmke (Kiel / GER) Lenni Haapasalo (Joensuu, FIN) Martti E. Pesonen (Joensuu, FIN) NBE ’05 , Rovaniemi , Finland.

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References 1

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  1. Building a bridge between school and university- critical issues concerning interactive appletsTimo Ehmke (Kiel / GER)Lenni Haapasalo (Joensuu, FIN)Martti E. Pesonen (Joensuu, FIN) NBE ’05, Rovaniemi, Finland

  2. Based on the project From Visual Animations to Mental Models in Mathematics Concept Formation(sponcored by DAAD and the Academy of Finland)Martti E. Pesonen (Department of Mathematics, University of Joensuu, FIN) Lenni Haapasalo (Department of Applied Education, Joensuu, FIN)Timo Ehmke (Leibniz Institute for Science Education / IPN, University of Kiel,GER)

  3. References 1 • Haapasalo, L. & Kadijevich, Dj. (2000). Two Types of Mathematical Knowledge and Their Relation. Journal für Mathematikdidaktik 21 (2), 139-157. • 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. • Kadijevich, Dj. & Haapasalo, L.(2001).Linking Procedural and Conceptual Mathematical Knowledge through CAL.Journal of Computer Assisted Learning 17 (2), 156-165. • Kadijevich, Dj. (2004) Improving mathematics education: neglected topics and further research directions.  University of Joensuu. Publications in Education 101. NBE '05 Ehmke, Haapasalo & Pesonen

  4. References 2 • Pesonen, M., Haapasalo, L. & Lehtola, H. (2002) Looking at Function Concept through Interactive Animations. The Teaching of Mathematics 5 (1), 37-45. • 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). http://www.joensuu.fi/mathematics/MathDistEdu/MAA2001/index.html NBE '05 Ehmke, Haapasalo & Pesonen

  5. References 3 • 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. • 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. NBE '05 Ehmke, Haapasalo & Pesonen

  6. ... 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. • Holton, D. (2001) The Teaching and Learning of Mathematics at University Level. An ICMI Study. Dordrecht: Kluwer. NBE '05 Ehmke, Haapasalo & Pesonen

  7. Background • Mathematics is considered as organized body of knowledge. • Students are largely passive, practicing old, clearly formulated, and unambiguous questions for timed examinations. •Theory is abstract and depends on an unfamiliar language. These features leave students dispirited and bored, and their performance in more advanced courses is poor because the foundations are weak. The assessment is reduced to bookwork and stereotyped questions, to be remembered without becoming a vital part of the student. (Joint European Project MODEM; http://www.joensuu.fi/lenni/modem.html NBE '05 Ehmke, Haapasalo & Pesonen

  8. School vs. University • The main problem:how students could develop their procedural school thinking towards abstract conceptual thinking? • Neglected topics: • promoting the human face of mathematics • relating procedural and conceptual knowledge • utilizing mathematical modelling in a humanistic, technologically-supported way • promoting technology-based learning through multimedia design and on-line collaboration NBE '05 Ehmke, Haapasalo & Pesonen

  9. Aims / 1st step • To generate hypotheses, what special benefits do the dynamic interactions offer and what new types of difficulties in conceptual thinking arise. • What advantages are there in manual dragging by the students (within the applets) and what in automatic animation? • How students use the tracing function and what significance do the given hints have? NBE '05 Ehmke, Haapasalo & Pesonen

  10. Aims / 2nd step • To analyse whether different representations (symbolic, verbal, graphic) given through interactive applets) lead to different test performance. • To consider possible explanations to these difficulties (e.g. why conceptually identical but functionally slightly different implementations lead to diverging interpretations). NBE '05 Ehmke, Haapasalo & Pesonen

  11. Ingredients • mathematical: the concept definitions • pedagogical: framework of concept building • technical: dynamic Java applets, WebCT test tools Example of a dynamic applet NBE '05 Ehmke, Haapasalo & Pesonen

  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) NBE '05 Ehmke, Haapasalo & Pesonen

  13. Interactive Graphical Representations (IGR) NBE '05 Ehmke, Haapasalo & Pesonen

  14. Interactive Graphical Representations (IGR) http://www.joensuu.fi/mathematics/MathDistEdu/ Animations2MentalModels/RovaniemiNBE2005/index.html NBE '05 Ehmke, Haapasalo & Pesonen

  15. Theoretical background • Interplay between conceptual (C) and procedural knowledge (P) (cf. Ref #1) • Multiple representations of concept attributes (cf. Ref #1) • Interactive Graphical Representations (IGR)(cf. Ref #2 - #4) NBE '05 Ehmke, Haapasalo & Pesonen

  16. Interplay between P and C • Procedural knowledge(P) denotes dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation forms. This usually requires not only knowledge of the objects being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them. • Conceptual knowledge(C)denotes knowledge of and a “skilful, conscious drive” along particular (semantic) networks, the elements of which can be concepts, rules (algorithms, procedures, etc.), and even problems (a solved problem may introduce a new concept or rule) given in various representation forms. NBE '05 Ehmke, Haapasalo & Pesonen

  17. . Developmental approach assumes that P enables C development. The term reflects the philogenetic and ontogenetic nature of knowledge. NBE '05 Ehmke, Haapasalo & Pesonen

  18. Educational approach . is based on the assumtion that P depends on C. The term refers to educational needs, typically requiring a large body of knowledge to enable transfer. NBE '05 Ehmke, Haapasalo & Pesonen

  19. Which one of the situations represents conceptual or/and procedural knowledge? NBE '05 Ehmke, Haapasalo & Pesonen

  20. Utilising MODEM theory… … emphasis being on (D), (I) , and (P) NBE '05 Ehmke, Haapasalo & Pesonen

  21. Multiple representations of concept attributes . http://www.joensuu.fi/lenni/programs.html NBE '05 Ehmke, Haapasalo & Pesonen

  22. Methods The focus: to concentrate on students’ difficulties to utilize sketches that contain special technical or mathematical features. Cognitive findings were represented just for considering possible explanations to these difficulties. NBE '05 Ehmke, Haapasalo & Pesonen

  23. Study#1 First semester Introductory Mathematics (N = 42) • a 2-hour exercise sessions in 2 groups • interactive sketches to introduce the function concept • answers were sent directly to the teacher • students’ actions were recorded by a screen capturer • the material was analyzed with qualitative methods. NBE '05 Ehmke, Haapasalo & Pesonen

  24. Study #2 Second semester Linear Algebra (N = 82) • Test items were posed to the students using WebCT • focus on an exceptionally poorly solved problem containing an IGR in the plane (52 students) • a query soon afterwards asking about reasons for poor performance (43 answers) • An open-ended feedback question expressions interpreted and classified NBE '05 Ehmke, Haapasalo & Pesonen

  25. Results by Study #1 .... .... and Study #2 NBE '05 Ehmke, Haapasalo & Pesonen

  26. Study #1 results: drag/animate Q: What advantages are there in manual dragging, what in automatic animation? • dragging is very popular throughout the tests... • ... and in some problems it is crucial • dragging is useful when studying what happens in special places, and when controlling values • animation is useful in getting students’ attention to special situations • most students use animations when it is helpful or necessary NBE '05 Ehmke, Haapasalo & Pesonen

  27. Example of tracing NBE '05 Ehmke, Haapasalo & Pesonen

  28. Study #1 results: tracing Q: What can be said about tracing? • one half of the students used tracing if available • tracing facility was not well guided, 2/3 did not clear the traces  messy figure • students with totally wrong ideas did not use tracing NBE '05 Ehmke, Haapasalo & Pesonen

  29. Study #1 conclusions The role of the applet hints? • Hints must be offered only when crucial; students stop using hints as soon they find them useless. • Link to formal definition is practically useless ... • ... because of students’ pure Cunderstanding (cf. the concept image vs. concept definition in Vinner 1991) NBE '05 Ehmke, Haapasalo & Pesonen

  30. Study #2 results:Students’ explanations http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/RovaniemiNBE2005/NBE05_Figure1JSP.html NBE '05 Ehmke, Haapasalo & Pesonen

  31. Study #2 results: Generalopinions of the tests NBE '05 Ehmke, Haapasalo & Pesonen

  32. ... in more detail NBE '05 Ehmke, Haapasalo & Pesonen

  33. ... expressions • Tasks suitable for testing mastery of the function concept. (girl, 90 %) • Hard to get information out of applets and to understand… • … but they are nice, different from ordinary exercises. (girl, 50 %) • Especially the figure-based tasks are difficult, because nothing alike was done before. (boy, 38 %) • Some problems easy, some not. Especially the problems concerning two variable functions were not easy. (boy, 75 %) • Problems were difficult, since the concepts are just sought. Training, training! (girl, 34 %) • Terrible tasks, even many of the questions are too difficult. (boy, first trial, 40 %) • Well, it was moderately easy on my second trial. Many problems were similar. (the same boy, second trial, 95 %) NBE '05 Ehmke, Haapasalo & Pesonen

  34. Defects in metacognitive thinking(cf. Haapasalo & Siekkinen in this NBE) • experts’ vs. novices’ strategies • essential vs. irrelevant elements & actions • easily too many dimensions: mathematical, technical, observational • example: one variable ignoreddynamical picture NBE '05 Ehmke, Haapasalo & Pesonen

  35. Technical problems • conflicts in using e.g. Javascript in the questions and orientation module in WebCT • browser problems with Java • browser problems with mathematical fonts NBE '05 Ehmke, Haapasalo & Pesonen

  36. Advantages of interactive applets • students become engaged with the content and the problem setting • students get a ”feeling” of the relation between the given parameters • dynamic pictures offer new possibilities to solve problems (e.g. trace or use scaling) • automatic response analysis provides feedback and ”learning when doing” NBE '05 Ehmke, Haapasalo & Pesonen

  37. Disadvantages 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 NBE '05 Ehmke, Haapasalo & Pesonen

  38. The need of pedagogical tutoring • Concerning teacher’s tutorial measures: • a) face-to-face tutoring is best for metacognitive defects, at least for less experienced students • b) for technical guidance also audio solutions should be taken into account. • Concerning appropriate pedagogical framework: • It is the students’ social constructions that lead to a viable definition for the concept (- ideal case!) NBE '05 Ehmke, Haapasalo & Pesonen

  39. Example (from Haapasalo & Siekkinen in this NBE) NBE '05 Ehmke, Haapasalo & Pesonen

  40. Novice learner (“Alien”)(cf. Haapasalo & Siekkinen in this NBE) NBE '05 Ehmke, Haapasalo & Pesonen

  41. Expert learner (cf. Haapasalo & Siekkinen in this NBE) NBE '05 Ehmke, Haapasalo & Pesonen

  42. Expert learner (cf. Haapasalo & Siekkinen in this NBE) NBE '05 Ehmke, Haapasalo & Pesonen

  43. Advantages of the WebCT… • questions can be authored using plain text style or html code (mathematics, pictures, applets) • easy to use for the students • quizzes can be corrected automatically, or at least by making minor revisions • data can be examined, manipulated and stored in many ways • 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 … might be objectivist / behaviorist loaded. NBE '05 Ehmke, Haapasalo & Pesonen

  44. Disdvantages of the WebCT… • technical solutions can become expensive • the lack of support for (higher) mathematics • not easy to use for the authors, e.g. navigation is complicated and running slowly • it is not possible to correct all the answers to a certain problem manually in a row • the assessment and teacher’s comments cannot be seen before answering all the questions • Therefore the test system cannot be used efficiently for “exam as a learning tool” …can be fatal regarding constructivism. NBE '05 Ehmke, Haapasalo & Pesonen

  45. Conclusions (1/3) • To shift from paper and pencil work towards technology-based interactive learning, an adequate pedagogical theory is needed. • Applets alone are not a big step to shiftprocedural school teaching to the university mathematics aiming 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. NBE '05 Ehmke, Haapasalo & Pesonen

  46. 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 of P and C 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. NBE '05 Ehmke, Haapasalo & Pesonen

  47. Conclusions (3/3) • University mathematics can be learned outside institutions by utilising web-based activities. • Most students’ difficulties appear in the steps of mathematising and interpreting. To validate this result, the correlation between test performance in IGR vs. paper-and-pencil problems are to be examined. • The on-going research in the DAAD project will focus on qualitative research of students’ thinking processes. NBE '05 Ehmke, Haapasalo & Pesonen

  48. 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.” NBE '05 Ehmke, Haapasalo & Pesonen

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