2. 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

3. 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)

4. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen References 1 l Haapasalo, L. & Kadijevich, Dj. (2000). Two Types of Mathematical Knowledge and Their Relation. Journal für Mathematikdidaktik 21 (2), 139- 157. l 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. l Kadijevich, Dj. & Haapasalo, L. (2001). Linking Procedural and Conceptual Mathematical Knowledge through CAL. Journal of Computer Assisted Learning 17 (2), 156-165. l Kadijevich, Dj. (2004) Improving mathematics education: neglected topics and further research directions. University of Joensuu. Publications in Education 101.

5. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen References 2 l Pesonen, M., Haapasalo, L. & Lehtola, H. (2002) Looking at Function Concept through Interactive Animations. The Teaching of Mathematics 5 (1), 37-45. l 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

6. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen References 3 l 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. l 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/dot 1991f-mdnor-function-pme.pdf http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot 1991f-mdnor-function-pme.pdf l 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.

7. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen... References 3 l Vinner, S. & Dreyfus, T. (1989). Images and Definitions for the Concept of Function. Journal for Research in Mathematics Education 20 (4), pp. 356-366. l Vinner, S. (1991). The Role of Definitions in Teaching and Learning. In D. Tall (ed.): Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer. l Holton, D. (2001) The Teaching and Learning of Mathematics at University Level. An ICMI Study. Dordrecht: Kluwer.

8. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen 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

9. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen School vs. University 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

10. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Aims / 1 st step l To generate hypotheses, what special benefits do the dynamic interactions offer and what new types of difficulties in conceptual thinking arise. l What advantages are there in manual dragging by the students (within the applets) and what in automatic animation? l How students use the tracing function and what significance do the given hints have?

11. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Aims / 2 nd step l To analyse whether different representations (symbolic, verbal, graphic) given through interactive applets) lead to different test performance. l To consider possible explanations to these difficulties (e.g. why conceptually identical but functionally slightly different implementations lead to diverging interpretations).

12. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Ingredients l mathematical: the concept definitions l pedagogical: framework of concept building l technical: dynamic Java applets, WebCT test tools Example of a dynamic applet

13. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Features of the interactive tasks l dragging points by mouse l automatic animation/movement  dynamic change in the figure l tracing of depending points l hints and links (text) l hints as guiding objects in the figure l response analysis (in Geometria applet)

14. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Interactive Graphical Representations (IGR)

15. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Interactive Graphical Representations (IGR) http://www.joensuu.fi/mathematics/MathDistEdu/ Animations2MentalModels/RovaniemiNBE2005/index.html

16. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Theoretical background l Interplay between conceptual (C) and procedural knowledge (P) (cf. Ref #1) l Multiple representations of concept attributes (cf. Ref #1) l Interactive Graphical Representations (IGR) (cf. Ref #2 - #4)

17. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Interplay between P and C l 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.

18. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Developmental approach assumes that P enables C development. The term reflects the philogenetic and ontogenetic nature of knowledge..

19. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen 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..

20. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Which one of the situations represents conceptual or/and procedural knowledge?

21. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Utilising MODEM theory… … emphasis being on (D), (I), and (P)

22. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Multiple representations of concept attributes. http://www.joensuu.fi/lenni/programs.html

23. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen 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.

24. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Study #1 First semester Introductory Mathematics (N = 42) l a 2-hour exercise sessions in 2 groups l interactive sketches to introduce the function concept l answers were sent directly to the teacher l students’ actions were recorded by a screen capturer l the material was analyzed with qualitative methods.

25. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Second semester Linear Algebra (N = 82) l Test items were posed to the students using WebCT l focus on an exceptionally poorly solved problem containing an IGR in the plane (52 students) l a query soon afterwards asking about reasons for poor performance (43 answers) l An open-ended feedback question expressions interpreted and classified Study #2

26. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Results by Study #1........ and Study #2

27. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Study #1 results: drag/animate l dragging is very popular throughout the tests... l... and in some problems it is crucial l dragging is useful when studying what happens in special places, and when controlling values l animation is useful in getting students’ attention to special situations l most students use animations when it is helpful or necessary Q: What advantages are there in manual dragging, what in automatic animation?

28. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Example of tracing

29. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen l one half of the students used tracing if available l tracing facility was not well guided, 2/3 did not clear the traces  messy figure l students with totally wrong ideas did not use tracing Q: What can be said about tracing? Study #1 results: tracing

30. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Study #1 conclusions l Hints must be offered only when crucial; students stop using hints as soon they find them useless. l Link to formal definition is practically useless... l... because of students’ pure C understanding (cf. the concept image vs. concept definition in Vinner 1991) The role of the applet hints?

31. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Study #2 results: Students’ explanations http://www.joensuu.fi/mathematics/MathDistEdu/Animations2MentalModels/Rovanie miNBE2005/NBE05_Figure1JSP.html

32. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Study #2 results: General opinions of the tests

33. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen... in more detail

34. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen... expressions l Tasks suitable for testing mastery of the function concept. (girl, 90 %) l Hard to get information out of applets and to understand… l … but they are nice, different from ordinary exercises. (girl, 50 %) l Especially the figure-based tasks are difficult, because nothing alike was done before. (boy, 38 %) l Some problems easy, some not. Especially the problems concerning two variable functions were not easy. (boy, 75 %) l Problems were difficult, since the concepts are just sought. Training, training! (girl, 34 %) l Terrible tasks, even many of the questions are too difficult. (boy, first trial, 40 %) l Well, it was moderately easy on my second trial. Many problems were similar. (the same boy, second trial, 95 %)

35. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Defects in metacognitive thinking (cf. Haapasalo & Siekkinen in this NBE) l experts’ vs. novices’ strategies l essential vs. irrelevant elements & actions l easily too many dimensions: mathematical, technical, observational l example: one variable ignored dynamical picture dynamical picture

36. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Technical problems l conflicts in using e.g. Javascript in the questions and orientation module in WebCT l browser problems with Java l browser problems with mathematical fonts

37. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Advantages of interactive applets l students become engaged with the content and the problem setting l students get a ”feeling” of the relation between the given parameters l dynamic pictures offer new possibilities to solve problems (e.g. trace or use scaling) l automatic response analysis provides feedback and ”learning when doing”

38. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen l new kind of representation form is unfamiliar for many students l computer activities are time consuming l problems in embedding to traditional curriculum l problems in measuring the results l students are conservative in new situations Disadvantages of interactive applets Disadvantages of interactive applets

39. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen The need of pedagogical tutoring l Concerning teacher’s tutorial measures: l a) face-to-face tutoring is best for metacognitive defects, at least for less experienced students l b) for technical guidance also audio solutions should be taken into account. l Concerning appropriate pedagogical framework: l It is the students’ social constructions that lead to a viable definition for the concept (- ideal case!)

40. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Example (from Haapasalo & Siekkinen in this NBE)

41. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Novice learner (“Alien”) (cf. Haapasalo & Siekkinen in this NBE)

42. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Expert learner (cf. Haapasalo & Siekkinen in this NBE)

43. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Expert learner (cf. Haapasalo & Siekkinen in this NBE)

44. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Advantages of the WebCT… l questions can be authored using plain text style or html code (mathematics, pictures, applets) l easy to use for the students l quizzes can be corrected automatically, or at least by making minor revisions l data can be examined, manipulated and stored in many ways l 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.

45. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Disdvantages of the WebCT… l technical solutions can become expensive l the lack of support for (higher) mathematics l not easy to use for the authors, e.g. navigation is complicated and running slowly l it is not possible to correct all the answers to a certain problem manually in a row l the assessment and teacher’s comments cannot be seen before answering all the questions l Therefore the test system cannot be used efficiently for “exam as a learning tool” l …can be fatal regarding constructivism.

46. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Conclusions (1/3) l To shift from paper and pencil work towards technology-based interactive learning, an adequate pedagogical theory is needed. l Applets alone are not a big step to shift procedural 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.

47. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Conclusions (2/3) l 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.

48. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen Conclusions (3/3) l University mathematics can be learned outside institutions by utilising web-based activities. l 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. l The on-going research in the DAAD project will focus on qualitative research of students’ thinking processes.

49. 15.09. '05 NBE '05 Ehmke, Haapasalo & Pesonen 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.”