Solving Mathematical Problem

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

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

Introduction : 

14.10.2008 ProMath 2004 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

Ingredients : 

14.10.2008 ProMath 2004 3 Ingredients mathematical: the concept definitions pedagogical: concept formation technical: dynamical Java applets, WebCT test tools

Theoretical background : 

14.10.2008 ProMath 2004 4 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) Theoretical background ) combined

Theoretical background : 

14.10.2008 ProMath 2004 5 Theoretical background Verbal - symbolical - graphical representations (see Haapasalo 1993, 1997) verbal: mean value of two numbers n and m symbolical: 1/2 (n + m) Example with dynamical picture graphical:

Features of the interactive tasks : 

14.10.2008 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)

General remarks : 

14.10.2008 ProMath 2004 7 General remarks 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” Advantages

General remarks : 

14.10.2008 ProMath 2004 8 General remarks 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.” Disadvantages or weak points

Study 1 : 

14.10.2008 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

Research Questions in Study 1 : 

14.10.2008 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”?

Conclusions 1 : 

14.10.2008 ProMath 2004 11 Conclusions 1 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 What advantages are there in manual dragging, what in automatic animation?

Dragging in special situations : 

14.10.2008 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

Example of tracing : 

14.10.2008 ProMath 2004 13 Example of tracing

Conclusions 2 : 

14.10.2008 ProMath 2004 14 Conclusions 2 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 What can be said about tracing?

Conclusions 3 : 

14.10.2008 ProMath 2004 15 Conclusions 3 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) What significance do the hints have, how much and what kind of guidance is ”optimal”?

Research Questions in Study 2 : 

14.10.2008 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?

Different graphical representations : 

14.10.2008 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)

Definition Identification (internal BO) : 

14.10.2008 ProMath 2004 18 Definition Identification (internal BO) Descriptive statistics Correlations (also with ”pre-knowledge” Function Tests 1&2) Student performance

Definition Identification (int & ext BO) : 

14.10.2008 ProMath 2004 19 Definition Identification (int & ext BO) Descriptive statistics Correlations Student performance

VSG Identification (internal BO) : 

14.10.2008 ProMath 2004 20 VSG Identification (internal BO) Descriptive statistics Correlations (also with ”pre-knowledge” Function Tests 1&2) Student performance

VSG Production (internal BO) : 

14.10.2008 ProMath 2004 21 VSG Production (internal BO) Descriptive statistics Student performance

VSG Production (internal BO) : 

14.10.2008 ProMath 2004 22 VSG Production (internal BO) Correlations (also with ”pre-knowledge” Function Tests 1&2) Student performance ** Correlation is significant at the 0.01 level (2-tailed) * Correlation is significant at the 0.05 level (2-tailed)

Student performanceaccording to total achievement : 

14.10.2008 ProMath 2004 23 Student performanceaccording to total achievement

Student performanceaccording to total achievement : 

14.10.2008 ProMath 2004 24 Student performanceaccording to total achievement

Student performanceaccording to total achievement : 

14.10.2008 ProMath 2004 25 Student performanceaccording to total achievement

References 1 : 

14.10.2008 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

References 2 : 

14.10.2008 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.

References 3 : 

14.10.2008 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.