3 Theoretical background
3.3 Interactive Graphical Representations (IGR)
Binary operations are two-variable functions, whose variables in linear algebra are usually vectors. Therefore the traditional static graphical representations become inadequate. Thus, a completely new “learning dimension” can be added by using dynamic figures. We utilized the MODEM framework above in our Interactive Graphical Representations (IGR). These allow - and mostly require - the learner to interact with the figures by dragging with the mouse or using control buttons. In our case the IGR pictures are implemented using dynamic geometry Java applets (JavaSketchpad and Geometria, see Pesonen 2001; Ehmke 2001). The advantage of IGR is that students become engaged with the content and the problem setting and 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). Also an automatic response analysis, which enables immediate feedback, can support concept understanding and ”learning when doing”.
Disadvantages are that computer activities are time consuming (especially in developing) and also the integration into a traditional curriculum may be problematic. The measuring of students’ achievements must be thoroughly reconsidered.
By a sketch we mean a dynamic interactive applet construction containing text parts, figures, and geometric elements (points, lines, rays, segments, circles and more advanced constructions), which is meant to be manipulated with the mouse. Control buttons can be used for showing, hiding, moving and animating the sketch elements (see Figure 2 in Chapter 5.1).
4 Methods
For the analysis of our research aims, two studies were conducted in the Mathematics Department of the University of Joensuu. The pilot study was done within a first semester Introductory Mathematics course in November 2002 (N = 42).
The students were mainly mathematics majors and the course lectures had already dealt with logic, sets and relations, but not yet functions. This study comprised a 2-hour exercise sessions in 2 groups. The first part of the learning material was implemented by the Geometria applet and it contained interactive sketch tasks about sets and relations, with focus on the introducing the function concept. The second part consisted of an html form containing dynamic JavaSketchpad figures (see Figures 2 and 3) together with the appropriate problem sets. The answers were sent directly to the teacher.
In both parts of the working periods the students’ actions were recorded by a screen capture program, and later the material was analyzed with qualitative methods. The second study was done during the first course on Linear Algebra (N = 92) in March 2004. These students were heterogeneous in their background: most of them were first year mathematics or physics majors but about 20% of them were 1st to 3rd year computer science majors. The test items were posed to the students using the course management system WebCT. We represent the most important affective findings of our studies, concentrating mostly on students’ difficulties to utilize certain sketches that contain special technical or mathematical features. Some cognitive findings are represented just for considering possible explanations to these difficulties3.
5 Results
5.1 Study 1
The aim of the first study was to analyse how students use the special advantages of interactive applets. In the following we summarize the main findings. A more detailed description of the results is given in Pesonen et al. (2003).
What advantages are there in manual dragging, what in automatic animation? Dragging was very popular throughout the tests. All students used this feature to interact with the learning content. Dragging is advantageous when studying what happens in special positions in the sketch and in the controlling of parameter values. The study confirms that animation is useful in attracting students’ attention to special situations. Most students used animations when it was helpful or necessary, but only 40% when it was not crucial. In some problems dragging was crucial. For example, when the students had to find the special places themselves, not all managed in this. A typical difficulty could be seen in the Problems concerning the Figure 2: The base value a can be controlled on the parameter axis in the bottom, where also the Neper number e is marked. For which values a is the function x → ax increasing? Varying the parameter a causes the whole curve y = ax change. Therefore, dragging a around the number 1 was crucial in finding out the values for which the function is increasing. On the basis of the screen capture analysis we found empirically some relationships between using the dragging mode for exploring special cases and solving the task correctly.
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Figure 2. A screen shot of the sketch concerning exponential functions
What can be said about tracing? The tracing facility, that was available in some tasks, plots the positions of the function value in the co-domain. This feature can be switched on and off by two labelled buttons inside the sketch. As long as the function is switched on, every point visited by the function value is marked. If the function is switched off, no new points are marked, but the old ones are still visible. They can be deleted by a small button, marked by a red X. An example of the trace after moving x from 1 to 0 is given in Figure 3. Our results show that about half of the students used tracing when it was available. In our learning material tracing facility was not well guided. The students had difficulties in applying the feature in a fruitful way. It could be seen that 67% did not clear the traces, which caused problems with messy figures. However, the tracing command could be used very fruitfully for the visualisation of the image of an interval (Figure 3). ). In this task some students showed a misconception about the range or image of a given function. They seemed to determine the image of a real interval simply by taking as image the interval between the image points of the left and right endpoints of the domain interval. The video analysis confirmed that these students had only examined the endpoints and did not use the possibility of tracing, which would have led them to the correct answer.
Figure 3. The dark dotted line visualises the image of the interval [0, 1]
What significance do the hints have, how much and what kind of guidance is ”optimal”? Hints were available by pressing a hint button within the sketch. This button shows a modal window with a helping text. By the close button, the hint disappears and the work can continue at the point where it was interrupted. Hints were available in five applets.
The results concerning the use of hints were alarming. Applet hints must be offered only when crucial; the students stopped using hints as soon as they found them not useful.
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5.2 Study 2
The learning material of the second study contained problems concerning verbal, symbolic and graphic representations of mathematical binary operations. A first descriptive report (Pesonen et al. 2005) shows that students who could identify a binary operation given through a verbal description are also able to identify a binary operation in a symbolic representation form (r = 0.31). This was in accordance to the findings in the MODEM project (Haapasalo, 2003, p. 15), where especially the verbalisation was an important step for the concept building. In contrast to this, identifying binary operations given in an interactive graphical form, do not stand in correlation with these two “classic” representation types (verbal-graphic: r = 0.04, symbolic-graphic: r = 0.05). To analyse this remarkable result deeper a qualitative study was applied post-hoc. The students were interviewed in a web-based questionnaire about their understanding of the interactive graphics. We chose from the test the most difficult task (p = 0.18) represented in Figure 4. The sketch contains three point objects: movable u, fixed v and the image uov. Because v is a fixed point, the function o cannot be a binary operation on the whole plane.
Figure 4. A sketch where the student interpretations were most diverging
We asked a group of students (N = 29), who could not solve this item correctly, some question about the item to find out why the difficulties arise. The student’s responses show that all students have recognized that the point v can be dragged (which would be necessary for a binary operation in the plane). Therefore, we can exclude that someone has not used dragging to solve the question. Further the students were asked:
a) I thought it is irrelevant, because u moved and the result was visible and moved.
b) I thought it is irrelevant, because u moved and the result was visible.
c) I thought it is irrelevant, because u and v and the result are seen.
d) I was confused and answered positively, just for sure.
About 70% agreed to statement a), about 40% to b), about 30% to c) and about 20% to d). This shows that students have difficulties to interpret the mathematical meaning of the applet constellation. Point v is fixed, so it cannot reach the whole domain. However, student did not have difficulties to identify analogous problems when represented in a symbolic or verbal form. This difference can be seen if we compare the average solution rates for verbal, symbolic and graphic identification tasks (Table 1). The group showed only small differences in the solution rates for symbolic and verbal tasks (symbolic: +4.3 %, verbal: –3.7 %) but a higher deficit in solving graphical tasks (–12 %).
Also an open text field was offered for their verbal explanations. For several students it had been enough to see that when u moves also the result moves in the plane.
Table 1. Solution rates for verbal, symbolic and graphic identification tasks
Symbolic tasks Verbal tasks Graphical tasks
All students 74.4 % 55.8 % 71.8 %
Students with item wrong (Figure 5) 78.7 % 52.1 % 59.8 %
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In the questionnaire we also showed these students three slightly different sketches about a certain binary operation on the real line interval [-c, c] (namely relativistic addition of speeds, but this was not told). We asked which of them suited best, and which worst, for illustrating that it really is a binary operation on the interval. Clearly worst was voted the one in which the result uov is seen even if the operands are dragged outside the interval. The other two were rated nearly equal; in the first the operands are tied to the interval and in the second the result vanishes whenever one of the operands is outside the interval.
6 Conclusions
Kadijevich (2004) points out four areas, which have been neglected in the research of mathematics education: 1) promoting the human face of mathematics; (2) relating procedural and conceptual mathematical knowledge; (3) utilizing mathematical modelling in a humanistic, technologically-supported way; and (4) promoting technology-based learning through applications and modelling, multimedia design, and on-line collaboration. These findings give special challenges to utilize modern technology in all its forms. Today even small pocket computers allow the use of the drag-and drop technology, where the student can easily manipulate mathematical objects between two windows, illustrating two different forms of mathematical representation. In many cases this means forming links between conceptual and procedural knowledge, which is a relevant perspective to evaluate the sustainability of educational technology, as done in Haapasalo & Siekkinen (2005). Their five implications fit the exemplary results of our two studies, which show that interactive learning modules offer new features and challenges not only for teaching and learning of mathematics but also metacognitive skills. Especially the possibility of dragging, tracing and animation provide a new aspect of representing mathematics. But as we have seen in the second study, the new possibilities easily come together with cognitive problems for a considerable part of students. Especially the interactive graphical representation can become problematic, as reported in Sierpinska et al. (1999). They found that students showed quite surprising ways in interpreting dynamic figures concerning plane vectors and basis. The interactive component seemed to differ from the classical representation formats (verbal, symbolic), which are traditionally used in school mathematics.
Our experiences show that moving from old studying culture towards modern technology-based one is full of cognitive, emotional and social problems (cf. Pesonen et al. 2005). The using interactive applets, for example, would not bring special advantages without an appropriate pedagogical framework. Even though they seem to bring interesting and useful elements in learning and assessment (as we have seen in study 1), it is especially the reflective tutoring that needs more consideration. Perhaps the most promising aspect of technology-based learning is to utilise the principle of simultaneous activation of conceptual and procedural knowledge. This 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. Our examples hopefully show that more or less systematic pedagogical models connected to an appropriate use of technology can help the teacher to achieve this goal. Interactive applets can be used not only for learning but also for assessment and for increasing new kinds of complexity for the content – being an essential element when building a bridge between school and university.
Acknowledgements
We would like to express our gratitude to two funding organizations for supporting our work. Developing of the applets is based on MODEM activities (www.joensuu.fi/lenni/modemact.html) during the teacher exchange program between the Universities of Bielefeld and Joensuu within the SOCRATES funding programme. Der Deutsche Akademische Austauschdienst (DAAD) has made possible an intensive co-operation between the University of Joensuu and the Leibniz Institute for Science Education (IPN), University of Kiel. Finally, we are grateful to the editors and the anonymous NBE reviewers for valuable comments, which have helped us to improve the presentation of our paper.
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INTRODUCING ICT IN HIGHER EDUCATION: 1 THE CASE OF SALAHADDIN/HAWLER UNIVERSITY
2005 - NETWORK-BASED EDUCATION 14th-17th SEPTEMBER 2005, ROVANIEMI, FINLAND 1
INTRODUCING ICT IN HIGHER EDUCATION: 1 THE CASE OF SALAHADDIN/HAWLER UNIVERSITY
2005 - NETWORK-BASED EDUCATION 14th-17th SEPTEMBER 2005, ROVANIEMI, FINLAND 1