Applying verbal, symbolical and graphical representations to studying basic mathematical concepts in interactive distance learning material

Martti E. Pesonen, Timo Ehmke, Thilo Wünscher

Summary We discuss the experiences and results of testing interactive tasks for Discrete Mathematics course teaching material in Autumn 2002. This was only a preliminary study, the material does not cover all the stages of the pedagogic method, so it has not been possible to make any comparing learning results. One important objective for the testing was to look for a suitable method through their action for the examining of the students' working. The material has been tested in three parts: in the beginning of the course briefly in AIM www-environment and in November-December more widely on participants of an Introductory Math Course and of Discrete Mathematics. The presentation is, to some extent, an extension to the articles Lehtola, Pesonen & Puumalainen 2002 (LPP) and Kähkönen, Lehtola & Pesonen 2002 (KLP). We can conclude, that interactive learning material helps the student to find fruitful ways to handle certain problem types. There is still some work to do in the technical and pedagogical implementations of the tasks, for example the instructions and hints concerning the dynamical sketches must be planned carefully, especially for distance learning.

1. Introduction

Studying mathematical concepts by reading and merely with the conventional problems is successful only for very few students. Thus, especially much is required of distance learning material, in addition to the impeccability of its subject content, it should be self-guiding and –explaining. The motivation can be tried to be improved with interesting and attracting ways of representation. Interactivity is one central effect, but in some mathematical visualisation situations it is even necessary; dynamic figures can be used to present many such situations that are very difficult to describe with static figures. Also totally new types of tasks can be produced in this way. Department of Mathematics of the University of Joensuu has the main responsibility in creating Discrete Mathematics course teaching material within an Applied Mathematics and Statistics sub-project of the Eastern Finland Virtual University Network. Attempt 2 has been made to integrate interactive visualisations and tasks to this material. With distance learning in mind these tasks, often based on dynamic figures, have been planned so that they will operate in an ordinary www environment, but also in course management systems like WebCT, which support html presentations.

2. Pedagogical background

We first describe problems in concept formation process, the chosen pedagogical framework and the interactive features of our teaching material.

2.1. Problems in learning mathematical concepts

The numerous research results show that in studying mathematics the most demanding step is to progress from concrete examples to an abstract concept, the so-called abstraction process. In mathematics matters to be abstracted are above all the concepts and their definitions. In understanding and use of definitions serious problems have been perceived; most students build their own mental model, a concept image about the concept, by collecting more or less relevant properties and characteristics as a substitute of the definition. Instead of the definition they tend to use this substitute even if they were familiar also with the definition (Vinner 1991). In the usual tasks of basic studies one can manage fairly well with the substitute and often the shortcoming will not be recognised even by the teacher. Haapasalo (1994, 1991) talks in this context about attributes which are related to the concept, that is, the different individual properties of the concept which can be verbal (V) symbolical (S) or graphical (G) and which make up the student’s mental model at the elementary level. For example Tall and Bakar (1991) write about the learning of the function concept: “the learner cannot construct the abstract concept of function without experiencing examples of the function concept in action, and the students cannot study examples of the function concept in action without developing prototype examples having built-in limitations that do not apply to the abstract concept”. The support to this claim is found also in many other sources as Breidenbach & al. (1992), Tall (1992), Vinner & Dreyfus (1989) and Brown & al. (1997). Because functions have been studied already at school, the dispelling of the vague concept image out of the way of the definition itself does not usually succeed without 3 special care. Therefore we have tried to include in our teaching materials plenty of surprising and unconventional examples and tasks that require careful thought, in order to create a cognitive conflict and shake the student's understanding. As a pedagogical study framework we chose a system which supports structured interactive learning environment together with multiple VSG representations.

2.2. Pedagogic frame – multiple representation

The pedagogic approach is based on the learning conception which is characteristic to so-called systematic constructivism (in more detail see Haapasalo 1994). Therein the learner is provided with versatile opportunities to construct information, problem solving, handling concrete models, social communication and information processing between different representation forms; however within a carefully structured learning programme. Adapted for example for learning conceptsit it is useful to separate five consecutive stages: 1) orientation: suitable learning opportunities are arranged, a desire to learn is aroused 2) defininition: sharpening the collection of properties, culminating in the exact definition 3) identification: learning to identify differences/similitudes between a model and the definition 4) production: inventing own examples and maturing to produce multiple representations 5) reinforcement: securing the understanding, e.g. through applications Multiple representations can refer to verbal describing, symbolical expression (for example as a mathematical formula or expression) and to graphical expression (for example a picture or graphical figre), which we use the combined notation VSG. With the different representations we can also bring out mutual dependence and interplay between procedural and conseptual knowledge (see Haapasalo, 2003). In this study material the problem sets concerning the concept of function are mainly related to the identification of the definition or between the different presentations, but also some GV and GS production and reinforcement tasks are included. 4

2.3. Interactivity in our teaching material

The pedagogical approach that has been used in the study on learning mathematical concepts through their multiple representations leans strongly on the interaction between dynamic figures and the user. The interactive visualisations have been made either with Java applets called JavaSketchpad or another called Geometria. With the help of the applet, a dynamic figure is brought to the www page; the user can drag some of the figure’s basic points, within predefined limitations, with the help of the computer mouse, and see how the figure is changing in corresponding way. Each figure, “sketch”, is associated to one or more tasks which are usually not easy to solve without “playing” with the figure. To the sketches based on the Geometria applet we can add even more interaction: the so-called response analysis. This means programmed feedback, the content of which depends on the user's actions, and which is shown with a push of the evaluate button (“Arvioi”), see Figure 1.

Figure 1. Geometria sketch with its feedback

The figure techniques itself has been described, e.g. in presentations Pesonen 2001, Ehmke 2001 and 2002, KLP and Lehtola 2002. Problem sets equipped with JavaSketchpad sketches concerning functions and binary operations have been experimented during several years, and experiences and results have been described in the articles LPP and Pesonen, Haapasalo & Lehtola 2002, and in the Master thesis of Lehtola 2002. From these we obtained very encouraging 5 experiences, but because of the additional features of Geometria applet the material made with it must be examined separately. Geometria-based problems and first experiences concerning set theory and the function concept were described already in the Master thesis of Lehtola. Here we try to present some more sophisticated analysis of the study material.

3. Purpose of the study

The study is a preliminary study which deepens our earlier experiments and in which the use of the tasks based on interactive dynamical sketches in studying the function concept is clarified. Based on its results and the achieved experience we try to develop such test material and test design, with which students' ways of operation and their working strategies when struggling with dynamical problem sets can be observed and classified reliably.

3.1. Final objective

The main task of the research project is to develop a model classification system of operating and learning styles, which could be used to look for an optimum in different working dimensions, and to develop prototype learning material which is in accordance with the model. With working dimensions we mean here the following properties:  problem setting (task type, the connection of the sketch to the problems)  dynamics of the figure (the variables, controls, interaction, dependencies)  freedom of action (dragging, animation, buttons, controls)  guidance (subject theory, hints, feedback, links)

3.2. Research problems

In the study we searched for clarification to the following problems:  What special benefits do the dynamical interaction offer, what new types of difficulties rise forth?  What advantages are there in manual dragging, what in automatic animation?  What significance do the hints have, how much and what kind of guidance is ”optimal”?  What kind of faulty ideas or misconceptions come into light with the help of interactive presentation? 6

4. Description of tasks and study design

Interactive units of the distance learning material for Discrete Mathematics were tested in three parts: in the beginning of the course in September 2002 briefly in AIM-www environment and in November more widely in Introductory Mathematics course and Discrete Mathematics course. In the pre-test arranged in the AIM1 environment we had a few dynamical tasks of set theory and logic, and also some calculating problems which the AIM system checked automatically using Maple2 computer algebra system. Both the dynamical sketches and the Maple checking functioned well and the students liked the tasks. However, only the tests in November, which were in an open www environment on html forms are described here. Dynamical Geometria problems were created by Lehtola and Pesonen, assisted partly by Ehmke, and Pesonen constructed the JavaSketchpad tasks. The problems dealt with elementary set theory, relations, functions and graphs, and they were tested in two different parts.

4.1. Problems concerning sets, relations and functions

As the testing population we had Autumn 2002 Introductory Mathematics course first year students (N = 42), who are supposed to become teachers or researchers of mathematical subjects. The tasks dealt mainly with definition identification. At the test moment the concept of function had not yet been taught to them at the university, but logic and set theory and a little about relations had already been studied. So their function image depended on their school mathematics experiences. Test material was divided into two parts: the first part (”pre-test”) comprised identification tasks and production tasks based on Geometria applet sketches, and the latter part (”post-test”) consisted of JavaSketchpad-based problem sets concerning identification and production tasks about functions and their properties. The interactive pre-test was intended to serve as learning module. The tests were performed as two hour computer activity in a computer lab in two groups. The students were requested to work in pairs but only small part did so. During

1AIM = Alice Interactive Mathematics, an ascetic free course management system that has been built in Gent University. 2Maple is a commercial computer algebra system (CAS) from Waterloo Maplesoft. 7 the work the screen actions were video-recorded to the computers hard disks with the screen capture program Camtasia Recorder. Furthermore, the students' post-test answers were sent as email to the researchers with the help of the www form. In the pre-test there were 17 tasks concerning sets, relations and functions. In each of them there was one dynamical sketch and a question related to it. Because this kind of work was new to most students, four example tasks were worked through together at the beginning of the test: one about sets, one about relations and two about functions. In the actual pre-test there were two SG production tasks about set theory and two about relations, and 13 definition identification tasks concerning functions. The tasks were presented as a series of short www pages in which links led forward and backward. The students were ordered to begin by adjusting and starting the Camtasia recording before opening the first task. The post-test was on one www form which started by the definition of function and different notations and representations. The test comprised 7 task groups, each one built around one dynamic sketch. The first one included definition identification and GS and GV production tasks. The others dealt with properties of real and vector valued functions of one real variable. The tasks were multiple choice or open-ended questions and the answers were sent through e-mail to the researchers. The main page of the test is http://www.joensuu.fi/mathematics/kurssit/JohdantokurssiTKD.html The statistical coding of the video material of both tests was made at the IPN (in Kiel) by Ehmke & co (see Figures 2 and 3). Combining and interpretation of the material and test results was made together In Summer and Autumn 2003. Harjoitus 1: Joukko-oppia Clicks on P correct? Q correct? R correct? Answer correct? button Arvioi 1 O Yes O No O Yes O No O Yes O No O Yes O No 2 O Yes O No O Yes O No O Yes O No O Yes O No 3 O Yes O No O Yes O No O Yes O No O Yes O No 4 O Yes O No O Yes O No O Yes O No O Yes O No Comments: ______

Figure 2. Example of a form used in coding the pre-test video material

Tehtävä 4 Time: 8 first click at: ___ last click at: ___ duration (sec): ___ Type ‘R’ (reset): ______Graph visible (button “Kuvaaja” clicked): O Yes O No Dragging x several times slowly between 1 and 2: O Yes O No If yes, was the graph visible? O Yes O No Dragging x several times slowly between 4 and 5: O Yes O No If yes, was the graph visible? O Yes O No

Comments: ______Figure 3. Example of a form used in coding the post-test video material

4.2. Properties of relations and graph theory problems

The second test entity contained identification and production problems and some constructions and visualisations of relations and graph. The test population of this part were students near the end of Discrete Mathematics 2002 course students in Joensuu (2nd to 4th year, N = 10). They students of mathematics or computer science. The students used the material as repetition of the course contents and at the same time they were supposed to evaluate the material: - clarity of problem settings or visualisations - importance dynamical features and hints - instructiveness of the tasks - how many trials were needed for solving each problem

The tasks and student evaluations of this part are found at Internet page (in Finnish) http://www.joensuu.fi/mathematics/ DidMat/Ehmke/JOENSUU2002/discrete/DMMateriaalinTestausJaTuloksia.html

5. Experiences and results

We present here some of the most interesting details from the extensive pre-test and post-test research material. The percentage values are only indicative in the results of the post-test because from its video material sure information was not obtained in about ten cases in each task, likewise towards the end about ten did not answer the tasks at all.

5.1. Hints and their importance

The hint button is present in pre-test problems 4-8. The hint texts are displayed in a 9 separate small window which opens by pressing the hint (“Vihje”) button in the sketch. The hint window disappears by pressing its close (“Sulje”) button. From Table 1 it is seen that hints were not much used.

Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Use of hints 45 % 5 % 0 % 0 % 0 % Table 1 Development of the use of hints It is surprising that only 45% of students looked the first hint they met with. We can probably conclude from the rapid decrease of interest that the students did not feel they would benefit from the hints. On one hand, nearly all were able to solve the task 4 by already first try (88%), and on the other hand the content of the hint may have remained vague because of its laconic character: ”In position (X,x) of the table is 1, if xRX, otherwise 0.” Perhaps the most interesting is that already one disappointment - an unsuccessful hint - would seem to be turning off the interest in hints. The identification tasks 5-8 were managed excellently, except for task 8 which 17% answered wrong. Another reason for not using hints can indeed be the easiness of the tasks.

5.2. Use of the definition link

A link to a short www page containing the definition of function is found below the function sketch in exercises 5-17. In each task there were no more than three students visiting the page, in most of them only one or none. Also this refers to the fact that the tasks were not difficult enough.

5.3. Using the tracing facility

Tracing button was available in post-test sketches 2-7. The tracing function draws for example the image points of a function to the co-domain when the variable moves in the domain. Figure 4 shows an example of the result trace when the variable x is moving from point 1 to point 0 on the upper real axis. 10

Figure 4. Trace in the sketch of post-test problem 2

Tracing is switched on and off with the sketch button pair Trace/Stop (”Jäljitä”/”Lopeta”). When being on, the trace facility draws the track of all points that are defined to be traced with a certain colour, and the stop button prevents from drawing new points. However, the drawn points stay visible until the tracing is cleaned with the small red cross button on the lower right corner of the sketch. The trace does not disappears even on resetting the sketch internally by the keybord ‘R’ button. Reloading the whole document would help, but this would lead losing all the input that has already been entered. About half of the students used tracing in the tasks 2-7. The use of tracing did not decrease towards the end of tasks, so it was obviously regarded as useful (cf. the use of hints). It seemed that the cleaning of the trace was a problem to several students even though it had been discussed just before task 2 of the worksheet. This instruction should perhaps be in the sketch itself and we have indeed added it to later sketch versions. Actually this can be seen to be a shortcoming in the design of JavaSketchpad. Now 67% of the students did not use the cleaning possibility - for one reason or another - and in some tasks this caused mess which hindered solving. A typical way of action leading to wrong answers gives reason to doubt a misconception of the range and image of functions. Several students 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. Analysis of video material of the five students’ work who gave these kinds of wrong solutions confirmed that they had examined only the endpoints. Also they did not use tracing. Here the 11 tracing would have been of significant help.

5.4. Dragging points by mouse

In dynamic geometry computer programs dragging a point with the mouse causes the depending parts of the figure to be transformed accordingly. In most of these sketches this part is a point, a line segment or a straight line. Dragging was popular throughout the test. But indeed, it was nearly impossible to solve many problems without manual interaction with the sketch. Special actions were possible with dragging: controlling values, dragging the variable slowly over or around an important point. An interesting question is whether dragging was used intentionally. It seems that this took place quite generally, as Table 2 concerning the post-test shows.

Post-test Tehtävä 1 Tehtävä 2 Tehtävä 2 Tehtävä 5 Tehtävä 5 More than 3 x → 2 0 ↔ x ↔ 1 a pisteen 1 x → 3 cases lähistöllä Dragging 65 % 100 % 91 % 60 % 43 % Table 2 Dragging points by mouse in special situations The differences can be explained on the variable difficulty level of the tasks. The students had to find out themselves where the special targets are. This seems to be the restricting factor, not the use or the desire to use dragging. In sketch 5 the transformed part is a whole curve f(x) := ax depending on the draggable base value a. So we can set a hypothesis that it is advantageous for the learner to be able to examine curve families with the help of dynamical figures which depend on a parameter. This question is tried to be answered by investigating the problem 5a ): ”For which values a is the curve y = ax increasing?”. According to the test results, only 38% of those who reported something gave the right solution. On the basis of the video observations we get a more exact analysis. Cross table 3 separates the students with respect to solving the problem and using successful dragging.

Problem 5a Solution Curve increasing correct Wrong 12

Dragging yes 10 3 a near 1 no 1 9 Table 3 Dragging and success in problem 5a The result supports the supposition that the dynamic dragging possibility helps when examining the behaviour of one parameter curve families. On the other hand, some students may have been aware in advance about exponential functions.

5.5. Using animations Dynamical geometry programs are usually provided also with the possibility to animate the figure in some pre-defined way. A simple animation can be interpreted as automated dragging but with the animation facility also very complex dynamic visualisations can be achieved. Several animations can take place simultaneously and animation and dragging can be combined; so animations can be used to facilitate showing complicated phenomena. Therefore, it is useful to know how diligently the students use the animation facility. In Table 4 we see how often the students used the animation button in the pre-test.

Pretest Animation not used Animation used Animation used more problem once than once

9 61 % 22 % 17 % 10 68 % 17 % 15 % 11 58 % 30 % 12 % 12 56 % 15 % 29 % 13 63 % 18 % 19 % 14 41 % 46 % 13 % 15 71 % 12 % 17 % 16 73 % 10 % 17 % 17 66 % 15 % 19 % Table 4 Frequencies of animations used in the pre-test The tasks were simple function definition identification tasks in which there were both the dragging and animating facilities. In some of them it was important to look the the animation (or use dragging), while in others it was no big advantage. From the results one can see that the majority of the students used animation when it was necessary, but only about 40% when they felt that they were able to solve the task also without. Task 14 was merely an example where animation was supposed to be used to see a visualisation. 13

There was an animation possibility also in the post-test tasks 1, 2 and 7, which was obviously an advantage, see Table 5.

Post-test Animation not used Animation used Animation used more problem once than once

1 13 % 36 % 51 % 2 31 % 43 % 26 % 7 47 % 34 % 19 % Table 5 Frequencies of animations used in the post-test Task 1 was succeeded excellently with a mere animation, which also is seen from the fact that animation was used many times. But in problems 2 and 7 it was worthwhile to drag with the mouse instead.

5.6. Animation or dragging?

Many tasks can be solved with the help of animation or dragging. We can see from Table 6, concerning task 1, that the majority used either dragging and alittle animation (60%) or animation and not at all dragging (17%). Post-test problem 1 Animation: special situations 0 - 2 3 - 6 Dragging: 0 10 % 17 % special situations 3 - 5 60 % 13 % Table 6 Animations and draggings in post-test problem 1 The animation facility is worth offering at least when something surprising happens in special areas which are not found easily on the manual dragging.

5.7. Use of the scaling

In some sketches it is possible to change the axis scales by dragging the unit point 1. Scaling facility was used only by few people (see Table 7), even though it would have been necessary at least in problem 5b. Post-test Problem 5 Problem 7 Use of scaling 40 % 0 % Table 7 Use of scaling in post-test problems 5 and 7 From cross table 8 one can see that without scaling only 5% solved the task 5b. The fact that scaling possibility was not advertised anyhow was a conscious choice; we wanted 14 to see whether the students can find ”unreported properties ”.

Post-test Use of scaling problem 5b yes no Solution correct 33 % 5 % wrong 24 % 38 % Table 8 Advantage of the scaling facility in problem 5b

6. Conclusions and comments

Generally speaking the students succeeded well in the tasks. The results were quite good in the definition identification tasks concerning functions. In the post-test one could feel tiredness, the students did not have enough patience to concentrate on the numerous problem sets. Teaching material which requires constant attentiveness and thinking has been found to be strenuous already in earlier experiments, and now there was furthermore the screen capture program which slowed down the functions of the computer. Perhaps this caused also troubles with the technique of the material. Several students had difficulties with loading the applets and from some sketches horizontal and vertical lines were missing, at least according to the video captures. Sketch hints were very seldomly used, so the hint system must be re-considered carefully. In earlier experiments graphical sketch hints have been used to clarify the constructional dependencies. For example crucial points or lines can been shown by a push of a hint button, and this has shown to be useful. On the other hand, the response analysis feedback in the pre-test sketches served as guidance at the same time. The link to the definition was not really used at all. Does this reflect the attitude that no help was even expected from the definition (see Section 2.1 and.Vinner 1991)? In the function tasks the students used dragging by the mouse more preferably than animation. So animations should not be present if there is no real advantage from them. And of course, dragging is far more activating operation. In problems where it was possible to examine the function also as a two-dimensional xy-coordinate presentation about one third not do so. Here some additional guidance would have been necessary. In real distance learning material variable working and material must be sequenced so 15 that there is left enough time to chew the new matters, in order to prevent information clots. On the other hand, material should not be too splintered either, but a certain thematic entity should be fairly intensively studied; not forgetting repetition which could take place for example with automatically checked self tests.

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Acknowledgements The Department of Mathematics of the University of Joensuu and the Eastern Finland University Network have supported the work of Martti Pesonen and Hanna Lehtola and the seminar visits of Timo Ehmke.