Solving mathematical problems with dynamical sketches: a study on binary operations 

Martti E. Pesonen
University of Joensuu, Finland
E-mail: martti.pesonen@joensuu.fi

Timo Ehmke 
Leibniz-Institute for Science Education (IPN), University of Kiel, Germany
E-mail: ehmke@ipn.uni-kiel.de

Lenni Haapasalo
University of Joensuu, Finland
E-mail: lenni.haapasalo@joensuu.fi

Abstract:  This paper examines the using of computer technology for mathematics teaching on tertiary level, especially for distance learning and assessment.  The emphasis is in dynamical graphics, which is suitable for representing elements in the abstract theory of Linear Spaces. The study concerns with learning environments and assessment for binary operations within WebCT management.  

Key words: binary operation, function, computer-based, conceptual, procedural, interactive, dynamic geometry, technology-based 

1.	Introduction

One of the main goals of mathematics teaching is to promote solving of mathematical problems. In this educational task, new concepts or new procedures can be (re)invented by the students – or vice versa – the old ones can be at least implicitely applied in a new way. For being able to make pedagogical conclusions for comprehensive understanding of concepts, procedures and the links between the two, it is often reasonable to start to analyse familiar concept pairs. Concept building itself can offer different kind of problem solving processes as well at its productive as reproductive stages. 

Binary operation seems to be a difficult concept for students at the beginning of their studies. It is a special kind of two-variable function within a set: it maps two arguments of the given set to a value in the same set. Because the domain and co-domain are the same set, we therefore call it an internal binary operation. For an external binary operation the two arguments can be from different sets. The first argument is aimed for “scaling” the second argument and the results are in the same set as the second argument. The formal definitions can be expressed as follows:
• Internal binary operation: Let A be a non-empty set. We call an internal binary operation every function AA A.
• External binary operation: Let A and K be non-empty sets. We call an external binary operation (or scaling function) every function KA A.

To be able to fully appreciate these definitions we must have full understanding of the function concept. A great deal of research on this field has been done during the last 25 years even in tertiary level context (e.g. Tall & Bakar 1991, Breidenbach et al. 1992, Tall 1992, Vinner & Dreyfus 1989, and Brown et al. 1997). Using a quite recent analysis of conceptual and procedural knowledge by Haapasalo & Kadijevich (2000), Pesonen et al. (2002) present computer-based activities for enhancing university freshmen’s understanding of the function concept.  It seems that simultaneous activation of conceptual and procedural knowledge (to be defined later) can enhance learning and it can be realised by using dynamical interactive learning materials, as Java applets, for example. This kind of approach, utilizing dynamical geometric ideas is missing in the studies concerning the learning of binary operations.

Our study is a product of a teacher change activity between the two partner universities within the German Academic Exchange Service (DAAD). It is based on dynamical Java applets, which were developed and tested in both universities and administrated within a WebCT environment. 

2.	Theoretical background

The planning and evaluating of learning processes are based on the theory of a large empirical project MODEM (Haapasalo 2003; see also http://www.joensuu.fi/lenni/modemeng.html). For introducing it, we use a recent analysis of conceptual and procedural knowledge (Haapasalo & Kadijevich 2000), and make the following distinction:
• Conceptual knowledge (abbreaviated to C through our presentation) denotes knowledge of and a skilful “drive” along particular 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. 
• Procedural knowledge (abbreaviated to P) denotes dynamic and successful utilization of particular rules, algorithms or procedures within relevant representation forms. This usually requires not only the knowledge of the objects being utilized, but also the knowledge of format and syntax for the representational system(s) expressing them. Furthermore, P often calls for automated and unconscious steps, whereas  C  typically requires conscious thinking. However, P (like instrumental understanding) may also be demonstrated in a reflective mode of thinking when, for example, the student skillfully combines two rules without exactly knowing why they work.

The dominance of P over C seems quite natural both in the development of scientific and individual knowledge. So, an appropriate pedagogical idea in any topic could be to go for spontaneous procedural knowledge. The logical relation between P and C in this so-called developmental approach (Haapasalo & Kadijevich 2000, 147) is based upon genetic view (i.e. P is necessary for C) or simultaneous activation view (i.e. P is necessary and sufficient for C).  On the other hand, it seems appropriate to claim that the goal of any education should be to invest on conceptual knowledge from the first beginning. If so, the logical basis of this so-called educational approach is dynamic interaction view (i.e. C is necessary for P), or again the mentioned simultaneous activation view. The latter means that the learner has opportunities to activate conceptual and procedural features of the current topic simultaneously. By “activating” we mean certain mental or concrete manipulations of the representatives of each type of knowledge. Being in the intersection of two complementary approaches, the simultaneous activation view is loaded with some expectations concerning the planning of learning environments. Modern technology, of course, offers natural solutions for these kinds of activities.

If we accept the view that a viable understanding of the function concept constitutes a necessary background for higher mathematics, we have to try to allow students to come up with their procedural ideas and finally help them to understand the conceptual features of the function, as well. This means searching a certain balance between developmental approach and the educational one, building a bridge between more or less procedural school thinking and conceptual academic thinking. The leading question, coming up implicitly from the criticism of tertiary mathematics above is the dilemma between conceptual (C) and procedural (P) knowledge: Does the student have to understand for being able to do, and vice versa. A comprehensive theoretical framework for the approaching to this basic question can be found in Haapasalo (2003). It seems appropriate to underline that these two types of knowledge must be somehow related when the learning process is our focus. However, it is the variables in the assessment of this process that promote or obstruct possible qualitative and quantitative links between the two knowledge types. By assessment, in the global sense, we mean the planning, realization and control of the learning processes, made not only by the teacher but especially by the learners and learner teams themselves. 

We are now ready to finish the representation of the MODEM framework used for our assessment (see Figure 1). 

Figure 1.  Sophisticated interplay between developmental and educational approach within the MODEM framework

When searching convenient ways to introduce binary operation in a verbal, graphic or symbolic form, teacher offers students opportunities to get oriented into this concept. It is this kind of Orientation (O) that forms the first phase of the systematic concept building in the right hand box of Figure 1. It basically utilizes a developmental approach: the interpretations of the situation can be based on mental models of the pupils, coming, more or less, from their naive procedural ideas. These act like a wake-up voltage in an electric circuit that triggers another, much more powerful current to be amplified again. The procedural and conceptual knowledge types start to accelerate each other, offering a nice opportunity to use the SA method, for example. The SA method, being at the intersection of the logical definitions of the two approaches, links the developmental approach and educational approach in a most natural way.

The role of the Concept Definition (D) is to offer for students opportunities to make their own investigations, to express the investigation results especially in verbal forms in each case, and to argue about these results inside the collaborative teams and between the teams. As a result of social construction, a definition for the concept is born, meaning that students try to fix the relevant determiners of the concept in verbal, symbolic and graphic forms. Especially in the phases of orientation and definition, creative thinking and productive work is needed.  The next phases of concept building utilize the principle of the dynamic interaction. The idea is to give students a sufficient number of opportunities to construct concept attributes and procedural knowledge based on them. 

In the phase of Identification (I) we have to allow students opportunities to train themselves to identify concept attributes in verbal (V), symbolic (S) and graphic (G) forms. For this we need six kinds of tasks (I): IVV, IVG, IVS, IGG, ISS and ISG. During the learning process, the teacher must be ready, if necessary, to begin with tasks that require distinguishing between only two elements before going on to the identification of several elements. 

In the phase of Production (P) we have to give pupils the possibility to produce from a given presentation of the concept another representation in a different form. The development of production (P) requires nine combinations: PGV, PGS, PGG, PSG, PSV, PSS, PVS, PVV and PVG. The tasks of identification and production must be achievable without any complicated processing of information on the student’s part.  
In the phase of Reinforcement, the goal is to train and utilize concept attributes and to develop procedural knowledge to be used in problem solving and applications. 
The interaction between Verbal, Symbolic and Graphical forms described above gives from the first beginning an excellent framework not only for learning but also for assessment of students’ conceptual understanding (we speak about VSG-task hereafter).  
Since binary operations are two-variable functions, whose variables in linear algebra are usually vectors, a static graphical representation becomes inadequate. Thus, a completely new “learning dimension” can be added by using dynamical 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 dynamical 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. Figure 2 illustrates the mathematising cycle by the solving of mathematical problems (Blum 1996; Schupp 1998).

	

Figure 2.  Four step cycle by solving of mathematical problems

The process starts with a situation, which contains problem in a more or less authentic context. In our study the situation is given through an IGR connected with the question “find out if this is a binary operation”. The second step is labelled as “mathematising”. The student tries to build a (mental) mathematical model of the content given through the situation. In our case within the study this is the most complex situation. The familiarity of students with IGR is not as strong as the familiarity with the symbolical language. The student must interact with the IGR and collect as much information about the coded mathematical situation as possible. The third step is to select and organize the information to a coherent mental model of the mathematical situation. To decide if the given mathematical IGR represents a binary operation, the student has to check if all necessary conditions of the definition are fulfilled. In the fourth step the student gives the answer to the question and validates again if it fits to the given problem.

3. Design and research questions

Our study was made during the first course (n = 92) on Linear Algebra at the Mathematics Department of the University of Joensuu in Spring 2004. The study was focused on a small part of a sequence of computer-based exercises on the course. Altogether 25 problems were delivered to and answered by each student, using the course management system WebCT on a 2-hour exercise session.

In the partial study reported here we wanted to look for answers to the following research questions:

1.	Do different kinds of interactive graphical representations of the same     operation lead to differences on the students’ performance?
2.	Which kinds of interactive graphical representations of binary operations are easy and which are difficult to identify for students?

In a forthcoming wider report we shall deal with e.g. correlations between the students’ performances in identification and production problems.

4. Results

Let us take a look at the WebCT-questions 21-25, which we call here problem sets 1-5, respectively. Each problem set consisted of 3-4 alternative items, each of which were answered by about 30 or 23 students.

Problem set 1 deals with operations in R, coded in the IGR as a line object with dragable and non-dragable points. The points u and v are dragable on the line. The position of the points u and v on the line should be seen as two real numbers. On the line there are two points -c and c which define an interval; see Figure 3. The three problems differ as seen in Table 1. The question to the student was to decide whether this is a binary operation in R. 

Figure 3. Interactive graphical representation of a binary operation in R

Table 1. Description and results of problem set 1

Problem	Description
	Binary
Operation	Frequency, n 	Difficulty, p	
1A	u and v fixed to [-c, c]	no
	31	0.55 
	
1B	u and v free, 
u o v vanishes outside [-c,c]	no
	31	0.81 
	
1C	u and v free, 
u o v always defined	yes
	30	0.67 

Only 55 % of the students can identify 1A to be not a binary operation. Perhaps they had not checked carefully if u and v are dragable on the whole line (i.e. if the result is valid for all values of the domain) or they were not able to interprete the boundedness of u and v to the line segment as an interactive graphical representation with the domain [-c, c]. Students probably proved for u and v inside the interval [-c, c], but could not link the situation in the IGR with the mathematic definition of binary operation. More obvious (p=0.81) was the situation if the result vanishes when dragging u or v outside the interval. One third of the students did not identify the graphical representation correctly as a binary operation (problem 1C). Maybe the visible part of the line was not recognized as a representation of R. 

Problem set 2 is in a technical sense identical with problem set 1 (Figure 4). The question for the student was to find out if the given IGR represents a binary operation in the interval [-c, c]. The results in Table 5 show high solutions rates for the problem 2B (p = 0.85) and problem 3B (p = 0.86), whereas problem 2A was slightly more difficult (p = 0.71). The only difference between the problem 2A and the other two is that the points u and v in problem 1A are not dragable on the whole line. In mathematical sense u, v  [-c, c]. Probably one third of students assumed that only the result must be in the co-domain, whilst the domain is the whole line (R).
Restricting the domain and the co-domain to the interval [-c, c] reduces difficulty, at least in the first and third problem item. It seems that in problem set 1 the students get confused by the unknown representation of a restriction of the domain. However, in problem set 2 this confusion is less high, because the task description is giving already information about the restricted domain.

Table 2. Description and results of problem set 2

Problem	Description
	Binary
Operation 	Frequency, 
n	Difficulty, p	
2A	u and v fixed to [-c, c]	yes
	28 
	0.71	
2B	 u and v free, u o v vanishes outside [-c,c]	yes
	27 
	0.85	
2C	u and v free, u o v always defined	yes
	37	0.86	

Problem set 3 concentrates on operations in R2. In all four problem items the points u, v and u o v are given (see Figure 4 and Table 3). Problem 3A contains a conventional binary operation in R2. The points u and v are dragable in the whole plane and u o v is always the midpoint of u and v. In item 3B we restricted the image and in item 3D the domain. In IGR 3B u and v are dragable in the plane, but u o v stays on the initial position. In IGR 3D only u is dragable and v is fixed to the initial coordinates. In problem item 3C two results exist and move when changing the position of u or v, so the definition of a binary operation is not fulfilled. The last column in Table 3 shows the percentage of correct responses: 86 % of the students identified IGR 3A as a binary operation. All changes from the “normal” situation of 3A lead to a reduced percentage of correct responses significantly. Only 40 % accepted the image to be constant, while half of the students regard two values as correct.  Most strikingly, less than one out of five, can recognize the inadequate restriction of the domain in 3D, namely that v is not defined in all R2.

		
Figure 4: Four interactive graphical representations of operations in R2

Table 3: Description and results of problem set 3

Problem
	Description
	Binary
Operation 	Frequency, 
n	Difficulty, p	
3A
	u o v = 1/2 (u + v)	yes
	22	0.86	
3B	u o v = constant	yes
	25	0.40	
3C	two results u o v	no
	23	0.52	
3D	v is fixed	no
	22	0.18 

Problem set 4 deals with binary operations in discs, Figure 5 shows the IGR. The graphical representation is similar in all three problem items D1-D3, but differs in the behaviour of the point objects. Table 4 shows the properties of x, y and x o y. The question for the students was to find out whether the IGR is a binary operation in the disc. 
Students are not very familiar with this kind of visualisation, where they had to check in all three versions if x and y are dragable in the whole circle and if the result x o y stays within the circle, too. What happens with x o y, if x and y are outside the circle, is not relevant for the answer, but the results (last column of table 5) show that it does matter.
About 83 % of the students are sure to see a binary operation if x o y stays always inside the circle, even when x or y are outside (problem D1). But only 70 % say this is a binary operation when they drag x or y outside and x o y follows outside the circle, too (D2). Problem D3, in which the points x and y are bounded inside the circle, was solved correctly by 73 %.  The decreased solution rates of D2 and D3 lead us to suspect that some students do not take care about the definition of the domain and the co-domain. Probably they had the whole plane in mind when giving their response. 

Figure 5. Interactive graphical representation of a binary operation in a disc.

Table 4. Description and results of problem set 4

Problem
	Description
	Binary
Operation	Frequency, n	Difficulty, p	
D1	x and y free, 
x o y stays inside	yes
	30	0.83 
	
D2	x and y bounded inside, 
x o y stays inside	yes
	22 
	0.73	
D3	x and y free, 
x o y can be outside	yes
	27	0.70	

5. Discussion

We would like to quote the IBMT (Interaction Between Mathematics and Technology) principle by Kadijevich et al. (2004): “When using mathematics, don’t forget available tool(s); when utilizing tool, don’t forget the underlying mathematics.” In other words, mathematics can not only direct the tool utilization, but also help us to achieve it in a more efficient and suitable way. In such a way, mathematically-grounded “button pressing” would not compromise thinking but rather enhance it for the benefit of the learner.  Even though it is well known that technology can shift mathematics teaching from paper and pencil work towards interactive learning, an adequate pedagogical theory is needed for planning and realizing of the learning environments. While the focus of school teaching often is to reach sufficient procedural knowledge, teaching of university mathematics aims to high conceptual understanding. For both of these, we have to be ready to handle the following dilemma: Should the student need to understand in order to be able to do, or vice versa? (cf. Haapasalo 2003). Perhaps the most promising aspect by technology-based learning (as IGR), is to utilize the principle of the simultaneous activation. This allows the teacher to get free from the worry about in which order student’s mental models develop when interpreting, transforming and modelling of mathematical objects. Our examples hopefully show that more or less systematic pedagogical models connected to appropriate using of technology can help teacher to achieve this goal. Interactive applets can be used not only for the learning but also for the assessment and for increasing new kind of complexity for the content. It is evident that even university mathematics can be learned outside institutions by utilizing web-based interactivities.  Our IGR studies suggest that most of students’ difficulties appear in the steps of mathematising and interpreting. To validate this result the correlation between test performance in IGR problems and in problems represented in symbolic form should be examined.  Furthermore, qualitative research of students’ thinking processes would probably give valuable information for developing of appropriate learning environments. The ongoing research in the DAAD project will focus on these questions. 

Literature

Blum, W. 1996. Anwendungsorientierter Mathematikunterricht – Trends und Perspektiven. In G. Kadunz et al. (eds.), Trends und Perspektiven. Schriftenreihe Didaktik der Mathematik, vol. 23, Hoelder-Pichler-Tempsky, Wien, Austria, pp. 15-38.

Breidenbach, D., Dubinsky, E., Hawks, J. & Nichols, D. 1992. Development of the process conception of function. Educational Studies in Mathematics 23, pp. 247-285. 
  
Brown, A., De Vries, D., Dubinsky, E. & Thomas, K. 1997. Learning binary operations, groups, and subgroups.  Journal of Mathematical Behavior 16 (3), pp. 187-239.

Ehmke, T. 2001. Eine Klasse beweglicher Figuren für interaktive Lernbausteine zur Geometrie. Dissertation. University of Flensburg.

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

Haapasalo, L. & Kadijevich,  Dj. 2000. Two types of mathematical knowledge and their relation. Journal für Mathematikdidaktik 21 (2), pp. 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. Proceedings on the IXX Symposium of the Finnish Mathematics and Science Education Research Association. University of Joensuu. Bulletins of the Faculty of Education 86, pp. 1-20.

Kadijevich, Dj., Haapasalo, L., Hvorecky, J. 2004. Using Technology in Applications and Modelling. Paper presented in the International Congress of Mathematics Education 10. Copenhagen, Danmark. July, 2004.

Pesonen, M. E. 2001. WWW Documents With Interactive Animations As Learning Material. In  the Joint Meeting of AMS and MAA, New Orleans, January 2001. Internet: 
	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), pp. 37-45. 

Schupp, H. 1988. Anwendungsorientierter Mathematikunterricht in der Sekundarstufe I zwischen Tradition und neuen Impulsen. Der Mathematikunterricht 34 (6), pp. 5-16.

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 (pp. 495-511). NY: MacMillan.
 
Tall, D. & Bakar, M. 1991. Students’ Mental Prototypes for Functions and Graphs. Internet:
http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1991f-mdnor-function-pme.pdf

Vinner, S. & Dreyfus, T. 1989. Images and definitions for the concept of function. Journal for Research in Mathematics Education 20 (4), pp. 356-366.

Contact addresses:

Timo Ehmke, Leibniz-Institute for Science Education (IPN), University of Kiel, Olshausenstr. 62, D-24098 Kiel, Germany
e-Mail: ehmke@ipn.uni-kiel.de
homepage: http://www.ipn.uni-kiel.de/ehmke/

Lenni Haapasalo, University of Joensuu, Department of Applied Education, P.O. Box 11, 80101 Joensuu, Finland
e-mail: lenni.haapasalo@joensuu.fi
homepage: http://www.joensuu.fi/lenni

Martti E. Pesonen, University of Joensuu, Department of Mathematics, P.O. Box 11, 80101 Joensuu, Finland
e-mail: martti.pesonen@joensuu.fi
homepage: http://www.joensuu.fi/mathematics/personnel/PesonenM.html