Solving Mathematical Problems with Dynamic Sketches: a Study on Binary Operations

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

Martti E. Pesonen

University of Joensuu, Finland

E-mail:

Timo Ehmke

Leibniz-Institute for Science Education (IPN), University of Kiel, Germany

E-mail:

Lenni Haapasalo

University of Joensuu, Finland

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Abstract: This paper examines the use of computer technology for mathematics teaching on a tertiary level, especially for distance learning and assessment. The emphasis is on dynamic graphics, which are suitable for representing elements in the abstract theory of linear spaces. The study concerns 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 the 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 implicitly applied in a new way. In order to be able to make pedagogical conclusions for comprehensive understanding of concepts, procedures, and the links between the two, it is often reasonable to start by analysing familiar concept pairs. Concept building itself can offer different kinds of problem-solving processes at its productive as well as at its 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:

• Internal binary operation:Let A be a non-empty set. We call an internal binary operation every function AAA.

• External binary operation:Let A and K be non-empty sets. We call an external binary operation (or scaling function) every function KAA.

To be able to appreciate these definitions we must have full understanding of the function concept. A great deal of research has been done in this field during the last 25 years, even in the 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 the simultaneous activation of conceptual and procedural knowledge (to be defined later)can enhance learning and it can be realised by using dynamic interactive learning materials, such as Java applets, for example. This kind of approach, utilising dynamic geometric ideas, is missing in the studies concerning the learning of binary operations.

2. Theoretical background

The planning and evaluating of learning processes are based on the theory of a large empirical project MODEM[1] (Haapasalo 2003; see also In order to introduce it, we use a recent analysis of conceptual and procedural knowledge (Haapasalo & Kadijevich 2000) and make the following distinction:

• Conceptual knowledge (abbreviated toCthroughout 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 (abbreviated to P) denotes dynamic and successful utilisation of particular rules, algorithms or procedures within relevant representation forms. This usually requires not only knowledge of the objects being utilised, but also 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.

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 a genetic view (i.e. P is necessary for C) or a 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 in conceptual knowledge from the very beginning. If so, the logical basis of this so-called educational approach is the dynamic interaction view (i.e. C is necessary for P), or again the simultaneous activation view already mentioned. The latter means that the learner has opportunities to activate conceptual and procedural features of the current topic simultaneously. By “activate” 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 for a certain balance between the 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 above criticism of tertiary mathematics, is the dilemma between conceptual (C) and procedural (P) knowledge: Does the student have to understand in order to be able to do, and vice versa. A comprehensive theoretical framework for approaching this basic question can be found in Haapasalo (2003). It seems appropriate to emphasise 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, realisation 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[2] used for our assessment (see Figure 1).

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

When searching for convenient ways to introduce binary operation into a verbal, graphic or symbolic form, the teacher offers students opportunities to orientate themselves in 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 utilises 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 support 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 students the opportunity to make their own investigations, to express the investigation results especially in verbal forms in each case, and to argue about these results within 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 utilise the principle of 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 give students opportunities to train themselves in identifying 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 utilise concept attributes and to develop procedural knowledge to be used in problem-solving and applications.

The interaction between verbal, symbolic and graphic forms described above gives, right from the beginning, an excellent framework not only for learning but also for an assessment of students’ conceptual understanding (we will 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 dynamic figures. We utilised 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. To decide whether the given mathematical IGR represents a binary operation, the student has to check if all necessary conditions of the definition are fulfilled.

3. Design and research questions

Our study was done during the first course (n = 92) on Linear Algebra in the Mathematics Department of the University of Joensuu in Spring 2004. Altogether 25 problems were posed to the students using the course management system WebCT.

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

1. Which kinds of differences in the identifying of binary operations do different representations of a mathematical relation cause?
2. Which kinds of differences in students’ performance do different kinds of interactive graphical representations cause?
3. Which kinds of interactive graphical representations of binary operations are easy and which are difficult to identify for students?

4. Results

To analyse the first research question, Table 1 shows the descriptive results of the three test subscales. The symbolic (p=0.73) and graphic (p=0.72) identification problems have the lowest difficulty. Identifying a binary operation given through a verbal description is more difficult (p = 0.56).

Table 1. Problem type statistics and bivariate correlation

Concept / Items / Mean / SD / n / Verbal / Symbolic / Graphic
Verbal / 5 / 0.56 / 0.21 / 92 / 1 / 0.31 / 0.04
Symbolic / 4 / 0.73 / 0.20 / 92 / 0.31 / 1 / 0.05
Graphic / 5 / 0.72 / 0.22 / 92 / 0.04 / 0.05 / 1

Concerning the correlations between the three problem types, we found the highest correlation between symbolic and verbal definition identification problems. This is in accordance with MODEM-findings that the verbalisation seems to have an important role in the concept building, and in the linking of conceptual and procedural knowledge (cf. Haapasalo 2003, p. 15).

Identifying a graphically represented binary operation is not correlated with the two other types. This result is remarkable; it seems that the interactive component given through the graphical representation contains a new dimension of complexity. Therefore, we shall analyse these kind of problems in more detail.

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

Problem set 1deals 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 2. The three problems differ as seen in Table2. The question to the student was to decide whether this is a binary operation in R.

Figure2. Interactive graphical representation of a binary operation in R

Table2. 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,
uov vanishes outside [-c,c] / no / 31 / 0.81
1C / u and v free,
uov always defined / yes / 30 / 0.67

Only one half of the students could identify that 1A is not a binary operation. The situation was more obvious 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).

Problem set 2 is in a technical sense identical with problem set 1 (Figure 2). 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 Table3 show high solutions rates for the problems 2B and 2C, whereas problem 2A was slightly more difficult. The only difference between problem 2A and the other two is that the points u and v in problem 2A are not dragable on the whole line. In a mathematical sense this means u,v[-c,c]. Probably one third of the 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 the 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 lower because the task description already gives information about the restricted domain.

Table3. Description and results of problem set 2

Problem / Description / Binary
Operation / Frequency,
n / Difficulty,
p
2A / u and vfixed to [-c, c] / yes / 28 / 0.71
2B / uand v free, u ov vanishes outside [-c,c] / yes / 27 / 0.85
2C / u and v free, u ov 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 ov are given (see Figure 3 and Table4). Problem3A contains a conventional binary operation in R2. The points u and v are dragable in the whole plane and u ov 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 ov stays at 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 Table3 shows the percentage of correct responses: Most students identified IGR 3A as a binary operation. All changes from the “normal” situation of 3A led to a significantly reduced percentage of correct responses. Less than one half of the students accepted the image as being constant, while half of the students regarded two values as correct. Most strikingly, less than one out of five were able to recognise the inadequate restriction of the domain in 3D, namely that v is not defined in all R2.

It is appropriate to assume that, when leaving school, mathematics students do not have a good understanding of two-variable functions. When changing just u, they do not realise that they cannot change v at all. Obviously, the statement "in whole R2" is quite mysterious for the students at this stage because it means a requirement for the domain as well as for the co-domain. Another explanation might be purely psychological: It is easier for students to guess "yes" in the situation 3B than "no" in the situation 3D.

Figure3: Four interactive graphical representations of operations in R2

Table4: Description and results of problem set 3

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

Problem set 4 deals with binary operations in discs, Figure4 shows the IGR. The graphical representation is similar in all three problem items 4A, 4B and 4C, but differs in the behaviour of the point objects. Table5 shows the properties of x, y and x oy. 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 also if the result x oy stays within the circle. What happens with x oy, if x and y are outside the circle is not relevant for the answer but the results (last column of Table5) show that it does matter.

Even though most of the students are sure to identify a binary operation if x oy always stays inside the circle (even when x or y are outside, as in problem 4A), they make a false identification when dragging x or y outside and x oy following outside the circle (4B).

Problem 4C, in which the points x and y are bound inside the circle, was solved correctly by 73%. The decreased solution rates of 4B and 4C lead us to suspect that some students do not pay attention to the definition of the domain and the co-domain. They probably had the whole plane in mind when giving their response.

Figure4. Interactive graphical representation of a binary operation in a disc.

Table5. Description and results of problem set 4

Problem / Description / Binary
Operation / Frequency,
n / Difficulty,
p
4A / x and y free,
x oy stays inside / yes / 30 / 0.83
4B / x and y bounded inside,
x oy stays inside / yes / 22 / 0.73
4C / x and y free,
x oy 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 utilising tools, don’t forget the underlying mathematics.” In other words, mathematics cannot only direct the tool utilisation, 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 realising learning environments. While the focus of school teaching is often to reach sufficient procedural knowledge, the teaching of university mathematics aims at 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 of technology-based learning (as IGR) is to utilise the principle of simultaneous activation. 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. It is evident that even university mathematics can be learnt outside institutions by utilising web-based interactivities. Our IGR studies suggest that most 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 even more thoroughly. Furthermore, qualitative research into students’ thinking processes would probably give valuable information for developing appropriate learning environments. The on-going research in the DAAD project will focus on these questions.