Designing technology-based mathematics lessons: a pedagogical framework.

         To integrate technology into mathematics teaching and learning
         effectively, teachers could create a technology-based learning
         environment that provides students with opportunities to
         experience the process of mathematical investigation. These
         opportunities range from exploring using mathematical ideas to
         making and testing conjectures, as well as extending their
         conjectures to a general form, if possible. Additionally, the
         learning environment should support students in ways that
         encourage them to articulate not only what they know about the
         mathematical ideas in their exploration, but also how they
         arrive at their conjectures and how they generalise the ideas.
         This article offers a framework that encompasses the processes
         of exploring, conjecturing, verifying, and generalising to help
         mathematics teachers plan and design effective technology-based
         lessons to create an environment that engages students in
         meaningful learning in the mathematics classroom. An
         interactive spreadsheet template, based on a popular
         mathematics problem commonly found under the topic of calculus
         and involving finding the maximum area of a rectangular
         enclosure given a fixed perimeter, was designed to illustrate
         the framework.

INTRODUCTION

The Ministry of Education in Singapore has introduced two Master-plans for Information Technology (IT) in Education since 1997. The intent was to encourage teachers to harness the numerous benefits of IT in teaching and learning. However, anecdotal evidence gathered through interactions with local practising mathematics teachers suggests that the potential of IT seems to remain unrealised in the classroom. So, why are local teachers still not tapping into the potential of IT despite the fact that they have been provided with training to equip them with the basic proficiency to integrate IT into the curriculum? Reasons they typically cite, which are consistent with those reported in the research literature (Heid, 1997; Oppenheimer, 1997; Kaur & Yap, 1998; Manoucherhri, 1999), include a lack of curriculum time to allow pupils to learn through exploration and investigation, a lack of outside curriculum time for teachers to plan and design appropriate technology-based lessons, and the inadequacy of teacher training. Although the teachers' reasons may appear to be rational justifications, the main reason seems to indicate a lack of pedagogical knowledge and confidence in implementing technology-based lessons. Teachers often feel reluctant or uncomfortable because their pedagogical knowledge perhaps does not include a framework for conducting technology-based activities in their lessons.

Drawing on this problem faced by teachers, this paper aims to offer a framework to help mathematics teachers plan and design effective technology-based lessons. The paper begins with a discussion of the framework and follows with a description of the mathematics problem selected to illustrate it as well as a typical approach for solving it. A pre-designed Excel template is developed to guide pupils through this problem. A description of the tasks, which pupils are expected to perform in this activity by using this template, and an elaboration of how the framework underpins the teaching and learning of these tasks, are provided in the subsequent sections.

THE FRAMEWORK

Characterised as a psychological perspective on knowledge and learning (Jaworski, 1994), constructivism refers to the idea that learners generate meaning as they make sense of their world. Although opinions are diverse within the philosophy of constructivism, all seem to share one common view that knowledge is not passively attained by learners from their teachers, but rather, it is constructed by learners through their own experiences (Mestre, 1989; Olivier, 1989; Davis, Maher, & Noddings, 1990; Cobb, 1994). Therefore, it is logical to state that learners need to explore and test ideas through relevant activities in order to construct new ideas. This belief underpins the essence of the framework in this paper. The framework, developed from a constructivist point of view, aims to create a technology-based learning environment where students can construct their mathematical knowledge through interactive activities with computers. Such a learning environment can also provide students with opportunities for social interaction where they share and discuss ideas with their peers as well as their teachers. The social context constructed in the course of their interaction helps to enhance the students' thinking and learning in the classroom (Vygotsky, 1978; Cobb, Wood, & Yackel, 1990).

The framework involves four key components: exploring, conjecturing, verifying, and generalising. Of the four components, the last three are fundamental processes of mathematical thinking. Given that the current literature considerably advocates the exploring and conjecturing processes (Schoaff, 1993; Olive, 1998; Leong & Lim-Teo, 2002), these processes are perceived to be important for developing technology-based mathematics lessons. The exploring process can promote pupils' inquiry and investigation of the task and the conjecturing process provides a means for pupils to construct their own mathematical knowledge. The verifying component is deemed essential because pupils should be encouraged to cultivate a good habit of testing and checking the appropriateness and reasonableness of their conjectures, especially when problem solving is emphasised strongly in the Singapore mathematics curriculum (Ministry of Education, Singapore, 2000). The generalising process is crucial because it involves pupils' constructing of mathematical knowledge when they articulate what they have obtained from a specific case to a general situation. When pupils make connections between different mathematical ideas, they can be considered to have learned with understanding (Carpenter & Lehrer, 1999). Figure 1 provides a visual representation of the four components that make up the framework, each of which is then discussed in detail in the following paragraphs.

Under the exploring process, pupils inquire into a given task and then conduct their own investigation of the task based on instructions, which may suggest a particular heuristic to aid their investigation rather than directly informing them what to do. The conjecturing process requires pupils to make an inference or a judgement about a given task based on their intuition or evidence from an exploration, which may still be inconclusive. However, their inference or judgement may not be mathematically correct. At this point, pupils will have to substantiate the truth of their conjecture by showing their reasons in the verifying process. By undergoing these three processes, pupils can actually experience the process of mathematical investigation of a particular task as well.

[FIGURE 1 OMITTED]

As illustrated in Figure 1, the three components of exploring, conjecturing, and verifying are related in the form of a triangle (referred to as an E-C-V triangle). The structural connections of the components in the E-C-V triangle suggest that there is no linear order in which the components will occur. Therefore, when given a task, pupils normally start with exploring first, and then follow by conjecturing. However, if the conjecture is flawed, they may backtrack to exploring again instead of progressing to verifying. Pupils can then modify their earlier conjecture and proceed subsequently to verifying. Alternatively, pupils might make a conjecture based on their intuition first before they explore any given task.

Once pupils have finished exploring the given task, as well as making and verifying conjectures, teachers can encourage pupils to extend the given task to a new problem situation. In other words, the given task becomes a specific case of the new problem situation. While doing so, pupils may generate further cases that eventually lead to the articulation of the characteristics generic to these cases. The whole process of detecting and articulating the common characteristic from some cases is known as generalising. Generalisation plays an important role in the development of mathematics concepts, especially when generic characteristics are common in mathematics. Thus, being able to recognise them from the numerous concrete examples in mathematics and express them in a general form is important. This explains the importance of developing pupils' ability to generalise in mathematics.

The generalising process however, appears to be more viable after the given task has been successfully solved. Given that the completion of the given task seemingly marks the beginning of the generalising process, it is therefore deemed appropriate to consider generalising as a separate component in the framework rather than as a part of the E-C-V triangle.

As Figure 1 clearly shows, the framework has a hierarchical structure with learning taking place in phases and culminating with the generalising process. Each phase of learning involves the three components of exploring, conjecturing, and verifying and progresses upward. This upward progression corresponds to the increasing degree of complexity of the tasks to be performed by pupils. An example of how the framework may be applied is provided in the next two paragraphs.

Phase 1 can be, for instance, the stage of manipulation where pupils are involved in some exploratory activities related to a given mathematical problem. Here, pupils may acquire a sense of what the given problem is about as well as what the answer might be. From Phase 2 onwards, pupils can explore the same problem from different perspectives. For example, pupils may tackle the problem from a numerical approach in Phase 2; in Phase 3, they may attempt it using the graphical approach. Then, in Phase 4, pupils may be asked to work out the problem from a symbolic perspective.

It is important to highlight the fact that these different phases can lead pupils to build up their knowledge about the problem from the concrete to the abstract in a gradual manner. However, not all four phases will be expected to be experienced within a typical technology-based lesson. The number of phases depends on the sophistication of the given problem. For some problems, one phase might be enough, whereas other problems might require the use of multiple phases.

The framework described in this paper appears to share some similarities with the mathematics learning cycles outlined by Fleener, Westbrook, and Rogers (1995) and Frid (2000). The learning cycle by Fleener et al., which essentially illustrates how students can learn a mathematical concept through an ongoing sequence of learning activities, comprises the three stages of exploration, conceptual invention, and expansion. However, Frid's learning cycle, with the same role but achieved in an iterative manner, has five stages in the following order: finding out about the learners, exploration, formalisation, consolidation, and application. The exploration stage in both learning cycles, like the exploring component in the framework, is where students examine and discover a mathematical idea either on their own or with teachers' guidance. Another common feature involves promoting mathematical communication among the framework and the two learning cycles. Although the articulation of ideas may not be a key component in the framework, it is encouraged throughout all the learning activities. Similarly, students are also encouraged to articulate their ideas explicitly mainly under conceptual invention and formalisation in the learning cycles by Fleener et al. and Frid respectively.

On the other hand, the framework differs from the learning cycles in that the components of the E-C-V triangle do not necessarily have to occur in a linear order. However, as a cycle, the different stages in a learning cycle need to follow sequentially in a prescribed manner. In addition, the framework places an emphasis on two important mathematical thinking processes, viz. conjecturing and verifying; however, they are seemingly not a focus in the learning cycles.

The framework is best illustrated by an example. Therefore, the next section of this paper provides a description of the mathematical problem used to elaborate it.

THE "FENCING A GARDEN" PROBLEM

The mathematics problem that is used to illustrate the proposed framework involves finding the maximum area and the dimensions of a rectangular enclosure given a fixed perimeter. For example,

        A florist has 80 metres of fencing to make a rectangular garden.
        Find the maximum area of the garden and its dimensions for which
        the area is a maximum.

In Singapore, this problem is commonly found under the topic of Calculus in Additional Mathematics and comes up occasionally when studying the graphical solutions of quadratic functions in Elementary Mathematics. The problem is usually solved by using an analytical approach, beginning with the formulation of an equation connecting the area of the rectangular garden and one of its sides. Denoting the area of the rectangular garden by A square metres, the length by L metres, and the breadth by B metres, and applying the fact that 80 metres of fencing is used to enclose the garden, it follows easily that the perimeter of the garden is 2L + 2B = 80, which reduces to

L + B = 40 (Equation 1)

Using Equation 1, the equation for the area of the rectangular garden in terms of L is then readily established as

A = Lx(40 - L) = 40L - [L.sup.2] (Equation 2)

After deriving Equation 2, the first derivative of A is needed in order to find the value of L that produces the maximum area A. Therefore, by differentiation, the first derivative is

dA/dL = 40 - 2L

For maximum or minimum values of A, the first derivative is equated to zero and solved, thereby producing L = 20. Since the first derivative changes sign from positive to negative as L increases through 20, A is a maximum when L = 20. Hence, by substituting L = 20 into Equation 2, the maximum area of the rectangular garden is found to be 400 square metres and it occurs when the dimensions are 20 metres by 20 metres.

Despite being considered a standard calculus problem, the "Fencing a Garden" question does not necessarily have to be solved by using the analytical approach as presented above. Other approaches can actually be adopted in seeking solutions to such a problem. For instance, a pre-designed Excel template can be developed to illustrate how this problem can be solved numerically, graphically, and algebraically.

Excel is a popular and widely used spreadsheet of the Microsoft Office series and is readily available on school computers. An interactive and versatile piece of software, Excel allows pupils to enter values into a spreadsheet and then check immediately to see if they are correct. Even if pupils do not get a correct result, they can rely on Excel for help (Sutherland & Rojano, 1993). In addition, Excel features strong graphical capabilities and useful mathematical formulae that ease graph drawing and computation. All these benefits give ample reasons for selecting Excel as a suitable IT tool for conducting technology-based lessons in the classroom. The tasks assigned to pupils, the teaching sequence, and the learning outcomes pupils are expected to manifest when working on the Excel template are discussed in the next section.

TASKS AND TEACHING SEQUENCE

The investigation of the "Fencing a Garden" problem can be carried out in phases. The pre-designed Excel template comprises three worksheets, which are labelled as Exploration 1, Exploration 2, and Generalisation. In the Exploration 1 worksheet, pupils explore the problem and conjecture about the solutions to the problem by observation. In the Exploration 2 worksheet, pupils attempt the same problem using different representations--numerical, graphical, and symbolic. Here, incorporating multiple representations of solving the problem not only helps pupils learn in an effective way but also facilitates their thinking process. Finally, in the Generalisation worksheet, pupils are required to consolidate the patterns they have observed in the previous worksheets, and then generalise and formalise the solution for the problem.

Exploration 1

Within this teaching sequence, pupils will be introduced to the mathematical problem--Fencing a Garden--as well as the idea of using a spreadsheet to seek its solutions (see Figure 2). They are not required to enter any rule or construct any graph, except to explore the problem to acquire a sense of what the answer to the problem might be, and perhaps, at the same time to develop a sense of why the answer is likely to be correct.

[FIGURE 2 OMITTED]

To facilitate pupils' understanding of the problem, this worksheet includes a figure of a rectangle, which varies in size and shape corresponding to the change in the length. Therefore, by clicking on a button on the scroll bar, pupils can manipulate the length of the rectangular garden and explore how a change in length affects the area and the shape. Pupils can immediately observe the corresponding changes in the value of the area as well as the shape of the garden as displayed by the figure. When pupils go through this process, they are engaged in an interactive process of simulation where the length and area of the figure undergo continuous changes and yet it retains its perimeter. It is important for pupils to realise that when the length increases, the area also increases but reaches a maximum value at one instant and then decreases thereafter. To check that pupils observe this pattern, teachers can encourage them to describe their observation, followed by an explanation of why they think this is so. This facility allows the figure to serve as a dynamic and valuable visual aid for pupils to understand and then explain the mathematical situation.

After some exploration, pupils should have acquired a sense of the maximum area for the problem when the given perimeter is 80 metres. With this tentative idea, they may want to conjecture as it turns out that the maximum area is 400 square metres. However, it is crucial that teachers do not immediately confirm the conjecture. Instead, they can invite pupils to share with the class how their conjectures come about. Discussions such as this help teachers to ascertain whether the conjecture is a valid one or simply a guess. Once pupils can offer their justifications correctly, they are considered to have verified the conjecture. In the case of an incorrect conjecture, pupils can be asked to backtrack to the initial stage of exploration, re-examine the problem, and then conjecture again.

Exploration 2

Within this teaching sequence, pupils will be introduced to three representations of the solution to the "Fencing a Garden" problem: numerical, graphical, and symbolic (See Figure 3). In contrast to the more intuitive exploratory approach adopted in the previous worksheet, the approach adopted in this worksheet generally involves a considerable amount of computation, manipulation, and graphing. However, the use of a spreadsheet simplifies these procedures in such a way that the basis of the different representations remains explicit, and yet the nature of the solution process can still be fully understood and appreciated.

[FIGURE 3 OMITTED]

The first task for pupils is to derive an algebraic expression for the breadth, B, of the rectangular garden in terms of its length, L, using the given perimeter of 80 metres. Stating that the expression is B = 40 - L should be easy for pupils. The objective of this task is twofold. First, the expression can be used subsequently in the numerical representation to facilitate the computation of the breadth given any value of length; and, secondly, the expression can lead pupils to derive the equation relating the area and the length in Part 3(c) later.

After obtaining B = 40 - L, the exploration of the numerical representation of the problem begins with pupils entering values for the length, the corresponding breadth, and the resulting area into the table in Part 3. For instance, if a length of 10 metres is entered, the corresponding breadth is 30 metres (since B = 40 - 10 = 30) and the area of 300 square metres is the product of the length and the breadth. As soon as this set of inputs is entered into the table, the point (10, 300) appears on the graph grid provided where the horizontal axis represents the length and the vertical axis represents the area. This feature of the Exploration 2 worksheet offers the added benefit of seeing the graphical representation at the same time that the inputs for the length, breadth, and area are entered into the table under the numerical representation. Therefore, when pupils completely fill in the table, the pictorial representation of the relationship between the length and the area will have been traced out on the grid, thus clearly depicting how the two parameters are connected (see Figure 4).

[FIGURE 4 OMITTED]

Based on the data represented in the tabular and graphical forms, pupils can now conjecture about the maximum area and the dimensions of the rectangular garden for which the area is maximum. This idea can be followed by getting pupils to compare their conjecture about the area with the one predicted earlier in the Exploration 1 worksheet. In the case where the two conjectures are identical, the verifying process is considered to have occurred when pupils justify their conjectures successfully. On the other hand, if the conjectures contradict, pupils will have to review their approach, reflect on their conjectures to see if they make sense, and then make another conjecture if necessary. Upon reaching a conclusion regarding the maximum area and the dimensions, pupils can subsequently proceed to identify the shape of the garden formed as required in Part 3(b) of the worksheet. In this case, a square garden generates the maximum area.

The next part of the Exploration 2 worksheet involves the symbolic representation of the problem. The aim is to get pupils to establish an algebraic equation for the relationship connecting the length and the area, corresponding to the data displayed in the tabular and graphical forms. Although the relationship is known to be a quadratic function of the form A = p[L.sup.2] + qL + r, where p, q, and r are constants, the constants are not given, and pupils are thus left with the task of finding them. In this task, pupils are again involved in the processes of exploring, conjecturing, and verifying.

The exploring process occurs when pupils input arbitrary values for the constants into the boxes and check to see if the curve that is produced passes through all the points on the grid. Pupils will soon realise that acquiring the correct equation in such a manner is not a trivial matter, however. Given the fact that the correct equation, which is A = -[L.sup.2] + 40L, contains a large coefficient for the term in L and no constant term, it is likely that pupils will still be unsuccessful in establishing the correct equation despite numerous attempts. Certainly, there must then be a more efficient way of obtaining the equation! At this juncture, teachers can lead pupils to study the inputs in the table again and examine how the values for the area are computed from the corresponding values for the length. The conjecturing process is said to have occurred when pupils attempt to generalise and articulate an equation for determining the area when given the length. In this case, the expected equation for the area is L(40 - L), given that the length is L and the breadth is 40 - L. Subsequently, pupils can verify the equation by testing it through entering its coefficients into the boxes in the worksheet. If the equation is correct, the curve that is produced will then pass through all the points on the grid. Further verification may also be performed by substituting any one set of values taken from the table into the equation.

Generalisation

Before pupils work on this final worksheet, teachers can reiterate the three representations of the problem encountered in the previous two worksheets by altering the values of the given length of the fencing, in addition to consolidating and summarising the various findings. Next, teachers can lead and engage pupils in a meaningful discussion of the two questions given in the worksheet. The process of generalisation is incorporated because it is both essential and crucial to provide opportunities for pupils to apply skills previously learned and to extend their learning to solve new problems not explicitly covered formerly by instruction (Carpenter & Lehrer, 1999). Even more importantly, this process facilitates the teachers' evaluation of whether or not pupils can adapt their knowledge to solve new problems and can therefore be considered to have learned with understanding (See Figure 5).

[FIGURE 5 OMITTED]

To investigate the case under which the rectangle formed with a given perimeter will have the maximum area, pupils can easily explore the problem with different values of the perimeter. The Exploration 1 worksheet is designed to allow pupils to alter the perimeter of the rectangular garden and then to work on the whole activity again. By working with more cases of the problem using different perimeters, pupils may acquire a better sense of the problem and develop a deeper understanding of it as well. Consequently, they may notice a particular commonality among the various cases. Generalisation is said to have occurred when pupils can articulate that a square formed with a given perimeter produces the maximum area.

With regard to the second question, there are at least two approaches that teachers can adopt when asking pupils to determine the maximum or minimum value of a quadratic function. One of the approaches is to graph the function and read off the maximum or minimum value from it, as in the Exploration 2 worksheet. This is no longer a tedious task because commonly accessible graphing software can reduce the drudgery of manual plotting, thus allowing pupils to focus purposefully on interpreting the graph.

The other approach requires some algebraic manipulation, and may be more abstract than the first. Hence, the latter approach could be a stumbling block to many pupils who find algebra difficult. In the form of y = a(x - p[).sup.2] + q, the maximum or minimum value of the function is determined by the term q. When the coefficient a is negative, q is maximum and conversely, it is minimum when a is positive.

In the case of the "Fencing a Garden" problem, when the area has a perimeter of 80 metres, the quadratic equation after the transformation is given by the equation A = -(L - 20[).sup.2] + 400. Clearly, since the coefficient of (L - 20[).sup.2] is negative, it then follows that the maximum value of A is 400. Regardless of the approach adopted, pupils are considered to have articulated the generalisation for the second question when they are able to explain how the maximum or minimum value can be determined from the quadratic equation.

CONCLUSION

In this paper, we have presented a pedagogical framework anchored in constructivism to help mathematics teachers plan and design effective technology-based lessons as well as create an environment, which engages pupils in a mathematics investigation that allows them to experience the process of problem solving. The use of IT helps pupils to perform many tasks in the lesson--from exploring the possible multiple representations of a mathematical problem to generating and verifying conjectures about the solution to the problem, as well as making generalisations. Lessons such as this one can cater to the different pupils' mathematical abilities and allow pupils to learn at their own pace. For instance, academically strong students can progress all the way to the last stage of generalisation while weaker pupils can stop at any of the earlier phases.

Additionally, it is human nature to be curious. It is this curiosity that compels one to explore--to go beyond the mundane and the ordinary into the realm of uncertainty. Within the classroom, the pupils' innate curiosity can often be tapped in technology-based lessons. Therefore, when integrating technology into mathematics lessons, it is essential that teachers provide opportunities for pupils to explore mathematical ideas, to make and verify conjectures, and then finally to generalise the ideas by extending the conjectures. Technology-based lessons such as this one precisely encapsulate the essence of the framework. Only when IT is exploited to provide pupils with a means of investigating mathematical ideas and developing a deeper conceptual understanding of mathematics can it be said that IT has been used pervasively and effectively as outlined by the vision of the Singapore Ministry of Education's Masterplan II for IT in Education (Ministry of Education, Singapore, n.d.).

Notes

(1.) The authors contributed equally to the writing of this paper.

(2.) The authors thank Associate Professor Eric Wood for his helpful suggestions in the process of preparing this paper.

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BOON-LIANG CHUA 

National Institute of Education

Singapore

[e-mail]

YINGKANG WU

East China Normal University

People's Republic of China

[e-mail]