INTEGRATING NUMBER, ALGEBRA, AND GEOMETRY WITH INTERACTIVE GEOMETRY SOFTWARE Kate Mackrell Institute of Education, University of London In order to compare the potential for the integration of number, algebra and geometry using interactive geometry software, a series of tasks related to finding the area of a circle was performed using Cabri II Plus, Cinderella, GeoGebra and Geometer’s Sketchpad. It was found that, while each program had the facility to perform the tasks, there were differences in the design of the programs that could lead to either facilitating or impeding the development of student understanding. INTRODUCTION There is an increasing awareness that the details of the design of pedagogical tools, are significant and should be researched (Jackiw, 2010). This paper arises out of research on the design of four different interactive geometry (IGS) programs: Cabri II Plus (Cabri), Cinderella, GeoGebra and Geometer’s Sketchpad (GSP). The focus has been on the identification of affordances and design decisions as a basis for further research on the impact of such differences on student learning. Current IGS software provides a means by which algebra, geometry, and number can be meaningfully linked. Falcade (2007) showed that geometric construction could be used to enhance student understanding of the concept of function. Laborde (2010) explored the creation of dynamic graphs. Jackiw (2010) suggested that dynamic number provides an appropriate link between algebra and geometry. A particular series of tasks involving the integration of number, algebra, and geometry were hence used to compare the four programs. The topic, finding the area of a circle, is universal, and each task (e.g. constructing a geometric object or an algebraic expression) involved processes common to other topics. The pedagogical approach (involving exploring and gathering information about a mathematical situation, making and testing conjectures, then generalizing and proving results) has been promoted extensively in the UK since the 1980’s: the specific tasks were based upon the principle stated by Laborde (2010, p. 218) that “the teaching of mathematics must help students learn how to adequately use various representations and to move between them if needed.” As the aim of the research has been to identify rather than study the impact of design decisions, the tasks were performed only by the researcher in order to: a. ensure that the same tasks were performed with each program as a basis for comparison. The pedagogical approach emphasizes student choice and, in a classroom, will result in different tasks being performed by different students.

b. address the issue of familiarity. The researcher was in communication with the developers of all the programs to ensure that any initial lack of understanding was not reflected in her conclusions. c. ignore the effects of any differences in task presentation necessitated by differences in the programs and in student familiarity with the programs METHOD: Each task was performed several times with each program as questions concerning the affordances of the programs arose and were answered by the software developers, or when more detail was required. RESULTS: Task 1: Create a circle and a segment to represent its radius In this task, the basic mathematical situation to be explored was set up. In Euclidean geometry, a circle is of a set of points equidistant from a given point, and does not depend on location. Cabri and Cinderella each had a tool by a circle could be created and explored simply by clicking to create a centre point and define an initial radius. Dragging the centre point of this circle moved the circle without changing its size; dragging on the circle itself changed its radius. In the other programs objects such as a point on the circumference or a segment or number giving the length of the radius needed to be chosen. The simplest option offered by all the programs (and hence the option used in further tasks) was to create a circle given a centre point and a point on the circumference (referred to as the radius point). Dragging the centre of this circle changed both its location and its size: only by dragging the circle as a whole was its radius maintained. In Cinderella and GSP the same motion, dragging, is used to create the circle and to move it. Making the circle by clicking at the centre, holding down and pulling felt analogous to pulling one arm of a pair of compasses away from the other. Cabri and GeoGebra required a click – release - move – click motion, a different motion from dragging, but with the same visual effect and hence potentially confusing. In GeoGebra, the coordinates of the centre point and radius point appeared after clicking and before releasing which was distracting, unnecessary, and potentially off-putting to learners who had not yet encountered coordinates. In the geometry of Euclid, distances, areas and volumes could be compared but were never assigned a number. Introducing a number moves away from the origins of geometry, but, if the number is variable, enables a move toward algebra. In each program, a circle may be created from its centre and a number to represent its radius. Possibilities for the radius of the circle can be ordered by variation and naming. In Cinderella, the number, input through a dialogue box, cannot be edited or used for any purpose other than defining the circle. GeoGebra gives the same option, although

the number may be edited. In Cabri and GSP, the number is selected by clicking on any number displayed on the page: the result of previous measurement or calculation, or entered directly. Cabri allows numbers to be placed on the page without being assigned names. GSP requires a new parameter to be defined, which will be assigned a name. GeoGebra also allows previously defined numbers to be used, but only by typing in the name of the variable, such as the name of a slider, representing the number. A slider, whereby a number changes as a point is dragged, gives a visual representation of the variability of a number, and enables numbers to be changed by the same operation, dragging, that changes geometric figures. It has the potential to be an important link between geometric and algebraic representations. GeoGebra sliders have the appearance of points on segments, but the points and segments are not actual geometric objects, however. In the other programs, geometric sliders, which give measurements such as the relative position of a point on a segment, may be created. The numbers defined by such sliders will not necessarily have names. A conjecture is that the process of manipulating numbers which are linked to objects that change may be important in developing the idea of a variable. Changing a number such as the radius of the circle and noticing the effect may be a useful introduction to the idea of variation without algebraic terminology and notation (through the use of geometrically-defined sliders). Using names for variable numbers may be a later step in developing the idea of the variable. Task 2: Measure the area and radius of the circle. In this task, information is found about the mathematical object being studied. With measurement, the introduction of numbers becomes necessary. In each program, it is straightforward to find the area of the circle and the distance between the centre and radius points. The table below gives a screenshot for each program with a list of construction steps so far. The algebra window is also displayed for GeoGebra. Cabri objects (but not measurements) have been labeled. Other labels were supplied by the programs.

Cabri II Plus

Geometer’s Sketchpad

Cinderella

GeoGebra Table 1: Constructed figures and figure descriptions

An issue with measurement is that GeoGebra and Cinderella do not display units (although this can be changed in Cinderella). Although not apparent above, trailing zeros are also not displayed. For example 3.004 to two decimal places would be written as 3 rather than 3.00, which can lead to statements such as “3 x 4 = 12.02”. Unlike Cabri and GSP, Cinderella and GeoGebra distinguish names, which appear in the figure descriptions, and labels, which appear on the page. For example, the area of the circle in Cinderella has the name “A0” but the label “|C0|”, and in GeoGebra has the name “areac” and the label “Area”.

A confusion concerning objects and algebraic variables is evident in GeoGebra. Geometric objects and measurements of these objects may each be given a name, and the names may look equally “algebraic”, but only the name of the measurement refers to a variable (assuming that the measurement is not fixed), as only numerical quantities may be variables. Every program identifies the segment from A to B as a geometric object, a segment. However, GeoGebra treats it as a variable, and assigns it the value of the length of the segment: apparently “the algebraic representation of a segment is its length” (Hohenwarter, 2010, personal communication). This is also a puzzling misuse of the phrase “algebraic representation”: the length is a measurement rather than an algebraic expression, it is not the only measurement which can be made of a segment, and it does not determine the segment in the way the equation of the circle can determine the circle. A further issue here is the naming of coordinates and equations as “values” in Geogebra. Particularly in interactive geometry, the coordinates of a free point are an indication of its (temporary) location relative to certain coordinate axes: equating a point to its coordinates is simply wrong. There is also the issue that the information concerning coordinates and equations is displayed at all. For Cinderella, such information must be shown. In GeoGebra, such information must be hidden. Task 3: Change the radius of the circle and observe the effect on its area. In this task, further information concerning the mathematical object was obtained, and a conjecture was made about the relationship between two variables. Dragging is one of the chief links between geometry, number and algebra in an IGS. By dragging the radius point, a static circle, with a fixed radius and area, becomes a circle whose radius and area are now variables, capable of being related. GeoGebra shows the coordinates of the radius point as it is dragged: the other programs enable a focus on the way in which the area changes as the radius is changed without distraction. The measurements move with the figure in Cabri and Cinderella and can be attached to the figure in GeoGebra and GSP (although this is not straightforward). It is clear that as the radius increases, so does the area. A conjecture is hence that the area is some multiple of the radius. Task 4: Test the conjecture by calculating area/radius and seeing how this changes as the radius is changed. In this task, the conjecture about the relationship between the two variables was tested, and refuted. An advantage of Cabri and GSP is that the general division of area by radius could be achieved simply by using a calculator tool to divide the existing area value by the existing radius value, with numbers entered into the calculator by clicking on them. As the radius and area changed, the calculation was continually updated. It is

unnecessary for the student to deal with the abstract idea of dividing one variable by another. However, the calculator can also act as an introduction to this idea. In GSP, when a number is selected on the page, its name appears in the calculator, making it clear precisely what is being calculated, and the label “area/radius” will appear next to the completed calculation. In Cabri, calculation involves more algebra; when a number is selected on the page, the number is assigned a variable name, starting with “a”. This name appears both on the page next to the number and in the body of the calculator. An expression is built up in the calculator, and the variable names on the page indicate which number will be substituted for each variable in the expression when the expression is evaluated. The use of the function tool for calculation in Cinderella immediately made calculation seem more daunting. Numbers could be selected either by clicking on on them on the screen to place their names in the calculation box, or by typing in the names. The possibility of dual input means that the user could either see the calculation as just involving numbers or as involving a relationship between variables. In contrast, GeoGebra required the names of variables be typed in the input bar in order to perform any arithmetic operations, hence demanding the awareness that one variable might be divided by another with no means to build this awareness. The algebra window, with a large amount of distracting information, needed to be open to find the names of the variables and the text input requirement created issues with syntax, made more difficult by the confusion regarding the segment name, which, it turned out, would behave as a variable in the calculation. Having performed the calculation, it is clear that the result changes as the radius changes: the conjecture concerning a linear relationship was incorrect. Task 5: Using the measurements to create a graph of area against radius In this task, a new representation of the mathematical situation was created to give further insight on the relationship between the two relevant variables. One of the most powerful features of IGS is the ability to visually represent the way in which measurements vary: a graph of area against radius may be constructed directly from the existing measurements. It is possible in all programs to show inbuilt coordinate axes and directly plot the point representing (radius, area), but with Cabri or GSP the basic idea of coordinate representation may be explored. A number can be transferred to a linear object which acts as an axis. For example, the radius measurement of 2.5 cm may be used to create a point which is 2.5 units from a fixed point along a line functioning as the x axis. As the radius changes, this point will move along the axis accordingly. The corresponding area of 19.4 cm2 may be represented by a point along another axis. Parallel axes form a dynagraph (Goldenberg, 1992); axes at an angle enable the

construction of the point that is reached by travelling 2.5 units along the x line followed by 19.4 units along a line parallel to the y line. Once the point is plotted, each program enabled it to be traced, to create a visual record of the way in which area varies as radius is changed. The set of all possible points representing (radius, area) could then be obtained by creating a locus, which represents the graph of area against radius. This graph was constructed through an understanding of coordinate representation with no recourse to algebraic equations. In order to make more of the graph visible, it would be useful to reduce the scale on the y axis. This is unproblematic in Cabri and GSP. However, Cinderella does not allow axes to be rescaled (although zooming in or out is possible). Geogebra allows rescaling, but with the consequence that the circle changes shape, as shown below.

Figure 2: GeoGebra circle with unequal axis scaling

In Cabri and GSP, a fundamental design decision was to treat the screen as a simulation of a Euclidean plane, where distance is measured by a rigid ruler. Coordinate axes provide a reference frame relative to which objects such as the circle have a location and possibly an algebraic equation. When axes change, the relative location and equation of objects will change, but the objects will not. In contrast, Cinderella and GeoGebra use a coordinate system related to the intrinsic screen coordinate system to define all objects: selecting the centre point and radius point for the circle define it by an equation referring to this coordinate system. This is why these programs give coordinates and equations in the figure descriptions. This provides an equally acceptable model of the Euclidean plane – provided that the coordinate axes cannot be scaled independently. Hence Cinderella does not allow such scaling. In GeoGebra, when the axes change, the fixed defining equation means the circle visible on the screen needs to change. In an investigation concerning the area of an object, the experience of the visible area of the object changing might be highly confusing to the learner.

Task 6: Graph an algebraically defined curve to fit the locus and hence find a formula for the relationship between radius and area. In this task, a further construction is made to test the conjecture that the graph is quadratic and to find its coefficient. Cabri is the only program that will find the equation of a locus directly (GeoGebra does not even list the locus as an object in the algebra window). All programs will fit to the locus a graph defined by means of an algebraic equation. Cabri II Plus requires an expression to be defined and applied to an axis, whereas GSP, GeoGebra and Cinderella require the definition and plotting of a function. A parameter p can be introduced to create the graph of y = p x2. Manipulating this parameter will give the curve of best fit as y = 3.14 x2. Task 7: Test the formula found In this task, the specific conjecture represented by the formula found in the previous task is tested. This was achieved by editing the calculation from task 4 to area/radius2, which now gave a constant value of about 3.14. It was also possible to create a function or expression 3.14*r^2, substitute the radius for r and compare the result with the measured area. Cabri used the simpler language of evaluating an expression, and the other programs used the language of functions, with text input needed for GeoGebra. The final stage pedagogically would be to prove this result, or at least give some reasons why the area of a circle has this particular relationship to its radius, but this has been beyond the scope of this paper, although not beyond the scope of IGS, which could, for example, be used to compare the area of the circle to that of the square containing it, or to “unfold” the circle into an approximate parallelogram. CONCLUSION The series of tasks shown here illustrate the ways in which IGS could be used to develop links between number, algebra and geometry through representation of a mathematical situation in different ways. In particular, dynamic number has emerged as particularly important, as predicted by Jackiw (2010). Dynamic number can serve as an introduction to variation, and naming such numbers as an introduction to algebraic variables. Relationships between dynamic numbers may be explored and expressed as graphs or algebraic formulae. A circle could be created according to its fundamental definition, without numbers, showing the primacy of geometry in this context. However, a circle could also be created by using a number to define its radius. Changing this number and noting its effect on the circle might be important in developing the idea of variation, whereas naming the number might be a move toward the idea of variable. Measuring the area and radius of the circle involved introducing number as a description of geometry. In an environnment where geometric objects may be

changed by dragging, measurements are variables between which relationships may be conjectured. Such relationships may be tested by calculation, ostensibly involving just numbers, but in fact involving variables. Calculation itself may be a means of developing awareness of general expressions. Creating a graph meant creating a visual representation of the relationship between the variables of radius and area. This was done first without algebra by using the basic definition of a graph as a locus (a unique feature of IGS environments), and then by means of an algebraic definition, showing that the relationship between radius and area could be expressed algebraically. The relationship found could be tested algebraically by substituting measurements into a formula. The relationship was not proven, however, although some justification would have been possible with IGS. Although quite different in some respects, Cabri, GSP, and Cinderella were each well suited to the tasks, giving scope for students to explore the idea of variability before requiring the use of specific variables. What was critical in these programs is that text entry involving the names of variables was not necessary. Each had specific perceived strengths and weaknesses, which will be reported in more depth in further research. What was surprising, however, is the number of problems that arose using GeoGebra, specifically “developed as a tool to support dynamically linked multiple representations of mathematical objects” (Hohenwarter, 2010, personal communication). Jackiw (2010) pointed out a number of the problems in GeoGebra’s algebraic representation, and similar problems were found in this study. GeoGebra had mathematical errors in number (rounding of decimals), algebra (treating a segment name as a variable and assigning it a value) and geometry (not consistently representing the Euclidean plane). In addition, information was continually given which was irrelevant and distracting (the display of coordinates whenever an object was created or dragged). Its reliance on text entry made it slow to use, liable to syntax errors, and meant that students needed to understand the meaning of a variable in order to use it, an understanding that could be developed in the process of using the other programs. Text entry also meant that the algebra window, with unnecessary and distracting coordinate information, needed to remain open in order to access the names of variables. It lacked the functionality that would enable students to understand coordinate representation from first principles. The “algebraic” representation of objects given by GeoGebra was at best irrelevant in exploring connections in the task described here. At worst it was distracting and misleading. Future research will compare student responses to similar tasks using Cabri, GSP and Cinderella, testing conjectures made here concerning the role of dynamic numbers linked to geometric objects in facilitating the development of algebraic concepts and looking more closely at the effect of differences between these programs. The programming language available within Cinderella has been beyond the scope of this

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