Dynamic representations of angle
with a Logo-based variation tool:
a case study


Chronis Kynigos
University of Athens, School of Philosophy,
Dept. of Philosophy, Education and Psychology,
Section of Education and Computer Technology Institute,
PatraPanepistimiopolis, Ilissia, Athens, Greece,
tel: 7248118, fax: 7248979,


Dynamic manipulation of a conventional representation of angle is made possible through a variation tool which in Turtle Geometry enables parametric procedures to become descriptions of evolving geometrical objects by means of dragging a slider and observing how the object changes in relation to the value of the variable. On going research is reported regarding the understandings formed by 12 year old pupils working with the concept of angle in a microworld incorporating procedures designed for its dynamic representation. A variety of geometrical and numerical conceptions of angle and turn were articulated by the pupils during their explorations.


angle, dynamic representation, programmability, variation, concept

1 A variation tool in Logo

The conventional means with which variation can be expressed in Logo are either through parametric procedures executed once at a time or through multiple executions of mathematical structures where the value of the parameter varies each time like, for example, in a recursive procedure. The functionality of sliders appearing in Logo environments such as LCSI's Microworld Project Builder has mainly focused on iconic interfaces for procedure execution. Sliders work as iconic interface devices providing a value to procedures or primitives followed by the name of the slider as numerical input. This means that a) the respective procedure or primitive is executed with the value currently portrayed on the slider, and b) when the slider button is dragged during the running of a procedure, the computer pauses and then continues to run the procedure with the new value.

This paper describes an example of the implementation of a variation tool which is being developed within the framework of the YDEES project. The project involves the building of a component oriented architecture for programmable exploratory software with a wide range of functionalities and the concurrent use of the software in school communities literate in Logo-like learning ([4], [13], Koutlis et. al. YDEES report, T13). In its broader function, the variation tool can be attached to any component designed to incorporate systematic variation and when this happens, it provides the user with a direct manipulation metaphor for sequentially changing the numerical value of the varying parameter and simultaneously observing the behaviour of the varying parts in relation to each other and to the invariant ones.

In Turtle Geometry, the variation tool behaves like this: clicking the mouse on any part of the trace of a graphical representation of the execution of a parametric procedure with a specific value "energises" a variation component which provides a slider for each of the parameters of the procedure (12). Each slider has an editable range and step and a tracking feature. Dragging the button on the slider "continually" erases and re-executes the procedure with the value corresponding to each position of the button. The user observes the graphical representation of the parametric object as the parameter changes sequentially. For example, applied to a procedure for a square with a variable for its sides would have the effect of the square growing and shrinking. Variation is thus conveyed not as a set of instances of a varying object but as an evolution of that object.


state A


state B

Figure 1 The effect of the variation tool

The effect is in fact impossible to represent by a static figure on paper. In figure 1, moving the slider from state A to state B causes the effect of a square continually growing from A to B, and the highlighted numerical input to roll sequentially with a step of 1.

This paper describes preliminary research on the kinds of meanings formed by a group of 12 year olds working with a microworld based on the variation tool, which was designed to provide the means to dynamically manipulate conventional representations of angle (Kynigos and Siouti, YDEES report T15B).

2 Angle concepts and representations

In mathematics, angle has been defined in two different ways stemming from two different geometrical systems, Plane Geometry and Analytic Geometry. Respectively, definition one states that angle is the area space formed by the intersection of two semi-planes each formed by two lines on a plane. There is no emphasis on the measure of angle since by definition it is an infinite area space and since in Plane Geometry the interest is on properties of figures or on relative measures, rather than on absolute measure quantities. Definition two refers to the quantity and the direction of a rotation of a vector, where the measure is all-important and a vehicle to the algebrisation of the concept of angle.

From the early years in mathematics curricula, this distinction is unclear, since definitions and representations refer mostly to the plane conception of angle, but angle measure retains an all-important role. The usual graphic representations of angle are, within this perspective, an incomplete "matching" of the two since the two lines become two line segments, the area space is ignored and in its place, the representation of angle measure takes the form of an arc close to the segment intersection point (figure 2).

Figure 2 Conventional Angle Representations

Pupils have a lot of trouble with the concept and the representation of angle. They perceive angle in various ways, such as the size of the area space between the line segments, the size of the linear trace of the arc, the length of the line segments, the distance between the line segments and the orientation of the pair of line segments [3]. Angle A in figure 2 would thus be easily perveived as larger than angle B because its shape is larger, the lines are larger and the "size" of the arc is larger. In any case, they cannot be equal because their orientation is different. Pupils' intuitions regarding angle seem to derive from the perception of the plane and when they focus on angle-as-turn they seem to prefer static representations of angle [16],[7]. They also seem to have intuitions about the concept of "turn" derived from experiences of bodily motion [9].These two sets of intuitions, however, are fragmented and pupils have a lot of difficulty in forming relations between [10],[14],[5].

It could be argued thus, that the way angle is "taught" and represented in school contributes to the confusion caused by the conceptual difficulties pupils seem to have in relating two different perceptions of angle.

3 The Variation tool and Angle

Through exploratory software for Geometry like Cabri, Logo and the more focused Turtle Math [1], there has been a clear attempt to provide pupils with dynamic means for representing angles and turns. In the microworld environment we are currently developing, we employed the variation tool as a means of dynamically manipulating the graphical output of symbolic descriptions of classical angle representations. The "technical" part of the microworld [6] is based on a series of such descriptions in the form of parametric Logo procedures, where the variables refer to the parts of the representation we aim for the pupils to dynamically manipulate in each case.

Figure 3 illustrates an example of such a procedure where the first two variables represent the lengths of the line segments and the third represents the measure of the internal angle. The graphical output of its execution with a specific value for each of the three variables is shown.

Figure 3 A procedure for an Angle Mircoworld

The graphical representation in itself is an instance of the concept of angle with a value of 60, something which resembles angle A in figure 2 and which would often be found in a book. The procedure and the variation tool, however, enable a dynamic handling of this representation where the lengths of the segments and the size of the angle can be changed in a continuous way. Dragging the slider of any of the variables causes continuous change in a) the numeric values on the slider and on the highlighted input to the executed procedure in the Logo code and b) the graphical representation. At each moment during the dragging of a slider, the current value in its position is shown. Furthermore, the procedure is designed so as to represent a turtle walking along the path of a static representation of angle [15]. The pupils could thus enact the procedure as they do when they play turtle. This was a specific pedagogic decision in designing this procedure and there could be many alternative approaches. The intention was that the pupils would be able to both use the procedure as a tool and understand its contents enough so that they could change them.

4 A preliminary case study

4.1 Pupil and activity profile

A group of three 12 year olds worked with the microworld for four hours in the computer room of their school. For almost two years before the study, they had been carrying out collaborative investigations as part of a weekly classroom activity with their normal teacher [10], on a variety of topics involving the use of Logo, a wordprocessor and a graphics software. In each such investigation, they would typically experiment and construct something with Logo, transfer it to the graphics software and add colour, and freehand drawing and then transfer everything to a wordprocessor file writing a report of their activity and discussing it with the rest of the class. Their teacher classified them as a little above average in their overall performance in school. During the research sessions, care was taken to discuss the activity with the same terms as their work with their class. A typical example is that their teacher would "give" them a procedure with which they would experiment and suggest "theories" regarding its behaviour. They would then discuss these and subsequently make a construction using the procedure, such as. for example, a bridge based on an arc procedure. In the same sense, the "angle" procedure was written by the researcher and the pupils were initially asked to experiment with it using the variation tool. In a subsequent phase, the pupils constructed a procedure for a rectangle with one set of opposite variable sides and substituted the straight lines in the angle procedure with their rectangle procedure which they called "Mithridatis" (figure 4).

Being the initial part of a larger on-going study within the YDEES project, this activity was not extensive enough to yield information about the pupils' development of the concept of angle, nor could it provide a breadth of approaches to the problem. The data, however, did reveal the quality and the breadth of meanings the pupils formed (a discussion regarding the forming of meanings in exploratory mathematics situations can be found in Noss and Hoyles [17]) regarding the concept of angle, turn and turn measure (verbatim transcription from a video recording). The pupils discussed angle as both a dynamic and a static entity and used both its geometrical and its numerical facets.

4.2 Angle as a dynamic entity

Angle as a moving object

Throughout the activity, an important characteristic in the pupils' perception whenever the issue of angle was discussed was that of motion. In previous studies of pupils working with turn [8],[2],[10], their focus seemed to be on the action and on the intitial and final state of turning rather than on the process of turning. Mitchelmore and White, [16], go so far as to suggest that pupils' intuitions lie rather with the static features of turn representation rather than on the action of turning. In using a tool directly manipulating turn, the pupils in this study seemed to focus on the process of angle change rather than on one specific amount of turn each time. Here is an example of some comments suggesting this (these were interspersed and not part of one dialogue).

Michalis: "if we move slowly, it turns round and round, 90 degrees, how many it says here" (points at the numerical value on the slider)
Kostas: "It opens and closes with the angles"
Mikchalis: "It grows and shrinks"

However, whenever the pupils were asked what an angle is they would revert to the static description, as in the following two comments.

Researcher: "what is an angle?"
Michalis: "when two lines meet on a point, its the distance between those two lines"

Researcher asks which is the angle between the two rectangles drawn by the "Mithridatis" procedures (see figure 4):

Sofia: "an angle between rectangles is the angle between these lines" (points to the variable sides of the rectangles).

Figure 4 An Angle Between Rectangles

Researcher asks whether there is an angle in the shape in front of them:

Michalis: "the angle apart from the ones inside the rectangle is this one, the one which turns here"

The final comment indicates that Michalis perceived the "angle - Mithridatis" procedure as having two kinds of angles, the ones in constructing the rectangles and the one between the rectangles which he defined by means of its motion. In his talk there was no indication of perceiving the procedure as an angle object in itself, based on the latter angle and equivalent to the first angle procedure, despite the intention of the researcher. In fact, this is not surpsising since the pupils had previously constructed many parallelograms in a two month - long investigation, so they had a lot of experience with the parallelogram angles.

Angle as a changing quantity

Another view portrayed by the pupils, apart from angle as an evolving object, was that of a changing measure. They noticed the connection between dragging the button, graphical change and numerical changes, both on the slider and on the interpreter. The angle slider was the one they tried last and Michalis pointed out:

"the third slider is the degrees, i.e. the closing and the opening of the shape"

connecting measure and graphical representation. In fact, the pupils talk indicated that they were using the connection to describe the bahaviour of the tool as shown in the following two comments:

R: Why does the bottom slider make the shape open and close?"
Kostas: "because the smaller they get (the numbers) the more the angle is narrower:
R: So, whatăs happening with the degrees? If you change a number, what will happen?"
Kostas: "it will grow"
Michalis: "if it (the number) increases, it will open up more".

4.3 Angle and Number

Modulo 360

The pupils did not make a specific connection to the 360-degree rotation even though they changed the end of the range to 1000 and discussed the effect of dragging the slider. Even though they had worked with angles in their curriculum maths and rotations in their Logo investigations for some time, they only referred to the notion of a whole turn by calling it "double", in the sense of overlapping after completing one turn.

Sofia: It turned 180 that's a half turn and because we put more than 180 (end of the range) it made a whole turn"
Kostas: "A double turn"

Although no comments were made to indicate the pupils' feeling of the size of the angle unit measure, it is interesting to consider the effect of observing the angle figure change as the slider is dragged in varying unit steps.

The meaning of range

The pupils did play with changing the range and the step. They tried giving various values to the end of the range and observed the way in which the numbers changed as the slider was dragged. In fact for large numbers, the visible changes in the counter were larger that unit changes due to machine response. This may have been one of the reasons the pupils assumed that there was a connection between the step and the range:

Kostas: "and if you put less than 100, then over there (in the step slot) there will be a point and you will get point five" (they had tried a step of 0.5 before).

Perhaps due to the short time they investigated range, they did not relate the rate of change to the size of the range when the speed of dragging remains constant.

Special cases of geometrical objects and their representations are often not perceived as such by pupils, such as, for instance, a straight line being in fact the representation of a 180, or a 360 or a 0 degree angle. The effect of the object moving to and from a special case was noticed by the pupils and their comment indicates perceiving an angle of 1 degree as a special case of angle:

Michalis: (they had inserted a range from 1 to 10000) "When its one it disappears but the larger the number, the larger it gets"

Negative and positive values

Negative numbers are initially used and understood by pupils within the activity of subtraction and the minus sign as denoting subtraction [18]. When the researcher suggested they put a negative initial range value, the pupils stated that there's no such thing, which is consistent with research indicating that perceiving the notion of negative number on its own represented by placing the minus sign is hard to understand. In trying to make sense of dragging the slider across a range of -200,200, the pupils formed a notion of inverse values, stating that a negative number is behind a positive one. They also related negative values to the inverse action to the one represented by the positive values (left turn).

R: "what if the angle starts with a smaller number than zero?"
Kostas: "there is no number smaller than zero"
R: "no?"
Michalis: "its minus one.. (after inserting a range of -200,200 and dragging the slider several times) minus 200 is behind zero"
Michalis: "its like starting from 200 but instead of right, left"
Kostas: "when its at zero, its normal. Its one on top of the other".

5 Conclusions

Observation of this group of pupils indicates that dynamic manipulations of symbolic and graphical angle representations could be an interesting design feature of angle microworlds encouraging pupils to express and explore a variety of ideas related to the concept. These pupils informally used geometrical concepts of angle, turn and angle change and numerical concepts related to range, multiplicity and negative - positive numbers. They also linked iconic and symbolic representations and used the language feature of the software to change the angle procedure to an angle-between-rectangles representation. Given more time and experience with the tool, the pupils could have inserted any object to denote an angular relation. No claims can be made yet on the potential development of such concepts through work with the microworld, nor on the breadth and variety of meaning formed by different pupils. The range and quality of ideas articulated by the pupils suggests, however, that more research into the development of angle concepts by working with such representations and tools would be interesting and could well lead to the design of rich microworld environments and curriculum focused activity.

Designing tools for dynamic representations of geometric and algebraic concepts and relations is important and something which was not possible before the availability of direct manipulation interfaces. The reason is that maths is about relations, abstractions and processes and the only conventional means for representing such things are either symbolic or iconic instances of generalised entities. Without means for dynamic representations we can only have a set (however large) of instances and no effect of continuity. The variation tool may be an interesting vehicle for linking symbolic and graphical mathematical representations and through observing their evolution by directly manipulating parameter change, acquire a sense of the generality and abstraction underlying some static instances of mathematical structures. The programmability feature of the software allows and encourages creativity and specificity in the design of such representations and tools by mathematics educators, curriculum developers and of course pupils (figure 4).


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