1 Introduction: Role of Digital Technology in Education

Whether to use or not digital technology in mathematics classrooms is not an issue anymore nowadays, the question rather shifted to how to use it more efficiently and how to benefit the best from its affordances.

Since 1980s, researchers question the role technology should play in education. Two distinct roles have been highlighted by Pea (1985) and described in terms of amplifier and reorganizer metaphors. The amplifier metaphor suggests that technology changes “how effectively we do traditional tasks, amplifying or extending our capabilities, with the assumption that these tasks stay fundamentally the same” (p. 168), while the reorganizer metaphor posits that technology changes “the tasks we do by reorganizing our mental functioning, and not only by amplifying it” (ibid.). A simplified vision of the two metaphors leads to considering the use of digital technology either to do traditional tasks although in a different way or to do new tasks that cannot be done without this technology (Ripley, 2009). Likewise, Thomas and Lin (2013) point out that key affordances of technology emanate from the tasks that are used with it. However, designing tasks incorporating technology and having an epistemic value (Kieran & Drijvers, 2006) is not trivial for mathematics teachers.

In this chapter, we aim at highlighting that a given (mathematical) digital tool can be mobilized in manifold ways with different learning potential. We illustrate these considerations on the example of dynamic geometry (DG). The choice of dynamic geometry is motivated by a discrepancy between its potential to support students’ learning evidenced by numerous research (e.g., Arzarello et al., 2002; Baccaglini-Frank & Mariotti, 2010; Leung, 2015) on the one hand, and its limited use in mathematics classrooms (e.g., Bretscher, 2010; Kriek & Stols, 2011; Molnár & Lukáč, 2015).

Jones (2005) claims that “carefully designed tasks” with their appropriate enactment by the teacher are necessary for an efficient use of DG fostering students’ learning:

Overall, research in this area [use of DG software] indicates that successful access to geometrical theory does not happen without carefully designed tasks, professional teacher input, and opportunities for students to conjecture, to make mistakes, to reflect, to interpret relationships among objects, and to offer tentative mathematical explanations. (p. 29)

This chapter therefore proposes an analysis of selected DG tasks aiming at highlighting their differences in terms of student’s cognitive activity, thus showing the range of potential use of DG.

The tasks, presented in Sect. 3, are categorized according to the SAMR framework (Sect. 2.2) and are analyzed referring to the instrumental approach (Sect. 2.1). The concluding Sect. 4 discusses implications of the analyses for the teaching and learning mathematics with technology.

2 Theoretical Framework

Instrumental approach (2.1) is the main theoretical framework that we use to analyze mathematical tasks under consideration in Sect. 3. These are organized according to the SAMR model presented in Sect. 2.2.

2.1 Instrumental Approach

The instrumental approach (Rabardel, 2002) was elaborated to understand processes by which a user transforms a (digital) tool—an artifact, into an instrument enabling her to achieve her goals. While the artifact (material or symbolic) is available to the user, the instrument is a personal construct elaborated by the user during her activity with the artifact in the course of the so called instrumental genesis. The process of instrumental genesis comprises two interrelated sub-processes: instrumentation leading to the constitution and the evolution of schemes of use of the artifact in the user, and instrumentalisation during which the user adapts and personalizes the artifact according to her knowledge and beliefs. The development of schemes of use manifests itself in a user’s invariant behavior in a given class of situations (Vergnaud, 1990).

2.2 SAMR Model

Several frameworks have been elaborated to understand the role of technology in teaching and learning, such as RAT framework (Hughes, Thomas, & Scharber, 2006) described as “an assessment framework for understanding technology’s role in teaching, learning and curricular practices”.Footnote 1 According to this framework, technology can be used as a Replacement when it “serves as a different (digital) means to same instructional practices”, as an Amplification when it “increases efficiency, effectiveness, and productivity of same instructional practices”, and as a Transformation when it “invents new instruction, learning, or curricula” (ibid.).

SAMR framework developed by Puentedura (2006) is another framework suggesting four roles of technology ranging from substitution to redefinition (Fig. 1).

Fig. 1
figure 1

SAMR modelFootnote

http://hippasus.com/resources/tte/puentedura_tte.pdf.

(Puentedura, 2006)

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Let us explain the different uses of technology from substitution to redefinition levels on the example of a quiz. An online version of a traditional paper-based quiz, where the student checks what she considers as correct answers, does not offer her any functional change. Technology in this case is thus used as a substitute of a paper quiz, although it can facilitate administration of the quiz (via a url instead of paper copies for example) and collection of students’ responses. If the environment within which the quiz is implemented can provide feedback about the correctness of the answer, this functional improvement (augmentation) fosters learning. Indeed numerous studies suggest that feedback is most effective when it is provided immediately, rather than days or weeks later, and seems to positively impact both students’ achievement (e.g. Razzaq, Ostrow & Heffernan, 2020) and engagement (e.g., Sancho-Vinuesa et al., 2013). If the environment provides not only true–false feedback but a more elaborated feedback such as hints (e.g., link to lessons) in case of incorrect answers, the student’s task is deeply modified: such feedback supports learning by orienting the student toward appropriate remedial activities (modification). Finally, a quiz that personalizes student’s path through the items according to her answers can only be developed with technology (redefinition).

As we show in the following section, the four levels of the SAMR model align with the four roles of dynamic geometry identified by Laborde (2001). For this reason, we refer to this framework when considering various uses of dynamic geometry.

3 Various Uses of Dynamic Geometry

Laborde (2001) identified four different roles of dynamic geometry in the tasks:

  • DG is used “mainly as facilitating material aspects of the task while not changing it conceptually” (p. 293). These are for example construction tasks in which the only difference “lies in the drawing facilities offered” by dynamic geometry (ibid.). Dynamic geometry can be seen as a substitute of traditional tools.

  • DG “is supposed to facilitate the mathematical task that is considered as unchanged”. In this case, “DG is used as a visual amplifier […] in the task of identifying properties” (ibid.). Indeed, geometric properties of a figure being preserved while dragging its free elements, their visual recognition is facilitated. Dynamic geometry substitutes traditional tools, but brings certain functional improvement (augmentation).

  • DG “is supposed to modify the solving strategies of the task due to the use of some of its tools and to the possibility that the task might be rendered more difficult” (ibid.). Whereas a construction of a geometric figure with traditional tools can result in a visually correct drawing although controlled by perception, the same task in DG environment requires using geometric properties to obtain a figure that resists while dragging its free elements. Solving strategies in DG environment are thus deeply modified.

  • The task only exists in DG environment. Laborde (2001) refers to the so-called “black box” tasks in which students are asked to reconstruct a dynamic figure provided in a DG environment that preserves geometric relations when its free elements are dragged (redefinition).

In the following sections, organized according to the levels of the SAMR framework, we discuss various possible uses of DG and analyze them from the instrumental perspective. Following Lopez-Real and Leung (2006), who consider that

dragging in DGE can open up some kind of semantic space (meaning potential) for mathematical concept formation in which dragging modalities (strategies) are temporal-dynamic semiotic mediation instruments that can create mathematical meanings (p. 666),

we focus on the dragging functionality of dynamic geometry, considered as artifact, to highlight its semiotic potential.

3.1 Substitution Level Tasks

Inviting students to make a free drawing using DG tools, without paying attention to geometric relations is perhaps the “simplest” task (Fig. 2). Such a task can offer an opportunity to the students to get acquainted with DG menus and tools when they are introduced to this technology. Such a task can also be an occasion to exploit the semiotic potential of DG tools by comparing and contrasting them with traditional tools. For example, the fact that, in order to draw a straight line, the user needs to click to two different spots on the screen, which results in creating two distinct points and subsequently a line passing through these points, conveys the idea that a straight line passes through two distinct points. This is not necessarily the case with using a ruler to draw a straight line, which rather emphasizes the straightness of the line.

Fig. 2
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Example of a freely drawn figure

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Another example of a task at the substitution level is constructing a geometric figure following a construction program (i.e., a series of instructions). In the example shown in Fig. 3, the task is proposed to primary school pupils. The use of DG presents several advantages. The figures pupils construct can be quite complex, not only usual ones, since the task is facilitated within the DG environment. Pupils with motor difficulties of drawing with traditional instruments can succeed the task. Self-evaluation is also easier as drawings are more accurate and pupils can modify elements of the figure without deleting the correct steps.

Fig. 3
figure 3

Construction program (left) yielding a geometric figure (right)Footnote

Task retrieved from http://www.ac-grenoble.fr/ien.st-gervais/spip.php?article1420.

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Drag mode in these tasks is used to a limited extent, if at all: to adjust elements of a drawing either for the purposes of perceptive satisfaction (free drawing) or to separate elements of a figure to ease its construction (for example, when two points are too close to each other that they may be confused when selecting one of them). Restrepo (2008) classifies this dragging instrument as dragging without mathematical purpose.

It is not rare to find resources in which DG is used as a mere substitute of traditional tools although its potential could have been exploited to a greater extent. An example is given in Fig. 4 showing a task aiming at discovering that the area of a triangle ABC does not change when one of its vertices, say A, belongs to a line (d) parallel to the opposite side [BC]. Instead of dragging the vertex A on the line (d), the task invites to construct three distinct points A1, A2 and A3 on (d) (instruction 4 in Fig. 4 below), construct four triangles ABC, A1BC, A2BC, A3BC (instruction 5), display their areas (instruction 6) and observe the property (instruction 7).

Fig. 4
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Task aiming at discovering a property of the area of a triangle

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In this task, the drag mode is not exploited at all. The only contribution of dynamic geometry is the accuracy of measures of lengths of the segments and of the areas of the triangles.

The same goal, namely observing the above-mentioned property of the area of a triangle, can be achieved with a simpler construction: triangles ABC and ABD with D on a line parallel to [BC]. One can either display areas of the two triangles or ask the software for checking the relation between the two triangles (Fig. 5). In both cases, dragging D along the parallel line allows verifying that the observed fact (i.e., equality of the areas) is not a coincidence.

Fig. 5
figure 5

Checking the relation between t1 and t2 (areas of triangles ABC and ABD) with the Relation command

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In this case, DG offers a functional improvement; it is therefore used at the augmentation level, discussed in what follows.

3.2 Augmentation Level Tasks

Tasks analyzed in this section fall under the robust construction paradigm. Laborde (2005) characterizes robust constructions as those “for which the drag mode preserves their properties”. The author provides the example of an angle AMB inscribed in a circle (Fig. 6). When the point M is dragged along the circle, one can easily observe that the angle AMB remains right. This robust construction shows that “for any point of the circle (except A and B) angle AMB is a right angle” (Laborde, 2005).

Fig. 6
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Angle AMB inscribed in a semi-circle

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Typical tasks within this paradigm consist in exploring robust constructions. These may be either constructed by a teacher or by students who follow detailed instructions. The students are then invited to vary elements of the figure (point M in the example presented in Fig. 6) in order to recognize or to discover a geometric property based on the observation of the figure: the property at stake remains invariant (the measure of the angle AMB). Referring to the Marton et al.’s (2004) framework of variation, described in (Leung, 2017), the epistemic function of variation enabled by dragging in a robust construction is to allow separation of aspects of a figure that vary from other aspects that remain invariant. In the above-mentioned example, dragging points A and B allows observing for example that the segment [AB] is always a diameter of the circle, but its horizontal direction is not a necessary condition for the angle AMB to be right. Laborde (2005) sums up the contribution of this robust construction to the learning of the associated geometric theorem as follows:

The robust construction contributes to a better identification in action of the elements of [the theorem] for several reasons:

  • The construction requires to take into account two conditions to get a right angle: AB must be a diameter and M a point on a circle […].

  • It allows contrasting the invariance of the angle and the varying nature of point M.

  • It exteriorizes the variable nature of point M and the set in which it varies (Laborde, 2005).

Dragging elements of a robust construction allows producing quickly a number of different drawings sharing the same geometric property, which helps students “extend their visual images of a property […] and reject some spatio-graphical properties” that they can attach to the figure. Thus, “the drag mode is used as tool for distinguishing between contingence and necessity” (Laborde, 2005), which constitutes a clear functional improvement comparing to traditional tools. From the instrumental perspective, Restrepo (2008) ranges this modality of dragging among exploratory dragging instruments: its purpose is to look for invariants in a given figure, which facilitates identification of its geometric properties.

3.3 Modification Level Tasks

Robust construction tasks are another kind of tasks falling under robust construction paradigm. Students are asked to construct geometric figures that satisfy given conditions even when their elements are varied by dragging, for example, construct a square given its side or given its diagonal. As Laborde (2005) specifies,

Eye ball constructions are invalidated by the drag mode since it becomes visible that some of the conditions are not satisfied. The drag mode is a critical factor in robust construction tasks that makes the difference with a paper and pencil environment. In such construction tasks in dynamic geometry, the drag mode provides a visual feedback from the fact that the construction does not meet all the required conditions. The strength of DGE lies in this possibility of showing at the spatio-graphical level the theoretical weakness of the construction.

The necessity to resort to geometric properties when constructing a figure modifies deeply the construction task in comparison to the same task realized in paper and pencil environment, where the students “very often stay at a graphical level and try only to satisfy the visual constraints” (Laborde, 2005). The drag mode provides students with a visual feedback about the correctness of their construction; it is therefore used as an instrument for validating constructions (also called dragging test, e.g. by Arzarello et al., 2002) and helps students gaining awareness of the distinction between a drawing (material diagram representing a geometric object) and a figure (theoretical object defined with its properties) (Laborde & Capponi, 1994).

Less common tasks at the modification level are those in which students are asked to look for conditions under which certain configurations are obtained. In the example taken from Laborde (2005), a circle with segment [AB] as a diameter and a point M not belonging to the circle are given. Students are asked to find a position of M outside the disk such that the angle AMB is obtuse (Fig. 7). The purpose of this task is to let students explore the situation and notice that the angle AMB is acute when M is outside the disc and obtuse when it is inside and eventually discover the relationship between the measure of AMB and the position of the point M.

Fig. 7
figure 7

Searching for particular position of the point M

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This construction is coined soft because, as the point M is not constructed as a point on the circle, the targeted geometric property, namely the fact that the angle AMB is right when M belongs to the circle, is not directly visible, as it is the case in the robust construction (see Fig. 5). Rather, this property is inferred from observing that “the circle is the border between two regions, one in which angle AMB is obtuse and one in which angle AMB is acute” (Laborde, 2005), hence AMB must be right when M in on the circle.

Soft constructions present several features that offer interesting learning opportunities. First, tasks that exploit soft constructions are more engaging than their robust versions. Indeed, they offer genuine problems to be solved and dynamic geometry is a support for exploring given situations. Pea (1985) evokes

dynamic what-if capacities of such systems [that] make it possible to display immediately the consequences of different approaches to a problem (p. 171).

We claim that soft construction dynamic geometry task is significantly modified as it offers support for generating and testing various conjectures given different hypothetical conditions. Dragging plays a crucial role in this exploration. Moreover, students’ exploration of a soft construction leads to putting more emphasis on the link between the condition (in our case, M is on the circle) and the consequence (the angle AMB is right), which facilitates grasping the meaning of a geometric property as an implication, which makes it particularly relevant in proof oriented tasks.

3.4 Redefinition Level Tasks

Among the tasks that cannot exist but within a dynamic geometry are the so called “black box” tasks. Clerc (2006) described a black box in dynamic geometry as a geometric figure made up of initial objects and final objects the construction and displacement of which are linked to the initial objects. The construction of these final objects is hidden. A mathematical task that can be set up with a black box consists in asking the students to solve them, that is to say to find out how to construct the final objects from the initial ones, the construction must of course resist when the free objects in the figure are dragged. Figure 8 shows a black box where the initial objects are three distinct points A, B, and C (or a triangle ABC), and the final objects are points X and Y.

Fig. 8
figure 8

A black box task

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The student is expected to explore the figure, make conjectures, verify them experimentally and eventually reconstruct the points X and Y. While dragging, the student can observe for example that the point X remains inside the triangle whereas the point Y can get outside (Fig. 8, center). She can draw lines, circles, midpoints… to enrich the figure; she can measure distances or angles (Fig. 8, right). Dragging clearly plays a critical role in searching for the hidden relationships. It is used both for exploring the figure and verifying conjectured geometric properties by highlighting their invariance.

4 Concluding Remarks

The purpose of this chapter was to show that a given digital educational technology can be used in many different ways, ranging from a mere substitute of traditional tools to offering unique learning opportunities in novel tasks. Dynamic geometry has been taken as an emblematic example and the tasks analyzed have been taken from past research or from available curricular resources.

The various uses of digital technology illustrated on the example of dynamic geometry can apply to other tools. Let us consider spreadsheets (Lagrange & Erdogan, 2009). At the substitution level, spreadsheet can be used as a traditional double entry table, to organize data. The use of formulas and their dragging adds a functional improvement to performing calculations (augmentation level). Tasks mobilizing spreadsheet functions are deeply modified compared to traditional approaches as they require modeling and generalization. Finally, tasks mobilizing advanced functionalities, such as conditional formatting, charts or programming macros fall under redefinition level.

Our analyses highlight that tasks at the transformation levels (modification and redefinition) show a greater potential for a student-centered approach, engaging students in inquiry-based problem-solving activity, compared to the tasks at the enhancement levels (substitution and augmentation), which are rather teacher-centered, requiring less important cognitive activity from students.

Therefore, digital technology itself is not transformative; it is the way how it is used that can be transformative. As the teachers’ role in technology-based education is crucial, it is urgent to provide them with support aimed at raising their awareness of the manifold uses of technology and to help them develop practices at the transformation levels.