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Nevertheless, it has often been argued that using the drag mode to test myriad cases may reduce or even eliminate any need for deductive proof (Allen, 1996; Chazan, 1993a; Hadas et al., 2000; Hoyles and Jones, 1998; Laborde, 2000). When students can generate their own empirical evidence Allen (1996) is one of the strongest critics of DGS. For him DGS tends to blur the distinction between illustration and proof. A dissection proof of the Pythagorean Theorem exemplifies this. In this DGS-based construction, students can cover the largest square by using the parts of the smaller squares, which are partitioned (Figures 4, 5 and 6). Allen argues that such an illustration cannot be considered as proof, since no previous theorems are invoked. Figure 4 The dissected Pythagoras illustration first screen

Figure 4 Nevertheless, it has often been argued that using the drag mode to test myriad cases may reduce or even eliminate any need for deductive proof (Allen, 1996; Chazan, 1993a; Hadas et al., 2000; Hoyles and Jones, 1998; Laborde, 2000). When students can generate their own empirical evidence Allen (1996) is one of the strongest critics of DGS. For him DGS tends to blur the distinction between illustration and proof. A dissection proof of the Pythagorean Theorem exemplifies this. In this DGS-based construction, students can cover the largest square by using the parts of the smaller squares, which are partitioned (Figures 4, 5 and 6). Allen argues that such an illustration cannot be considered as proof, since no previous theorems are invoked. Figure 4 The dissected Pythagoras illustration first screen

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This view of learning has been enabled by the immediate availability of a very powerful set of computer tools, namely DGS, in classrooms. DGS realises the vision described above by enabling heuristic explorations through its most defining ‘dragging’ feature (Goldenberg and Cuoco, 1998). Due to this feature, the construction dynamically responds to the changes made by maintaining the invariant. Kaput (1992) states that conjecturing and more inductive approaches to geometric knowledge are facilitated by this aspect, as students can recognise patterns and easily check the generality of their hypotheses. Figure | The Geometer’s Sketchpad construction of a right triangle ABC and three squares on its sides
This view of learning has been enabled by the immediate availability of a very powerful set of computer tools, namely DGS, in classrooms. DGS realises the vision described above by enabling heuristic explorations through its most defining ‘dragging’ feature (Goldenberg and Cuoco, 1998). Due to this feature, the construction dynamically responds to the changes made by maintaining the invariant. Kaput (1992) states that conjecturing and more inductive approaches to geometric knowledge are facilitated by this aspect, as students can recognise patterns and easily check the generality of their hypotheses. Figure | The Geometer’s Sketchpad construction of a right triangle ABC and three squares on its sides
One might think this is merely one occasion. However, what Sketchpad allows is that one can easily drag vertices 4, B or C to different places and still see that this relationship holds. Although it is not possible to represent the dynamic experience on paper, the reader can imagine that Figure 2 Same construction, vertex A is dragged
One might think this is merely one occasion. However, what Sketchpad allows is that one can easily drag vertices 4, B or C to different places and still see that this relationship holds. Although it is not possible to represent the dynamic experience on paper, the reader can imagine that Figure 2 Same construction, vertex A is dragged
Figure 3 Same construction, vertices C and B are dragged
Figure 3 Same construction, vertices C and B are dragged
Figure 5 The dissected Pythagoras illustration second screen
Figure 5 The dissected Pythagoras illustration second screen
Figure 6 The dissected Pythagoras illustration third screen
Figure 6 The dissected Pythagoras illustration third screen
The proof activities were sampled by first counting the number of lessons meant to be covered in one class period. On average, the textbooks in the study have about 102 lessons that cover geometry content. (The term
The proof activities were sampled by first counting the number of lessons meant to be covered in one class period. On average, the textbooks in the study have about 102 lessons that cover geometry content. (The term
Figure 7 Experimental/explorative activity percentages across the three types of curricula Below I examine the role of DGS using three research questions within the three types of curricula defined earlier:
Figure 7 Experimental/explorative activity percentages across the three types of curricula Below I examine the role of DGS using three research questions within the three types of curricula defined earlier:
paper and pencil-based explorations. It is even implied that paper-pencil is preferred to DGS-based explorations: “Using the Dissection of Squares worksheets or the Sketchpad demonstration will speed up the investigation, but the use of many different triangles drawn by the students [by hand] strengthens the inductive conclusion” (p. 462). Secondly, DGS use mostly appears in the teacher’s edition (in 15 of the activities, or 48%). DGS is included in the student text in only six activities, mostly as a link to an online Geometer’s Sketchpad sketch and its drawing. For instance, in one of these activities students investigate the theorem “the sum of the lengths of any two sides of a triangle is greater than the length of the third side” (p. 214). Using a compass and a straightedge, students are to construct triangles with given sets of segments. Prior to conjecturing, students are asked if they were able to construct these triangles and discuss their results. After students are told to state their conjecture, the text reads: “You can also think of it [the triangle inequality conjecture] in another way: The shortest path between two points is along the segment connecting them. In other words, the path from A to C to B can’t be shorter than the path from A to B (p. 214). This explanation is followed by four screen shots of a DGS construction showing a triangle ABC, the lengths of the sides, AB, AC and CB, and the sum AC + CB, as one of the vertices (point C) is being dragged to different places (see Figure 8 below representing one of these screens). By looking at these, students are expected to notice that the distance from A to C to B is always greater than or equal to the distance from A to B.
paper and pencil-based explorations. It is even implied that paper-pencil is preferred to DGS-based explorations: “Using the Dissection of Squares worksheets or the Sketchpad demonstration will speed up the investigation, but the use of many different triangles drawn by the students [by hand] strengthens the inductive conclusion” (p. 462). Secondly, DGS use mostly appears in the teacher’s edition (in 15 of the activities, or 48%). DGS is included in the student text in only six activities, mostly as a link to an online Geometer’s Sketchpad sketch and its drawing. For instance, in one of these activities students investigate the theorem “the sum of the lengths of any two sides of a triangle is greater than the length of the third side” (p. 214). Using a compass and a straightedge, students are to construct triangles with given sets of segments. Prior to conjecturing, students are asked if they were able to construct these triangles and discuss their results. After students are told to state their conjecture, the text reads: “You can also think of it [the triangle inequality conjecture] in another way: The shortest path between two points is along the segment connecting them. In other words, the path from A to C to B can’t be shorter than the path from A to B (p. 214). This explanation is followed by four screen shots of a DGS construction showing a triangle ABC, the lengths of the sides, AB, AC and CB, and the sum AC + CB, as one of the vertices (point C) is being dragged to different places (see Figure 8 below representing one of these screens). By looking at these, students are expected to notice that the distance from A to C to B is always greater than or equal to the distance from A to B.
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