Working with accurate representations: the case of preconstructed dynamic geometry sketches.
The tendency of some study students to miss or disregard measurements and visual evidence of relationships is of concern because exploring invariant properties with dynamic software requires focused attention on all aspects of an object as it updates under dragging. The study results suggest that, for students to reap the benefits of working with an accurate image, they must learn to notice details, use dragging in a systematic way, and communicate visual ideas. But first, they must set aside, during the exploration phase, the logic-based bias towards diagrams that they may bring from traditional geometry.
This article is based on the results of a case study undertaken to evaluate the benefits and limitations of the use of preconstructed, web-based, dynamic, geometry sketches in activities related to the teaching of deductive geometry at the secondary school level. The data showed that the students responded enthusiastically to the use of the sketches, and that the environment encouraged communication and collaboration between partners. In addition to these positive findings, results about students' responses to sketch affordances, mathematical knowledge and skills, and ability to reason visually, led to a deeper understanding of: (a) the relationship between web-based activities and the development of geometric thinking skills, and (b) the relationship between the design of preconstructed sketches and the exploration process (Sinclair, 2003). Results indicate that preconstructed, web-based, dynamic, geometry sketches can illuminate the nature of geometric objects for a diverse group of students; however, they also suggest that students require explicit teaching of exploratory and visual strategies to interpret a dynamic mathematical image they did not create.
The data analysis also uncovered an important sub-theme--how students respond to a visually accurate image. The ability to display an accurate image is commonly assumed to be a benefit of dynamic geometry software--it appears reasonable to conclude that the task of noticing and interpreting relationships between objects is easier if figures are drawn to scale. However, the results showed that many students either do not realize or ignore the fact that the onscreen image is accurate, and that others, who recognize and might want to use visual evidence, lack the tools to do so. The tendency of study students to miss or disregard measurements and visual evidence of relationships is of concern because exploring invariant properties with dynamic software requires focused attention to details that update under dragging. The researcher contends that this tendency is, in part, caused by traditional but limiting attitudes towards diagrams that students bring to the dynamic environment.
Recent studies have given us a deeper understanding of how students interact with dynamic software programs. For example, we know that a geometry problem cannot be solved simply by perceiving the images on the Cabri screen, even if these are animated. The student must bring some explicit mathematical knowledge to the process (Arzarello et al., 1998). That is, an intuition about a generalization involves more than observed evidence (Fischbein, 1987). With respect to Geometry Inventor (1994) we know that contradiction and uncertainty spur the student's need to explain "why" something is true (Hadas, Hershkowitz, & Schwarz, 2002). However, these studies have focused on activities in which students construct, or work with objects they have constructed. The case study that informed this article analysed student use of preconstructed sketches.
Preconstructed sketches created with Cabri Geometre (Baulac, Bellemain, & Laborde, 1992), or The Geometer's Sketchpad (Jackiw, 1991) as well as preconstructed, web-based sketches created with Java Sketchpad (Jackiw, 1998), or Cinderella (Richter-Gebert & Kortenkamp, 1999) can be used as an alternative to having students construct their own dynamic diagrams. In preconstructed dynamic sketches (whether web-based or not), points can be dragged; pre-set relationships, such as measurements and ratios, update as a consequence of dragging; and action buttons to hide or show details, to move and to animate objects can be included. Angles and lengths are represented accurately to within a small error. Preconstructed web-based sketches created with JavaSketchpad do not permit the user to construct or delete objects; however, those designed with Cinderella can include options for constructing a limited number of objects.
Based on his analysis of the role of pictures in proof, mathematical philosopher Brown (1999) postulated that some diagrams are "instruments ... which help the unaided mind's eye." Barwise and Etchemendy (1998) went even further, to conclude that diagrams play an integral role in reasoning. But what about preconstructed dynamic diagrams?
A preconstructed dynamic sketch, like a textbook diagram, presents a geometric situation to the student in visual format. Interpreting such a sketch is similar to interpreting a picture that someone else has drawn--students must use the available affordances to investigate and draw conclusions about how items are connected. This can be a challenging task because mathematical pictures and diagrams contain a great deal of information represented in a concise but "nonsequential" format (Goldenberg, Cuoco, & Mark, 1998). In other words, we must first organize the visual information that we receive. In addition, the meaning we take from an image depends in part on what we already know about what we are looking at (Roth & Bowen, 2000; Wheatley, 1998; Whiteley, in press). If we are the ones who created a mathematical diagram, we are aware of the underlying relationships it attempts to present. If, on the other hand, we are interpreting someone else's diagram, we have no prior knowledge and must rely on the diagram, the context, and our visual skills.
Motion adds another layer of complexity to the dynamic geometry environment. Dynamic sketches are put in motion by animation controls or by dragging, a provision that allows students to explore objects by moving them at a controlled speed. In 1998, Arzarello et al. classified modalities of dragging as: "dragging test," "wandering dragging," and "lieu muet" (dummy locus). They found that students who produced good conjectures made use of "lieu muet" dragging, a purposeful mode, which "can be seen as a wandering dragging which has found its path" (Arzarello et al., 1998, p. 37).
Motion has several implications for the use and design of preconstructed sketches. Recent research into how we take meaning from an image that is moving, has shown that we are only able to grab four to six visual objects at once and that focused attention is needed to notice change (Rensink, 2000). This implies that any visual detail in a sketch must catch and hold students' attention to be useful. Another, subtler issue, is that a dynamic software image can be deformed through motion into a special case. Geometer's use this capability deliberately to examine extremes, but unsuspecting students can use it accidentally to create a special case, which then serves as their model. Such a situation cannot occur with textbook or teacher prepared diagrams that display a general case.
Geometry diagrams fall into three categories: (a) special case, (b) general case, and (c) inaccurate. One might think that the last category is unnecessary, however, most textbooks contain many diagrams that include measurements but are not drawn to scale. Moreover, most mathematicians have collections of "bogus" diagrams deliberately designed to trap unsuspecting students. They provide an opportunity to surprise--and thereby to encourage mathematical thinking skills. For example, according to the diagram in Figure 1, we can "prove" that all triangles are isosceles!
[FIGURE 1 OMITTED]
In contrast, dynamic geometry sketches cannot be inaccurate unless deliberately edited. The situation illustrated in Figure 1, when constructed (and dragged) in Sketchpad (Figure 2) demonstrates that the bisector of angle A and the perpendicular bisector of BC can never intersect in a single interior point. When triangle ABC is scalene, intersection point P is an exterior point, and when ABC is isosceles or equilateral, the angle bisector, AD, and the perpendicular bisector of BC are concurrent.
[FIGURE 2 OMITTED]
Despite the ubiquitous use of dynamic sketches in elementary and secondary mathematics, it is the researcher's contention that deductive geometry teaching has not moved to address the benefits and problems of accurate models.
Many researchers have recommended an increased emphasis on the use of visual reasoning in mathematics (cf., Dreyfus, 1991; Duval, 1998; Goldenberg et al., 1998; Presmeg, 1986, 1999). With respect to computer images, Sutherland and Balacheff (1999) reported that visual images displayed on computer screens allow students to gain access to mathematical knowledge by "rendering more visible the nature of the objects with which a student is engaging" (p. 2), and in a 1986 study of visualisation in high school students, Presmeg found that dynamic imagery--although used by only a few "visualisers"--was effective in helping students generalise (Presmeg, 1986). The implication is that visually accurate images are beneficial in helping students understand geometric ideas.
This notion, however, may be inconsistent with the way geometry is traditionally taught. In the typical classroom students studying geometry are instructed: (a) to ignore what they "see" in a diagram and focus only on given and deducible information, and (b) to avoid drawing a special case when they create their own diagrams.
The results of this study suggest that the practice of teaching students to ignore "reality" and consider diagrams as mere placeholders for logical "givens," coupled with a lack of attention to visual interpretation skills impairs students' ability to interpret preconstructed, "accurate" (i.e., to within a measurable error), dynamic sketches.
Since a particular paper and pencil sketch will usually display a general case, it is difficult for students to appreciate the significance of an admonition about special cases. However, students using dynamic software are very likely--in fact, almost certain--to drag a figure past a special case, and thus much more likely to stop on a special case and be faced with the consequences.
The research used a case study approach and multiple sources of information--observation field notes, videotape, audiotape, a student questionnaire, and interviews with teachers. Collected data was transcribed, then analysed by coding, developing categories, describing relationships, and applying simple statistical tests where appropriate.
Three mathematics classes from two different secondary schools participated in this study. The 69 students were enrolled in the Ontario grade twelve advanced mathematics program (replaced in 2002), which covered topics in algebra, geometry, analytic geometry, and trigonometry (Curriculum Guideline: Mathematics Intermediate and Senior Divisions, 1985). The study focused on congruence and parallelism, the first section in the geometry unit. Although the students had done introductory work on deductive geometry related to congruence and parallelism in grade 10 and on similarity in grade 11, none had worked with dynamic geometry software.
Three 75 minutes sessions or four 45 minutes sessions were held with each class. During this time, students worked in pairs. In each class, several pairs were studied in more depth by audiotaping or videotaping their activities. Teachers rated student achievement as weak, average, good, very good, or excellent. Students in any particular pair were not necessarily at the same level. In the article, where appropriate, student achievement level is noted. Students quoted in the article are identified by pseudonyms.
JavaSketchpad, was used to prepare four web-based, dynamic geometry sketches for student pairs to explore during the sessions, two extra sketches for those who finished early, and one sketch for a group discussion. The labsheet that accompanied each sketch provided directions for opening and manipulating the sketch, a statement of the problem, and questions related to the task.
Problems chosen as the basis for the web-based sketches were similar in difficulty to those in the student text, Mathematics: Principles and Process, Book 2 (Ebos, Tuck, & Schofield, 1986) and related to triangles and quadrilaterals. Each of the sketches supported the possibility of arriving at a solution from a transformation perspective as well as from a straightforward application of congruency theorems. The intention was to allow students to use symmetry considerations, (a) to visually confirm or negate conjectures, and (b) to develop a new perspective on geometric relationships.
In the prestudy interview the three study teachers identified difficulties that their students experience in the geometry strand. Teachers mentioned that students constructing congruency proofs frequently select sides or angles that do not correspond to one another, or, in fact, do not even belong to the subject triangles. They noted that this problem usually occurs when figures overlap or are presented in rotated, reflected, or translated form.
To the researcher, these student difficulties revealed an inability to "see" each overlapping figure separately or to mentally transform a figure to a new orientation to compare it with another. To address these difficulties, sketches were designed to include action buttons or provisions to highlight particular figures, to toggle details on and off, and to rotate or reflect shapes so that they could be superimposed, or viewed from the same orientation.
An overview of two tasks is included here to help the reader understand the context for the discussion.
Task 1. This task was developed to introduce students to JavaSketchpad, and specifically, to address student difficulties with overlapping figures and selection of triangles. It provided an opportunity to view a problem as an application of a reflection.
In the sketch (Figure 3) [DELTA] ABC and [DELTA] FCB are reflections of one another in the perpendicular bisector of BC, which can be toggled on or off using the "Show perpendicular through H" and "Hide" buttons. When the sketch is first opened, point A is red, indicating that it can be manipulated. As point A is dragged, its reflection, point F undergoes opposite motion.
The two triangles ABC and FCB can be separated using an action button with label, "Separate ABC and FCB," so students can examine the triangles apart. The button, "Show reflection and mirror," reflects triangle ABC in a red mirror line. The button, "Match FCB and A'B'C'" causes triangle FCB to move on top of the reflection of triangle ABC, demonstrating congruency. A reset button is provided to move triangle FCB to its original position.
In addition, the "Show Given Information" button controls the display of the markings that indicate the equality of AB and FC, and [angle]ABC and [angle]FCB, as well as the measures of these lengths and angles. As the figure is dragged, the measurements update.
[FIGURE 3 OMITTED]
The labsheet presented the problem: "Prove triangle ABC congruent to triangle FCB." As a method of proof, students could use a straightforward application of SAS (side, angle, side) congruency. Alternatively, [DELTA]ABC is congruent to [DELTA]FCB because when the triangle is flipped over FC will lie on AB, CB will lie on BC and [angle]FCB will lie on [angle]ABC. Thus, students could use the idea of reflection to explain the congruency.
The students were also asked: "How can the information provided by these images be used to explain why [DELTA]ABC is congruent to [DELTA]FCB?"; "What additional information can you deduce about point H from the diagram?" and "Find another pair of congruent triangles in the figure."
Task 2. This task was the second activity. Students were asked to prove that BA = BC using various pairs of triangles. In all cases students needed to deduce at least one piece of information from the given information before developing the proof, (i.e., all options involved at least two steps--a problem identified by the teachers).
[FIGURE 4 OMITTED]
For example, in the case shown in Figure 4, students needed to deduce that [angle]DEF = [angle]EDF by the isosceles triangle theorem, and that [angle]BED + [angle]DEF = [angle]BDE + [angle]EDF before employing ASA to prove that [DELTA]ABE is congruent to [DELTA]CBD.
To provide the basis for a transformation proof, all chosen pairs of triangles in the sketch were reflections, and the relationship between members of each pair could be established or not established by considering what would happen if one member of the pair was flipped over. For example, with [DELTA]BDF and [DELTA]BEF, BE would lie on BD (equal sides of an isosceles triangle), EF would lie on DF (given equal) and BF would lie on BF (common sides). However, for pair #2, [DELTA]BFA and [DELTA]BFC, only the behaviour of BF could be predicted.
The sketch for this task was designed to address student difficulties with overlapping triangles, and selection of triangles. To help students notice details: the four chosen pairs of congruent triangles were shaded in four different colours; given equal angles were shaded red; information could be toggled off and on to allow details to stand out; triangle pairs could be separated; measurements for the given angles and lengths were displayed; and measurements updated as the sketch was dragged. To help students pick out a shape within the larger diagram: overlapping figures could be separated; colour was added to emphasise the shapes; and colour was used to overlay angles and sides within the shape.
The sketch also included a pair of triangles that could not be proven congruent by deducing directly from given information. In this case, students needed to prove one pair congruent then apply the results.
In addition, the labsheet included instructions/questions such as: Drag each red point and observe the measurements; Write two additional facts that you know and explain why they are true; If you proved the pair congruent, how would this help you prove BA = BC? What is an alternative explanation for the congruency of triangle ABC and triangle FCB?
OBSERVATIONS AND DISCUSSION
It was assumed that measurement detail and the visual evidence of relationships (e.g., two segments appearing to be the same length) would be used by students in addition to information gleaned through traditional geometric markings, to explore the sketches, and to formulate and check conjectures; however, there were a variety of responses, which offer interesting and sometimes conflicting perspectives on the use of "reality information" in geometry.
Only two student pairs mentioned the onscreen measurement data explicitly, although (unconnected) comments such as the following could have been based on measurement data.
Pat: Oh the sides are equal [gazing] Barb: Well this equals--yeah, they're pretty close ...
In general, however, observations showed that students gave only passing attention to the measurements. It is not clear whether study students had difficulty keeping track of measurements while watching objects move, as Rensink (2000) might have suggested (in Java Sketchpad lengths and angles are in a list and not attached to the object), or whether they were simply unaccustomed to using measurement data in deductive geometry. The lack of attention to measurement information was in stark contrast to frequent mention of colour and geometric markings. It is of concern because the ability to explore how a figure has changed requires focused attention to details that update under the operation of dragging.
While ignoring (or not noticing) measurements, students frequently used the appearance of the onscreen image to make conclusions. There were over 30 annotated examples of students relying on "looks like" evidence. Here are three (unconnected) examples in which students explicitly gave this as a reason:
Doug: Angle E is 90[degrees]. Well, I'm thinking this is--cause it looks like it, right? Barb: I wrote that opposite angles will be equal and so are the triangles because they look the same. Sue: Well, this has got to be--[pointing] this looks equal.
Although these conclusions were not based on deductive logic, the students quoted were taking the visual evidence into consideration. In contrast, the researcher was surprised to observe that students sometimes offered conclusions that were in direct conflict with the visual evidence. For example, Barb and Sue--two of the same students quoted above, made the following (unconnected) comments:
Barb: Maybe cause it's slanted you can't tell it's a square. [The shape was a parallelogram--which could be dragged to a square, but had not been.] Sue: If this is equilateral these sides would have to be equal. [In this instance, the triangle was clearly not equilateral.]
These two above-average students appeared to treat the onscreen image as if it were a pencil sketch--as if the diagram represented objects and their relationships, but was not drawn to scale. They used their knowledge of deductive theorems to correctly solve the problem, but gained nothing from being able to use an accurate model.
The researcher hypothesised that such responses might stem from prior use of textbook diagrams. Geometry teachers frequently warn their students not to make conclusions based on the appearance of diagrams because they are not necessarily accurate. The results reported here suggest that the "diagram bias" thus created may act as a roadblock when students examine a preconstructed, and "accurate" (i.e., to within a measurable error), dynamic sketch.
On the other hand, Doug, an average student quoted previously, was attentive to visual detail. Since he and partner Sal, also an average student, had difficulties with deductive geometry, their first inclination was to examine the sketch. Perhaps because of this, excerpts of their discussions, such as the following (Figure 4), focus on what they see.
Sal: So we have--what? Doug: We have FD. Sal: No, no that's one big triangle. Sal: This splits the triangle in half, right? So, FD is a common? A common side. They have a common side here, right? Sal: But F's not--Oh. All right.
Another pair's poor grasp of geometric concepts led to slow progress through the task questions, but their conversation, as shown in this excerpt (related to a task not shown), also focuses on what they see.
Pat: No, it says rotate ... Dave: How do you rotate it? You can't unless it's round. You can only rotate it-- Pat: Oh, it rotates on one point. Dave: Yeah so--so it stays in one point. Pat: It goes in a circle. It goes around the midpoint. Dave: Yeah, it goes in a circle ... Pat: It goes around the midpoint and doesn't stretch. Dave: How do you know it's a midpoint? How do you know it's a midpoint? Huh? And why does this keep on going? It goes around on one point, right? Pat: Yeah. [They gaze intently at the screen]
Thus Barb, and Sue, whose mathematics achievement was rated very good and excellent, respectively, by their teachers, allowed traditional deductive geometry teaching (e.g., ignore the way a diagram looks) to interfere with their analysis of the image before them. In contrast, Pat, Dave, Doug, and Sal, (achievement levels: average, weak, average, average, respectively), did not draw conclusions that contradicted the evidence they saw.
The Special Case
Despite these comments, the use of "looks like" evidence in geometry can lead to error, even if the student is afforded a measurement tool. The problem is related to the possible existence of a special case. Teachers and textbook writers avoid the difficulty by ensuring (unless they want to deliberately confront students) that diagrams display a general case. Students, however, when using dynamic geometry software, can unwittingly create a special case and proceed to use this as a model for analysis.
Since points in preconstructed sketches can be dragged, one might assume that students would take advantage of this affordance throughout an investigation; however, study students, initially intrigued by the ability to drag points, usually stopped dragging after a short time and concentrated on interpreting a static figure. If this static figure was a general case, the situation was similar to that faced by a student using a textbook diagram; if the static figure was a special case, the situation was problematic.
In the following example, the abandonment of dragging led to an erroneous conclusion. Doug and Sal were looking at a triangle in task 2 (see Figure 4). Angle BEA may have been very close to a right angle on their diagram, but if they had dragged the sketch they would have seen it change.
Doug: BEA--angle E is 90[degrees] ... Sal: There's no thing [referring to the symbol for a 90 degree angle]. Doug: Well, you can't see it. Sal: That's right Doug: Well, I'm thinking this is a n[sic]--cause it looks like it, right?
In this case, Doug and Sal treated the sketch as accurate--they drew a conclusion based on what the angle looked like. Certainly many informal conjectures suggest themselves to mathematicians because they "look like" they are true. Some withstand further investigation; others prove false. The students' problem was not in attempting to interpret the visual evidence before them but in failing to realise that they were observing a specific case.
Paul and Sue (both excellent students) provide an example of a pair that made a conjecture based on a special case, and then discovered their error. In a task on the second day (not shown), they were asked to investigate the question: "When do the diagonals of a parallelogram right bisect one another?" The sketch included a parallelogram ABCD, with diagonals AC and BD. Opposite sides were marked as parallel lines. Measurements of the sides, diagonals, and semi-diagonals were provided. The pair had decided the following:
Sue: Ok, if DB right bisects AC then the parallelogram will become a square. Paul: Two diagonals bisect each other at right angles then the parallelogram becomes a square.
Sometime later, after slowly dragging the sketch to examine several instances of the figure, they were surprised to find that their conclusion was not true:
Sue: Obviously it's not a square now ... it's a parallelogram Paul: Still a parallelogram--so we were wrong
Usually, when study students realized that dragging was not furthering their understanding, they stopped and looked for details in a static image, that is, they directed their attention to relationships within one instance of a figure. As the example with Paul and Sue demonstrates, to avoid basing conclusions on a special case students must compare relationships across a sequence of images by dragging in a systematic way.
After students have noticed and conjectured, the next step is to develop a proof--to explain why. But if study students had wanted to construct a final proof without abandoning the visual image how would they have done so?
In the following example from task 1 (Figure 3), Katy and Bea (both average students) recognised that a reflection could explain the congruency relationship but they were unable to construct a formal transformation-based proof so they reverted to developing a traditional congruency proof on their labsheet.
Katy: Can't I just put BG equals CG? Bea: What? Katy: Because, when you flip this over--right? Bea: I know. Katy: BG's going to equal CG. Bea: Right!
In another example, a sketch (not shown) allowed students to use rotation to explain why two segments were equal. The question was: "How can the information provided by these images be used to explain why DM equals BN?" A student named Clara (achievement level, very good) intuitively realized that when the triangle was rotated, DM would fall on BN but she could only remark lamely, "Because the triangle fits--the triangle fits both."
Clara did not follow up with a step-by-step analysis using visual references. She reverted to interpreting the geometric markings and used a traditional congruency proof. This was not surprising. Although visual evidence of symmetry relationships could have provided the basis for a proof, transformation geometry, and deductive proof are usually taught separately. Study students recognized relationships between reflected and rotated objects but did not have the tools to construct a proof based on these ideas.
The foregoing examples also highlight students' unfamiliarity with describing visual information in precise terms. We cannot intuit if we do not perceive and the researcher contends that students are not taught to perceive visual details related to measurement concepts--they are taught to select given information from diagrams--even if this information is false to the eye.
Accuracy is an attribute of dynamic software images. This article has presented evidence to show that students who do not take this into consideration may use ineffective strategies to investigate relationships and to develop proofs.
The study students who treated dynamic sketches like textbook models ignored visual evidence such as appearance clues and measurements that might have provided support for a deeper understanding of geometric relationships. Students, who had always been provided with diagrams that carefully displayed a general case, did not recognize the pitfalls of creating and analyzing a special case. Their past experiences, rooted in the study of a single image, prevented them from recognizing the powerful information to be gained by dragging to examine across cases.
Students who focused on visual information and drew conclusions based on the appearance of the onscreen image did not have the tools to develop and communicate a proof based on visual concepts. Some eventually eschewed the dynamic capabilities of the software and focused attention on constructing traditional congruency proofs. Those with poor deductive geometry skills were unable to complete the tasks successfully.
It is hoped that the findings of this study will raise awareness of the impact of accuracy in relation to dynamic geometry activities, and that it will lead to discussions about how to nurture students' ability to notice and describe what they see, to broaden geometry practices to deal specifically with methods of exploring accurate images, and to develop a proof structure that embraces visual concepts.
Arzarello, F., Micheletti, C., Olivero, F., Robutti, O., Paola, D., & Gallino, G. (1998). Dragging in Cabri and modalities of transition from conjectures to proofs in geometry. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 32-39). Bellville, South Africa: Kwik Kopy Printing.
Barwise, J., & Etchemendy, J. (1998). Computers, visualization, and the nature of reasoning. In T. W. Bynum & H. M. James (Eds.), The digital phoenix: How computers are changing philosophy (pp. 93-116). London: Blackwell.
Baulac, I., Bellemain, F., & Laborde, J. (1992). Cabri Geometre. Pacific Grove, CA: Brooks-Cole Publishing.
Brown, J. R. (1999). Philosophy of mathematics: An introduction to the world of proofs and pictures. London: Routledge.
Curriculum Guideline: Mathematics Intermediate and Senior Divisions. (1985). Toronto, Ontario: Ministry of Education.
Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Conference of the International Group for the Psychology of Mathematics Education, Assisi, Italy (Vol. 1, pp. 32-48).
Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century: An ICMI study (pp. 37-51). Dordrecht, The Netherlands: Kluwer Academic.
Ebos, F., Tuck, R., & Schofield, W. (1986). Mathematics: Principles & process, book 2. Scarborough, Ontario, Canada: Nelson Canada.
Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, The Netherlands: Kluwer Academic.
Geometry Inventor [Computer software]. (1994). Cambridge, MA: Logal Educational Software Ltd.
Goldenberg, E. P., Cuoco, A. A., & Mark, J. (1998). A role for geometry in general education. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 3-44). Mahwah, NJ: Lawrence Erlbaum.
Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2002). Analyses of activity design in geometry in the light of student actions. Canadian Journal of Science, Mathematics and Technology Education, 2(4), 529-552.
Jackiw, N. (1991). The Geometer's Sketchpad (Version 3) [Computer software]. Berkeley, CA: Key Curriculum Press.
Jackiw, N. (1998). Java Sketchpad [Computer software]. Berkeley, CA: Key Curriculum Press.
Presmeg, N. C. (1986). Visualisation in high school. For the Learning of Mathematics, 6(3), 42-46.
Presmeg, N. C. (1999). On visualization and generalization in mathematics. In F. Hitt & M. Santos (Eds.), Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (21 st, Cuernavaca, Morelos, Mexico, October 23-26, 1999 (Vol. 1, pp. 23-27).
Rensink, R. A. (2000). The dynamic representation of scenes. Visual Cognition, 7, 17-42.
Richter-Gebert, J., & Kortenkamp, U. H. (1999). Cinderella [Computer software]. Berlin, Hiedelberg: Springer-Verlag.
Roth, W.M., & Bowen, G. M. (2000). Learning difficulties related to graphing: A hermeneutic phenomenological perspective. Research in Science Education, 30, 123-139.
Sinclair, M.P. (2003). Some implications of the results of a case study for the design of preconstructed, dynamic geometry sketches and accompanying materials. Educational Studies in Mathematics, 52(3), 289-317.
Sutherland, R., & Balacheff, N. (1999). Didactical complexity of computational environments. For the Learning of Mathematics, 4, 1-26.
Wheatley, G. H. (1998). Imagery and mathematics learning. Focus on learning problems in mathematics, 20(2 & 3), 65-77.
Whiteley, W. (in press). Teaching to see like a mathematician. In G. Malcolm (Ed.), Proceedings of Visual Representation and Interpretation. Amsterdam: Elsevier.
York University Canada email@example.com
|Printer friendly Cite/link Email Feedback|
|Publication:||Journal of Computers in Mathematics and Science Teaching|
|Date:||Jun 22, 2004|
|Previous Article:||Complex quantification in Structured Query Language (SQL): a tutorial using relational calculus.|
|Next Article:||Combing online and paper assessment in a web-based course in undergraduate mathematics.|