RIDGE is a research project on mathematical reasoning in dynamic geometry environments. Such environments are becoming increasingly common in schools. There is some debate, however, concerning the effects of this software on students' mathematical reasoning. Some researchers have pointed out that the graphical nature of such software can undermine the need for some kinds of mathematical reasoning. Others see dynamic geometry as helpful for the development of students' reasoning. RIDGE seeks to increase our understanding of the ways students reason in this context, and the benefits and dangers of the use of dynamic geometry software in schools.
Three questions guide the research:
- What patterns of reasoning are revealed in the context of students' investigations in dynamic geometry?
- What do these patterns indicate about the underlying inclinations of the students to reason in particular ways?
- What does detailed analysis of patterns of students' reasoning in this context reveal about the nature of, and preference for, mathematical reasoning?
Given the increasing usage of dynamic geometry software in schools it is important to come to a better understanding of students' reasoning when using such software and how the software, the task given the students, and the teacher's interventions combine to influence reasoning. Such an understanding would provide a basis for the development of materials and principles to guide teachers in their use of the software. In a larger context, the relationship between the strongly visual nature of dynamic geometry software and traditionally verbal and symbolic ways of representing mathematical reasoning can be seen as analogous to the relationship between visual and verbal modes of communication in society as a whole. How people think when communicating through these media has become an important consideration as visual media have become more prevalent.
What is Dynamic Geometry?
Dynamic geometry is geometry in which the figures are not fixed, but can move, constrained only by the way in which they were created. For example, the figure to the right shows a rhombus, constructed from four equal segments. Because it is displayed in CabriJava, a dynamic geometry environment, its size and the sharpness of the angles joining the segments can change, but the equality of the segments forming it remains invariant.
You can experiment with the figure by "dragging" points. Click and drag point A. What changes? What remains the same?
Click and drag point C. What changes? What remains the same?
What do you notice about the measures of the angles or the intersection of the diagonals?
Dynamic geometry has recently increased in importance as computer software has been developed that makes it more accessible to students. Two important examples are Cabri Geometry and Geometer's Sketchpad. Use of such software is endorsed by recent policy documents (for example, the National Council or Teachers of Mathematics' Principles and Standards) and is becoming increasingly common in schools.
What effect does Dynamic Geometry have on Reasoning?
There is disagreement among researchers in mathematics education concerning the effect on students' reasoning of working with dynamic geometry software. Some research has suggested that use of this software might undermine students need for proof, as it provides strong visual evidence for the truth of conjectures. Hadas, Hershkowitz and Baruch (2000) note that:
The appearance of dynamic geometry environments raised a question concerning the role of proof in the curriculum, since conviction can be obtained quickly and relatively easily by dragging. Dragging a geometrical object enables students to check the invariance of a conjectured attribute on a whole class of objects. Such an operation leads students to be convinced of the truth of the conjectured attribute. Therefore DG environments may prevent students from understanding the need and function of proof. (p. 129)
Laborde (2000) echoes this concern: "it has often been claimed that the opportunity offered by such environments to ‘see’ mathematical properties so easily might reduce or even kill any need for proof and thus any learning of how to develop a proof" (p. 151).
On the other hand, a number of studies have suggested that dynamic geometry software can be useful in developing students' reasoning, if the motivation for reasoning is shifted from verification to explanation or discovery. DeVilliers (1997) incorporated 'what if' questions into students' explorations with dynamic geometry software, giving rise to new discoveries. In such an environment the challenge becomes one of explaining why what has been discovered is the case, rather than verifying that it is so. Similarly, Dreyfus and Hadas (1996) conducted a study in which dynamic geometry software was used to produce unexpected or surprising situations that motivated students to try to explain and verify conjectures deductively. Jones (2000) also describes a dynamic geometry environment in which students progressed thorough reasoning using the software to more mathematical explanations.
All of these researchers note the difficulty of studying students' reasoning within the complexity of the environment offered by the software, made even more complex when the students' activity is embedded in a classroom. Healy and Hoyles (2001) offer a striking example of this, a teaching sequence using dynamic geometry software which, for some students, supported a shift from everyday argumentation to deductive reasoning, while for others the software acted as a constraint on their thinking that left them less able to make use of what they already knew about geometry. The debate among researchers as to the effect of the use of dynamic geometry software on students' reasoning, and especially Healy and Hoyles study, suggests that a more fine grained examination of students' reasoning in such contexts is needed.
In the above discussion of existing research in dynamic geometry software and reasoning, it will have become clear that two aspects of reasoning are especially important in such research: the identification of kinds of reasoning used in mathematical activity, and investigation of the needs that motivate proving in mathematics. These two aspects of reasoning are central to the PRISM model for describing reasoning, which will be used in the RIDGE project.
What is the PRISM model?
The PRISM model was developed through a research project called the Psychology of Reasoning in School Mathematics (PRISM, funded by SSHRC). The model distinguishes reasoning across a number of dimensions, four of which are most relevant to the RIDGE: Need, Kind of Reasoning, Formulation, and Formality.
- Need refers to the personal or social need that the students feels and which motivates them to reason. Explaining, Exploring, and Verifying are the three needs described and subcategorised in the PRISM model.
- Kind of reasoning includes the categories of reasoning deductively, inductively, abductively and by analogy, and subcategories of these.
- Formulation refers to the students' awareness of their own reasoning.
- Formality refers to the extent to which their expression of their reasoning conforms to outside norms for expressing reasoning.
What does the RIDGE project involve?
In 2005-2006 the research team collected and analysed data of students working on dynamic geometry activities. The students were participating in an after school mathematics club which will provide an occasion for exploration of additional aspects of dynamic geometry beyond what is possible in the traditional classroom. Patterns of reasoning were characterised using the PRISM model and based on those characterisations elements of mathematical reasoning will be identified by the research team. Aspects of the activities that seem to be related to students' mathematical reasoning, based on the task analysis and characterisations by the PRISM model, will be identified.
In 2006-2007 the research team collected and analysed data of students working on revised dynamic geometry activities in their regular classrooms, where we worked with local teachers who are seeking to integrate dynamic geometry more fully into their teaching. The students' reasoning when involved in these activities is being analysed using the same methods as in the previous year, and compared with results from the previous year.
What problems and activites were used?
Problems and activities have been developed for use in classrooms and in after school activities. These have been based on problems used in prior research on dynamic geometry and reasoning (see RIDGE problems) and have been developed to address curricular requirements (see RIDGE activities).
References
deVilliers, M. (1997) The role of proof in investigative, computer-based geometry: some personal reflections. In J. King and D. Schattschneider (eds.), Geometry Turned On! Dynamic Software in Learning, Teaching, and Research, The Mathematical Association of America, Washington, DC, pp. 15–24.
Dreyfus, T. and Hadas, N. (1996) Proof as answer to the question why, Zentralblatt für Didaktik der Mathematik 28(1), 1–5.
Hadas, Nurit, Hershkowitz, Rina and Schwarz, Baruch B., (2000) The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44, 127–150.
Healy, Lulu and Hoyles, Celia, (2001) Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6, 235–256.
Jones, Keith (2000) Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44, 55–85.
Laborde, Colette, (2000) Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44, 151–161.
See also: Problem
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