The role of dynamic software in the identification and construction of mathematical relationships.

       What features of mathematical thinking do students exhibit when
       they use dynamic software in their problem solving approaches? To
       what extent does the systematic use of technology favour students
       development of problem solving competences? What type of
       reasoning do students develop as a result of using a particular
       tool? This study documents features of mathematical practice that
       students display when they systematically use dynamic software in
       problem solving. In particular, the use of the software
       involves the construction of simple geometric configurations that
       become a platform to formulate questions that lead them to the
       construction or recognition of mathematical relationships. In
       this process, students can build their own repertoire of
       mathematical results and also utilize their previous knowledge to
       support, justify, or explain their conjectures. Here, it becomes
       important for students to develop methods and strategies to
       observe particular relationships, express them using specific
       notation, and provide arguments to demonstrate and communicate
       their results.

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Recent curriculum reforms suggest that students need to utilize distinct technological tools in their process of learning mathematics. It is well recognized that students need to develop distinct strategies to identify and examine relevant information to deal with problems or situations that involve the use of mathematical resources. Given this perspective, it is common that they pose questions, explore particular conjectures, use distinct representations, and develop ways to communicate their ideas or results. The presence of technological tools in mathematical classrooms tends to influence not only the content and organization of the curriculum studied, but also ways in which students approach and learn it ((National Council of Teachers of Mathematics (NCTM), 2000). It is also recognized that there are multiples ways in which students can employ those technological tools and, as a consequence, there is a need to investigate what aspects of mathematical practice actually enhance students' learning as result of using particular tools. The intent of this study is to investigate aspects of reasoning exhibited by high school students while using dynamic software to construct and examine a set of geometric configurations. In particular, mathematical results emerge from exploring the behavior of parts of a certain configuration as a result of moving other elements within such a figure or configuration. Hence, it becomes important to characterize types of questions, conjectures, explanations, and forms of communication that students develop during their problem solving experiences.

DISCUSSION

A Problem Solving Scenario: Elements of a Conceptual Framework 

Mathematical problem solving has become a central activity in students learning of the discipline (NCTM, 2000). However, it is important to recognize that there are different scenarios in which students can be engaged in problem solving practices. Thus a problem solving approach may be based on creating a mathematics microcosm in the classroom where students openly discuss a set of well-selected non-routine problems (Schoenfeld, 1998). What happens when students are asked to participate in the process of formulating questions? What is the role of the use of technology in achieving this goal? To what extent are students' methods for solving problems enhanced when they systematically utilize technological tools? A fundamental principle to frame students understanding of mathematical ideas is that they need to conceive of their learning as an opportunity to pose questions or problems to be explored and solved. That is, students not only generate questions as a means to understand situations or problems but, in addition, some of these questions may eventually become problems to be solved.

       Once you have learned how to ask questions--relevant and
       appropriate and substantial questions- you have learned how to
       learn and no one can keep you from learning whatever you want or
       need to know (Postman & Weingartner, 1969, p. 23).

Thus, the process of understanding particular themes, contents or situations involves the formulation and exploration of significant questions. In this context, the use of dynamic software seems to offer students a powerful tool not only to identify potential relationships but also to explore and visualize their behaviors.

Eighteen high school students participated in a problem-solving course during one semester, meeting four hours a week. An important goal was to ask the participants to use Dynamic Software to work on a series of activities that involve:

(i) Routine problems that appear in textbooks. The idea here was to discuss the extent to which the use of the software helps transform the original nature of the task. That is, with the help of the software students were encouraged to explore connections to or extensions of the initial problem.

(ii) Problems proposed by the instructor in which the participants were asked to compare traditional approaches with those achieved by the use of the software.

(iii) The use of the software to construct simple configurations that were used as platforms to identify and explore mathematical relationships.

During the development of the sessions, students had the opportunity to formulate questions that led them to reflect on their previous knowledge to understand and approach tasks in different ways. Using this perspective, students can also find new approaches to present, structure or support knowledge previously studied. In addition, what students learn needs to be communicated or expressed through the use of some media. It becomes important for students to develop a particular language and notation to organize and present what they learn. Likewise, students knowledge should be constantly revised or scrutinized openly by other students.

Participants had access to a computer but they were encouraged to work in pairs or small groups of three. We also suggested a particular pedagogic approach to encourage students to learn through an inquiry process. This instructional approach has emerged from systematic implementation of tasks in which it becomes relevant for students to examine mathematical themes by asking questions. Thus, the development of the sessions consistently showed the following structure:

1. The instructor introduces the task to the students and asks them to work on the task in groups of two or three students for about 20 minutes. The role of the instructor is to monitor students' work and help them clarify (via questions) the statement of the task. Each small group hands in a written report showing the students' approach to the task.

2. The instructor asks some small groups to present their work to the whole class. During each small group presentation, the rest of the group, including the instructor, asks questions to clarify what may not be clear or may need some elaboration from the small groups presentation.

3. The instructor identifies strengths and limitations associated with each small group's presentation and discusses with the whole class mathematical ideas, strategies, concepts and distinct representations that are relevant to students' solutions to the task. In addition, the instructor may introduce a new concept or analyse extensions or possible connections to the original statement of the task or problem.

4. Students are asked to individually revise the initial task. Here each student has the opportunity to incorporate new ideas, concepts, or strategies that he/she has judged to be relevant during the development of the session.

This paper focuses on analyzing what occurred during two two-hour sessions in which one of a small group presented a simple construction that eventually led to the recognition of basic mathematical results. It is important to mention that even when the small group proposed the task and initiated the discussion, more mathematical results became apparent from the participation of the whole class.

Introducing the Task: The Importance of Posing Questions and Results

In a problem-solving environment there are distinct events that appear during the development of the session. What features of mathematical thinking do students exhibit when they use dynamic software in their problem solving approaches? To respond this question, it was decided to focus on the behavior of a small group of students during the presentation of its work to the whole class. We are interested in identifying global students' approaches and particular questions that appeared during the development of the session. In this context, the use of the software became a powerful tool for students to pose and respond to series of questions that led them to identify and construct relevant mathematical relationships.

[FIGURE 1 OMITTED]

A small group commenced the session by drawing (with the use of dynamic software) two points on a Cartesian plane. Group members then asked the class "what can we do with two points? And the presenter responded, "you can draw, for example, segment AB, line AB and measure the length of segment AB" (Figure 1). You can also observe that the length of the segment varies when point A or B is moved. At this stage, the presenter added new elements to this simple initial configuration and begin to identify and explore particular relationships among the components of the emerging construction.

[FIGURE 2 OMITTED] bisector, the triangle ACB is always isosceles. The immediate reaction from the class was "why?" At this point the whole class began to examine and justify this statement and eventually accepted it, since it was shown that triangles AMC and BMC are congruent. Thus, the first result associated with this construction was:

       Given a segment AB and its perpendicular bisector n, then the
       triangle ABC (where C is any point on n) will always be an
       isosceles triangle.

This result emerged from observing that the lengths of sides AC and CB are always the same for any position of point C. Students were then encouraged to present an argument based on congruence of triangles. That is, they eventually showed that triangles AMC and BMC are congruent (SAS postulate).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

A second result was identified:

     The triangle formed by points A, B of segment AB, and a point C on
     its perpendicular bisector located at the intersection of that
     bisector and a circle with its center on either point A or B and
     radius AB is equilateral.

[FIGURE 5 OMITTED]

How can we draw a right angle with vertex C? Can we move vertex C in such a way that the two sides of the angle pass through A and B respectively? If we draw a right angle ACB, then what properties does this angle hold? These are examples of the types of questions that the class discussed to eventually agree to draw a circle with center point M and radius MA and locate C as the intersection point between the circle and the perpendicular bisector. At this point they stated that triangle ABC is a right triangle since side AB is the diameter of the circle (Figure 5). Thus, another result was:

      Triangle ABC is a right triangle when point C is located at the
      intersection of the perpendicular bisector of segment AB and the
      circle with its center as the midpoint of AB and its radius half
      of that segment.

It is interesting to mention that the argument the students used to support this result was based on using a result that they had previously studied: If one side of an inscribed triangle is the diameter of the circle, then the triangle is a right triangle. At this point, students from the small group mentioned that they have completed their presentation; however, this task was again addressed during the following session. The idea was to add more elements to this configuration and explore the behavior of particular points as a result of moving, in this case, point C along line n.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

At this stage, it was evident that students recognized that the software is a powerful tool to identify and explore the behavior of particular relationships. In particular, they realized that the general properties of the parabola described previously could also be verified by quantifying that the distance from any point P on the locus to point B (focus) and to line n (directrix) was always the same (Figure 8).

[FIGURE 8 OMITTED]

When any of the small groups identified a particular curve as a result of exploring the behavior of dynamic representations that included new elements added to the initial configuration, they were encouraged to present their construction to the whole class. Thus, the class not only carefully examined the construction, but also participated in the process of justifying properties that first appeared only visually. That is, students eventually recognized that it was important to provide arguments to support their conjectures. In addition, it was observed that students developed a certain kind of ability to add other elements to the configuration when there was a possibility of generating interesting relationships. For example, in general, they noticed that point C, located on the perpendicular bisector, was a key point to use in searching for potential relationships. The fact that there was a triangle in the configuration, for example, led them to think of adding to the configuration, elements that include heights, perpendicular bisectors, and angle bisectors in that triangle and to observe the behavior of particular intersection points of these lines.

Students observed that the locus seemed to be a hyperbola. Again the class began to examine whether the generated locus held basic properties attached to this figure; for example, they tried to identify its foci, vertices, center, etc.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

By introducing a Cartesian system, students commenced the identification of possible candidates to be the elements of this figure. For example, some students selected the X-axis to be the hyperbola's focal axis, located candidates for foci F1 and F2 and showed that any point S of the hyperbola satisfies d([F.sub.1]S)-d([F.sub.2]S) as always being a constant. In this trial and error strategy, it helps to measure particular parts of the configuration and observe whether any selected point of the hyperbola holds its definition. Another strategy was to identify the main elements of the figure through the conic command. That is, to associate the shown locus with the corresponding conic (selecting five points of the locus and the command conic). Here, the software can also provide the corresponding equation (Figure 11).

[FIGURE 11 OMITTED]

Here, another result emerged:

      In an isosceles triangle ABC (where C is any point on bisector n),
      line MN (where M is the midpoint of side BC) and the perpendicular
      line to segment BC that passes through vertex A are intersected at
      point R. The path (locus) created by point R when point C moves
      along the bisector is a hyperbola.

It is also important to mention that in some cases, students' attempts to identify properties associated with particular locus failed, since what students had visually recognized as a particular conic did not correspond to their conjecture. For example, a small group drew the angle bisector of angle BAC and the perpendicular bisector of BC. These objects are intersected at P and when students identified the locus of point P when C moves along the perpendicular bisector n, they thought that the locus was a parabola (Figure 12). However, when they constructed the corresponding conic that passes through five point of the locus, they observed that the associated conic with those points did not overlap the original locus (Figure 13). Students recognized that it was always important to examine whether what they had visually identified as a particular figure actually held mathematical properties that define that figure. In particular, the use of the software seemed to help students orient the process of searching for arguments to support their conjectures. Figure 13, for example, was an argument to stop thinking that Figure 11 was a parabola.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Looking Back

There are aspects of mathematics practice that appeared as important during the development of the activity:

(i) There is no one established or well-defined problem to be initially solved by the students. Instead, questions or problems emerge through the process of constructing a particular configuration that involves points, segments, bisectors, triangles, heights, etc. Thus, students have an opportunity to observe and identify properties attached to different components of the figure in order to pose and pursue particular questions.

(ii) Contents or theorems that students used to relate particular subjects (triangles, bisectors, with Euclidean geometry and conics with analytic geometry) together now seem to appear connected. In fact, students not only reconstruct some particular relationships but also investigate and document new ways to generate particular figures. In addition, students are able to study properties attached to those figures. For example, by measuring particular parts of the figure, students can verify properties attached to the conic. In this case, any point located at the generated locus must satisfy the definition of hyperbola.

(iii) Students get involved in cycles of mathematical understanding that include the importance of posing questions or conjectures, exploring them, providing mathematical arguments to support them, and communicating their results. Hence, students realize that it is not only important to observe a particular relationship but also to provide arguments to support it. In addition, they value the need to develop a language to communicate their results.

(iv) The rule of dynamic software becomes an important tool for students to guide the exploration of mathematical relationships. In some cases, the use of the software provides evidence about the existence of particular relationships. The goal here is to show a mathematical argument to justify such existences. In other cases the software itself functions as a tool to generate figures (loci) that later need to be examined in terms of their properties. Another important factor is that the use of the software allows students to quantify or measure elements involved in the figure (lengths, areas, perimeters) and document their behavior as a result of moving or changing other other components within the same configuration.

(v) In general, the process of analyzing parts of certain geometric configurations represents a challenge for students allowing them to observe and document the behavior of families of objects (segments, lines or points) within a dynamic representation. Students themselves get the opportunity to reconstruct or discover new theorems or relationships. A crucial aspect that emerged in students' problem solving instruction is that with the use of dynamic software they had the opportunity to engage in a way of thinking that goes beyond reaching a particular solution or response to a particular problem.

Remarks

An important goal of mathematical instruction is to provide an environment for students in which they have an opportunity to demonstrate their own ideas on ways to deal with mathematical problems. These initial ideas need to be challenged and expanded and the use of technological tools seems to offer an important means to meet this goal. In particular, students can construct distinct types of representations with the help of technological tools that are then able to be studied in terms of answering or discussing questions or issues posed by the students themselves. In this context, students can reconstruct mathematical relationships previously studied and also have the opportunity to identify new set of relationships.

Note

The author acknowledges the support received by Conacyt, project #42295, during the development of this work.

References

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston VA: The Council

Postman, N. & Weingartner. (1969). Teaching as a subversive activity. New York: A Delta Book.

Santos-Trigo, M. (2002). Students' use of mathematical representations in problem solving. Mathematics and Computer Education Journal, 36(2), pp. 101-114.

Schoenfeld, A H. (1998). Reflections on a course in mathematical problem solving. Research in Collegiate Mathematics Education III, pp. 81-113.

MANUEL SANTOS-TRIGO

Center for Research and Advanced Studies

Mexico

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