The role of technology in students' conceptual constructions in a sample case of problem solving.

It is well recognized that an important component in mathematical instruction is to provide students with an opportunity to develop and use diverse representational systems in order to solve a variety of mathematical tasks (National Council of Teachers of Mathematics, 2000). Representations play an important role during the construction of models that help students to solve problems. What type of problem-solving activities do students need in order to develop ways of thinking that value the use of distinct representational systems to understand, and solve, and eventually propose problems? Teachers often state that it may be difficult for them to formulate new problems or situations that actually help their students search for different ways to approach them. It is also recognized that textbooks are the main source of examples that teachers use in their classrooms. Can textbook exercises be transformed into problem-solving activities that encourage students to develop mathematical thinking? This study documents what high school students showed when they were explicitly asked to use technological tools to examine and solve a set of routine problems from different angles or perspectives.

A fundamental instructional principle used to organize and structure the learning activities implemented in this study was to encourage students to think of different ways (construction of models) to solve problems and to discuss strengths and limitations associated with each solution method. Goldenberg (1995) states that:

  In current practice, the great bulk of mathematics teaching takes
  place within a single representational system. Much time and effort
  are spent in building students' skills in manipulating the formal
  symbolic language of traditional classroom mathematics, while
  relatively little time is devoted to other representations of the same
  ideas (p. 156).

Thus, it is important to acknowledge that students' mathematical understanding involves not only the use of various representations but also being able to transit, in terms of meaning, from one representation into another. That is, it becomes important that students' learning experiences not only focus on reporting solutions but also on identifying features of the models used to solve the problems. Here, the process of finding different methods to approach the tasks requires that students use several types of representations that help them develop appropriate conceptual systems. These systems tend to be expressed by students through models that involve the use of descriptions, explanations, and the use of diverse representations (Lesh, in press). In particular, the use of technology often offers students an important window to observe and examine connections and relationships that become relevant during the solution process.

Lines of Mathematical Inquiry

There are different learning trajectories for students to take in order to achieve mathematical competence; however, a common ingredient is a need to develop a clear disposition toward the study of the discipline. Such a disposition includes a way of thinking in which students value: (a) the importance of searching for relationships among different elements or components of the tasks in study (expressed via mathematical resources), (b) the need to use diverse representations to examine patterns and conjectures, and (c) the importance of providing and communicating different arguments (Santos, 1998). Thus, it becomes important to encourage students to think of the discipline in terms of dilemmas or challenges to be met and resolved. This means that they need to conceptualize their learning experiences in terms of activities that involve posing questions, identifying and exploring relationships, and providing and supporting their answers or solutions (NCTM, 2000). It is necessary to value the students' participation and persuade them about the power of reflecting on what they do, in mathematical terms, during their interaction with tasks or mathematical content. "To be able to guide students' inquiry toward the learning of the mathematical content in the syllabus, teachers must first convince students that inquiry is a legitimate, safe, and productive way to learn in school (Lampert, 1995, p. 215). Here, students' view of mathematics involves accepting that it is more than a fixed, static body of knowledge; it includes that they need to conceptualize the study of mathematics as an activity in which they participate actively in order to identify, explore, and communicate ideas attached to mathematical situations.

  ... Students themselves become reflective about the activities they
  engage in while learning or solving problems. They develop
  relationships that may give meaning to a new idea, and they
  critically examine their existing knowledge by looking for new and
  more productive relationships. They come to view learning as problem
  solving in which the goal is to extend their knowledge (Carpenter &
  Lehrer, 1999, p. 23).

Duval (1999) recognizes that students need to coordinate diverse registers of representations in order to achieve mathematical competence. That coordination involves transformations within the same register of representation (arithmetical and algebraic transformations) and coordination among different representations (examination of a function through symbolic, table, diagrams, and dynamic representations).

Students need to use different representational media to express their ways of thinking while dealing with tasks or problems. The students' constructions of powerful representational systems play an important role in developing distinct artifacts to understand and explain complex systems (Lesh, in press). The use of different tools offers students the possibility of examining situations from perspectives that involve the use of various concepts and resources. As a consequence, each representation might become a platform to identify and discuss mathematical qualities attached to the process of solution. Thus, during this process, a table might shed light on trends displayed by discrete data while an algebraic approach focuses on continuous behavior and general tendency (infinity). Geometric and dynamic approaches to the problem might provide a means for students to visualize and examine relationships that are part of the depth structure of the task. Specially, dynamic constructions help students focus their attention on common properties that appear while moving elements within the same configuration or representation. Lesh (in press) argues that "useful ways of thinking usually need to develop iteratively and recursively, with input from people representing multiple perspectives." In this context, solving the task goes beyond reporting a particular solution, it is a process of constructing, investigating, representing, applying, interpreting, and evaluating several ways to solve the problem (Schoenfeld, 1998).

General Procedures and the Initial Problem

Twenty-four students (grade 12) worked on a series of tasks that included textbook problems (in addition to assignments of the course). Students were taking an introductory integral calculus course (they had already taken algebra, Euclidean geometry, analytic geometry and an introduction to differential calculus). Students met twice a week during 2.5 hours. All the students had previous experience in using Excel, Cabri-Geometry software and the symbolic calculator. They all volunteered to participate in the weekly technology sessions (problem-solving approach) in addition to attending their regular calculus class. Both the class teacher and the researcher selected the problems and worked together during the development of the sessions. In this report we document features of mathematical thinking that students exhibited while interacting with one problem. In general, each student had access to a computer (Excel and dynamic software) and a calculator. In each session, students had opportunity to work individually, in small groups, and as a part of the whole group discussions. Some students had some experience in using the tools and often those students taught other students relevant features of the software during and out of the regular sessions. The example used to illustrate the students' ways of thinking (models) involves a routine problem that regularly appears in calculus textbooks. Thus, models that students exhibit during their interaction with this task illustrate mathematical processes and contents that appear when students use distinct representations to explore mathematical qualities attached to various methods of solutions. Ideas from arithmetic, algebra, geometry, and calculus emerge naturally as a means to analyze relationships that appear in each student's approach. It is important to mention that there is no intention to provide a detailed analysis of students' work, rather each student's approach to the task is shown to highlight the type of representation used to solve the problem. In particular, attention is paid to the variety of ideas and strategies that emerged when students are encouraged to use different technological tools to represent and approach even routine exercises. Thus, the initial nature of the task can be transformed into sequences of students' mathematical explorations.

The Initial Problem (1). The distance between two telephone poles is 10m as shown in the figure. The length of each pole is 3 and 5 meters respectively. To support the poles, a cable from the top of each pole will be tied to a point on the ground between the two poles. Where must that point be located in order to use the minimum length of cable? (Figure 1).

[FIGURE 1 OMITTED]

Students worked on this problem first individually and later as a part of a group of three students. When some small groups presented their approaches to the entire class, it was common that new ways to solve the task emerged from the class discussion and students had opportunity to rewrite their initial approaches. At the end of each session, the teacher directed the class to discuss advantages and limitations of what students had presented. So, in general, students became aware not only of the power of their own methods but also of the strengths of other students' approaches.

Students' construction of models to solve the problem.

The term model is used to characterize ways in which students identify and employ ideas, concepts, representations, operations, and relationships to solve problems. So the construction of models is a process that involves constant exchange and refinements of students' ideas.

Students' initial interaction with the problem focused on identifying key ideas to detect particular relationships. Thus, understanding the task involved the introduction of a representation and notation that led students to discuss a set of questions.

(i) An important mathematical idea embedded in this task is to recognize that the length of the cable varies when P is moved along the segment between the two poles. Here, it is also important to quantify that change. Students, in general, introduced particular notation that helped them identify key elements of the task (figure 2).

[FIGURE 2 OMITTED]

(ii) How can we know that the length of the cable changes when point P is moved along the line between the two poles? How can we determine the distance between one pole and point P? What data do we have? Can we use the Pythagorean theorem? These were initial questions that helped students identify relevant information and explore relationships between the length of the cable and location of point P.

One first stage involved analyzing particular cases to calculate the length of the cable at different locations of P (exploration of the idea of variation). Some students indicated that to find the length of the cable they could apply the Pythagorean theorem. They recognized that the hypotenuse in each triangle represented part of the length of the cable. That is, the sum of the two hypotenuses represented the length of the cable. Some students also recognized that the procedure used to determine the length of the cable for a particular case could also be employed for other locations of P. Excel became a powerful tool to organize the information and to explore the relationship between the length of the cable and the location of point P. A challenge for students was to organize the data in a meaningful way to keep track of particular relationships between point P and cable length.

I. A Discrete Model. Some students focused on calculating particular cases that emerge when point P is moved along segment AB. Although they initially divided the segment of length 10 into two arbitrary segments, later they organized the lengths systematically into a table arrangement. AP represents the distance from the length of the shorter pole to point P; PB the length between point P and the other pole. P1 and P2 represent the lengths of the poles, D1 and D2 the corresponding hypotenuses and D1 + D2 the length of the cable. Students completed a table (figure 3) that includes a refined partition of segment AB and the hypotenuses of the two triangles.

Students observed that the length of the cable decreases up to some value and then increases again. Here, they identified that when the point is 3.75 from the point A, then the cable gets the minimum length.

A Visual Representation. After students generated a table, they presented the corresponding graph (figure 4). Here, they observed that when the point gets closer to one of the poles the length of the cable gets larger. Indeed, they observed that when the point was 3.75 the length of the cable reaches the minimum value. Here, students asked: When or under which conditions does the midpoint of the segment determine the minimum length of the cable? Students reported that when the lengths of the poles were the same then the midpoint of AB was the point at which the cable reaches its minimum length. Otherwise, the point will be located on the segment and close to the pole with shorter length. The key ingredients of this model include the idea of analyzing particular cases, the process of refining the segment partition, the use of the Pythagorean theorem, the use of the tool (Excel) and the visual representation of the data.

[FIGURE 4 OMITTED]

II. A Symbolic Model and the Use of a Calculator. A small group of students expressed the hypotenuse in each right triangle as [square root of ([x.sup.2] + 9)] and [square root of (25 + (10 - x)[.sup.2])] respectively and the length of the cable as the sum of these two expressions. That is, l(x) = [square root of ([x.sup.2] + 9)] + [square root of (25 + (10 - x)[.sup.2])]. How can we find the minimum value of l(x) in this expression? Can we graph this function? With the help of a calculator, some students graphed the function l(x), and identified the value in which the minimum value is reached (figure 5). Other students, who decided to follow algebraic procedures to find the minimum value of l(x), realized that they could contrast their results with those obtained through the use of the symbolic calculator. So the calculator functioned as a monitor of the students' work (figure 6).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

In this model, students focused on representing the function that related the position of the point to the length of the cable. Here, the fundamental components of this model include the use of particular notation, algebraic procedures, and establishing connections between the graph representation and the problem. The calculator had a dual purpose: to graph the function and identify the solution and to verify algebraic operations (derivative, roots of equation).

III. A Geometric Model. Another method suggested by the instructor was to examine the case in which one pole was reflected on its vertical line with respect to segment that joins the two poles (figure 7). They observed that angles APC and APE are congruent. Students recognized that in this case the segment ED that intersects AB at P is the minimum length of the cable. They argued that any other point different P' will generate a triangle EP'D in which the sum of EP' and P'D will be always longer than ED (figure 8). The argument was based on using the triangle inequality. That is, they showed that in [DELTA]EP'D, EP' + P'D > ED.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

To find the distance AP, some students recognized that triangles APE and BPD were similar. Therefore, the corresponding sides held proportionality, that is, x/3 = [10 - x]/5 which led to x = 15/4.

Slope Approach. Some students also realized that the minimum distance of the cable is obtained when the slopes of the two lines CP and PD are the same but with opposite sign. That is, when the angles APC and BPD are congruent. Here, they introduced a coordinate system with A as its origin point. Thus, they calculated the slopes of the line that passes by (0, 3) and (x, 0) and the line that passes by (x, 0) and point (10, 5).

[m.sub.1] = -3/x and [m.sub.2] = 5/[10 - x]; to hold the condition, students observed that:

[-3/x] = [5/[10 - x]] [left and right arrow] -30 + 3x = -5x [left and right arrow] x = [15/4].

The components attached to this model involve the use of properties of triangles (congruence and similarity) to identify the solution. Supporting the solution was also a key part of this model. In addition, the use of a coordinate system played an important role to introduce basic ideas of analytical geometry.

IV. A Dynamic Model. Yet another approach that students showed while dealing with this problem was to represent the problem through the use of dynamic software. This software allowed students to determine graphically the relationship between the distance AP and the length of the cable (CP + PD). Here, by moving point P along segment AB a graph of the behavior of the length is generated. It is also important to mention that a table including some values of distance CP and the corresponding value of CP + PD can also be obtained. In this case what students reported was an approximation of the minimum length of the cable, that is, 12.81 (figure 9). In addition, students could drag basic parameters and generalize their results (for example, varying distance between poles or pole lengths).

[FIGURE 9 OMITTED]

A key component of this model was to represent dynamically the relationship between the point of the segment and the length of the cable. The use of a coordinate system to show the graphic representation of that relationship was also an important ingredient. In addition, students in this model could explore other cases in which they change parts of the initial representation (lengths of poles and segment).

An extension:

Students had the opportunity to discuss advantages and disadvantages attached to each model. In particular, they noticed that the geometric and dynamic model did not involve algebraic procedures to find the solution. Here, the teacher posed the following problem:

  Let C be a given point in the interior of a given angle. Find points A
  and B on the sides of the angle such that the perimeter of the
  triangle ABC is a minimum (figure 10).

[FIGURE 10 OMITTED]

Students' initial strategy was to use the software to represent the situation. In using the software it was also important to introduce a particular notation. Let OR and OR' be rays with a common point O and C a point on the interior of angle ROR'. Thus, their first goal was to find the minimum length from point C to either of the angle sides. They drew from point C a perpendicular to ray OR'. This perpendicular cuts ray OR' at Q. Then they chose point A on ray OR and drew a perpendicular from A to ray OR'. This perpendicular cuts ray OR' at P. Students observed that the configuration AP, CQ, and PQ is similar to the initial problem and located point B on segment PQ to minimize AB + BC. They also observed that when point A is moved along ray OR a family of triangle is generated. That is, for each distance OA there is a particular triangle assigned. To find which triangle from that family reaches the minimum perimeter, students drew the graph (using the Locus command) that describes the perimeter associated to each value of distance OA. They observed that when distance OA is equal to 6.20 cm. then the minimum perimeter is 9.04 cm (figure 11).

[FIGURE 11 OMITTED]

They also checked that when the initial perpendicular from point C is drawn to ray OR the triangle with the minimum perimeter is the same as the previous case (figure 12).

[FIGURE 12 OMITTED]

Second Extension.

  Suppose that the lengths of the poles are a and b and that the
  distance between them is d. The cables are now tied from the top of
  each pole to the base of the other pole. The cables will get
  intersected at point P. What is the height of that intersection point
  with respect to the line that joins both poles?

When students discussed diverse qualities attached to each approach, it was natural to think of a general case in which the particular numbers used for the poles and distances between them were replaced by any quantity. In this process, a student suggested also to examine the above extension. Here it was evident that students relied on previous approach to work on this extension. In particular, they focused on identifying similar triangles to determine the length of the height (figure 13).

[FIGURE 13 OMITTED]

Students represented the information of the problem and introduced pertinent notation. In the figure, EF is parallel to AB and they observed that triangles AQP and PFQ are similar. Triangles EDP and FBP are also similar. If AQ is x, QP is h, then by using properties of similarity they wrote:

x/h = [d - x]/[b - h], this implies that x = [d X h]/b

Similarly from triangles EDP and FBP they expressed that:

[a - h]/x = h/[d - x], from which they obtained that x = [a X d - d X h]/a = [(a - h)d]/a

By combining the above expression:

[d X h]/b = [a X d - d X h]/a

That is, h = [a X b X d]/[d/(a + b)] = [a X b]/[a + b]

They also reported that for the case when a = 3 and b = 5 and d = 10, then h = 15/8.

Mathematical Qualities of Students' Models. An important concept that students examined through their approaches to this problem was the concept of variation. Using a table to present different instances became a powerful strategy to introduce the idea of approximation and limit. For example, students who initially identified examples of two integers whose sum was 10 to calculate different values of the cable length, later realized that these operations could also be done with the help of Excel. At this moment, they not only proposed decimal numbers but also introduced a systematic way to organize the data. When students moved from considering integers to decimal numbers, it was evident that they realized that it was important to refine their original way of dividing the segment. Graphing data from the table allowed students to visualize the behavior of the cable length. In particular they observed that when the point P is near one of the poles the length of the cable gets longer. Here, they realized that the length of the poles was relevant to find the location of P. For example, they recognized that when the poles have the same length then the cable reaches the minimum at the midpoint of the segment that joins both poles. Here, it was evident that students were not only interested in finding the solution of the task, but also exploring other possible cases.

During the discrete model, students identified key ideas that were necessary to express the length of the cable in terms of the given information. For example, they recognized that there were two right triangles in which they could calculate their corresponding sides and the length of the cable was the sum of the two hypotenuses. Students who focused on finding a function to express the length of the cable in terms of the distance x eventually gave a formula. However, at the beginning they did not know what to do with that formula. The use of a hand calculator became an important tool to graph the function and to find the minimum length of the cable. In addition, this tool was used to find the derivative of such function.

The geometric model was also important to relate this problem with contents that include triangle inequality, Cartesian system, equation of a line, and similar triangles. Students recognized that the reflection of the pole was a powerful strategy that does not rely on a series of operations as the other methods. In addition, they recognized that the segment joining the tip of one pole to the reflected tip of the other pole represented the minimum length of the cable. To support this, they introduced the idea "What about if not." That is, students argued that the hypotenuse was the minimum length by showing that any other point on the segment that joins the bases of the two poles will produce a larger distance. Finally, the dynamic model offered students the opportunity to explore directly what happens with the length of the cable when point P is moved along the segment. In addition, students constructed a table in which values of the distance from one pole to point P and their corresponding cable lengths were shown. The main difference between this dynamic representation and the one achieved with the calculator is that students with the help of the software actually constructed the representation and they were able to move the point while the calculator graph was given directly. The extensions of the problem were also important in terms of employing resources previously used to represent and solve them. In addition, they recognized that the distance from one pole to the intersection point was the same as the distance to the point that defined the minimum distance.

Remarks

When students openly search for various approaches while working on mathematical tasks it is common to identify different types of representations that help them examine and use different problem-solving resources and strategies. Some tasks or problems that often appear in regular textbooks can be taken as a platform to engage students in mathematical practice. In particular, the use of technology became a powerful tool to explore properties and relationships that did not appear in paper and pencil approaches. Here, the teacher recognized that he had used this problem with other classes and their solutions mainly involved the use of algebra; however, during the development of these sessions, he observed that, in particular, the dynamic approach helped students understand the concepts of variation and change involved in the problem. It was evident that students' ideas about solving routine problems get enhanced when explicitly they search for various ways to represent and solve the tasks. That is, routine problems are seen as a means to encourage students to extend and reflect on their mathematical thinking. Thus, teachers might use initially some of their textbook problems as a way to engage students in the search of powerful representations to elicit and refine their previous mathematical ideas. These ideas eventually are transformed into models that are useful to solve problems.

AP     PB     P1  P2  D1          D2           D1+D2
1      9      3   5   3.16227766  10.2956301   13.4579078
1.25   8.75   3   5   3.25        10.0778222   13.3278222
1.5    8.5    3   5   3.35410197   9.86154146  13.2156434
1.75   8.25   3   5   3.473111     9.64689069  13.1200017
2      8      3   5   3.60555128   9.43398113  13.0395324
2.25   7.75   3   5   3.75         9.22293337  12.9729334
2.5    7.5    3   5   3.90512484   9.01387819  12.919003
2.75   7.25   3   5   4.06970515   8.80695748  12.8766626
3      7      3   5   4.24264069   8.60232527  12.844966
3.25   6.75   3   5   4.4229515    8.40014881  12.8231003
3.5    6.5    3   5   4.60977223   8.20060973  12.810382
3.75   6.25   3   5   4.80234318   8.0039053   12.8062485
4      6      3   5   5            7.81024968  12.8102497
4.25   5.75   3   5   5.20216301   7.61987533  12.8220383
4.5    5.5    3   5   5.40832691   7.43303437  12.8413613
4.75   5.25   3   5   5.61805126   7.25        12.8680513
5      5      3   5   5.83095189   7.07106781  12.9020197
5.25   4.75   3   5   6.04669331   6.89655711  12.9432504

Figure 3. A discrete approach to the problem.

(1) The general statement of this problem is known as Heron's Problem. It is often stated as "given two points on the same side of a line, find a point on that line such that the sum of its distances to the given points is minimal."

Note: The author acknowledges the support received from Conacyt (reference #42295-S) during the development of this study.

REFERENCES

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

Santos, M. (1998). On the implementation of mathematical problem solving: Qualities of some learning activities. In E. Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), Research in collegiate mathematics education. 111, pp. 71-80. Washington, DC: American Mathematical Society.

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Manuel Santos Trigo

Center for Research and Advanced Studies, Mexico