Students’ use of mathematical representations in problem solving
Santos-Trigo, Manuel
INTRODUCTION
Recent mathematics curriculum reforms [5, National Council of Teachers of Mathematics (NCTM), 2000] have pointed out the importance of students becoming engaged in the process of formulating and solving problems. Students’ life experiences provide examples of events that can be examined or analyzed via mathematical formulations. Thus, tasks or problems might be embedded in situations in which students may be required to provide or gather information in order to deal with the task via mathematical resources, i.e., “How much water does your family use in a week?” This concept led us to conduct the study that is the subject of this paper.
During the process of working on tasks, technology often becomes a powerful tool, enabling students to achieve different representations. These can be the source from which students pose or formulate their own questions or problems. Students’ use of various representations to approach tasks can help to promote and enhance mathematical skills that include the tendency to formulate conjectures, the use of diverse representations and several methods of solutions, and the use of a variety of arguments, including counterexamples, to support and communicate mathematical relationships or results.
SUBJECTS, TASKS, METHODS, AND PROCEDURES
Twenty-five first-year university students who were taking a calculus course participated in the study. The instructor allotted two sessions per month (1.5 hour each session) during one semester to implement a series of tasks that required the use of various representations and problem-solving strategies. Students worked individually and in small groups. An important tool available to them was the use of a graphic and symbol-manipulating calculator. Students also had access to the use of dynamic computer software in the computer lab. The information gathered for the analysis comes from the student interviews and written reports. Ten tasks were implemented throughout the development of the course, and the work shown by students while working individually, in small groups, and during the whole class interaction was used to demonstrate the results. In particular, the students’ interviews were seen as a means to enhance their own learning (in addition to providing data for the analysis). Initially, students worked on each task individually, later they shared their ideas within a small group, and eventually some of the small groups’ approaches to the task were used to engage students in a whole class discussion. The role of the instructor was to monitor both students’ individual and small group interaction with tasks. In some cases, the instructor asked questions to help students consider a particular representation of the tasks or provided explanations regarding terms or statement of tasks. During the whole group discussion, the instructor coordinated and motivated the students’ participation.
FEATURES AND EXAMPLES OF TASKS
The tasks used in the study did not require sophisticated resources for students to understand their statements, and it was easy to identify goals to pursue. Another factor was that students’ previous knowledge played an important role in constructing methods to approach the task. In this process, students became engaged in discussions where new and powerful mathematical ideas appeared as a need to solve the task. Three tasks given to students in the first, third, and fifth sessions of the course were chosen to demonstrate what students did during the problem-solving sessions.
The first task was within a mathematical context and involved dealing with quadratic equations of the form chi^sup 2^ + p(chi) + q = 0, where p and q are real numbers. Students were told that all equations of this form could be represented as points in the plane. For example, the quadratic equation chi^sup 2^ + 2(chi) + 3 = 0 can be represented as (2, 3), and a point in the plane as (0, 4) represents the equation chi^sup 2^ + 4 = 0 if we let the coordinates of the point be represented as (b, c) from the general form of the quadratic equation chi^sup 2^ + b(chi) + c = 0. Here, students were given a set of quadratic equations and were asked to identify their corresponding points in the plane; they were also asked to write the corresponding quadratic equations associated with a set of points. The main purpose of this task was to explore what types of ideas and resources students displayed from topics, such as quadratic equations, that they had previously studied.
The second task represented a situation in which a physician explains to his patient how his prescribed drug will work during a period of time (treatment). The patient is given some tablets and receives the following information:
(i) Dose or amount of active substance of each tablet: 16 units
(ii) When the patient takes the tablet, his body begins to assimilate the active substance. Ten minutes later his body will have assimilated the total 16 units.
(iii) When the patient has assimilated the total dose of the tablet, his body begins to eliminate the medication. The patient must take the next tablet when the previous dose has been reduced to half that amount. The physician tells the patient that this reduction takes place every four hours. Thus, the patient will take the second tablet when the previous amount (16 units) assimilated from the first tablet is reduced to 8 units. This dose reduction will occur four hours after the patient takes the first tablet. Thus, when the patient registers 8 units, he will take the second tablet and his body will again assimilate the 16 units ten minutes later; at this point the amount of units stored in his body will be 8 + 16. Here, his body again commences the elimination process and when it reaches half of this amount (12 units after four hours), the third supply takes lace, and so on. The patient will follow this treatment for a week.
Based on the treatment description, interest was generated in analyzing the amount of medication stored by the patient’s body at different stages during the treatment. In particular, what amount of medication does the patient store when he/she takes each tablet, and ten minutes later? Is it possible to discuss the evolution of the patient treatment in terms of mathematical resources?
The third task dealt with a geometric construction in which students used dynamic software (Cabri Geometry) to explore and review some concepts and ideas studied previously. An attractive feature of this task was that a simple construction was used as reference to generate all the conics studied in an analytic geometry course.
PRESENTATION OF RESULTS
An important goal during the learning of mathematics is for students to work out problems, situations, or contents in terms of questions that eventually lead them to look for patterns, describe processes, make conjectures, and support and communicate results [3]. Thus, inquiring seems to be a key activity during the students’ examination of mathematical situations from different angles through the use of different means and tools. A proper representation often becomes a vehicle or tool with which to engage students in mathematical thinking. To present the relevant points of the study, we followed the general problem-solving phases identified in the Introduction (understanding the task, using diverse strategies, reporting results, and looking for generalization and extensions of the tasks). The analysis initially includes the identification and discussion of particular problem-solving activities or strategies that emerged from the students’ interaction with the task. For instance, it became important to document the extent to which students used tables to organize the information; the ways they interpreted graphic representations (relationship between the phenomenon and representations); the examination of particular cases to explore patterns that led them to study general cases, and extensions or connections of the task. Indeed, this part of the analysis provided information to evaluate the actual potential of the tasks. A second goal in the analysis of the information was to examine qualitative differences shown by students during the use of problem-solving strategies.
It is important to notice that the visual representation of this task helped students observe relationships among the points (equations). Another observation that emerged from examining the graphic representation was that “if a quadratic equation is given randomly, it is highly probable that it has two real solutions”. This statement was based on observing the regions in which the corresponding points are located.
While working on the second task, students initially realized the need to identify relevant information associated with the treatment. Data that they judged to be relevant included the amount of active substance of the tablet (16 units), the time in which the patient assimilated the active substance (ten minutes), frequency of dosage, and amount of active substance eliminated by the patient during each drug supply. It is important to mention that the identification of key information for this task came from students’ discussion in which they tried to explain variations in the amount of drug during the treatment. In general, this phase became crucial to proposing ways to organize and present the information.
(a) Use of a table. Initially, the students proposed two ways to organize the information provided in the task: a systematic list and a table. Later, they decided to use a table to present the information. An important aspect was to determine the entries that could help display the behavior of the relevant information during a period of time. Thus, the entries of the suggested table included frequency of dosage, total time (hours) elapsed from the beginning of the treatment at each take, and amount of active substance at each supply and ten minutes after each supply. Some students recognized that there was a recursive method to generate each datum in the table. Here, they used Excel to present the information.
Based on this type of table, all students were able to describe the amount of active substance stored in the patient’s body at the moment of each supply and ten minutes after. They noticed that in both cases the amount of substance stabilizes in the patient’s body. Indeed, they mentioned that the amounts get close to 16 and 32 units, respectively. It is important to note that students were surprised to observe that the amount of substance in both cases gets stable rapidly, since they thought that any process that involves adding a certain amount would grow larger and larger in the body. Here, they failed to recognize that the process involves a reduction factor that influences the remaining amount of substance.
(b) Graphic representation. By using the data shown in the table, it was possible to present a visual approach or graphic representation of relevant information. One student proposed using Excel to show the data from the table in a graphic representation (Figure 2).
The question was asked, “What differences do you see between data represented in the table and data displayed in the graphic representation?” Students again observed that both representations show that the amount of active substance at each taking and ten minutes after converges to 16 and 32, respectively.
The group discussion focused on the advantages provided by the algebraic approach. In particular, the students expressed that what they expected to occur in the patient body did not match what they observed in the table. Thus, the table representation enabled them to see a balance between the elimination factor and the amount of substance in each supply. They recognized that the formula allowed them to calculate the amount of active substance at any supply. The students also mentioned that it was important to work with the rational representation of data to detect a pattern or formula. It was observed that the use of a calculator in some cases functioned as a way to validate part of their results and to represent and visualize mathematical information.
To introduce the third task, students were asked to construct (via Cabri Geometry) an initial geometric configuration [8]. The software functioned as a tool to find and examine diverse properties or relationships that emerged from moving points within the same configuration. Steps (i), (ii), and (iii) define the initial figure as follows:
(i) Draw a line L and select point C on that line, then draw a circle with center C and radius CA (Figure 5).
Based on the above construction, the first question for students was: “What is the locus of point S when point Q is moved across the circle?”
It may be difficult to imagine and describe the locus of point S when moving point Q within the above configuration. However, with the help of the software, students should be able to carry out this task (Figure 8). Thus, by applying the command “Locus” to point S when point Q is moved across the circle, the following figure appears:
(iv) Now by moving point P along line L, it is observed that the locus changes into other figures. For example, when P gets close to point A, the following figure appears (Figure 9):
It is also observed that when point P is moved to the right along L, at one point the figure (Figure 10) will look like:
Students at this stage realized that by moving point P along line L, the locus produces figures that seem to be ellipses, hyperbolas, parabolas, circles (when P goes to infinity along L), and right lines (when P becomes point Q. The next task was to argue that the figures shown meet properties that define those figures. Indeed, the software again became a powerful tool to enable identification of the main components of each figure (vertex, focus, directrix, center, etc). That is, it was necessary to explain, via specific arguments, what the students observed while moving part of the configuration. In general, they introduced a coordinate system and selected point C as the origin (Figure 11). Thus, based on this representation, the students proved that:
(i) The locus is an ellipse when the absolute value of P (distance from the origin to P) is greater than 2r (r is radius CA)
(ii) The locus is a parabola when CP is equal to 2r
(iii) The locus becomes a hyperbola when CA is less than 2r
(iv) The locus is a circle when P is moved far away to infinity along L
During this process, students compared measurements and eventually formulated diverse conjectures to identify each relevant component of the figures. Throughout the process it was clear that the software became a powerful tool, enabling them to verify whether a particular locus held the definition they had studied for that figure. For example, an ellipse is the set of points on a plane whose distances from two fixed points (foci) on the plane have a constant sum.
Thus, since it was also easy to assign measurements to particular parts of the figure, the task offered students an opportunity to verify properties of those figures that they had previously studied. Besides, they were surprised that from a simple construction they could generate all the figures they had examined in an entire course. Throughout the interaction with the task, it was evident that the software encouraged students to develop a spirit of inquiry and a sense of mathematical power, in particular, their ability to discern mathematical relationships.
CONCLUDING REMARKS
was eventually connected to a parabolic equation. In the solution process, students also realized that it is important to examine tasks from different perspectives. In particular, the connection between quadratic equations and points in the plane was new for students and showed them that their previous knowledge related to this type of equation was improved through this connection. In this direction, learning a concept or idea is a continuous process in which students constantly need to examine such concepts or ideas from diverse perspectives.
When dealing with the second task, students observed mathematical qualities associated with the use of different representations. For example, they noticed that the use of a table offered discrete information for identifying the stability of the amount of substance stored in the patient’s body, while the graphic representation offered them a visual way to observe that behavior. In both tasks, dealing with general cases helped students go beyond particular instances considered in each task. For example, the first task gave students an opportunity to find geometric meaning to the discriminant expression that they had used before. Finding a general expression in the second task helped them select a suitable representation (rational) to find the corresponding pattern. Here, the use of a symbol calculator became a powerful tool for determining such expressions. It is also important to note that throughout the solution process, students were aware of the need to constantly pose questions, discuss qualities of data representations, look for diverse ways of developing solutions, and the need to provide explanations. In the third task, the software became an important tool first to identify familiar curves and second to remember and verify properties attached to each figure. It was evident that these types of tasks became basic to promoting the students’ approaches; but also the role of the instructor, who encouraged students’ participation, was a key component during the implementation of the tasks in the classroom. In general, the use of technology provided students with a powerful window to explore connections among different types of representations. In particular, students exhibited conceptual systems in which the use of representations achieved via the use of technology played an important role in eliciting their understanding of and solutions to the tasks.
REFERENCES
1. P. T. Carpenter and R. Lehrer, “Teaching and Learning Mathematics with Understanding”, in E. Fennema and T. A. Romberg (Eds.), Mathematics Classrooms that Promote Understanding, pp. 19-32, Lawrence Erlbaum Associates, Mahwah, NJ (1999).
2. Paul Goldenberg, “Multiple Representations: A Vehicle for Understanding Understanding”, in Software Goes to School. Teaching for Understanding with New Technologies, edited by David N. Perkins, Judah L. Schwartz, Mary Maxwell West, and Martha Stone, pp. 155-171, Oxford University Press, Oxford, England (1995).
3. Paul Goldenberg, “Habits of Mind as an Organizer for the Curriculum”, Journal of Education, Vol. 178, No. 1, pp. 13-24 (1996).
4. M. Lampert, “Managing the Tension in Connecting Students’ Inquiry with Learning Mathematics in School”, in D. Perkins, J. Schwartz, M. West, and M. Stone (Eds.), Software Goes to School. Teaching for Understanding with New Technologies, pp. 213-232, Oxford University Press, New York, NY (1995).
5. National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, Reston, VA (2000).
6. T. A. Romberg and J. J. Kaput, “Mathematics Worth Teaching, Mathematics Worth Understanding”, in E. Fennema and T. A. Romberg (Eds.), Mathematics Classrooms that Promote Understanding, pp. 3-17, Lawrence Erlbaum Associates. Mahwah. NJ (1999).
7. M. Santos, “On the Implementation of Mathematical Problem Solving: Qualities of Some Learning Activities”, in E. Dubinsky, A. H. Schoenfeld, and J. Kaput (Eds.), Research in Collegiate Mathematics Education, Vol. 3, pp. 71-80, American Mathematical Society, Washington, DC (1998).
8. M. Santos and H. Espinosa, “Searching and Exploring Properties of Geometric Configurations via the Use of Dynamic Software”, International Journal of Mathematical Education in Science and Technology (in press).
9. A. Schoenfeld, Mathematical Problem Solving, Academic Press, New York, NY (1985).
Manuel Santos-Trigo1
Center for Research and Advanced Studies
Cinvestav-IPN.
Av.IPN #2508, Sn Pedro Zacatenco
07360, D. F. Mexico
msantos@mail.cinvestav.mx
1Currently, Visiting Professor, Purdue University, Department of Curriculum and Instruction, School Mathematics and Science Center, 1442 Laeb, W. Lafayette, IN 47907-1442, msantos@purdue.edu.
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