DEVELOPING UNDERSTANDING IN MATHEMATICAL PROBLEM-SOLVING A STUDY WITH HIGH SCHOOL STUDENTS*
Armando Sepúlveda Manuel Santos
Universidad Michoacana - Cinvestav, IPN Cinvestav, IPN
[email protected] [email protected]
This study documents what high school students achieved when working with a set of tasks that involved different methods of solution in a problem-solving oriented course. During the
implementation of learning activities students had the opportunity to work in small groups, present and defend their ideas to the whole class, and constantly revise their work as a result of the criticisms and opinions that appeared during the development of the sessions. The models that the students constructed in the processes of solution were documented, focusing on how they use distinct resources, representation, strategies and ways of communicating their results.
Introduction
What type of problems or tasks favors the development and understanding of students’ mathematical ideas? What does it mean for students to learn mathematics? What type of instruction promotes students’ learning? These are some of the questions that have been part of the research agenda in mathematics education during the past fifteen years, and were used as a guide in the development of this study. In particular, recent proposals in mathematics curriculum suggest the organization of mathematics learning around problem-solving activities (NCTM, 2000), and it has been recognized the importance for students to develop distinct resources and strategies to pose and solve different types of problems. Also, it becomes relevant to consider learning scenarios where students have the opportunity to reflect over the use of resources and processes in working with mathematics and that allow them to extend and reinforce their methods of posing and solving problems (Santos and Sepúlveda, 2003). In these scenarios the students present their ideas and listen to and examine the ideas of other students in such a way that they constantly reflect over their own ways of understanding mathematical ideas. In this study we were interested in documenting the thought processes shown by the students when they worked with a set of problems. The problems were designed with certain principles in mind: Were easy to understand and interesting for the students, they involve fundamental concepts and ideas of the curriculum and were posed in a manner that the work of the students could be analyzed and documented (Balanced Assessment Package for the Mathematics Curriculum, 1999, 2000).
Conceptual Framework
Three important themes became relevant and helped organize and structure the development of this study:
1. The idea that learning mathematics involves the development of a disposition on the part of the students to explore and investigate mathematic relationships, to use distinct forms of representation in order to analyze particular phenomena, and to use distinct types of arguments and communicate results. The NCTM suggests that it is important that the students construct
*
Artículo publicado en: McDougall, D.E. & Ross, J.A. (Eds.). (2004). Proceedings of the twenty-sixth annual meeting of the North American Chapter of the Interntaional Group for the Psychology of Mathematics Education. Toronto: OISE/UT.
their own mathematical knowledge as a result of solving distinct types of problems. Thus, relevant features that are enhanced in this process include: the motivation to express what they know; the encouragement to be open to investigate what they do not know through discussion, experimentation and the exchange of experiences; and the recognition of the importance of the thought processes used in their attempts at problem solution.
2. The recognition that learning mathematics is a continual process that is favored in an atmosphere of problem-solving (Schoenfeld, 1998) wherein the students have the opportunity to develop ways of thinking that are consistent with the development of the discipline. In this context, the students conceptualize the discipline in terms of questions or dilemmas that must be examined, explored and solved through the use of distinct strategies and mathematic resources (Hiebert, et al., 1996).
3. The acceptance that significant problems can be incorporated in different contexts (Barrera and Santos, 2002): a context of pure mathematics where, from a simple presentation, the student must use basic concepts to analyze and solve the problem; an ordinary daily context where the student has to interpret a familiar event, use distinct mathematical resources, and establish a series of considerations to solve the problem; and an artificial context where the situation is constructed from a series of suppositions about the behavior of variables or parameters that explain the development of the situation which in the treatment of the problem the student must project the use of strategies and representations in the methods of solution.
Participants, Research Methods and Procedures
The present study is structured around six problems that were selected, and adjusted from those found the Balanced Assessment Package for the Mathematics Curriculum (1999, 2000). Twenty-four students in the eleventh grade of a public school participated in the course of 16 weeks that included the following phases of instruction:
i) Introduction to the activity. The teacher gave a brief introduction to explain the goals to the students and the importance of their participation.
ii) Discussion in small groups. Students were organized into teams of three students each, with students of distinct levels of mathematical skills. At the end of the group work the students turned in a report of their solution.
iii) Student presentations. Each group presented their solution and the other members of the group (including the teacher) had the opportunity to ask questions.
iv) Full class discussion. The teacher promoted collective discussion to analyze the different methods of solution presented by the students and when necessary, presented a summary of students work and discussed possible extensions of the task.
v) Individual work. The students had the opportunity to return to the problem and incorporate the ideas discussed during the session.
To work the task students were organized into eight small groups of three students. An attempt was made to assure that in every group the levels of mathematical skills and personalities of the students were different within each group, according to assessments made in the first sessions and the opinions of their previous teachers. The idea was that the small group
organization would guarantee the participation of the students in the interaction with the other members of the group and during the presentation of their ideas to the entire class.
The sources of information that were used to analyze students’ work came from:
i) the written reports from the students corresponding to the work in small groups and the individual work permitted the identification of the ideas, and application of resources used to solve the problem in two distinct moments: the first moment reflects the initial (spontaneous) way of thinking of the students about the questions that were posed when they worked in small groups; the second moment reflects the level of understanding that was acquired as a result of the interaction with the entire class and when they solved the problem individually. This permitted the determination of whether there was an evolution in the understanding of the problem on the part of the students.
ii) Videotapes of the small group work, the students’ presentations, the collective
discussions, and the students’ interviews. The videos permitted the analysis of the ways in which the students participated in the solution of the problem, in the presentations by the small groups, in the full class discussions and in the interviews. We could analyze with more detail the
behavior of the students that helped them to solve the task. We could also identify crucial moments in which there were changes in the thinking of the students which allowed them to solve the problem.
iii) The observations of the teacher. During the development of the course, it was important to identify students’ difficulties, ways of interaction in which the students used the resources and strategies that were important to analyze students’ competences.
Some questions that served as a guide to analyze students’ work were: Were there differences between the answers given by the students in the reports from the group work and the contents of the individual reports? How to characterize those differences? Is it possible to identify what refinement processes were presented in the initial responses of the students to the solution of the problems and those shown as a result of their participation in a learning community?
The Problem and Some Considerations
We will now present one of the problems called “Shadows” from the ordinary context which was posed along with Figure 1:
Figure 1 Figure 2
1. Alicia is 1.5m tall and is standing 3m from the base of a lamppost that is 4.5m high. How long is Alicia’s shadow?
2. How does the length of Alicia’s shadow vary when she gets closer or farther away from the lamppost? Draw a graph with a system of perpendicular axis. Can you find a formula for this graph?
3. Simon is 2m tall. How can the graph represent the change for his shadow? Compare this graph with the one you drew for Alicia.
An important aspect for the understanding and solution of the problem is for the students to construct a representation of the situation that will help them find the relationships in the information of the problem. For example, the height of the lamppost, Alicia’s height and the length of her shadow can be represented with the corresponding segments. In this manner, a figure composed of three similar rectangular triangles is obtained: one with the form of the lamppost, one with Alicia’s figure, and one that describes the distance between Alicia and the lamppost when a horizontal segment is drawn over Alicia’s head (Figure 2). The relationship of similarity leads to the solution to the problem, considering the segment that represents the shadow first as a fixed quantity and then as a variable. Effectively, the students drew graphs as described, with some variations; however, we observed diverse manners of employment of the mathematical relationships.
Presentation of Results
First we will present the analysis of the work with students in small groups through the written reports and applicable video segments. Then we will go to the presentations of the solutions from the different groups and collective discussions that students’ participation
generated. Thereafter we will present the results of individual work, making a global analysis of the interaction offered by this type of instruction.
The written reports turned in by the groups demonstrated distinct approaches in the students’ attempts to solve the problems. What ideas, concepts, strategies and representations were
relevant in these answers?
In analyzing the written reports and the video of the work in small groups, we observed that the approaches to question 1 were:
I) Those of groups D and G which assumed that there were two similar triangles (that which is formed by the lamppost and that of Alicia in the Figure 1; they remarked that the angles were equal: angles in the base of lamppost and Alicia’s feet; angles in the lamp of the lamppost), without clearly justifying why this relationship existed, establishing their proportions and operations to obtain the answer.
II) Group E constructed a rectangular triangle (on Figure 1) with the vertices of Alicia’s shadow, elongated the shadow, the hypotenuse, and drew it parallel to the lamppost in whose sides of the right angle they wrote as 2cm (citing the similarity of the triangle that is formed by the lamppost with that constructed; they marked the angles), calculated the acute angle as
1 2 2 = = cm cm
Tanθ , θ = 45°; thereby obtaining the length of Alicia’s shadow: 1.5 45 5 . 1 = ° = Tan x .
III) Group C drew a horizontal line over Alicia’s head, forming a triangle with the lamppost (as in the Figure 2), saying that it was the same as that formed by Alicia (there are repeated letters, they used the symbol of congruence; they marked the angles) and that the sides of the right angle were in relation to 1, and from this obtained their solution.
It is notable that without having expressly established the relation of similarity nor having precise arguments, Group D applied the proportionality between the corresponding sides of the two similar triangles, Groups E and C constructed similar triangles, one applied trigonometric resources and the other used the relation between sides of the right angle of similar isosceles triangles.
IV) Groups B and H assumed and used the expression:
x m m m 1.5 3 5 . 4
= without showing the elements that justified their answer. Group F came by this through the schematic application of the Rule of Three, writing in a rectangle the quantities 4.5m, 3m, 1.5m, x (obtained of the Figure 1). However, this approach was later analyzed and changed during the students presentations.
Based on those approaches, the students gave the following answers:
D, G, E C and A coincided in that Alicia’s shadow measured 1.5m. B and F affirmed that the shadow measured 1m, while group H reported verbally that the shadow of Alicia is the third part of the distance to the lamppost. That is to say that group H verbally agreed with groups B and F. During the presentation of Group H in the full class discussion, Andrés made the drawing on the board and verbally explained the solution that Alicia’s shadow measured 1m; without finding arguments to answer the questions of the teacher and the other students; for example, in the following discussion, illustrates Andrés’ confusion in this explanation:
Andrés: As this is 4.5 [points to the lamppost] and this is 1.5 [Alicia], then if from here to here it is 3 meters… [he stopped to think].
Teacher: Then how long is the shadow? Andrés: One,…one meter.
Julio: How can it be one? [Julio was a member of group D].
Andrés: That’s what I don’t understand, but the next part I does [he explains is a low voice].
Teacher: Let’s see, what do you think? Say before Andrés presents the solution to the next questions [some students say that the shadow measures 1m and other 1.5m]. Andrés: …Okay [insecure, he wants to write his answers to questions 2 and 3; he goes
back to his seat].
Given that various students showed that they were not in agreement with Andrés, Victoria (of Group E) goes to the blackboard, draws the figure as stated and constructs a rectangular triangle with sides of 2cm in the vertices of Alicia’s shadow on the floor, marking the right angles of the lamppost, from Alicia’s feet and in the constructed triangle; the acute angle in the prolongation of the shadow she calls θ. Then she writes what the group has come up with:
1 2 2 = = cm cm Tanθ ,⇒θ =45°; x Tanθ = 1.5; m Tan x 1.5 45 5 . 1 = °
= ; she then says: Victoria: This angle θ measures 45° [points to the angle]. Teacher: Explain, why forty-five?
Victoria: Because this triangle is similar to this one [points to the constructed triangle and Alicia’s form]. We measured these sides and used this function, the tangent, and it
gave us 45°. With this 45° we came up with another formula [points to the triangle that Alicia forms] that is tangent. The tangent of 45° is equal to…; 1.5 divided by the unknown which gives 1.5m.
Still, the members of Groups F and H are not convinced. Those of Group H, unsure, accepted what was wrong because they were confused and they have not completely considered the distance from Alicia to the lamppost along with the shadow. At the invitation of the teacher, Core of Group F goes to the front, makes a drawing and says:
Core: Well, I have these triangles with this angle [marks the angle where the shadow ends], these parallel lines and these triangles [marks the right angles of the lamppost and Alicia’s feet], then…4.5m is to 1.5m [speaks and writes in schematic form the rule of three] as 3m is to x; then…x=1m.
Teacher: What do you think?
Class: Wrong [The voices of Andrés (Group H) and Rubí (Group D) are distinguished]. Teacher: Why? Come up [to Group D]. Look people, there are divided opinions, some say that Alicia’s shadow measures 1m and other say that the shadow measures 1.5m [Rubí goes to the board]. Why is Group F’s solution wrong?
Rubí: Ah, well [he explains without writing] because they said that 3m was x…but it can’t be 3 meters because they lacked this piece [points to the shadow in Core’s drawing].
Teacher: Then explain to us your solution here in this part of the board.
Rubí: 4.5m is to 1.5m [speaks and writes. Assumes from the beginning that the triangles are similar without mentioning this explicitly] as 3m plus x is to x, because you have to include all of this part because it is not here where Alicia’s shadow ends but here. Then we have x, but first we do the division and it gives that 3 is equal to 3m plus x over x…, we have x equal to 3 meters divided by 2. That is 1.5m. Teacher: What do you think?
Class: Very good! [Applause].
At this time Groups B and H are convinced that their solution was wrong because they had not considered the side of the triangle that was formed by the lamppost; in particular, Andrés and Karla said that Alicia’s shadow is not 1m but measures 1.5m. “I said it wasn’t 1” commented Julio (Group D).
When Group D presented their solution to question 2 and the class seemed to agree on what this group presented, Rubí also changed the distance from Alicia to the lamppost and noted that Alicia’s shadow was half the distance to the lamppost, then drew a line that passed at the middle of the distance to the lamppost, then drew a line that passed through the points (1,0.5), (4, 2), etc., which they got making the substitution in the proportion
x x m m m + = 3 5 . 1 5 . 4 ; in place of 3m they put x (“it varies”), the shadow they now call y in place of x, to obtain 3y = x+y;⇒2y = x; finally
they wrote their formula 2
x
y= . Joel, another member of the group, asked to go to the front and
changed 1.5m for 2m, and wrote x x m m m + = =2.25 3 2 5 . 4
, and said that Simon’s shadow is x=2.4m. After drawing a line for Simon and writing the equation
25 . 1
x
y= (the same procedure as in
question 2, the distance from Simon to the lamppost he called x in place of 3m and the shadow he called y).
It seemed that the form in which Group D made their presentation, clear and orderly,
contributed to unify the criteria of the other groups, as their arguments were accepted; also, from the work in small groups, the Groups C, D, E and G coincided in the answers given for the questions.
Discussion of the Results
During the presentation of the groups it was appreciated that the students used arguments of proportionality or established proportions without justifying them with precise reasoning.
Although some students’ approaches showed serious inconsistencies, the students had an idea of how to identify similar triangles even though they did not provide arguments as to why this relation of similarity can be established in a determined pair of triangles and from this
established the proportionality between the corresponding sides. That is to say that they showed difficulties in the use of appropriate language. As a result of the interaction, some of the groups reaffirmed and defended their ideas and others modified theirs, as was the case with Groups B and H after the presentation of Group D.
With the presentations, in fact, the collective discussions began, and the students and the teacher questioned affirmations, made corrections or asked for clarification from those who were presenting. Many of these interventions came when the students explained their reasoning that brought them to consider certain relationships or the use of representations.
The class discussion was beneficial for the advances of the students, for example, Group D, after several questions, clearly posed in the proportion that solved question 1, the distance between Alicia and the lamppost is a fixed quantity (3m) but in questions 2 and 3 the distance varies, in the proportion they wrote x in place of 3m and they denoted the shadow y, whether it was Alicia or Simon. That is to say that they made the change in the designation of the variables to express the relationship proportional in the typical notation of the function. Here there was an evolution in mathematical understanding of the problem on the part of the members of this group; we think that this process could have contributed to the understanding of the students of the other groups.
Remarks
Two important aspects became relevant during the development of this work:
i) The importance of designing or reformulating activities in which the students have the opportunity to utilize previously studied mathematic resources and the process of solution demands from them the extension or consideration of new resources or concepts for the solution of problems. Here one must identify the mathematical potential of the activity before using it in the classroom. In particular, it was interesting to project the distinct potential methods of solution.
ii) The implementation of the activity in the instruction must consider the active participation of the students in the distinct phases of solution. In particular we recommend that initially the students work in small groups of three; afterwards each group should present their attempts at solution to the whole class. In such a way the group that is making the presentation has the opportunity to defend their methods of solution and the other students, along with the teacher can formulate questions and ask for explanations that help them understand and justify what they have presented. In particular the public presentations were a forum for discussing points related to the use of certain relationships and the necessity to justify the work in each of the groups.
In general, during the work the students on this group of problems they experienced difficulties as much in the use of the language as in the use of the resources to pose and
communicate their ideas, but the form of instruction permitted a refinement of their ideas in their approximations to the problems, which permitted them to get ever closer to the solution.
References
Balanced Assessment Package for the Mathematics Curriculum. High School Assessment
Package 1 & 2. (1999 & 2000). White Plains, N.Y.: Dale Seymours Publications.
Barrera, M. F., Santos, T. M. (2000). Cualidades y procesos matemáticos importantes en la resolución de problemas: Un caso hipotético de suministro de medicamento. En Seminario Nacional de Formación de Docentes. Uso de Nuevas Tecnologías en el Aula de Matemáticas. Ministerio de Educación Nacional de Colombia.
Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K., Human, P., Murray, H., Oliver, A., & Wearne, D. (1996). Problem solving as a basics for reform in curriculum and instruction: The case of mathematics. Educational Researcher, pp. 12-21.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School
Mathematics. Reston Va.: National Council of Teachers of Mathematics.
Santos, L. M.; Sepúlveda, A. (2003). Hacia la construcción de un ambiente de instrucción basado en resolución de problemas. En M. M., Socas; M., Camacho; A., Morales (Eds.) Formación
del Profesorado e Investigación en Educación Matemática V. Didáctica de las Matemáticas.
Departamento de Análisis Matemático. La Laguna, España, pp. 323-341.
Schoenfeld, A., H. (1998). Reflections on a course in mathematical problem solving. Research in
