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STUDENTS’ IDEAS ON FUNCTIONS OF TWO VARIABLES: DOMAIN, RANGE, AND REPRESENTATIONS Rafael Martínez-Planell Universidad de Puerto Rico – Mayagüez
[email protected]
Maria Trigueros Gaisman1 Instituto Tecnológico Autónomo de México
[email protected]
The study of student understanding of multivariable functions is of fundamental importance given their role in mathematics and its applications. The present study analyses students’ understanding of these functions, focusing on recognition of domain and range of functions given in different representational registers, as well as on uniqueness of function value. APOS and semiotic representation theory are used as theoretical framework. The present study includes results of the analysis of interviews to 13 students. The analysis focuses on student’ constructions after a multivariate calculus course, and on the difficulties they face when addressing tasks related with this concept. Introduction and Purpose of the Study The notion of a multivariable function is of fundamental importance in advanced mathematics and its applications. Even though its understanding is essential for mathematics, science and engineering, little is known about students’ ideas and difficulties. There are very few research based studies that probe student understanding of the particularities of a multivariable function. This lack of research findings limits our understanding of how students learn the main ideas of the multivariable calculus. The present study is a continuation of a previous study (Trigueros & Martínez-Planell, 2007) which reported on student understanding of graphs of functions of two variables. The focus of the present study rests on the following research questions: What are students’ conceptions of domain and range of functions of two variables when they finish a Multivariate Calculus course? How are these conceptions related to their abstract, general notion of function? Theoretical Framework Two conceptual frameworks inform the theoretical basis used in this study. Firstly APOS theory is used to model the development of the concept of two variable functions and, secondly, semiotic representation theory, provides the conceptual tools to analyze flexibility in the use of different representations and its role in the cognitive evolution of the mathematical ideas under consideration. As APOS Theory is a well known theory, only its application for the purpose of this study is described. For more detail the reader may consult Asiala et al. (1996), and Dubinsky (1991, 1994). The application of APOS theory to describe particular constructions by students requires researchers to develop a genetic decomposition—a description of specific mental constructions one may make in understanding mathematical concepts and their relationships. A portion of a preliminary genetic decomposition for the function of two variables concept given in Trigueros and Martínez-Planell (2007), is summarized below since it will be referred to throughout this paper: The Cartesian plane, real numbers, and the intuitive notion of space schemata must be coordinated in order to construct the Cartesian space of dimension three, R3, through the action Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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of assigning a real number to a point in R2, and the actions of representing the resulting object both as a 3-tuple and as a point in space and making conversions between them. These actions are interiorized into a process that considers all the possible 3-tuples and subsets of 3-tuples, and their representation in space, to construct a process that when coordinated with the respective verbal, analytic and geometric representations can be thematized as three dimensional space, R3. This space schema is coordinated with the schemata for function and set through the action of assigning one and only one specific height to each point in a given subset of R2, either analytically or graphically. This action is interiorized into the process of assigning a height to each point on a subset of R2 to construct a two variable function, and the process of conversion needed to relate its different representations. When the process of generalization of these actions to consider any possible function of two variables, as a specific relation between subsets of R2 and R is encapsulated, it can be considered that the notion of two-variable functions has been constructed as an object. Duval (1999, 2006), argued that thinking processes in mathematics require not only the use of representation systems, but also their cognitive coordination. In Duval’s analysis, understanding and learning mathematics require the comparison of similar and different representations. According to this author, there are two different types of transformations of semiotic representations: treatments, which are transformations of representations that happen within the same representation register, and conversions which consist of changes of representation register without changing the object being denoted. He argues that these two types of transformations are the source of many difficulties in learning mathematics, and that overcoming these difficulties needs to take them both into account: to compare similar representations and treatments within the same register in order to discriminate relevant values of the mathematical object so that students notice the features that are mathematically relevant and cognitively significant, and to convert a representation from one register to another to dissociate the represented object and the content of the particular representation introduced so that the register does not remain compartmentalized. Method An instrument was designed to conduct semi-structured interviews with students and test their understanding of the different components of a proposed genetic decomposition (Trigueros & Martínez-Planell, 2007). Nine students were interviewed. The students were chosen from a group of undergraduate students at a private university who had taken the equivalent of an introductory multivariable calculus course the previous semester. The instructor of the mathematics course they were currently taking chose what he judged three good, three average, and three weak students to be interviewed. On the basis of the results obtained, and in preparation for the present study, the researchers decided to conduct more interviews focusing on items in which it seemed more data would be useful. The instrument was once again revised to do this and five new interviews were conducted. These students were chosen from a group of undergraduate students at a public university who had just finished taking the equivalent of an introductory multivariable calculus course that same semester. The instructor of the multivariable course they took chose what he judged two slightly above average, and three average students to be interviewed. All interviews lasted for 45-60 minutes, were audio-recorded, and all the students’ work on paper was kept as part of the data. The results obtained in 13 of these interviews were independently analyzed by two researchers, and the conclusions negotiated. Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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Results Results of the analysis of students’ responses during the interview showed that none of these students was able to demonstrate all the constructions described in the genetic decomposition. Their description of what they consider a function seems to be more closely related to that of relation and associated mainly with its analytical representation. Students differed in the difficulties they faced during the interview but most of them struggled with the description of the domain of these functions in all the representations included in the tasks and in the conversion between representation registers. Eight of 13 students had difficulty with the arbitrary nature of the relational correspondence. For example, when Emily was asked if the rule: “Input: weight in kilograms and height in centimeters. Output: name of person with that weight and height” defined a function, she responded: Emily: ok … [nervous laugh] for me, the fact that you have the weight, a weight and a height in centimeters [nervous laugher], the name is not there anywhere, … I don’t know how one gets to … The same type of response is observed in Rodrigo: Rodrigo: …because it can’t be that with only the weight and the height we can obtain the name of the person. When asked to define a function of two variables Gaddis responded: Gaddis: …, for me a function of two variables is a function of the form f ( x, y) is equal to x … a term in x, a term in y, and a constant or any number, then the domain would be, the two variables, that would be independent … and the range would be the result of those two variables evaluated in the function. In the genetic decomposition we assumed that the construction of the notion of function of two variables requires the coordination of a schema for R2 with that of function through the action of assignation of one and only one value to each element in the domain of the function. Some students did not show to have made this coordination. Among them, there were 4 students who did not consider that a function is determined by the uniqueness of that assignation. Rodrigo is one such student: Interviewer: and if I were to give you a list with all students at this university with their weight and their height Rodrigo: ok, here you could, but you’d get several results Interviewer: and that, would be a function? Rodrigo: yes, hmm, yes, yes Another student, Gaddis, commented to this same question: Interviewer: …is that a function? Gaddis: …yes, I think so Interviewer: So if a give you the weight, say 60 kg and height 2 m, what could be the output? Gaddis: It would be the name of a person with that data Interviewer: and, if there were more than one person with that data? Gaddis: …the output would be the name of the person Later, while further exploring Gaddis’s uniqueness of value notion, the interviewer presented the equation x2 + y2 =1 to him, and asked: Interviewer: … y, is it a function of x? Gaddis: … it could be, if we solve for the y Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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Interviewer: ok, I solve for the y and get 1 x 2 , the y, is it a function of x? Gaddis: y, there it depends on x in that case Interviewer: then, is it a function of x? Gaddis: I think it is. Most of the students had difficulties when describing the domain of a two variable function. It was found that 3 students do not have a clear idea that the elements in the domain of this type of function are always ordered pairs. Even though they accept they need ordered pairs in order to find the value of a particular function, when they are asked how many elements are there in the domain of a function given in a table representation, they count the elements of each pair as different elements in the domain. For example: Fernando: … the elements, would be eight [he was using a 4 by 4 table] Interviewer: give me an example of an element in the domain of Fernando: 0 comma 2 Even though he uses “0 comma 2” as example of an element in the domain, he still counts the domain as a set of numbers. In terms of the genetic decomposition it seems these students have not interiorized the action of assigning one and only one specific height to each point in a given subset of R2; rather than applying the action to elements of R2, they seem to be acting on sets of two real numbers. In the case of María, she correctly listed the elements in the domain, but when asked to count them, she stated: Maria: ok, then, do I count them as pairs or separate? Another problem some students had in finding the domain of functions was that they were not able to restrict the domain to specific subsets. For example, when analyzing the function defined by f ( x, y) x 2 y 2 1, where the domain is restricted to the pairs ( x, y) that satisfy: 1 x 1 and 1 y 1 , Paola could not understand the meaning of that restriction. With some help, she eventually succeeded in drawing the domain. However, later on, when trying to find the range she said: Paola: the thing is that I’m not sure what it means that it is restricted to the pair of numbers that satisfy this Of the 8 students having difficulty representing the domain of the above function, 5 tried to draw the graph of f first, without considering any restriction. Emily and Gracielle are examples of this difficulty: Emily: I’m trying to do the graph, to know more or less what would be the domain … Gracielle: … [mumbles] …It says x goes from -1 to 1, and y goes from -1 to 1, a circle Interviewer: and why is it a circle? Gracielle: because the graph is a paraboloid … It was also found that students’ ability to find the domain of a given function was related to the representation register used to present the information. The following was one of the questions of the interview: The following is the complete graph of a function f:
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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a. Find the domain of f. b. Evaluate f (0,0) , f (2,0) , f (2,2) , f (0,2) . c. Find the range of f.
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In parts (a) and (c), the domain and range were found correctly by 10 students, and incorrectly by 3. As was already mentioned, some students tried to graph functions in order to find their domain, even when a restriction defining it was given, so it seems that students did not have much difficulty obtaining domain and range information from a graphical representation of a two-variable function. However, as was reported previously (Trigueros & Martínez-Planell, 2007), students do have great difficulty obtaining the graphical representation of functions of two variables, as well as obtaining other kinds of information from such surfaces. In this case Patricia, who treated domain incorrectly as a set of numbers when presented with functions in tabular or algebraic representations, was able to correctly find the domain in this case. This is one of the several examples found which confirm what was observed by Gagatsis, Christou, and Elia (2004): the cognitive demands for translating (converting) among representations are not the same, and so each one needs to be specifically attended. It was also observed that all students who could get neither the domain nor the range of a function when its graphical representation was given also had difficulties when asked to find them for functions given in tabular or algebraic representations. Some students showed some confusion between the domain of the function and the intersection of the graph of the function with the xy plane: Patricia: …yes, and if this (pointing to x 2 y 2 1 0 ) …is zero, if z = 0, then it is a circle with radio equal 1… All the above mentioned difficulties seem to be related to the coordination of students’ schema for space and that for function. Although they are able to assign values to specific points in R2, either analytically or graphically, they cannot consider sets of points in the plane as the domain of the function or do not clearly understand the role of the domain of the function. When they are able to consider sets of points, they have difficulty considering the result of applying a function to the whole set. Regarding the range of the function, most students’ responses showed that their difficulties were mainly related to the lack of interiorization of the actions needed to find values of the function into a process. Some students showed that their idea of range of a function was not clearly differentiated from that of graph of the function, but this was not a prevalent difficulty. Most of them were able to calculate specific values in the range or to read it from a given graph. This was also observed through a task where most students used this strategy to match the algebraic representation of six functions, with their corresponding graphs. Students showed many difficulties doing this task. We expected them to use sections in the analysis of the graphs but only one of them was able to do this. Gaddis was the only student to consistently, and without prodding, use sections to analyze graphs of two-variable functions: Interviewer: say, start with the second formula g ( x, y) sin( x) y Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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ok, we take that g ( x, y) is the function in z, if we put that the y is 0 , we have that z sin x , which is the normal function, and if we put that the x is 0, the sine of 0 is 0 and we have that the, that z y Interviewer: and you, could you identify, with what you’ve done, the graph of that function? Gaddis: wouldn’t it be the graph on the upper right hand corner? Interviewer: that’s the one … and let’s say that h( x, y) sin( xy) Gaddis: every time that x or y is 0, it will be 0, z is substituted equal to … one would have to assign values to the x and the y and see the behavior of … to me it would be the, the second on the left hand column [correct] He was asked more questions that showed that he understood what he was doing and used sections in his graph analysis. These results are also evidence that independently of the grades obtained by students during their course, they were not able to construct a process conception of function. They did not demonstrate having generalized the action of taking a point in the domain of the function and assigning it a height, or to have constructed the process of conversion needed to develop effective strategies to relate different representations of functions. We found that students’ definitions of two variable functions can be classified in essentially three groups. The first one contains definitions of the function machine sort: two inputs one output, or input output (5 students); the second one uses variable dependence, algebraic expression, or formula (4 students); and the third type of definition is given in terms of geometric images (4 students). Fernando’s definition, for example, was given in terms of an input output conception, which can be related to a process understanding of function: Fernando: so that for each, …, so that for my domain, for each element of my domain, I can only have one point in the range and no more points for each point in the domain Interviewer: and what is it that makes the function be of two variables? Fernando: of two variables? That I have two different … because my function, because my output depends on two inputs, precisely, and we have two variables Gaddis uses a formula; this can be related to an action conception of function: Gaddis: …, for me a function of two variables is a function of the form f ( x, y) is equal to x … a term in x, a term in y, and a constant or any number, then the domain would be, the two variables, that would be independent … and the range would be the result of those two variables evaluated in the function. Pablo, whose definition was very clear, seemed to be guided by a geometrical model: Pablo: …, a function of two variables means, is, that starting from a certain region in a plane one can, that is, of variables x y, a height function can be constructed and for each point in that x y region there exist only one defined height When considering only the definition for two variable functions given by students, it may seem that some of them have a process conception of these functions, however, when relating their definition with their responses to the other tasks it is shown that this is not the case: they have not interiorized their actions and they are not always able to do treatments and conversions on different representations.
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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Discussion All the interviewed students had successfully finished a course on multivariate calculus in their university, but in spite of this, showed a very shallow understanding of the concept of two variable functions. They were not able to coordinate the schema that were considered to be important in the construction of this concept. If we review the historical development of the function concept as summarized in Kleiner (1989) and Sfard (1992), we find the development of the concept can be divided in three stages: The first formal definition of function was given by Johann Bernoulli in 1718: “one calls here Function of a variable a quantity composed in any manner whatever of this variable and constants.” This definition is similar to that given by Euler in 1748: “a function of a variable quantity is an analytical expression composed in any manner from the variable quantity and numbers or constant quantities.” Latter developments required that the definition of function could include cases where functions were not expressed by equations. Further developments spurred by the need for rigor, together with the great growth experienced in all fields of mathematics in the late 19th and early 20th centuries, led to the prevalent view of function as a mapping between arbitrary sets and Bourbaki’s definition of a function as a set of ordered pairs. Results show that students’ idea of a function seems to be pre-Bourbaki, but this does not deter many of them from succeeding in an undergraduate multivariable calculus course. Objectives of teachers of these courses include helping students develop a deep understanding of the concept of function. In particular we consider teachers would like to help students construct an object conception of two-variable function, as a set of ordered pairs. We can conclude that results of this study show that students’ conception of domain of a two variable function is not clearly differentiated from that of real valued functions. Their difficulties finding or describing the domain of functions can be related to a lack of coordination between the schema of R2 and that of function, and with conversions between representations. These difficulties underline the difficulty involved in the generalization that takes place in the transition between functions of a real variable and multivariable functions. It seems that the assumption that this generalization is straightforward for most of the students is not valid. Results on domain and range of functions show that these students were not able to interiorize the notion of two variable functions into a process even though they had already finished a calculus course on multivariable functions. The relationship between students’ notions of domain and range of a two variable function and their construction of the general concept of function is less clear from the data of this study. We found students who showed good understanding of domain and range of functions of two variables but whose general definition of function did not enable them to consider functions defined on arbitrary sets, and uniqueness of image. Other students’ definition of two variable functions showed some aspects that could be related to interiorization of this concept, but demonstrated a poor understanding of domain and range of specific functions in different representations. We consider that more research is needed to gain a deeper understanding of this relationship. The results of this study show that the description of how a student may construct the notion of a function of two variables has many subtleties that need to be addressed in instruction if students are to achieve at least a process conception of the modern function concept. One difficulty that had been reported has to do with the structure of students’ schema for threedimensional space, which includes building a subschema of subsets of R3 to be able to analyze graphically functions of two variables (Trigueros & Martínez-Planell, 2007). The construction of Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
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the function concept as a process requires that students interiorize actions on sets. They also need to differentiate between functions of one and two variables and be able to consider subsets (in particular, restricted domains) of R2 (and ordered pairs) as domains of functions. Further, the action of assigning a unique value to each point on a subset of R2 needs to be interiorized into a process that includes the treatments and conversions needed to identify important elements in each representation and to relate different function representations. As seen repeatedly in the analysis of results, students frequently struggle when doing actions on representation registers and when converting between them. To address this, and agreeing with Duval’s position on cognitive development (2006) and Gagatsis, et al (2004), we consider that much work was to be done with functions in different representations. Representations constitute different entities and, as such, require explicit instruction; actions to perform treatments and conversions and opportunities to interiorize them as processes must be part of the instructional process. Endnotes 1. This project was partially funded by Asociación Mexicana de Cultura A.C. References Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and development in undergraduate mathematics education. In J. Kaput, E. Dubinsky, & A. H. Schoenfeld (Eds.), Research in collegiate mathematics education II (pp. 1-32). Providence, RI: AMS Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Dordrecht, NL: Kluwer. Dubinsky, E. (1994). A theory and practice of learning college mathematics. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp.221-243). Hillsdale, NJ: Erlbaum. Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. In F. Hitt and M. Santos (Eds.), Proceedings of the XXI Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 3-26. Columbus, OH: ERIC Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1), 103-131. Gagatsis, A., Christou, C., & Elia, I. (2004). The nature of multiple representations in developing mathematical relationships. Quaderni di Ricerca in Didattica, 14, 150-159. Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282-300. Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In E. Dubinsky & G. Harel (Eds.). The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). United States: MAA. Trigueros, M. & Martínez-Planell, R., (2007). Visualization and abstraction: Geometric representation of functions of two variables. In T. Lamberg & L.R. Wiest (Eds.), Proceedings of the 29th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, 100-107.
Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.
