ated with this concept (Janvier, 1987, p. 27), and the difficulties presented in the processes of articulating systems of representation of a concept involved in ...
/ I ! I D JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (1), 123-134 ISSN 0364-0213. t,/ I¥1 Copyright © 1998 Ablex Publishing Corp. All rights of reproduction in any form reserved.
Difficulties in the Articulation of Different Representations Linked to the Concept of Function FERNANDO HITT Departamento de Matemdtica Educativa, Cinvestav, Mdxico
The concept of function admits a variety of representations. Understanding the concept implies coherent articulation of the different representations which come into play during problem solving. Experimental studies with secondary school students have demonstrated that some representations are more difficult to articulate than others. Mathematics teachers also have problems of translation preserving meaning when passing from one representation to another. Some of these problems were identified in preliminary studies. On the basis of the latter, fourteen questionnaires were designed in order to explore these difficulties. The results show that, for a given task, the difficulties of teachers are not the same as those of their students.
1.
CONCEPT OF FUNCTION
The concept of function is of fundamental importance in the learning of mathematics. Eisenberg (1992, p. 174) states that "the notion of developing in students a sense for functions should be a major goal of the secondary and collegiate curriculum." Achieving this goal does not appear to be easy, given the diversity of representations associated with this concept (Janvier, 1987, p. 27), and the difficulties presented in the processes of articulating systems of representation of a concept involved in problem solving. Research carried out in recent years with respect to the concept of function shows certain levels of understanding of the concept. Thus, we can classify at the first level those students who demonstrate an incoherent mixture of different representations of the concept after undergoing a process of learning. Students who can identify different representations (see, for example, Vinner, 1983; Markovits, Eylon, & Bruckheimer, 1986) would be classified at another level. For the purposes of this article, we have adopted the idea of system of representation as used by Kaput (1987, 1991). A number of experiments involving specific tasks of translation of problems from one system of representation to another, with the preservation of meaning, have been carried out (see Duval, 1988; Vinner & Dreyfus, 1989; Ruthven, 1990; Dubinsky & Harel, 1992; Sfard, 1992). Monk (1992, pp. 181-182) studied the presentation to students of problems where the statement of the problem does Direct all correspondence to: Fernando Hitt, Departamento de Matematica Educativa del Cinvestav, Av. Instituto Politecnio Naciona12508, Col. San Pedro Zacatenco, C.P. 07360, Mexico, D.F..
123
Definition of the function concept (usual text book definitions)
C14.
Equality of functions
C9.
Mathematical statements
Construction of functions
C8.
C13.
Functions represented graphically
C7.
Functions expressed algebraically
Functions expressed algebraically
C6.
C12.
Evaluation of functions
C5.
Functions in a context
Statements of functions
C4.
C11.
Statement of problems in the context of real life
C3.
Functions represented graphically
Subeoncepts of concept of function (Domain, Image Set, Rank)
C2.
C10.
Graph of functions
C1.
Questionnaires
Presentation of 4 definitions: relationship between variables, ordered pairs, rule of correspondence, entry-exit
Mathematical statement
Algebraic
Geometric, pictorial
Graphic
Verbal, algebraic
Statement of properties of certain functions
Graphic
Algebraic
Algebraic
Verbal
Verbal
Graphs of functions with indication of points.
Curves on the plane
Presentation
TABLE 1. Description of the Questionnaires
Decision as to truth or falsity of definitions Classification for teaching purposes
Proof or search for counter-examples
Operation with functions (+, -,., o)
Articulation between representations: physical context, symbolic algebraic and pictoric
Articulation between representations: pictorial, symbolic-algebraic and physical context
Identity functions which are equal
Construction of functions with these properties in algebraic and/or graphic form
Traslation from the graphic to the algebraic representation
Traslation from the algebraic to the graphic representation
Given numerical points or letters, calculate value of the function at that point
Identification of functions and writing down their definition
Tabulation and graphic
Identification of points of Domain, Image Set, Rank
Identification of functions
Required task
CONCEPT OF FUNCTION
125
not indicate in a direct manner the system or systems of representation required to solve it. The studies mentioned above show different levels of understanding of the concept of function. This body of research allows us to identify the following levels in the construction of the particular concept of function. However, we propose that the same levels are valid for other concepts. Level 1. Level 2. Level 3. Level 4. Level 5.
Imprecise ideas about a concept (incoherent mixture of different representations of the concept). Identification of different representations of a concept. Identification of systems of representation. Translation with preservation of meaning from one system of representation to another. Coherent articulation between two systems of representation. Coherent articulation of different systems of representation in the solution of a problem.
We will look at a mathematical concept which is stable in an individual. Can he or she coherently articulate the different representations admitted by the concept in a specific task? A central goal of mathematics teaching is thus taken to be that the students be able to pass from one representation to another without falling into contradictions. Some of the learning problems produced by the way students are taught are left aside with this formulation. On the other hand, the intrinsic difficulty of the concepts themselves should not be forgotten. In effect, we try to measure such difficulties in our mathematics teachers. As Norman (1992, p. 229) points out, "while there is a considerable body of literature addressing issues of students' understanding of the function concept, little has been done that has addressed secondary teachers' knowledge of this crucial concept." This study looks at errors committed by some mathematics teachers when they are confronted with the concept of function. The study forms part of a research project on the mistakes committed by teachers and students when they carry out a task related to the function concept. To this end, a series of fourteen questionnaires were designed, which we refer to as C1, C2 . . . . . C14 (see Table 1); these instruments allow us to distinguish the levels we referred to previously. Their structure is summarized by the specification of three elements as indicated in Table 1; content (left-hand column), required articulation task (right-hand column) and different representations of functions (central column). The questionnaires were designed to include different representations used in teaching aimed at the construction of the function concept. Both teachers and students participated in our experiments. This study refers exclusively to the performance of the mathematics teachers on our questionnaires.
2. EXPERIMENTAL PROCEDURE (DESCRIIrI'ION OF THE STUDY) The participants were 30 mathematics teachers (secondary level) who were beginning a postgraduate course on mathematics education. Two questionnaires per week were presented to the teachers for seven consecutive weeks. The teachers, working individually, had one hour to answer each one.
126
HITr
f y
X
FIGURE 1.
y~
10 err. 2 abst.
9 err., 1 abst.
9 err., 0 abst.
12 err. 0 abst.
8 err., 1 abst.
FIGURE 2. Errors and abstentionslinked to conic curves
This article reports the results obtained on some of the questionnaires, those where the results had the greatest relevance to our present concerns. The teachers' responses to questionnaire C3, C5, C6, C7 and C12 (see Table 1) were correct in general terms. The problems revealed by responses to C13 (construction of non-examples and proof) were similar to those obtained by an earlier study (Hitt, 1989), for which reason we have excluded them from this report.
3. 3.1.
ANALYSIS OF T H E R E S U L T S
C o m m e n t s on Questionnaires C1, C4 and C14
Questionnaire C1 presented the teachers with 26 curves, some of which represented a graphs of functions while others did not. For each item, the subject was asked to indicate
CONCEPT OF FUNCTION
127
X
4 err., 5 abst. F I G U R E 3.
Errors and abstentions linked to an "irregular curve"
whether the graphic representation corresponded (true) or not (false) to a function. A reason for the response was required. The second item was the graph showed in Figure 1. Twenty nine teachers said that this curve did not represent the graph of a function; that a teacher was in error, without giving reasons. The argument of the teachers were distributed as follows: two teachers used a definition of ordered pairs, ten teachers wrote that there were more than one image in certain points; six teachers explicitly used a vertical line cutting the curve in more than one point, eleven teachers said that there is not a graph of a function (without giving arguments). When teachers were shown conic curves like those in Figure 2, the six teachers who used a vertical line followed the same strategy, answering correctly. Are errors related to conic curves due to existence of an analytical expression? It seems that the answer is affirmative. That is, it seems the existence of analytical expression is part of the internal representations of the concept of function teachers have. Moreover, it seems that belief is stronger in some teachers than the formal definition of function they have. Approximately 33% of our group committed the error of thinking that conic sections (with principal axis on the x-axis, see Figure 2) are functions. The existence of an algebraic expression associated with a curve led them to abandon their definition of function. None of them used the definition of function, or explicitly used a vertical line, in their reasoning. In contrast to the results obtained for the students of Vinner (1983) and Vinner & Dreyfus (1989, p. 363), the "irregularity of the curve" caused more abstentions than those related to conic sections. For the graph in Figure 3 there were twenty-one teachers who correctly labeled it as a function, ant nine who did not. Question 1 in questionnaire C4 asked for the definition of the concept of function. In C14, taking into account the classification given by Nicholas (1966), we presented the teachers with four different, standard definitions of the function concept (taken from the usual textbooks) in terms of: (a) a Rule of Correspondence, (b) a Set of Ordered Pairs, (c) a Relationship between Variables, and (d) Entry-Exit. The teachers had to decide whether the definition given was correct or incorrect. Later, they had to classify those that they had indicated were true in order of preference from the point of view of teaching. The results were as follows: in C4 there were eighteen teachers who gave their definition in terms of the Rule of Correspondence (see Table 2). Six of them changed their definition in C14 (from the perspective of teaching), five chose Ordered Pairs and one the Relationship between Variables). Ten teachers defined function in terms of Ordered Pairs
128
HFfT TABLE2. Definitionof Function: Questionnaires C4 and C14
Definition of Function
C4
Rule of correspondence Rule of correspondence Rule of correspondenceone to one Rule of correspondenceone to one and onto Rule of correspondence(erroneous) Set of ordered pairs Abstention Relationship between variables Input-Output
14 3 3 1 1 10 1 0 0
Change of Definition 5 + ~
C14 ~ 3
~ 92 3
1 1 2 1 ~ - ~ +
5 3
0
in C4. Two of these changed definition in C14 (one for the Rule of Correspondence, and the other for the Relationship between Variables). None of the definitions given by the teachers when answering C4 corresponded to a definition in terms of the Relationship between Variables or Entry-Exit. The results regarding preference from the teaching perspective (C14) were as follows: in the first place, Rule of Correspondence (14 teachers); in second place, Set of Ordered Pairs (13 teachers); in third place, definition of the Relationship between Variables (3 teachers). Nicholas (1966, p. 767) suggests that the most appropriate definition from the point of view of teaching is the Relationship between Variables. However, our results show that it is not the choice of the teachers in our study. 3.2.
C o m m e n t s on Questionnaires C8 and C9
We will now analyze the responses to the questionnaires which refer to the construction of functions. Firstly however, we must point out that the idea of function as the relationship between variables by means of algebraic expression arises at the very beginnings of the concept (Hitt, 1989; Youschkevitch, 1976). Questionnaires on the function concept (Hitt, 1989) have verified that where the construction of functions is concerned, teachers have a marked tendency to construct continuous functions defined by a single algebraic expression. On this basis, several questions were designed to explore this phenomenon. Questionnaire C9 (equality of functions) was designed in order to detect possible weakness of the teachers when comparing two functions. We wanted to see whether different ways of writing the same function could generate misunderstandings. In general the responses of the teachers was satisfactory. Thus, for example, in the case of question 2, which asked if f(x) = 2 was equal to g(x) = ~ for all x ~ R, there were 26 correct responses. However, this performance on the part of the teachers was not observed when the skill was put to the test. That is, the difficulty of a task provokes the emergence of intuitive ideas during the process of problem solving, some of which are erroneous; furthermore, the individual is not wholly conscious of these ideas, as we shall demonstrate.
CONCEPT OF FUNCTION TABLE 3.
129
Subconcepts: Domain, Image and Isolated Points on the Curve Identification of points
In the Domain Images of points Points on the curve
Correct
Partially correct
Incorrect
Abstention
13
4
11
2
6
2
16
2
24
1
3
2
Question 12 of questionnaire C8 on the construction of functions asks the respondent to construct three distinct real variable functions, such that Ifl(X)l = If2(x)l = If3(x)l = 2. The results were as follows: six teachers correctly constructed the three functions; ten teachers constructed the first two and repeated the third (for example, fl(x) = 2, f2(x) = - 2 and f3(x) = 4r4 or '~f3(x) = 3 ~ ); nine teachers correctly proposed the first function and committed errors in the other two. Formally, these nine responses could be considered as correct. For example, these teachers wrote f2(x) = (6x + 2)/(3x + 1) and f3(x) = (4x2-4)/(2x 2 -2). Thus, since f2 is not defined in x = -1/3 and neither is f3 defined in x = ±1, they are different in one or two points from the first function. But the answers given to question 18 show that, in fact, they believe that the difference between the algebraic expressions makes the functions different. All the answers given by one of the teachers were incorrect and there were four abstentions. Another verification of the importance given to the algebraic representation of a continuous function is found in the answers given to question 18. We asked for two examples o f real variable functions, each one such that (foy)(x) = flf(x)) = 1. Not one teacher gave a correct response, correctly constructing two functions with this property. There were twenty one correct responses for the first function (fl(x) = 1), which was repeated for the second one, writing it in a different way. For example, twelve teachers used f2(x) = sin2x + cos2x. The wording of the question led to the idea of function as a relationship between variable being forgotten. But the intuitive idea of function used by Bernoulli, Euler, etc., as an algebraic expression, is similar to that of the teachers in our research. It is in this way that this idea appears as an epistemological type of obstacle. The results show
that the teachers have a marked preference for continuous functions defined exclusively by a formula. This lack of options in problem solving can produce errors. 3.3.
Comments on Questionnaire C2
Like students at secondary school (Markovits et al., 1986, p. 21), the teachers exhibit errors when they confound the subconcepts of the concept of function (Domain, Image Set). For example, question 1 of C2 asks for the identification of some points on the graph of a function (see Table 4, item 3). Only five teachers correctly identified the points (Domain, Images and points on the curve). The difficulties concentrated around
HITT
130
TABLE 4. Subconcept: Domain Points on the domain which have A as image
Question c2-3 Correct
Partially
Incorrect
Abstention
COrreCt
(a) 15
0
7
8
6
8
8
8
4
7
10
9
21
0
5
4
21
0
4
5
21
0
4
5
(b)
(c)
(d)
I I
(e)
(t3
the identification o f D o m a i n and I m a g e Set in the graphic representation o f the func tion (Table 3).
131
CONCEPT OF FUNCTION TABLE 5.
Graphic Representation to Physical Context
Graphic Representation to Sketch Question
Correct
Incorrect
Most frequent error
Abstention
(a) 4
~
3
~
1
2
2
2
h
w
11
17
V = volume,
FIGURE 4.
12
h = height
Decomposition of container in two parts
Question 3 of C2 asks that by means of diagrams, given a point in the Image Set in each diagram, the points of the Domain that have this Image under the implied function be indicated (see Table 4). The results show that the diagrams of items (e) and (f) (in Table 4) favor success, possibly due to the ease with which the Domain and the Image Set can be located visually in these diagrams. It should be noted that the functions in these two items are not one to one. Item (d) has a high degree of success, like items (e) and (f). This time the reason is very probably that the function is one to one, and the point asked for in the Domain is zero. In contrast, the element of Domain asked for is not zero in item (a), and the number of correct answers fell to fifteen. Greater difficulty is found where the functions are not one to one. This is reflected in the number of correct responses to items (b) and (c). Markovits et al. (1986) reported similar performances for secondary school students but with higher percentages of error. 3.4.
C o m m e n t s on Questionnaires C10 and C l l
The questions on C10 address the articulation between graphic representations in a real context. The teachers were shown graphs where the independent variable represented the
132
HITT TABLE 6.
Physical Context to Graphic Representation
Sketch to graphic representation Question
Correct
Incorrect
2
28
Most frequent error
Abstention
(a) 17
7
0
12
6
0
13
3
0
h
h
Surface area v s height
5
25
~
h Surface area vs height
(c) 2
17 h
h
Surface area v$
height
{d) 28
2
!
2
0
h' volume v s height
(e) 16
13
8 t
'\
5
1
t
speed vs time
height of a liquid during the filling of an unknown container and the dependent variable represented the surface area of the liquid or its volume. They were asked to draw a container that fitted the proposed situation (see Table 5). The interpretation of the independent variable in the graphs in C 10 and C11 presupposes an articulation with the system of representation of symbolic algebra, as well as interpretation of the representation in a real context. In the case of graph (a) in Table 5, there were twenty four correct responses. The numbers of errors increased for graph (b). We obtained eleven sketches of four different correct containers and seventeen erroneous drawings. The most common incorrect responses are indicated. It can be said based on responses to these questionnaires that complicating the problem
does not necessarily lead to refinement of the process of finding a solution. Here the form of the graph calls up intuitive ideas that suggest the form of the container. This phenomenon has been called "Iconic translation" by Monk (1992, p. 176). Questionnaire C 11 asks for the inverse process of articulation, that is, given the drawing of a container, a graph should be drawn in which the independent variable represents the
CONCEPT OF FUNCTION
133
height of a liquid and the dependent variable, its surface area. The first three questigns were very difficult for the teachers. In the three activities, they all drew two lines, as if they had mentally cut the containers in two (see Figure 4). In the first two exercises, where the containers had straight sides, they drew graphs with straight segments (see Table 6). In the case of the third container, twenty six graphs were drawn which included curves; however, given the complexity of the question, only two responses were correct. Once again, twenty eight correct responses were obtained for the simplest case, that of container (d). The iconic translation made by five teachers when they answered question (e), produced five erroneous responses (see Table 6), which associated the physical context of a graph with a straight line in the same direction of the mouvement of the body. In summary, we can classify the errors according to the following: • •
That the form of the graph determined the type of container the teachers drew (iconic translation). That the independent variable was not analyzed in either an analytic or graphic context.
In general, the visual representations of some containers are not articulated correctly with the analytic and graphic representations.
4.
CONCLUSIONS
Geometric reasoning based on orthogonal lines in order to decide whether a curve represents a function is disrupted in one third of our population, as a consequence of the existence of an algebraic expression associated with the curve (ellipse, circle, etc.). The latter provokes errors. That is, when one of the latter representations is present in the teacher and plays a relevant role, at that moment it displaces other representations to another level. The teachers did not easily identify the subconcepts of the function, Domain and Image Set, in graphic representations with Cartesian axes. The definition of function related to the concept of variable is not favored by the mathematics teachers. They prefer a Rule of Correspondence or Set of Ordered Pairs. The teachers do not use the definitions; these are relegated to a subsidiary level. The task of identifying functions does not pose a problem for the teachers. However, in the more complicated activity of constructing functions, the cognitive obstacle constituted by limitation to continuous functions defined by a single algebraic expression does not allow the vast majority of the teachers to construct different functions. In the case of the questionnaires which asked for the interpretation of a graph in a physical context, the form of the graph (if it is not a single line) evoked the form of the container in a high percentage (34%) of responses. In the case of going from the physical context to the graphic representation, the same phenomenon occurred. The movement of an object on a sloping plane induced some teachers (16%) to produce a linear graph in the same direction of the movement (time against speed). This shows that the independent variable is not identified and isolated in order to put it into context in an analytic and graphic representation. The high percentage of correct responses on the questionnaires C3, C5, C6, C7 and C 12, described in Table I gives us an idea of what the teachers know of the function concept corn-
134
H1TI"
pared with their students' knowledge. For example, in contrast to Ruthven's results (1990), the good performance of the teachers on C12 (questions similar to those asked by Ruthven), allows us to say that they have a certain knowledge of forms, if they are representations of functions, which can be articulated with their counterparts in an algebraic system. The results show that in unusual learning situations, this group of teachers does not coherently articulate between the various systems of representation involved with the concept of function, due to various difficulties.
REFERENCES Dubinsky, Ed, & Hard, Guershon (1992). The nature of the process conception of function. In G. Harel & Ed Dubinsky (Eds.), The concept offunction: Aspects of epistemology and pedagogy (pp. 85-106). Mathematical Association of America. Dural, Raymund. (1988). Graphiques et equations: l'Articulation de deux registres. Annales de Didactique et de Sciences Cognitives, 1,235-253. Eisenberg, Theodore. (1992). On the development of a sense for functions. In Guershon Harel & Ed Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 153174). Mathematical Association of America. Hitt, Fernando. (1989). The construction of function, contradiction and proof. Proceedings of the 13th International Conference for the Psychology of Mathematics Education, Vol. 2, pp. 107-114. Janvier, Claude (Ed.). (1987). Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Erlbaum. Kaput, James. (1987). Representation systems and mathematics. The notion of function as an example. In C. Janvier (Ed,), Problems of representation in the teaching and learning of mathematics (pp. 67-71). Hillsdale, NJ: Erlbuam. Kaput, James. (1991). Notations and representations as mediators of constructive processes. In E. Von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 53-74). City, Country: Kluwer Academic Publishers. Markovitz, Zvia, Eylon, Bat-Sheva, & Bruckheimer, Maxim. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2), 18-28. Monk, Steve. (1992). Students' understanding of a function given by a physical model. In G. Harel & Ed Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175-194). Mathematical Association of America. Nicholas, C. P. (1986). A dilemma in definition. The American Mathematical Monthly, 73, 762-768. Norman, Alexander. (1992). Teachers' mathematical knowledge of the concept of function. In G. Harel & Ed Dubinsky (Eds.), The concept offunction: Aspects of epistemology and pedagogy (pp. 215-232). Mathematical Association of America. Ruthven, Kenneth. (1990). The influence of graphic calculator use on traslation from graphic to symbolic forms. Educational Studies in Mathematics, 21, 431-450. Sfard, Anna. (1992). Operational origins of mathematical objects and the quandary of reification: The case of function. In G. Harel & Ed Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 59-84). Mathematical Association of America. Vinner, Shlomo, & Dreyfus, Tommy. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366. Vinner, Shlomo. (1983). Concept definition, concept image and the notion of function, Internation Journal of Mathematics Education in Science and Technology, •4(3), 293-305. Youschkevitch, A. (1976). The concept of function up to the middle of 19th century. Archive for History of Exact Sciences, 16, 36-85.
