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Annita Monoyiou and Athanasios Gagatsis
Department of Education, University of Cyprus
This study aims to contribute to the understanding of the algebraic and coordinated approaches students develop and use in solving function tasks and to examine which approach is more correlated with students ability in problem solving. Participants were 135 preservice teachers divided in two groups according to their mathematical ability (Mathematics and General Group). Implicative statistical analysis was performed to evaluate the relation between students approach and their ability to solve problems. Results provided support for students intention to use the algebraic approach. Students who were able to use the coordinated approach had better results in problem solving. The Mathematics group used more often the coordinated approach and had better results in problem solving.
introduction and theoretical framework
The concept of function is central in mathematics and its applications. It emerges from the general inclination of humans to connect two quantities, which is as ancient as mathematics. The understanding of functions does not appear to be easy. Students of secondary or even tertiary education, in any country, have difficulties in conceptualizing the notion of function. The understanding of the concept of function has been a main concern of mathematics educators and a major focus of attention for the mathematics education research community (Dubinsky & Harel, 1992; Sierpinska, 1992). A factor that influences the learning of functions is the diversity of representations related to this concept (Hitt, 1998). An important educational objective in mathematics is for pupils to identify and use efficiently various forms of representation of the same mathematical concept and move flexibly from one system of representation of the concept to another.
The use of multiple representations has been strongly connected with the complex process of learning in mathematics, and more particularly, with the seeking of students better understanding of important mathematical concepts (DufourJanvier, Bednarz, & Belanger, 1987; Greeno & Hall, 1997), such as function. Given that a representation cannot describe fully a mathematical construct and that each representation has different advantages, using various representations for the same mathematical situation is at the core of mathematical understanding (Duval, 2002). Ainsworth, Bibby and Wood (1997) suggested that the use of multiple representations can help students develop different ideas and processes, constrain meanings and promote deeper understanding. By combining representations students are no longer limited by the strengths and weaknesses of one particular representation. Kaput (1992) claimed that the use of more than one representation or notation system helps students to obtain a better picture of a mathematical concept.
The ability to identify and represent the same concept through different representations is considered as a prerequisite for the understanding of the particular concept (Duval, 2002; Even, 1998). Besides recognizing the same concept in multiple systems of representation, the ability to manipulate the concept with flexibility within these representations as well as the ability to translate the concept from one system of representation to another are necessary for the mastering of the concept (Lesh, Post, & Behr, 1987) and allow students to see rich relationships (Even, 1998).
Duval (2002, 2006) maintained that mathematical activity can be analysed based on two types of transformations of semiotic representations, i.e. treatments and conversions. Treatments are transformations of representations, which take place within the same register that they have been formed in. Conversions are transformations of representations that involve the change of the register in which the totality or a part of the meaning of the initial representation is conserved, without changing the objects being denoted.
Some researchers interpret students errors as either a product of a deficient handling of representations or a lack of coordination between representations (Greeno & Hall, 1997; Smith, DiSessa, & Roschelle, 1993). The standard representational forms of some mathematical concepts, such as the concept of function, are not enough for students to construct the whole meaning and grasp the whole range of their applications. Mathematics instructors, at the secondary level, traditionally have focused their teaching on the use of the algebraic representation of functions (Eisenberg & Dreyfus, 1991). Sfard (1992) showed that students were unable to bridge the algebraic and graphical representations of functions, while Markovits, Eylon and Bruckheimer (1986) observed that the translation from graphical to algebraic form was more difficult than the reverse. Sierpinska (1992) maintained that students have difficulties in making the connection between different representations of functions, in interpreting graphs and manipulating symbols related to functions. Furthermore, Aspinwall, Shaw and Presmeg (1997) suggested that in some cases the visual representations create cognitive difficulties that limit students ability to translate between graphical and algebraic representations.
The theoretical perspective used in this study is mainly based on the studies of Even (1998) and Mousoulides and Gagatsis (2004). Even (1998) focused on the intertwining between the flexibility in moving from one representation to another and other aspects of knowledge and understanding. The participants were 152 college mathematics students who were also prospective secondary mathematics teachers. In the first phase of the study they completed an openended questionnaire. In the second phase ten of them were interviewed. This study indicated that subjects had difficulties when they needed to flexibly link different representations of functions. An important finding of this study was that many students deal with functions pointwise (they can plot and read points) but cannot think of a function in a global way. The data also suggested that subjects who can easily and freely use a global analysis of changes in the graphic representation have a better and more powerful understanding of the relationships between graphic and symbolic representations than people who prefer to check some local and specific characteristics. This finding cannot be generalized since in some cases a pointwise approach proved to be more powerful. In the case of problem solving a combination of the two methods is most powerful.
Mousoulides and Gagatsis (2004) investigated students performance in mathematical activities that involved principally the second type of transformations, that is, the conversion between systems of representation of the same function, and concentrated on students approaches as regards the use of representations of functions and their connection with students problem solving processes. The most important finding of this study was that two distinct groups were formatted with consistency, that is, the algebraic and the geometric approach group. The majority of students work with functions was restricted to the domain of algebraic approach. This method, which is a point to point approach giving a local image of the concept of function, was followed with consistency in all of the tasks by the students. Only a few students used an object perspective and approached a function holistically, as an entity, by observing and using the association of it with the closely related function that was given. Moreover, an important finding of the study was the relation between the graphical approach and geometric problem solving. This finding is consistent with the results of previous studies (Knuth, 2000; Moschkovich, Schoenfeld, & Arcavi, 1993), indicating that a geometric approach enables students to manipulate functions as an entity, and thus students are capable to find the connections and relations between the different representations involved in problems. Specifically, students who had a coherent understanding of the concept of functions (geometric approach) could easily understand the relationships between symbolic and graphic representations in problems and were able to provide successful solutions.
In this study the concept of function is viewed from two different perspectives, the algebraic and the coordinated perspective. The algebraic perspective is similar to the pointwise approach described by Even (1998) and the one described by Mousoulides and Gagatsis (2004). In this perspective, a function is perceived of as linking x and y values: For each value of x, the function has a corresponding y value (Moschkovich et al., 1993). The coordinated perspective combines the algebraic and the graphical approach. In this perspective, the function is thought from a local and a global point of view at the same time. The students can coordinate (flexibly manipulate) two systems of representation, the algebraic and the graphical one.
The purpose of this study is to contribute to the understanding of the algebraic and coordinated approach students develop and use in solving function tasks and to examine which approach is more correlated with students ability in solving complex problems. More specifically, the research questions were the following:
What approach (algebraic or coordinated) do preservice teachers prefer to use when they solve simple function tasks?
How able are preservice teachers to solve complex function problems?
Which approach (algebraic or coordinated) is more correlated with preservice teachers ability in solving complex problems?
Are the preservice teachers, who had a specialization in Mathematics, using more often the coordinated approach and therefore having better results in problem solving in comparison with other teachers?
method
Participants were 135 preservice teachers. The subjects were for the most part students of high academic performance admitted to the University of Cyprus on the basis of competitive examination scores. Nevertheless there are big differences among them concerning their mathematical ability. More specifically 34 of them have a special interest in mathematics and took four extra courses in the subject during their studies (Mathematics group). The other 101 teachers did not have a special interest or specialization on mathematics (General group).
A test was given. The test consisted of seven tasks. The four tasks were simple tasks with functions (T1a, T1c, T2a, T2c, T3a, T3c, T4a, T4c). In each task, there were two linear or quadratic functions. Both functions were in algebraic form and one of them was also in graphical representation. There was always a relation between the two functions (e.g. f(x)= 2x, g(x)= 2x+1). Students were asked to interpret graphically the second function. The other three tasks were complex problems. The first problem consisted of textual information about a tank containing an initial amount of petrol (600 L) and a tank car filling the tank with petrol. The tank car contains 2000 L of petrol and the rate of filling is 100 L per minute. Students were asked to use the information in order to give the two equations (Pr1a), to draw the graphs of the two linear functions (Pr1b) and to find when the amounts of petrol in the tank and in the car would be equal (Pr1c). The second problem consisted of textual and algebraic information about an ant colony. The number of ants (A) increases according to the function: A=t2+1000 (t= the number of days). The amount of seeds, the ants save in the colony, increases according to the function S=3t+3000 (t= the number of days). Students were asked to use the information in order to draw the graphs (Pr2a) of the quadratic and linear functions and to find when the number of ants in the colony and the number of seeds would be equal (Pr2b). The third problem consisted of a function in a general form of f(x) = ax2+bx+c. Numbers a, b and c were real numbers and the f(x) was equal to 4 when x=2 and f(x) was equal to 6 when x=7. Students were asked to find how many real solutions the equation ax2+bx+c had and explain their answer (Pr3). The test was administered to students by researches in a 60 minutes session.
The results concerning students answers to the four tasks were codified by an uppercase T (task), followed by the number indicating the exercise number. Following is the letter that signifies the way students solved the task: (a) a was used to represent algebraic approach function as a process to the tasks, (b) c stands for students who adopted a coordinated approach function as an entity. A solution was coded as algebraic if students did not use the information provided by the graph of the first function and they proceeded constructing the graph of the second function by finding pairs of values for x and y. On the contrary, a solution was coded as coordinated if students observed and used the relation between the two functions in constructing the graph of the second function. In this case students used and coordinated two systems of representation the algebraic and the graphical one. They noticed the relationship between the two equations given and they interpreted this relationship graphically by manipulating the function as an entity.
The following symbols were used to represent students solutions to the problems: Pr1a, Pr1b, Pr1c, Pr2a, Pr2b and Pr3. Right and wrong or no answers to the problems were scored as 1 and 0, respectively.
For the analysis of the collected data the similarity statistical method (Lerman, 1981) was conducted using a computer software called C.H.I.C. (Classification Hirarchique, Implicative et Cohsitive) (Bodin, Coutourier, & Gras, 2000). A similarity diagram and an implicative diagram (Gras, Peter, Briand, & Philippe, 1997) of students responses at each task or problem of the test were constructed. The similarity diagram, which is analogous to the results of the more common method of cluster analysis, allows for the arrangement of the tasks into groups according to the homogeneity by which they were handled by the students. This aggregation may be indebted to the conceptual character of every group of variables. The implicative diagram, which is derived by the application of Grass statistical implicative method, contains implicative relations that indicate whether success to a specific task implies success to another task related to the former one. It is worth noting that CHIC has been widely used for the processing of the data of several studies in the field of mathematics education in the last few years (e.g., Evangelidou, Spyrou, Elia, & Gagatsis, 2004; Gagatsis, Shiakalli, & Panaoura, 2003; Gras & Totohasina, 1995). An independent sample TTest was also performed to examine if there are statistically significant differences between different groups of students (Mathematics and General Group) concerning their approach in the four tasks and their efficiency in problem solving by using SPSS.
results
The main purpose of the present study was to examine the mode of approach students used in solving simple tasks in functions and to test which approach is more correlated with solving complex mathematical problems. According to Table 1, most of the students correctly solved Task 1 and 2. Task 1 involved a linear function and Task 2 the simplest form of an equation of a parabola (y= x2). Their achievement radically reduced in tasks involved complex quadratic functions (T3 and T4). More than half of the students chose an algebraic approach to solve the first three tasks. In Task 4 most of the students chose a coordinated approach. In this task a coordinated approach seemed easier and more efficient than the algebraic.
T1 T2T3T4Algebraic approach with correct answer 54.8*54.856.324.4Coordinated approach with correct answer 32.531.117.748.1Incorrect answer/
No answer12.714.12627.5Table 1: Students responses to tasks
*Numbers represent percentages.
In the case of Task 1 (y=2x, y=2x+1), some students who used an algebraic approach found the points of section with x and y axis and constructed the graph. Others constructed a table of values in order to help them construct the graph. The students who used a coordinated approach compared the two equations and mentioned that the slope was the same and the two functions are parallel. Then they referred to the fact that the points of the second function are one more than the points of the other. Some of them found a point in order to verify their assertion.
In the case of Tasks 2 (y=x2, y=x21) and 3 (y=x2+3x, y=x2+3x+2), students who used an algebraic approach found the real solutions of the second equation and the minimum point and constructed the graph without using the first graph. In contrast, students who used a coordinated approach first compared the two equations and realized that they are parallel. Then they mentioned that the minimum point in the first case is one down and in the second case two above. Some of them found another point in order to draw the graph more precise. In the case of Task 4 (y=3x2+2x+1, y=(3x2+2x+1)), the students who used an algebraic approach found the point of section with yaxis and the maximum point. The students who used a coordinated approach compared the two equations and mentioned that the two functions are opposite and symmetrical to the xaxis. In this task, an algebraic approach was more complicated due to the fact that the equation does not have real solutions. Most of the students, after an unsuccessful effort to find the points of section with xaxis drew the graph using a coordinated approach.
Table 2 shows students responses to complex problems. Students performance was moderate. In Problem 1 only 38.5% of the students managed to use the information given in order to give the two equations. A larger percentage constructed the two graphs correctly (59.2%) and found their point of section (70.4%). Many students were unable to give the equations but manage to construct the graphs by constructing a table of values for x and y. Some of the students did not construct the graphs but found their point of section by using the table of values. In Problem 2 almost half of the students managed to construct the graphs (46.6%). A smaller percentage (35.5%) found their point of section. In this problem in order to find the point of section the students had to solve a second degree equation and that caused difficulties. Problem 3 was very difficult for the students since only 37% of them managed to solve it correctly.
Problem 1 Problem 2 Problem 3 a bcabaCorrect answer 38.559.270.446.635.537Incorrect answer/ No answer 61.540.829.653.464.563Table 2: Students responses to problems
*Numbers represent percentages.
Students correct responses to the tasks and problems are presented in the similarity diagram in Figure 1. More specifically, two clusters (i.e., groups of variables) can be distinctively identified. The first cluster consists of the variables T1c, T2c, T3c, T4c, Pr1a, Pr1b, Pr1c, Pr3, Pr2a and Pr2b and refers to the use of the coordinated approach and the solving of problems. The second cluster consists of the variables T1a, T2a, T3a and T4a which represent the use of algebraic approach.
Figure 1: Similarity diagram of the variables
From the similarity diagram it can observed that the first cluster includes the variables corresponding to the solution of the complex problems with the variables representing the coordinated approach. More specifically, students coordinated approach to simple tasks in functions is closely related with effectiveness in solving problems. This close connection may indicate that students, who can use effectively different types of representation in this situation both algebraic and graphical representation are able to observe the connections and relations in problems, and are more capable in problem solving.
Figure 2: Implicative diagram of the variables
Figure 2 illustrates the implicative diagram of the variables. The results of the implicative analysis are in line with the similarity relations explained above. Two separate chains of implicative relations among the variables are formed, namely Chain A and Chain B. The two groups of implications correspond to the two similarity clusters of the diagram presented above. Chain A involves the variables concerning the use of algebraic approach. Chain B refers to variables concerning the use of the coordinated approach and variables concerning solution to the problems. Chain B indicates that students who used a coordinated approach to solve the Task 1, 2 and 3 and succeeded in those tasks also solved correctly the three problems. According to the above diagram, students who can coordinate two systems of representation and flexibly move from the one to the other, have a solid and coherent understanding of functions and therefore are able to solve complex problems.
In order to determine whether there are significant differences between the two groups (Mathematics and General Group) concerning the approach they used and their performance in problem solving, the independed samples TTest was performed. The mean value of Mathematics group concerning the coordinated approach (EMBED Equation.3= 2.65, SD= 1.23) was statistically significant higher (t=6.95, df =133, p=.000) than the mean value of the General group (EMBED Equation.3=1.00, SD=1.18). In contrast, the mean value of Mathematics group concerning the algebraic approach (EMBED Equation.3= 1.35, SD= 1.23) was statistically significant lower (t=5.26, df =133, p=.000) than the mean value of the General group (EMBED Equation.3=2.71, SD=1.33). As far as the problem solving concerns the Mathematics group (EMBED Equation.3=5.41, SD=0.86) outperformed the General group (EMBED Equation.3=2.02, SD=1.89) and this difference was statistically significant (t=10.12, df =133, p=.000).
The Mathematics group used more often the coordinated approach in order to solve the four simple tasks and had also better results in the complex problem solving.
discussion
A main question of this study referred to the approach preservice teachers use in order to solve simple function tasks. It is important to know whether preservice teachers are flexible in using algebraic and graphical representations in function problems. Most of the students used an algebraic approach in order to solve the simple function tasks. A coordinated approach is fundamental in solving problems even though many students have not mastered even the fundamentals of this approach. This finding is in line with the results of other studies that suggest that many students deal with functions pointwise (Even 1998; Bell & Janvier, 1981). Students can plot and read points, but cannot think of a function as it behaves over intervals or in a global way. These studies also indicate that a global approach to functions is more powerful than a pointwise approach. Students who can easily and freely use a global approach have a better and more powerful understanding of the relationships between graphic and algebraic representations and are more successful in problem solving. Students preference in the algebraic solution is probably the curricular and instructional emphasis dominated by a focus on algebraic representations and their manipulation (Dugdale, 1993). In their textbooks, students are usually asked to construct graphs from given equations using pairs of values. As a result, students fail to connect algebraic and graphical representations and therefore fail to develop a globalcoordinated approach.
Students performance in problem solving was moderate. Although problems used in this study are some of those taught at school, subjects had difficulties. This finding suggests that in order to give a correct solution to a complex function problem the students must be able to handle different representations of function flexibly and move easily from one representation to the other.
Furthermore, an important finding of this study is the relation between the coordinated approach and the problem solving. The data presented here suggest that students who have a coherent understanding of the concept of function (coordinated approach) can easily understand the relationships between symbolic and graphical representations and therefore are able to provide successful solutions to complex problems. Moreover, data provided support that there is a close relationship between the use of a coordinated approach in functions and better understanding of equations, graphs and functions in general.
Although the participants of this study were preservice teachers they had many differences concerning their mathematical ability. Some of the students had a special interest in mathematics (Mathematics group). The Mathematics group, who mainly used the coordinated approach to solve the simple tasks and therefore were able to use an algebraic and a graphical representation at the same time, were very successful in problem solving. On the other hand, the General group mainly used an algebraic approach and their performance in problem solving was poor. Its obvious that the students who dealt with mathematics systematically in University by taking extra courses had developed a conceptual and deeper understanding of the concept of function. They were able to handle different representations of the concept, easy translate one representation to the other and as a consequence they were more successful in problem solving.
References
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@Times New Roman@Times New Roman"2
Annita Monoyiou@ !q
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<and Athanasios Gagatsis
2
P2
.Department of Education, University of Cyprus
2
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@ @Times New Roman @Times New Roman@Times New Roman @Times New Roman@Times New Roman @Times New Romanw2
@HThis study aims to contribute to the understanding of the algebraic and
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@Uexamine which approach is more correlated with students ability in problem solving. o
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@Participants were 135 pre
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:service teachers divided in two groups according to their
w2
(@Hmathematical ability (Mathematics and General Group). Implicative statis
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:@Oanalysis was performed to evaluate the relation between students approach and
2
L@Ytheir ability to solve problems. Results provided support for students intention to use ,2
^@the algebraic approach:2
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@INTRODUCTION AND THE
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ORETICAL FRAMEWORK E
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@IThe concept of function is central in mathematics and its applications. Ih 2
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@Vfrom the general inclination of humans to connect two quantities, which is as ancient
2
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@Lof secondary or even tertiary education, in any country, have difficulties i
2
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@Uconceptualizing the notion of function. The understanding of the concept of function
2
@Thas been a main concern of mathematics educators and a major focus of attention for t2
(@Fthe mathematics education research community (Dubinsky & Harel, 1992;
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:K2). A factor that influences the learning of functions is the diversity of i2
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^@Xobjective in mathematics is for pupils to identify and use efficiently various forms of 2
p@representation
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@QThe use of multiple representations has been strongly connected with the complex r g2
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@Smathematical situation is at the core of mathematical understanding (Duval, 2002). G2
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