Teaching the concept of function: Definition and problem solving Areti Panaoura, Paraskevi Michael-Chrysanthou, Andreas Philippou

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Areti Panaoura, Paraskevi Michael-Chrysanthou, Andreas Philippou. Teaching the concept of func- tion: Definition and problem solving. CERME 9 - Ninth Congress of the European Societyfor Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.440-445. ￿hal-01286927￿

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Teaching the concept of function: Definition and problem solving

Areti Panaoura1, Paraskevi Michael-Chrysanthou2 and Andreas Philippou2

1 Frederick University, Department of Education, Nicosia, Cyprus, [email protected] 2 University of Cyprus, Department of Education, Nicosia, Cyprus, [email protected], [email protected]

The present study investigates students’ abilities to un- subjects, such as physics (Sanchez & Llinares, 2003). In derstand the concept of function. Secondary education fact, the results we present in this paper are a part of students were asked to (i) define the concept of function a large scale cross-sectional study examining the use and present examples of functions, (ii) translate between of different modes of representations in functions at different representations of a function and (iii) solve the secondary school level. Adopting a developmental function problems. Findings revealed students’ great perspective at different grades of secondary educa- difficulties in proposing a definition of function, in tion, we aim to trace students’ abilities in defining solving tasks of conversions between different modes functions, recognizing and manipulating them across of representation, and in solving function problems. representations and in problem solving, emphasizing Based on the students’ abilities and misconceptions the approach used (algebraic or geometric). Thus, our about functions, teaching practices for improving the main questions are: (i) What abilities do students have students’ understanding of functions are discussed. to define and flexibly manipulate functions and solve function problems?, and (ii) What are the differences Keywords: Function, definition, use of representations, in students’ performance at the 3rd, 4th and 5th grades teaching methodology. of secondary education? Based on our results, we provide suggestions for teaching practices that can facilitate students’ understanding of functions. INTRODUCTION THEORETICAL FRAMEWORK For more than twenty years, the concept of function has been internationally considered as a unifying The role of multiple representations theme in mathematics curricula (Steele, Hillen, & in the understanding of functions Smith, 2013). Students face many instructional ob- There is strong support in the mathematics education stacles when developing an understanding of func- community that students can grasp the meaning of a tions (Sajka, 2003; Sierpinska, 1992). Kieran (1992) mathematical concept by experiencing multiple math- questions whether students’ inability of conceptually ematical representations of that concept (Sierpinska, understanding functions is related to its teaching or 1992). One of the main characteristics of the concept of is due to students’ inappropriate way of approach- functions is that they can be represented in a variety of ing function tasks. Sajka (2003) indicates that stu- ways (tables, graphs, symbolic equations, verbally) and dents’ abilities in solving tasks involving functions an important aspect of its understanding is the ability are influenced by the typical nature of school tasks, to use those multiple representations and translate leading to the use of standard procedures. According the necessary features from one form to another (Lin to a standard didactic sequence, students are asked & Cooney, 2011). In order to be able to use the different to infer the properties of a function using the given forms of representations as tools in order to construct graph, by following a specific procedure. a proof, students have to understand the basic features of each representation and the limitations of using In relation to the above, our study examines students’ each form of representation. conceptions of function, as it is one of the most impor- tant topics of the curriculum and it is related to other

CERME9 (2015) – TWG03 440 Teaching the concept of function: Definition and problem solving (Areti Panaoura, Paraskevi Michael-Chrysanthou and Andreas Philippou)

According to Steele and colleagues (2013), it is typical at 3rd grade of gymnasium (15 years old), 258 students in the U.S. for the definition of a function and a con- at 4th grade of lyceum (16 years old) and 183 students nection to the graphic mode of the function concept to at 5th grade of lyceum (17 years old). The students at 5th occur in the late middle grades, while the formal study grade follow a scientific orientation and attend more of function with an emphasis on symbolic and graphi- advanced levels of mathematics courses. Students’ cal forms occurs in high school. According to Bardini, participation was due to the voluntarily participa- Pierce and Stacey (2004), in their brief overview of tion of their teachers, thus our sampling procedure Australian school mathematics textbooks, symbolic was not randomized. equation solving follows graphical work. A proper understanding of algebra, however, requires that stu- Procedure dents be comfortable with both of these aspects of func- Two tests including different types of tasks were de- tions (Schwartz & Yerushalmy, 1992). It is thus evident veloped. The tests were developed by the researchers that the influence of teaching is extremely strong, and and secondary mathematics teachers. The tasks were the promotion of specific tools or processes enforces mainly aligned to the level of the 3rd graders and the the development of specific cognitive processes and tests were piloted to almost 100 students at each grade. structures. For example, it has been suggested that At the middle of the school year, each test was adminis- one way to improve learners’ understanding of some tered to students by a researcher or by a teacher who mathematical concepts might be the use of graphic and had been instructed on how to correctly administer symbolic technologies (Yerushalmy, 1991). the test. The testing period was 40 minutes. The tests were scored by the researchers, using 1 and 0 for cor- The role of definition in the rect and wrong answers, respectively. understanding of functions In mathematics definitions have a predominant role The tasks focused on (a) defining and explaining the in the construction of mathematical thinking and concept of function, (b) recognizing, manipulating conceptions. Vinner and Dreyfus (1989) focus on the and translating functions from one representation to influence of concept images over concept definitions. another (algebraic, verbal and graphical), (c) solving Over time, the images coordinate even more with an problems. accepted concept definition, which in turn, enhances intuition to strengthen reasoning. A balance between Costas has €20 and spends €1 per day. His sister has €15 definitions and images, however, is not achieved by and spends €0,5 per day. all students (Thompson, 1994). Actually, pupils’ defini- i. Find the function expressing the amount of money (y) each person will have in relation to the number tions of function can be seen as an indication of their of days (x). understanding of the notion and as valuable evidence ii. Design the graph showing this function for each of their mistakes and misconceptions. Elia, Panaoura, person. Eracleous and Gagatsis (2007) examined secondary iii. In how many days will the two brothers have the pupils’ conceptions of function based on three indi- same amount of money? What will this amount of cators: (1) pupils’ ideas of what function is, (2) their money be? ability to recognize functions in different forms of Figure 1: Example of a problem solving task representations, and (3) problem solving that involved the conversion of a function from one representation The first test asked students to (i) present a definition to another. Findings revealed pupils’ difficulties in of function and an example, (ii) explain their proce- giving a proper definition for the concept of function. dure for recognizing that a graph does not represent Even those pupils who could give a correct definition a function and present a non-example of function, of function were not necessarily able to successfully (iii) write the symbolic representation of six verbal solve function problems. expressions (e.g. “the area E of a square in relation to its side”), (iv) draw a graph to solve a problem, (v) METHODOLOGY recognize graphs of functions, (vi) explain a graph in terms of the context, and (vii) examine whether Participants symbolic expressions and graphs represent functions. The participants were 756 secondary school students In the second test students had to (i) present their pro- from eight schools in Cyprus. There were 315 students cedure for examining whether a graph represents a

441 Teaching the concept of function: Definition and problem solving (Areti Panaoura, Paraskevi Michael-Chrysanthou and Andreas Philippou) function, (ii) draw graphs of given functions, (iii) rec- Correct definition Correct example ognize graphs of given verbal or symbolic expressions, Grade 3 8.2% 23.0% and (iv) write the symbolic equations of given graphs. (N=315) The reliability for the total of the items on both tests Grade 4 58.5% 57.0% was high (Cronbach’s alpha=0.868). (N=258) Grade 5 32.7% 55.7% RESULTS (N=183) Table 1: Students’ correct definitions and examples for each grade According to our research questions, we first present the results of students’ performance on defining the Tasks 3rd (%) 4th (%) 5th (%) Total concept of function and their ability to present an (%) example in order to explain the definition. Then, we Procedure indicating 2.2 38.4 42.0 24.3 concentrate our attention on the procedure students that a verbal expression follow in order to justify whether a verbal expression or an equation is not or an equation is a function. A crosstabs analysis gives function further insight into the performance of students in Procedure indicating 2.5 32.6 47.5 23.7 defining and recognizing functions. Finally, differ- that an expression is ences between students in the three grade levels are function presented. An example of a func- 19.4 50.4 55.7 38.6 tion Based on the results of our descriptive analysis, 159 A non-example of a 3.8 38.4 38.3 23.9 students were able to present a correct definition, 242 function presented a wrong definition, and 358 students did Table 2: Percentages of students’ correct answers at specific tasks not give any answer. Most of the students who pre- sented a correct definition (Table 1) were at the th4 We analysed, by using crosstabs analysis (Table 3), the and 5th grade. The results were similar to the task that performance of students who correctly described a asked students to present an example for explaining procedure for determining whether a graph repre- the definition of function. Only 401 of the students sents a function in relation to their ability to correctly presented an example, and of these, 323 were correct. identify functions presented graphically. Results indi- The highest percentages of correct examples (Table cated that less than half of the students, who correctly 1) were at the 4th and 5th grade. This result is the first described a procedure, correctly recognized the graph indication that students are perhaps more capable that represented a function (44.1%). of providing an example in order to explain a mathe- matical concept than of defining it verbally by using We further analysed, by using crosstabs analysis formal language and/or symbolization. (Table 3), the characteristics of students who were able to correctly define function. Consequently, these In the first test students were asked to explain their students seemed to have a more conceptual under- procedure for identifying a graph that does not rep- standing of function. Results indicated that 89.4% of resent a function and provide a relation that does the students presented both a correct definition of not represent function. In the second test, students function and a correct example, showing that they were asked to explain their procedure for identifying likely have a strong theoretical understanding of the a graph that represents a function and to provide an concept. At the same time, of the students that provid- example of a function. Table 2 indicates the percent- ed a correct definition of function, 74.2% also succeed- ages of correct answers for the specific tasks. In all ed in describing a procedure for determining whether cases, the results were higher in the 5th grade, as was a graph did not represent a function, and 62.7% also expected, but there were especially negative results succeeded in describing the procedure for recogniz- for students in the 3rd grade. The majority of the 3rd ing whether a graph did represent a function. These graders did not give an answer and many of those who initial results permit us to assume that the students answered presented a wrong procedure. who are able to define function and explain by pro- viding an example are the students with the highest conceptual understanding.

442 Teaching the concept of function: Definition and problem solving (Areti Panaoura, Paraskevi Michael-Chrysanthou and Andreas Philippou)

In the second test, students who were able to correct- significant difference between the two grades at the ly describe a procedure for determining whether a lyceum. Concerning the presentation of an example graph represented a function were generally able to in order to explain the definition of the concept, the present an example of function (90.4%). Of the stu- difference was statistically significant 2,398(F = 5.896, dents who were able to correctly describe a procedure p<0.01) only between the 3rd and the 4th grade (x = 0.72, 3 for determining whether a graph did not represent x4 = 0.88, x5 = 0.78). The same analysis was conducted a function, 83.7% were also able to present a symbol- concerning students’ ability to describe a procedure ic non-example of a function. It is important to note, for recognizing an equation which did not represent however, that of the total sample, only 153 students a function, in respect to grade level. There were sta- correctly presented a procedure for recognizing a tistically significant differences between therd 3 grade graph that represented a function, while only 135 stu- and the two other grades (F2,292 = 33.510, p<0.01) with a dents correctly presented a procedure for recognizing large difference between the means x( 3 = 0.12, x4 = 0.47, a graph that did not represent a function. Similarly, x5 = 0.55). only 241 students correctly presented an example of a function, and only 165 students correctly presented Items from both tests were grouped in order to be able a non-example. to further analyse students’ performance concerning the specific aspects which were the main interest of Example Describing a Describing this study: (i) Propose definition, (ii) present examples for procedure for a proce- to explain a concept or a procedure, (iii) recognition defini- determining dure for of the concept, (iv) translation of the concept from one tion whether a determining representation to another, (v) construction of graphs graph repre- whether a which represent functions, as an indication of ma- sents a func- graph does tion not repre- nipulating the concept and (vi) problem solving tasks. sent a func- Table 4 presents the means and standard deviations of tion students’ performance on these specific dimensions. Recognition — 44.1% — of function X SD Correct 89.4% 62.7% 74.2% Definition 0.718 0.261 definition Examples 0.837 0.231 Function — 90.4% — Recognition 0.449 0.152 Example Translation 0.619 0.212 Function Construction 0.648 0.354 Non- — — 83.7% Problem solving 0.393 0.169 example Table 4: Means and standard deviations of students’ performance Table 3: Crosstabs analysis for students’ (total sample) understanding of functions ANOVA analysis was conducted for examining the sta- tistically significant differences for each of the above The second objective of the study was to identify sta- aspects of a conceptual understanding of function in tistically significant differences (p<0.05) concerning respect to the three grade levels. Statistically signif- the concept of function in respect to students’ grade icant differences (p<0.05) were found only concern- level in order to investigate the developmental aspect ing three of the dimensions. There was a statistically of a conceptual understanding of function. ANOVA significant difference 2,182(F = 7.678, p<0.01) concerning analysis was used for comparing students’ means in the proposed definition of the concept. The Scheffé presenting the definition of the concept and giving analysis indicated that the difference was between the rd th th an example of a function in respect to the categorical 3 grade with the 4 and the 5 grade (x3 = 0.33, x4 = 0.75, variable of their grade. In the first case, Scheffé anal- x5 = 0.71). The second statistically significant differ- ysis indicated that there was a statistically significant ence was for the students’ ability to recognize func- difference (F2,397 = 33.396, p<0.01) between the students tions presented in different forms of representations rd th th at the 3 grade with the students at the 4 and 5 grade (F2,570 = 18.926, p<0.01). The difference was between the (x = 0.11, x = 0.53, x = 0.48). Although the highest mean students at the 3rd grade in comparison to the students 3 4 5 th th th was at the 4 grade, there was not any statistically at the 4 and 5 grade (x3 = 0.48, x4 = 0.41, x5 = 0.40). It

443 Teaching the concept of function: Definition and problem solving (Areti Panaoura, Paraskevi Michael-Chrysanthou and Andreas Philippou) was unexpected that in this case the performance at further examine whether their processing efficiency the 3rd grade was higher than the two other grades. The and cognitive maturity prevent us from teaching the third statistically significant difference was for the specific concept at that age. Despite the tendency to students’ ability to translate the functions from one use the spiral development of concept in the teaching type of representation to another (F2,165 = 11.077, p<0.01). process and the curriculum (Ministry of Education The students at the 3rd grade had a lower performance and Culture, 2010), we have to rethink the teaching than the older students (x3 = 0.343, x4 = 0.547, x5 = 0.702). methods we use at the different ages and the cognitive demands of tasks at each age. DISCUSSION We believe that it is not adequate just to describe the We have concentrated on students’ ability to define students’ knowledge of a concept, but it is interesting the concept of function and on their ability to han- to design and implement didactic activities and exam- dle flexibly the different modes of representation of ine their effectiveness. Brown (2009) suggests that the function. The results of the present study confirm pre- construction of concept maps will enable teachers vious findings that students face many difficulties in to have in mind all the necessary dimensions of the understanding function at different ages of secondary understanding of the concept. The distinction of the education (Sajka, 2003; Tall, 1991). Findings revealed procedural understanding and the conceptual un- serious student difficulties in proposing a proper defi- derstanding and the lack of interrelations between nition for function or a tendency to avoid proposing aspects such as the definition of the concept, the ma- a definition due to their possible belief that the intui- nipulation of the concept and problem solving related tive and informal presentations of their conceptions to a concept have as a possible result the phenome- cannot be part of the mathematical learning. Those non of compartmentalization (Elia, Gagatsis, & Gras, difficulties are consistent with Elia and colleagues’ 2005). Thus, the use of multiple representations, the (2007) findings. Although formal definitions of mathe- connection, coordination and comparison with each matical concepts are included in the mathematics text- other and the relation with the definition of the con- books for secondary education, mathematics teachers cept should not be left to chance, but should be taught do not focus on definitions. Instead they promote the and learned systematically. use of algorithmic procedures for solving tasks, as actions and processes (Cottrill et al., 1996), and un- The verification of the interrelations between the derestimate the word and meaning (Morgan, 2013) different aspects of understanding functions are as interrelated dimensions of the concept. within our next steps, which include the confirma- tory factor analysis (CFA), for confirming the theory Secondly, it seems easier for students to present an about the structural organization of the conceptual example for explaining a mathematical concept, rath- understanding of functions. We are going to examine er than using a non-example in order to explain the whether the students’ abilities in defining the concept, respective negative statement. At the same time, it is in recognizing and manipulating the concept and in easier for them to use an example for explaining the translating the concept from one representation to concept of function rather than explaining the pro- another are important dimensions of the conceptual cedure they follow for determining whether a graph understanding of functions. Specifically we will focus represents a function. This is probably a consequence on the interrelationship between these dimensions. of the teachers’ method of using examples in order to Special emphasis will be given to the role and influ- explain an abstract mathematical concept. Thirdly, ence of the definition on the remaining dimensions students had a higher performance on manipulating of the conceptual understanding of functions, as our the concept in graphical form and translating from results revealed students’ great difficulties in this par- one type of representation to another, than on rec- ticular aspect of the concept. ognizing functions in algebraic and graphical forms. Symbolic equation solving follows the graphical work Concluding, the present study enables us to know and (Llinares, 2000), and probably for this reason, the per- understand how students conceptualize the notion formance on graphical functions was higher. Finally, of function and to realize the students’ obstacles and the results of the students at the 3rd grade of secondary misunderstandings. Teaching processes and teaching education were especially negative, thus we have to materials need to be enriched with problem solving

444 Teaching the concept of function: Definition and problem solving (Areti Panaoura, Paraskevi Michael-Chrysanthou and Andreas Philippou)

situations. The given examples presented by students Elia, I., Panaoura. A, Eracleous, A., & Gagatsis, A. (2007). were mathematically structured by using a formal- Relations between secondary pupils’ conceptions about istic way and there was not any reference to a daily functions and problem solving in different representa- experience. In the context of the interdisciplinary tions. International Journal of Science and Mathematics social reality, the concept of function has to be related Education, 5, 533–556. to other relevant domains such as physics, engineer- Kieran, C. (1992). The learning and teaching of school algebra. ing, and technology. In A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council Limitations of Teachers of Mathematics (pp. 390–419). New York, NY, A limitation of our study concerns the administra- England: Macmillan Publishing Co. tion of the tests. In cases where the administration Lin, F., & Cooney, T. (2001). Making Sense of Mathematics was not performed by the researchers, there is no Teacher Education. Dordrecht, The Netherlands: Kluwer. certainty that the proper amount of time was giv- Llinares, S. (2000). Secondary school mathematics teacher’s en to the students. Thus we cannot be sure that the professional knowledge: a case from the teaching of the same conditions held in every classroom during the concept of function. Teachers and Teaching: theory and administration of the tests, and this may have affected practice, 6(1), 41–62. the reliability of the tests. The teachers who adminis- Morgan, C. (2013). Word, definitions and concepts in discours- tered the tests, however, were provided with all the es of mathematics, teaching and learning. Language and necessary instructions. A further limitation of the Education, 19(2), 102–116. study was our inability to control the teaching method Ministry of Education and Culture (2010). National Curriculum of which was used for the specific concept. In Cyprus, Cyprus. Nicosia, Cyprus: Author. however, teachers receive the same instructions by Sajka, M. (2003). A secondary school students’ understanding the Ministry of Education for the teaching methods of the concept of function – a case study. Educational they have to use in their classes and the same in-ser- Studies in Mathematics, 53, 229–254. vice training. There is also a common curriculum and Sanchez, V., & Llinares, S. (2003). Four student teachers’ peda- a common textbook for students. gogical reasoning on functions. Journal of Mathematics Teacher Education, 6, 5–25. REFERENCES Schwartz, J., & Yerushalmy, M. (1992). Getting students to func- tion in and with algebra. The concept of function: Aspects Bardini, C., Pierce, R., & Stacey, K. (2004). Teaching linear func- of epistemology and pedagogy, 261–289. tions in context with graphics calculators: students’ re- Sierpinska, A. (1992). On understanding the notion of function. sponses and the impact of the approach on their use of The concept of function: Aspects of epistemology and algebraic symbols. International Journal of Science and pedagogy, 25, 23–58. Mathematics Education, 2, 353–376. Steele, M., Hillen, A., & Smith, M. (2013). Developing mathemati- Brown, J. (2009). Concept maps: Implications for the teaching cal knowledge for teaching in a methods course: the case of function for secondary school students. In R. Hunter, B. of function. Journal of Mathematics Teacher Education, 16, Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings 451–482. of the 32nd annual conferences of the Mathematics Tall, D. (1991). Advanced mathematical thinking. Dordrecht, The Education Research Group of Australia, V.1 (pp. 65–72). Netherlands: Kluwer Academic Press. Palmerston North, NZ: MERGA. Thompson, P. W. (1994). Students, functions, and the under- Cottrill, J., Dubinsky, E., Nichols, D., Schwingedorf, K., Thomas, graduate curriculum. In E. Dubinsky, A. H. Schoenfeld, & K., & Vidakovic, D. (1996). Understanding the limit concept: J. J. Kaput (Eds.), Research in Collegiate Mathematics Beginning with a coordinated process scheme. Journal of Education, 1 (Issues in Mathematics Education, Vol. 4, pp. Mathematical Behavior, 15, 167–192. 21–44). Providence, RI: American Mathematical Society. Elia, I., Gagatsis, A., & Gras, R. (2005). Can we “trace” the phe- Vinner, S., & Dreyfus, T. (1989). Images and definitions for the nomenon of compartmentalization by using the implica- concept of function. Journal for Research in Mathematics tive statistical method of analysis? An application for the Education, 20, 356–366. concept of function. In Third International Conference Yerushalmy, M. (1991). Student perceptions of aspects of al- ASI-Analyse Statistique Implicative (pp. 175–185). Palermo, gebraic function using multiple representation software. Italy: Universita degli Studi di Palermo. Journal of Computer Assisted Learning, 7(1), 42–57.

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