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Teaching the Concept of Function: Definition and Problem Solving Areti Panaoura, Paraskevi Michael-Chrysanthou, Andreas Philippou

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Teaching the Concept of Function: Definition and Problem Solving Areti Panaoura, Paraskevi Michael-Chrysanthou, Andreas Philippou Teaching the concept of function: Definition and problem solving Areti Panaoura, Paraskevi Michael-Chrysanthou, Andreas Philippou To cite this version: 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 HAL Id: hal-01286927 https://hal.archives-ouvertes.fr/hal-01286927 Submitted on 11 Mar 2016 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
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