ERG SES G 09, Mathematics and Education
The notion of function, which is considered to be fundamental to the development of mathematical knowledge and knowledge in other subject areas in school-curriculum, is so abstract and it has been found notoriously difficult for most students (Cuoco, 1995). The understanding of function concept does not appear to be easy, since the diversity of representations refer to different ways of thinking and communicating with others including verbal (spoken), written, kinesthetic (enactive), colloquial (informal or idiomatic), notational conventions, numeric, symbolic, and geometric (visual) aspects (DeMarois &Tall, 1996). An essential goal of mathematics teaching is that students be able to pass from one representation to another without falling into contradictions (Hitt, 1998). Kaput (1994) mentioned that although the representation of functions is useful for interpreting physical, social and mathematical phenomena, it is complex for students to apply concepts or ideas learned in the school mathematics in a new situation in or out of school. Because, there is a gap between mathematical functions defined in algebraic formulas and empirical functions explaining everyday phenomena (Kaput, 1994; Richland & Begolli, 2016). Since their ability to transfer representations across contexts, analogies (colloquial representation) have long been thought to have a central role in mathematics learning and teaching. Although there are many definitions of analogy in the literature, basically, an analogy is defined as a mapping of knowledge from one domain (familiar) into another (unfamiliar) (Gentner & Gentner, 1983). In other words, they are used to make unfamiliar familiar (Treagust, Duit, Joslin &Lindauer, 1992). In the literature, the familiar domain is referred as the “vehicle”, “base”, “source”, or “analog”; and the less familiar domain or the domain to be learned is referred to as “topic” or “target”.
The use of analogies in teaching and learning of function concept have received attention from mathematics educators (see, for example, Bayazit & Ubuz, 2008; Ubuz et al., 2009; Ünver, 2009; Ubuz, Özdil & Çevirgen, 2013). Bayazit and Ubuz (2008) researched the effectiveness of analogies in the teaching and learning of function concept. Ubuz et al (2009) discussed whether or not pre-service teacher-generated analogies are epistemologically appropriate to show the essence and the properties of the functions. Ünver (2009) examined how analogies are used on the function concept in the ninth grade mathematics textbooks and classrooms. Although, there have been relatively few studies reported about the way of teachers and textbooks using analogies or about the effectiveness of pre-service and in-service teacher-generated analogies to teach functions, to date, no research has been done on how students use analogies to learn or on how they perceive and how much they remember about teacher-generated analogies used in lessons on the concept of functions. There is no doubt that teachers employ analogies with the intention of promoting student understanding of function concept, however, their uses of analogies may not always correlate with students or may not make sense to them.
Taking into account all these considerations, it is felt that it is essential to search how students perceive or remember teacher-generated analogies. The purpose of the current study is to determine the possible reasons for students remembering some teacher-generated analogies more than other teacher-generated analogies. Current study scrutinizes the results obtained as part of a larger study in which the following research questions were addressed: “ What are the 9th-grade students’ perceptions of the effectiveness and the utility of analogies that their teachers employed during function concept teaching? Here, the sub-question is dedicated: “Why do 9th-grade students remember some teacher-generated analogies more than other teacher-generated analogies?
To gain a deeper insight into why students recall some analogies more than others, one-to-one videotaped interviews were made with students. This study is a part of a larger research agenda containing an observation of 91 mathematics lessons of two ninth-grade mathematics teachers (T1 and T2) and their 121 students, of which the interview participants were a sub-sample to understand instructional analogies and their influence on analogy use of ninth grade students. Fourteen (7 males and 7 females) ninth-grade students (S1-S14) from five different classes of the same private high school located in the Anatolian part of Istanbul, voluntarily participated in the interviews. The first four were from the classes (A, B) of T1 and the remaining ten were from the classes (C, D, E) of T2. Interviews were held at the students’ school over five weeks observation period. The individual interviews were semi-structural, conversational and continued nearly one hour. Two questions were posed in the interviews in the order were: (1) Identify one analogy that was used by your teacher (2) Why did you select this analogy? During the interview, both of the questions were posed to the students in a flexible manner. All the interviews were transcribed verbatim and all responses were analyzed to yield data using an inductive qualitative process of review, coding, and identification of semantic themes (Bogdan & Biklen, 1992). Data analysis process first started with getting transcriptions then continued with reading transcripts through several times to search for similarities and differences between them. In this process, initial categories describing the different reasons for recalling analogies were developed. Then, to decide whether or not they are adequately indicative of the data, initial categorization was modified by addition or deletion some of the descriptions. This process was followed until forming a coding scheme containing more general categories. Following the analysis, the frequency of each category was identified. This coding scheme consisted of nine categories given in result part. Each student’s interview data was examined and classified separately by two researchers, with an original agreement was being achieved. The title of the mathematics unit during the study carried out was “Functions”. The unit of the study consisted of three main parts: (a) function concept (b) graphs of functions, and (c) function types (polynomial, rational, irrational, constant, identity, linear, piecewise, one to one and onto functions).
During the interviews, almost all students (except S8) identified one or two analogies with underlying reasons. Student-identified analogies (A1-A9) with their specified analogs and targets can be listed in order from most to least among students shown inside of the brackets: A1: “Calculation of water bills regarding to personal use with municipality fee” analog with “function concept” target (S5, S7, S9, S10)”, A2: “Specification profits of a hotel by month” analog with “function concept” target (S11, S13), A3: “Specification profits of a hotel by month” analog with “piecewise function” target (S12), A4: “Tomato juice machine” analog with “function concept” target (S2, S4), A5: “Tomato fabric” analog with “function concept” target (S1)”, A6: “Tomato sauce machine” analog with “function concept” target (S3), A7: “Constant speed car” analog with “constant function” target (S5), A8: “Function machine” analog with “function concept” target (S6) and A9: “Cab fare” analog with “function concept” target (S14). Categorization obtained from students’ expressed reasons consisted of nine categories: mathematical/containing numerical expressions (A1, A3, A5, A8), to think applicable (A1, A2, A5), simple/intelligible (A1, A4), to help understanding (A1, A3, A4, A6), to remember/recall information (A2, A6), repeated (A2, A4), useful for later learning (A1, A8), attention-grabbing (A8), as personal interest (A2, A8, A9). Based on the data presented above, the most common reason for remembering the selected analogy was that they helped to understand. In the second order, it followed by that they were mathematical or containing numerical expressions and were thought as applicable. Differently, S6, S13, and S14 mentioned the reason that they were related to their personal interests (business life/commerce, economy, and financial matters). Lastly, S6 mentioned that the reason for remembering analogies based their attention-grabbing property, on the other side; S8 based her for not remembering any analogies to the same reason.
Bayazit, İ., & Ubuz, B. (2008). Instructional analogies and student learning: The concept of function. In Proceedings of International Group for the Psychology of Mathematics Education, (pp.145-152). Morelia, Mexico: Centre for Research and Advanced Studies of Saint Nicholas. Bogdan, R., & Biklen, S. (1992). Qualitative research for education: An introduction to theory and methods . Nedham Heights, MA: Allyn & Bacon. Cuoco, A. (1995). Computational media to support the learning and use of functions. In A. diSessa, C. Hoyles, & R. Noss (Eds.), Computers and exploratory learning (pp. 79–108). Springer. DeMarois, P. and Tall, D. (1996) ‘Facets and layers of the function concept’, in Puig, L. and Gutiérrez, A. (eds), Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Education, 2, Valencia, Spain, pp. 297-304. Gentner, D., & Gentner, D. R. (1983). Flowing waters or teeming crowds: Mental models of electricity. In D. Gentner & A. L. Stevens (Eds.), Mental models (pp. 99-129). Hillsdale, NJ: Erlbaum. Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function. The Journal of Mathematical Behavior, 17(1), 123- 134. Kaput, J. (1994). The representational roles of technology in connecting mathematics with authentic experience. In Biehler, R. et. al. (eds.) Mathematics didactics as a scientific discipline. Richland, L. E., & Begolli, K. N. (2016). Analogy and higher order thinking: learning mathematics as an example. Behavioral and Brain Sciences, 3(2), 160-168. Treagust, D.F., Duit, R., Joslin, P. & Landauer, I. (1992). Science teachers’ use of analogies: Observation from classroom practice. International Journal of Science Education 14: 413– 422. Ubuz, B., Eryılmaz, A., Aydın, U., & Bayazıt, I. (2009). Pre-service teacher generated analogies for function concepts. In V. Durand-Guerrier, S. Soury-Lavergne ,& F. Arzarello (Eds.) Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp.1871-1879). Lyon: France. Ubuz, B., Özdil, U., & Çevirgen, A. E. (2013). Preservice teachers' knowledge and beliefs: Their association to practice in the context of teaching function with analogies. In B. Ubuz, Ç. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 3277-3286). Antalya: Turkey. Ünver, E. (2009). Analysis of analogy use on function concept in the ninth grade mathematics textbook and classrooms. Unpublished Master's Thesis, Middle East Technical University. Turkey.
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