The function is the single most essential concept throughout education from kindergarten to graduate school (Harel & Dubinsky, 1992). The concept of function evolved with mathematics has various aspects (definitions, representations, and conceptions), so it has a broad understanding. Understanding of these multiple aspects is required for modern mathematics (Selden & Selden, 1992). Since “Multiple representations of concepts yield deeper and more flexible understanding” (Keller & Hirsh, 1998, p.1), students’ transition from one representation to another without any contradiction is one of the main objectives of teaching modern mathematics (Hitt, 1998). At this point, analogies (colloquial facet/representation) generated in classrooms become a powerful tool for learning and teaching function concept due to their “ability to draw parallels across different representations” (Gentner, 1998, p.107) and their ability to transfer representations across contexts (Novick, 1988). That being the case, it is inevitable for teachers to use an analogy, consciously or not, while explaining function and function-related concepts. Teachers employ analogies overtly with the word “analogy” or covertly with expressions such as “it is like …”, “it is the same as …”. This case raises a question about analogies: “Do the students understand what the analogy tool means, as their teachers intended?”

In the simplest sense, an analogy is a mapping of knowledge from one domain (familiar) to another (unfamiliar) (Gentner, 1983). In other words, they make unfamiliar familiar (Treagust, Duit, Joslin, & Lindauer, 1992). In the literature, various terms were used to define them; in this study, the terms “analog” and “target” will be used for familiar and unfamiliar domains, respectively. 

An analogy is one of the strategies that can be used to help students achieve meaningful learning. However, the literature suggests that analogies are not always beneficial for students (see Glynn 1991). Because not all analogies are good, and even not all good analogies are helpful for all students (Orgill & Bodner, 2003). To get positive results from analogy instruction, the first thing to do is to teach students how to use analogies and help them to identify the role of analogies in learning (Gentner, 1980; Harrison & Treagust, 1993, 2000; Orgill, 2013; Treagust, Harrison, & Venville, 1996; Venville & Treagust, 1997). 

Although the use of analogies in teaching and learning of function concept has received increasing interest from mathematics educators (e.g., Bayazit & Aksoy, 2011; Bayazit & Ubuz, 2008; Espinoza-Vasquez, Zakaryan, & Yáñez, 2017; Ubuz, Eryılmaz, Aydın, & Bayazit, 2009; Ubuz, Özdil, & Çevirgen, 2013; Ünver, 2009), it reveals that there are no sufficient studies in the present literature that work on function and analogy together. Furthermore, most of this limited research tended to study with pre-service or in-service teachers and investigate their knowledge and beliefs about functions and their understanding of the function concept. To date, no study has been done on students’ perceptions of analogies.

Taking into account all these considerations, it is felt that it is necessary to investigate how students perceive analogies to get an idea of how analogies should be presented to be beneficial to students. The purpose of the current study is to learn students’ general views about analogies, what analogies mean for them, and their awareness of instructional roles of analogies. The present research scrutinizes the results obtained as part of a more extensive study in which the following research question was addressed: “What are the 9^th-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: “What does analogy mean for 9^th-grade students considering ones generated on the concept of function?

In this study, an interpretive research methodology (Erickson, 1986) was used. To gain a deeper appreciation of what students’ thought about the analogy strategy, one-to-one videotaped interviews were done 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. The interview participants were chosen from these students. Sixteen (seven males and nine females) ninth-grade students (S1-S16) from five different classes of the same private high school, located in the Anatolian part of Istanbul, voluntarily participated in the interviews. The first six were from the classes (A, B) of T1 and the remaining ten were from the classes (C, D, E) of T2. The title of the mathematics unit during the study carried out was “Functions”. This unit of the original study consisted of three main parts: (a) function concept, (b) graphs of functions, and (c) function types, but for this study, just the first section, “function concept,” was handled. Following the curriculum sections, both of the teachers taught the unit using their usual teaching methods. Interviews were held at the students’ school over five weeks observation period. The individual interviews were semi-structural, conversational, and continued for nearly one hour. Three questions were posed in the interviews in order were: (1) what is an analogy? (2) what are the advantages of analogies? (3) what are the disadvantages of analogies? The majority of the interviews were spent asking students’ general views about analogies, and all questions were posed to the students in a flexible manner. The data analysis process first started with getting transcriptions and then continued reading transcripts several times to search for similarities and differences between students’ explanations. A code from ST1 to ST16 was assigned to each student for reporting. In this process, initial categories were developed for common responses. Then, to decide whether or not they are adequately indicative of the data, initial categorization was modified by adding or deleting 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. Each student’s interview data were examined and classified separately by two researchers, with an original agreement was being achieved.

Observations took place in 9th-grade classes during the teaching of the function concept. None of the teachers explain how to use analogies as a learning tool or not mention overtly what an analogy is. Consequently, explorations about what an analogy is and how to use it as a learning tool left to the students. Results revealed that some of the students’ descriptions of analogies comprised similar properties to the actual definition. Properties saw students’ responses to the first question were as follows: analogies are: “exemplifying (S1, S16); associating (S2); simulations (S2); liking, correlating or relating (S4, S12, S13, S16); puzzles (S5); illustration (S6); descriptions with examples (S7); problems (S8); similes (S8); real-life or daily life applications (S9, S15); examples, instances or giving examples (S9, S12, S14); method (S10), similarities (S13); making easier understanding (S13, S14); applications (of functions) (S16, S11)”. Almost all interviewed students answered affirmatively to question asking the advantages of analogies, and detailed explanations attended the majority of the students. Not surprisingly, most students mentioned that they liked analogies during their teachers’ function teaching. Students had different reasons for liking and finding them beneficial. Approximately 25% of the students stated explicitly that analogies are valuable tools to help them understand and remember easier and relate abstract concepts with the real world. Based on the interviews, most students were also aware of many disadvantages of analogy use. Some students were reluctant to use an analogy since they mentioned that they could understand mathematics with numbers, not verbal expressions like analogies. Some noted that the use of analogies not worth the time and effort. Besides, one student pointed out that analogies could be hazardous and confuse him.

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