In this chapter, I describe some central concepts that are used in this investi-gation. More detailed description about the concepts can be found in connec-tion to the chapters 3–6 in the theoretical framework of the research.
In this investigation the phenomenographic research approach is ap-plied. The aim of phenomenographic investigation is to describe how various phenomena, or objects in the world around us are experienced, conceptual-ized, understood, perceived and apprehended in second-order perspective (Marton 1993, 4425). The second-order perspective refers to the underlying ways of experiencing the world, i.e. it describes individuals’ ways of experi-encing something (Marton 1994; Marton & Booth 1997, 118). The second-order perspective means taking the place of the respondent, trying to see the phenomenon through his eyes, and living his experience vicariously. The second-order categories of description that are the fundamental results of a phenomenographic investigation describe thus the variation in ways people experience phenomena in their world (see chapter 9).
The term conception is a central concept in the phenomenographic ap-proach. A person's knowledge of the world is regarded as a number of con-ceptions and relations between them (Marton & Booth 1997). In recent years, the term conception has been complemented with the term experience in phenomenographic studies; in essence, conception corresponds to experience according to Marton and Booth (1997). They describe (ibid, 100) a person’s way to experience a phenomenon as follows: ‘qualitatively different ways of experiencing something can be understood in terms of differences in the structure or organization of awareness at a particular moment or moments’.
Marton’s and Booth’s description about a person’s way to experience some-thing indicates that an individual’s experience of a phenomenon is context sensitive, and can change in time and situation.
In this investigation, the concept conception and the concept belief are regarded to be the same concept, although some researchers distinguish the meaning of the two terms (cf. Ponte 1994, 169; Pehkonen 2001 13–15; see chapter 6).
The concept conception is defined in literature in many different ways.
In this investigation, conceptions are regarded as a part of teacher knowledge
10 Iiris Attorps
structure (cf. Grossman 1990; Ponte 1994; see chapter 6). They are defined as individuals’ ‘underlying ways of experiencing’ something (Marton & Booth 1997, 118), as persons’ subjective ideas of a ‘concept’ or a phenomenon, ‘as a whole cluster of internal representations…evoked by the concept´(Sfard 1991, 3). The objective side of the term ‘concept’ is defined as a mathemati-cal idea in its official form, as a theoretimathemati-cal construction (ibid). Sfard’s inter-pretation about the concepts conception and concept is similar as Tall’s and Vinner’s (1981) interpretation about the concepts concept image and concept definition. Conceptions, similarly like concept images are internal representa-tions evoked by a concept or a phenomenon, and they are generated by previ-ous experiences or impressions and by tasks in which the concepts and their definitions have been tested in teaching and learning of mathematics (Tall &
Vinner 1981; Vinner 1991; Sfard 1991). The terms concept and concept defi-nition mean both in Sfard’s and Tall’s and Vinner’s interpretation the concept as it follows from its mathematical definition (see chapter 5).
Vinner’s and Hershkowitz’s (1983) research results indicate that learn-ers’ concept images often include only prototypes. Prototypes are defined in this investigation as the specific examples of equations (ibid), which are constructed first in teaching and learning of mathematics (see chapter 5).
Building on the findings in this investigation and extending Grossman’s and Shulman’s two types of content knowledge, the terms subject matter conceptions and pedagogical content conceptions are used in this study (see chapter 6). Both terms are regarded as a part of teacher knowledge. The term subject matter conceptions is used to encompass the range of mathematical conceptions that a teacher in mathematics holds about equations. Similarly, the term pedagogical content conceptions includes the range of conceptions, that a teacher needs to represent and formulate equations in order his pupils can comprehend them.
Various terms have been used in the literature in the case where concep-tions, constructed by the learners themselves, are deemed to be in conflict with the generally accepted conceptions (e.g. Driver & Easley 1978; Confrey 1990b). The term alternative conceptions about equations describe in this investigation the teachers’ contradictory conceptions about equations com-pared with generally accepted conceptions (see chapter 9).
In Malinen’s theory of personal experiential knowing (2000, 134–140) a learner has personal experiential knowing, which is a holistic entirety and which originates from acquired life experiences. Thus, the concept personal experiential knowing in Malinen’s theory can be compared with the concepts
Description of the main concepts used in the study 11
conception and concept image defined by Sfard and Tall and Vinner. The most fundamental conceptions in a learner’s personal experiential knowledge are called the first-order experiences, which have the same characteristics as personal knowledge. The first-order experiences and/or personal knowledge have similarly as the concepts conception and/or concept image a subjective character and they can therefore include incomplete and inadequate or even distorted conceptions, incorrect theories and limited perspectives. Through-out Malinen’s study (2000), the concepts knowledge and experience have been intertwined and have overlapped to a greater or lesser extent. Another category of experience in Malinen’s (2000) theory is named second-order experiences, which can be characterized by the terms doubt, negative feelings and continuity. The learning experiences of second-order cause a shift in the personal knowing. This shift usually arouses doubt and negative feelings, since a learner notices that something is wrong with her familiar way of thinking. This shift in personal knowing can be compared with a shift evoked by cognitive conflicts. Such situations are arranged by teachers demonstrating to the students that their conceptions in certain situations lead to conflict with the scientific view. However, all of the second-order experiences are not instructive for a learner. A non-instructive second-order experience can in-stead arrest learning. Therefore, it must always exist continuity between per-sonal experiential knowing and second-order experiences.
Mathematics teachers’ conceptions about equations 13