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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
Citation: URL: http://www.allacademic.com/meta/p117642_index.html Direct Link: HTML Code:

Bowers, Michelle. and Ellerton, Nerida. "Language & Belief Factors in Learning & Teaching Mathematics & Physics: A Study of Three Teachers" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117642_index.html>

Bowers, M. L. and Ellerton, N. F. , 2004-10-21 "Language & Belief Factors in Learning & Teaching Mathematics & Physics: A Study of Three Teachers" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117642_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: Language and Belief Factors in the Learning and Teaching of Mathematics and Physics: A Study of Three Teachers Purpose of the Study Students are conditioned to expect to learn mathematics in the mathematics classroom, and to learn physics in the physics classroom. But mathematical concepts are frequently encountered in the physics classroom, and physics applications are often appropriate for illustrating concepts encountered in the mathematics classroom. An understanding of any differences (and similarities) in the communication patterns and teacher beliefs that occur in mathematics and physics classrooms should help to identify teaching approaches and classroom discourse practices which are most likely to support meaningful mathematics learning. This study represents an exploration of “science talk” and “math talk” in nonintegrated secondary mathematics and physics classrooms focusing on the mathematical concept of functions. The specific questions guiding this research are: 1) What are the main characteristics of language genres found in their classrooms? 2) How do mathematics and physics teachers’ conceptions, beliefs and knowledge, about mathematical functions, school mathematics and school physics influence their classroom practice? Theoretical Framework Researchers have identified unique language genres or discourse practices in classrooms (e.g., Ellerton, 1999; Lemke, 1989; Wickman & Östman, 2002). A language genre is an agreed-upon form and style of language which is developed in and by a particular discourse community to facilitate communication in that community (Bickmore-Brand, 1997; Ellerton & Clements, 1991; Hasan, 1996; Lemke, 1989). By observing classroom discourse the researcher can begin to identify patterns that constitute contextually-based language genres. Classroom discourse can be thought of as instances of communicating that represent dynamic actions either between others or with the self as a reflective individual (termed “self as thinking” by Sfard [2000]) that occur in a classroom setting. This communication can include verbal utterances, written texts, physical gestures, and other social contexts by teachers or students (Roth & Lawless, 2002). Furthermore, discourse is seen as both a means of communicating and a means of learning. Studies on teacher knowledge and beliefs indicate that teachers are not well aware of student difficulties and that teachers may have a limited understanding of the concept of function (Hadjidemetriou & Williams, 2002; Norman, 1992, 1999). Norman (1992) also discussed the lack of cognition of mathematical functions by teachers and introduced the concept of functional reasoning. Method of Inquiry Three teachers were selected as cases for analysis and were examined as a group. The teacher’s conceptions (including beliefs and knowledge) and classroom instructional practices provided the boundaries for each case. Semi-structured interviews with three teachers were conducted, and the teacher’s discourse patterns were examined during classroom instruction. Their classrooms were observed four to six times throughout the course of the academic year. Pre-determined tasks and open-ended questions were organized around the concept of mathematical functions. The semi-structured nature of the interviews also allowed for flexibility in pursuing relevant ideas arising from the conversations. Interview tasks were selected and modified from current literature on functions. In addition, through an exchange of emails, further questions were pursued and clarification sought. The interviews (and email exchanges) were conducted after the completion of the classroom observations to limit their potential influence on instruction. Teachers were observed and audio recorded during regular classroom instruction. Given the exploratory nature of the study, the audio recording was limited to the teacher’s voice, although some student interactions were recorded in my field notes. These audiotapes were later transcribed to provide a more detailed record. The three high school teachers who volunteered to participate in this research study were a 2nd year mathematic teacher, Mrs. Agnesi and Mr. Newton, a 10-year veteran physics teacher, both from the same Illinois High School. The third participant, Mrs. Arc is an experienced classroom teacher of 7.5 years who has taught both mathematics and science courses: Algebra 1 and Physics. Results and Conclusions To address the two research questions, interview transcriptions, email exchanges, classroom observation field notes, and classroom transcriptions were analyzed by comparing the language associated with instruction on the concept of functions in mathematics and physics classrooms. The use of everyday language (ie. language used in situations familiar to students and non-technical terms) was the most common theme across all three cases. These teachers expressed the belief that students should be introduced to new material first through language that connects with the students, then possibly with a move towards more formal language. Each teacher referred to formal mathematical language in a negative way. The teachers used these expressions either in the interview, in class, or both: Mr. Newton used “mathwanese;” Mrs. Agnesi used “math garbage;” Mrs. Arc used “alphabet soup” implying that mathematics language is cumbersome. Each teacher used formal mathematical language correctly, but emphasized it in different ways, as will be discussed later. Both physics teachers approached their respective courses as a laboratory, where knowledge was built inductively from the phenomenon being investigated. The physics teachers specifically referenced mathematics as a means of analyzing phenomena. This was mentioned in their interviews and classroom discourse, yet physics remained the focus of instruction as illustrated by the following transcripts. Newton: It has lots of inertia. While the earth loves this thing very much it wants to give it a huge hug. This thing is just way to cool to just go screaming to the earth because it has some serious inertia. It is just hanging tough here with some inertia. Ok, mathematically when you look at this big force, big mass; small force, small mass. Ok can those two products be the same? Oh my goodness quotients, when do you get to use quotients in a sentence? Can those two quotients be the same? (from classroom episode) Arc: In my AP course, when we introduce a new formula, we talk a lot about how the formula is set up and direct and inverse relationships and unit analysis and what cases, … would influence it. Like, for the universal gravitation what happens when r approaches infinity, or when r approaches zero. And we can do in depth analysis of that. (1st interview) The physics teachers drew clear distinctions between what was mathematics and what was science. Mathematics was not simply used to obtain a numerical result, but was seen as helpful in further analysis of the physical situations; students were expected to learn the usefulness of mathematics in science. In this sense, the discourse relied heavily on the mathematical language used for developing further understanding of the science concepts and situations. Both physics teachers introduced material by connecting science concepts to realistic situations from the students’ lives. Once the informal connections had been made the material was presented or derived into the more formal scientific notation. Mathematics was used to analyze and therefore give more meaning to the physical situation at hand. Students were expected to interact with the teacher and each other during this presentation and when discussing new material or during laboratory discussions. The physics teachers were more likely to present new material by connecting it to previously learned material or to a realistic situation in students’ lives. These teachers’ knowledge of the overall science curricula helped them avoid repeating material and assisted them in making intra-science connections with old material. The mathematics teachers were less likely to gesture and refer to diagrams and formulae in their mathematics classrooms than the physics teachers during their instruction. These physics teachers used formal terminology, but students were encouraged to conceptualize them in more personal terms. Although mathematics and physics share the language of mathematics, this language is used differently in the two contexts. Mathematics teachers in this study perceived that physics teachers do not use mathematics in as rigorous a manner as they believed should be the case, and the physics teachers believe that mathematics teachers are overly abstract in their presentation of mathematical concepts. The concept of function was approached differently in the two disciplines. There is strong evidence to suggest that the mathematics teachers were better able to formalize the mathematics and move between different representations of function. With respect to Norman’s (1992) functional reasoning, the mathematics teachers were better able to provide generalizations for functions and identify patterns from algebraic representations. The mathematics teachers were also better able to communicate within functional situations. Mathematics was used in physics as an analysis tool, often through equations and graphing of data. Mr. Newton’s limited understanding of functional reasoning combined with the nature of school physics led him to use these ideas as equations and graphing tools with little exploration of the powerful mathematics involved that could extend his knowledge. Interestingly, all three teachers emphasized the need to introduce material informally in the classroom through situations relevant to their students. All three teachers found formal mathematical language less than appealing and clearly set this way of speaking and thinking aside as they taught the curriculum. They implied that there is more to mathematics than symbolic language and this was reflected in their classroom instruction.

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