University of Exeter



Investigating Difference and Repetition in Mathematics Teachers’ Professional DevelopmentConstanta Olteanu and Lucian OlteanuLinnaeus Universityconstanta.olteanu @ lnu.se, lucian.olteanu @ lnu.seABSTRACTThe availability of instructional examples in the mathematics classroom challenges the way teachers confront students with several aspects of the content, which can contribute to designate the generality of mathematical concepts. Using the theory of variation and Deleuze’s philosophical concepts as the main interpretative framework, this study investigates what is involved in repetition and the nature of its interiority, what we understand by conceptual difference and difference without concept in a development study. The data consisted of 36 teachers’ lesson plans. Qualitative analysis of these data led to the identification of events types and the characteristics of repetition. Keywords: repetition, difference, variation theory, function, patterns of variationINTRODUCTIONKrainer (2014) specifies that mathematics education deals with the learning and teaching of mathematics. Krainer also specifies that teachers have a crucial role in this process. Thereafter, a core question in educational research is what influences teachers’ professional development in teaching mathematics. Olteanu (2016) carried out a rigorous systematic review of the research which focuses on teachers’ challenges of practice so they directly enhance learning and teaching. Some important results can be summarized as follows: (1) design of research studies that more effectively address the needs of practitioners (e.g. Hamersley, 2002); (2) the importance of including practitioners as importance partners in research (e.g. De Vries & Pieters, 2007; Lo, 2014; Marton, 2015; Olteanu & Olteanu, 2013; Ruthven & Goodchild, 2008); and (3) the linkages of research studies to practice (e.g. Gore & Gitlin, 2004; Olteanu, 2014). Still, there is little consensus as to what role educational research should play in directly influencing educational practice (e.g. Nuthall, 2004; Vanderlinde & van Braak, 2010). Kieran, Krainer and Shaughnessy (2013) claim that a distinct gap between research and practice has been created because too often, in mathematics education, research is no connected to the needs of teachers and does not take into account teachers’ reality. Olteanu (2014) identified that one need that teachers have provided for is the use of examples to visualize aspects of the content that have a strong power in the students’ process of learning of mathematics.The purpose of this article is to investigate what in the use of examples enables a teacher’s professional development. Also, the purpose is to investigate what is involved in repetition and the nature of its interiority, what we understand by conceptual difference and difference without concept in a development study. The following questions guide this paper: What are the characteristics of repetition in teachers’ professional development? What is it that repeats in repetition?IMPROVE EDUCATION BY LOOKING AT PRACTICE One kind of study concerning teachers’ professional development is in focus for this article, namely development studies. These are studies that aim to solve an educational problem by using relevant theoretical knowledge (McKenney, Nieveen & van den Akker, 2006; van den Akker, 1999). Development studies integrate state-of-the-art knowledge from prior research in the design process and fine-tune educational innovations based on piloting in the field. [...] By unpacking the design process, design principles that can inform future development and implementation decisions are derived. (Nieveen et al., 2006, p. 153)The scientific output in development studies generates design principles for use in solving education problems, by unpacking the design process to inform future development and implementation decisions. Two principles and several stages are followed in development studies (McKenney, Nieveen & van den Akker, 2013; van den Akker, 1999). The principles are: procedural design (the design approach) and substantive design (the design itself). The stages are: (1) preliminary research (development of conceptual framework based on literature review); (2) prototype (optimizing prototypes through cycles of design, formative evaluation and revision); (3) summative evaluation (explores transferability along with effectiveness); (4) systematic reflection and documentation. Because development studies observe phenomena in their natural settings, it requires a long-term link with practice to fully explore and optimize an intervention (McKenney, Nieveen & van den Akker, 2013). Educational design research study is a way to carry out development studies using an iterative process with the aim of providing solutions to practical and complex educational problems (McKenney & Reeves, 2012). Educational design research is not a methodology and the knowledge generated is measured in terms of its ability to improve educational practice (Design-Based Research Collective, 2003). This improvement can be done by using examples in a progressive way with the primary aim being to give students the opportunity to be able to discern certain aspects of the object of learning at the same time (Marton & Booth, 1997; Marton & Tsui, 2004). The object of learning is the content that teacher intends to teach as well as how the students are expected to make sense of and make use of the content (Marton & Booth, 1997). The word example is used in this article in accordance with Watson and Mason (2005) definition, namely to be “anything from which a learner might generalize” (p. 3). Because examples are essential for the formation of concepts, generalization, abstraction, analogical reasoning, and proof, it is important to identify ways in which the teachers become more aware of the potential and limitations of using examples. In the teaching using examples, repetition can be used in order to help students be successful in their learning behaviour. According to Marton and Trigwell (2000), repetition is the mother of learning. The use of repetition in learning is a common finding in the literature (Entwistle & Entwistle, 2003; Marton et al., 2005).DIFFERENCE AND REPETITION An object of learning is formed in the communication that occurs in different events between teacher and students (e. g. Olteanu, 2014, 2016). Olteanu (2016) defined event as a process that begins with one aspect of the content that one or more students have not yet discerned and ends with one or more students distinguishing that the aspect of the content. For example, linear functions are characterized by a constant additive increase or decrease and in a graph of a linear function, the slope of the line represents the rate of change, and a line’s slope describes its steepness as well as its direction (aspects of linear function). If some students do not discerned what slope really means, an event begins with a sequences of examples aimed to developed the mathematical concept of slope. The event in the development study presented in this article is a problem or problematic idea rather than a truth or a solution. If the event is a sequence of mathematical examples, the problem or problematic idea would be the students’ capability to discern aspects related to the object of learning by experiencing them. If those aspects are not discerned they become critical aspects for the students (Olteanu, 2016). For example, if some students do not discerned what slope really means, the slope is a critical aspect for those students. Deleuze (1994, p. 44) writes that “learning is the appropriate name for the subjective acts carried out when one is confronted with the objectivity of a problem ... whereas knowledge only designates the generality of concepts or the calm possession of a rule enabling solutions. When using examples in a mathematics classroom, students are confronted with several aspects of the content, which can contribute to generalizing mathematical concepts. For example, the understanding of the role of a and b in a linear function, y = ax + b, where a is the constant rate of change (or slope) and b is the value of y when x = 0 (the y-intercept; sometimes called the “starting point”) is an important part of using linear functions to model both real-world and purely mathematical relationships. For this reason, the analyses of the event as it changes from being specific (immanent) to a convergence of parts or elements (Deleuze, 1994), can provide a new way of understanding of what the characteristics of repetition are in teachers’ professional development. The things shared in an event having mathematical examples as a starting point generate a variety of aspects of the object of learning that do not have a fixed form, but can be formed during the lesson. This ability to be formed can be expressed using different patterns of variation. Previous research (Marton & Tsui, 2004; Olteanu, 2014) mentioned five patterns of variations, which can facilitate students’ discernment of critical features or aspects of the object of learning. These patterns of variation are: (1) contrast (in order to experience something, a person must experience something else to compare it with); (2) separation (an aspect must vary while other aspects remain invariant); (3) fusion (several critical aspects need to be considered together); (4) similarity (the property of two or more expressions to adopt the same meaning); (5) generalisation (is to see variations in the use of the object to fully comprehend it and involves recognizing that some features are not critical to the identification of that phenomenon); For example, in a graph of a linear function, the slope of the line represents the rate of change and a line’s slope describes its steepness as well as its direction. To understand the notion of slop the students should be able to take into account horizontal distance, vertical distance and the angle of inclination in a simultaneous manner. The principle of contrast allow the students to experience what is meant by vertical and what is not. The principle of separation allow the students to experience that the ratio of any vertical change to its corresponding horizontal change is constant in a linear relationship (the vertical distance is kept invariant, but the horizontal distance varies; the horizontal distance is kept invariant, but the vertical distance varies). The principle of fusion is used to vary both the horizontal and vertical distances at the same time. To give the students the opportunity to understand the rate of change, the patterns of variation should give the opportunity to experience linear functions in multiple ways—in algebraic symbols, situations, graphs, verbal descriptions, tables, and so on—by keeping the meaning invariant (similarity). When examining what characteristics these situations have in common the students have the opportunity to generalise the idea of the rate of change to other occasions (generalisation). Here the art of the teacher intervention is as much a result of the use of patterns of variation as it is of the tools used and the students. Examples shared by events are their having become different by using several patterns of variation in the course of their creation in a teaching situation.Stagoll (2005) explains that an event arises from a set of particular forces. These forces can emerge in relation to a set of dialectical opposites (e.g. whole/part) for which resolution is never fully granted. For example, a dialectical opposite related to an algebraic expression are: (1) a whole ax + b and; (2) the parts that form the whole (e.g. a, x, b); (3) the relation between the parts (e.g. the operation); (4) the transformation between the parts (e.g. the rewriting ax + b as a(x + b/a), with a ≠ 0); (5) the relation parts–whole; and (6) the relation between different wholes (e.g. the relation between ax + b and a(x + b/a)). In such dialectical opposites there are qualitative differences. In the dialectical opposite “the transformation between the parts” a qualitative differences is to understand that two or more expressions are equivalent if they have the same value. Even two expressions have different symbolic forms, they represent the same quantity when each variable in both expressions has the same value. For example, the expressions 2x – 3 is equivalent to 21x-3 because x in both expressions have the same value for each value of the variable x. It is important to recognize that the linear functions f(x) = 2x – 3 and f(x) = 21x-3 have the same rate of change (or slope). Between the forces that can emerge in relation to a set of dialectical opposites, qualitative differences that are express by singularities can be identified. Deleuze (1990) describes singularities as turning points and points of inflection in a topological way. In topology, a series of singularities can undergo a variety of transformations that allow the study of singular points. A singular point is simply a point where the differential of the function is not surjective. The differential function f is surjective (f: A → B be an arbitrary function with domain A and codomain B) if every element y in A has a corresponding element x in B such that f(x) = y. For example, the image can be connected to intended critical aspects (Olteanu, 2016) and the codomain can by connected to lived critical aspects (Olteanu, 2016). The intended critical aspects refer to the aspects of the content that teachers intend to present in the classroom. The students’ distinguished critical aspects of the object of learning are the lived critical aspects. If the image of the intended critical aspects is not equal with its lived critical aspects, then a singularity can be identified. For example, if the teacher not intend to present in the classroom the equivalence between different rate of change although the students not discern this equivalence this leads to a singularity concerning the concept of slope, namely to discern both horizontal and vertical distance simultaneously. A mathematical singularity indicates a threshold and should be distinguished from the logical sense of singularity in which a logically unique or singular system changes behaviour patterns.Difference is a singularity at the level of ideas and repetition is always affected by an order of difference. Deleuze (1994) specifies that the concepts of repetition and difference will intersect, “one concerning the essence of repetition, the other the idea of difference” (p. 31). Repetition is never the reproduction of the same, but the repetition of the different (Deleuze, 1994). A multiplicity is an idea; in other words, it is a set of differential elements, differential relations, and singularities (Deleuze, 1994). For examples, in the classroom the differential elements would be the teacher, the students, the mathematical content, and the examples. They are differential elements because they are defined only in relation to each other. The differential relations are what the students are able to do with the examples and with each other. They are differential in that they are relations of change in the elements: how students are able to discern important aspects of the content taken in the examples, to put these aspects together, and to use the examples to solve some tasks. These relations are strewn with singularities, or sensitive points, when the aspects of the content move across a certain threshold of the field. Changes in the elements, relations and singularities will change the teaching and learning in the classroom, which in turn enables the teacher to develop professionally. METHODSThe results presented in this article are based on a three-year longitudinal study that involved eight secondary schools and 22 teachers. The teachers were voluntary participants’ selected using maximum variation and typical case sampling (Patton, 2002). To facilitate the development study, the schools provided resources and administrative support and scheduled time for meetings in teachers’ timetables. The researchers brought audio and visual equipment to record the lessons for analysis. The teachers in the development study community presented their findings on the staff development day each semester in order to disseminate the outcomes gained with colleagues from other schools. In this way, the design of the development study was progressively promoted to other teachers. The design principles used in the development study were: (1) interventionist: the aims were to designing an intervention in schools and in a real setting; (2) iterative: during the study the research team incorporated cycles of analysis, design and development, evaluation, and revision; (3) involvement of practitioners: the teachers participated in all the activities and lessons during the project; (4) process oriented: the focus was on understanding and improving interventions by analysing the students’ capability to discern aspects related to the object of learning after experiencing them; (5) utility oriented: the design was measured by its practicality for teachers in real contexts; and (6) theory oriented: the design was based on concepts from variation theory, namely the object of learning, critical aspects and patterns of variation. The design of the study had a strong focus on what the teacher should be able to do. The focal point for data analyses in this article was teachers’ planning, namely the examples that teachers had planned to use and had has been used as the intended object of learning in three-year study of an elementary classroom and an upper secondary school. It involved four mathematics teachers (two from elementary and two from upper secondary school), all of whom had received initial formal teacher education. For this article I choose to analyse the teachers’ planning connected to the process of building generalization concerning the concept of first-degree functions in classes 7–9 (the students were 13–15 years old) and the first course in upper secondary school (students are 16 years old). To investigate the research questions, 36 teachers’ planning was analysed. Drawing on Deleuze (1994) frameworks, we constructed a model for analysing the examples. The schema for codifying and analysing the information with which we obtained the results is based on a cyclical process that seeks to minimise discrepancies. The states of development that emerged from this process identified the combinations of values of the variables to which, as a whole, the observations for a given example were best adapted. This procedure enabled me to characterise the singularities at the level of ideas (difference) and repetition affected by an order of difference. The analysis focused mainly on the set of differential elements, differential relations, and singularities (multiplicity) in the observed work of the teachers participating in the development study. RESULTSThe results presented below consist of a global inventory of the differential elements (1) and the frequency distribution of the differential relations over the four teachers involved in the study (2). The differential elementsThe differential elements identified in the study are: the teachers, the students, the content and the examples. In the elementary school, the teachers had, in the first year of the study, students in grades 7. In the second year of the study these students were in grade 8 and the third year in grade 9. This means that the teachers and the students were the same during the three years of the study. In upper secondary school, the teachers were the same during the study, but taught new classes of students each year. The object of learning was the same for elementary and upper secondary schools during this period, namely to improve students’ ability to discern relationships between variables. The differential relations In analysing the intended object of learning, differential relations between aspects of the object of learning, relations between aspects of the object of learning, and relations between examples could be identified. Four types of event were identified through a combination of theory-driven and data-driven analysis. These were termed: variables change, slopes, changing rates of change, and multiple ways to represent and analyze first-degree functions.Variables change (Event 1) is a basic situation in which functions are presented as an input-output relationship, and there is no variety from a variation theoretical perspective. Most functions that teachers in secondary and upper secondary school choose as examples are described by an algebraic expression of first-degree. For example, the functions as y = 4x + 3 (characterized by a constant additive increase). Those functions belong to the same families of linear function, namely the rate of change are positive and are expressed by a positive integer. Slopes (Event 2) are a transition state. There is some complexity in the conceptual structure, and variety begins to appear in the systems of representation. However there is still no variety from a variation theoretical perspective. The examples in this situation are characterized by a positive rate of change (e.g. y = 5x + 4) and graphical representations, but they not focus attention on links between symbolic and graphical representations of linear functions. These functions belongs the same families of linear function as in Event 1.Changing rates of change (Event 3) shows an advance in all types of first-degree functions except those of domain and codomain. The examples belong to different families of linear functions (positive and negative rate of change), and the teachers stressed the relationship between what happens with positive and negative slopes and how they are represented in conventional mathematics. There was variety in the systems of representation and the number of connections (Table 1). Table 1. Changing rates of changeSequences 1Sequences 1y = xy = - xy = 3xy = -3xy = 14xy = -14xy = 34xy = -34xGraphs corresponding to the equations in sequences 1 have different slopes, all of which are positive. By contrast, the equations in sequences 2 correspond to negative slopes. These graphs are lines that slant downward instead of upward from left to right as x increases. In the graphical representations both the horizontal and vertical distances of the rate of changes vary at the same time (fusion as pattern of variation). Explorations of this sort are helpful because they focus attention on links between symbolic and graphical representations of linear functions and as well as between positive and negative rate of change. Multiple ways to represent and analyze first-degree functions (Event 4) achieves full complexity from a variation theoretical perspective, and the information was used consistently for the completion of examples. This situation concerns a class discussion about what happens when the students justify their own generalizations focusing on how the constant rate of change in a linear function appears in different representations of the function and the relationships between those representations (similarity and generalisation as patterns of variation). For example, in a table the value of variable increase or decrease by a constant amount as the value of the other variable increases by a constant amount. In a graph the rate of change is represented by the slope—the steepness and direction—of the line, and in the symbolic form written in the form y = ax + b, a is the constant rate of change. In this way, the students have the opportunity to discern difference at the level of ideas and repetition that is affected by an order of difference.Table 2 presents the final classification of the three years of the project of the two groups into each of the four events. Each row represents a group of teachers and their corresponding intended object of learning, organized chronologically. Thus, for example, the intended object of learning to the teachers in grades 7 was assigned successively to the following events: 1, 1, 1, 2, 2 and 2.Table 2. Final assignment of intended object of learningGradesIntended object of learning123456711122281212339333344Upper secondary school (year 1)111222Upper secondary school (year 2)333444Upper secondary school (year 3)333444Between the first and the second year of the project, a turning point occurred for the groups in upper secondary school that generated a singular point at the level of ideas. For an example, the teachers used contrast, separation and similarity to describe differences between positive and negative slopes in both symbolic and graphical representations of a first-degree function (Table 1). Something similar occurred in the third year of the project in upper secondary school, with the notions of variety of phenomena, variety in systems of representation, complexity and systems of representation as an organizer of the conceptual structure. The analysis showed that what was repeated between the second and third year concerned the use of rich patterns of variation, that is to say an order of difference concerning the process of representing an idea using similarity and generalization as patterns of variation. In an example, the teachers used representations by keeping the meaning of an aspect invariant and varying the form of representation (words, symbols, tables and graphs). In this way, the repetition was not a reproduction of what was the same, but of things that were different. The characteristics of repetition consisted of: (1) qualitative differences in relation to the function as a whole; (2) the parts that formed the whole (slopes and y-intercept); (3) the relation between the parts (the ratio of the change in the output variable to the corresponding change in the input variable of the function); and (4) the relation of parts–whole. In the secondary school, the teachers kept invariant the same family of a linear function and varied the situations that they modelled. For example, for instructional examples they used the change in distance that relates to a particular change in time or price per item. Those examples did not provide an opportunity for students to experience the difference at the level of ideas. The intended object of learning was repeated in the second year of the project, and thereafter the teachers added some examples concerning the difference in the direction of line. In this way, the repetition was not a reproduction of what was the same, but a use of a different idea. By contrasting the slopes (positive and negative), the examples provided an opportunity to explore several different relationships and generalize the rate of change. By using similarity as a pattern of variation, the examples provided the opportunity to explore linear functions and identify linear patterns of change in tables, graphs, and symbolic rules. In the third year of the project, repetition occurred at the level of idea, but the examples provided the opportunity for students’ to experience a statement that described a general mathematical truth about some family of function, which can contribute to designating the generality of mathematical concepts. A reflection of this set of four identified events led to the distinction between the forces that emerge in relation to whole/part. In variables change and slopes, the intended critical aspects refer to using the first-degree function as a model for describing how variables change together (the relation between the parts and the transformation between the parts) by using only a constant additive increase. The instructional examples were restricted and the focus was on the product, or result, of calculation of the output. In changing the rates of change and using multiple ways to represent and analyze first-degree functions, the instructional examples were used as a process of representing an idea. The forces that emerged in the relation whole/part led to a generalization in which the characteristics of repetition concerned making sense of patterns of variation. In repetition is repeats the characteristics of the object of learning by considering the relationship between the variables, and the representation of first-degree function as a process. CONCLUSIONSThe main result of the study reported in this article was the characterization of the events and classification of the intended object of learning based on an analysis of the instructional examples. This result confirms that the didactic knowledge of the groups of teachers evolved according to stable patterns. The groups progressed in the development of their choice of instructional examples at different paces and levels of advancement. Analysis of the discrepancies in each event shed light on which notions presented more difficulty to students. For example, the notion of connection presented a high number of discrepancies with positive difference. The numbers of examples which were chosen by the groups of teachers had a level of connection to each other higher than expected. The potential of repetition in the teaching and learning of school mathematics relies on the process of producing examples, which involves design and analysis. Design may include: (a) implementation of a known procedure to generate the successive terms in, for example, an arithmetic or geometric sequence or (b) modification of a given procedure to use a new but related example. Analysis is the process of determining the differences in repetition. To summarize, the potential of repetition lies in its ability to help students develop an understanding of mathematical concepts and. Spaced repetition is a learning technique that incorporates increasing the intervals of time between subsequent reviews of previously learned material in order to exploit the critical aspects effect.The notions of repetition and difference are not well understood and research is needed to identify the ways in which mathematics teacher develop their own knowledge. The findings could inform development of a set of recommendations for supporting professional learning that is grounded in the practices of mathematics teachers who have varying levels of experience and work in a variety of different contexts. Such recommendations may stimulate further research on the design of examples in development studies with focus on professional development of mathematics teacher.Over time, mathematics education research has shifted its focus from consideration of mathematical content and curriculum development, to the mathematical learning of students, to interactions between students and teachers, to the learning of teachers, and most recently to the learning of teacher educators (Krainer, 2008). This paper argues that philosophical research is needed to enhance theoretical understanding of how mathematics teachers learn and develop. Research of this type is important because it acknowledges the complex forms of knowledge needed by teacher educators and the multiple social settings in which their learning takes place. REFERENCESDeleuze, G. (1994). Difference and repetition. London: Athlone. (org 1968) Deleuze, G. (1990). The Logic of Sense. New York: Columbia University Press.Design-Based Research Collective. (2003). Design-Based Research: An Emerging Paradigm for Educational Inquiry. Educational Researcher, 32(1), 5–8. De Vries, B. & Pieters, J.M. (2007). Bridging the gap between research and practice: Exploring the role of knowledge communities in educational change. European Educational Research Journal, 6(4), 382–392. Entwistle, N. J. & Entwistle, D. M. (2003). The interplay between memorising and understanding in preparing for examinations. Higher Education Research and Development, 22, 19–42.Gore, J. M., & Gitlin, A. D. (Red). (2004). Visioning the academic-teacher divide: Power and knowledge in the educational community, Teachers and Teaching: Theory and Practice, 10(1), 35–58. Hammersley, M. (2002). Educational research, policymaking and practice. London: Sage.Kieran, C., Krainer, K., & Shaughnessy, M. J. (2013). Linking research to practice: teachers as key stakeholders in mathematics education research. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung, (Eds.), Third International Handbook of Mathematics Education, Vol. B (pp. 361–392). Dordrecht, The Netherlands: Springer. Krainer, K. (2014). Teachers as Stakeholders in mathematics education research. The Mathematics Enthusiast, 11(1), 49–59.Lo, M. L. (2014). Variationsteori – f?r b?ttre undervisning och l?rande. Lund: Studentlitteratur.Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: L. Erlbaum Associates.Marton, F., & Tsui, A.B.M. (2004). Classroom discourse and the space of learning. New Jersey: Lawrence Erlbaum Associates, Publishers.Marton, F. & Trigwell, K. (2000). Variatio est mater studiorum. Higher education research and development, 19(3).Marton, F., Wen Q., & Wong, K.C. (2005). “Read a hundred times and the meaning will appear ... “ Changes in Chinese University students’ views of the temporal structure of learning. Higher Education, 49, 291–318. Marton, F. (2015). Necessary Conditions of Learning. London: RoutledgeMcKenney, S., & Reeves, T. (2012). Conducting Educational Design Research: What it is, How we do it, and Why. London: Routledge.McKenney, S., Nieveen, N., & Van den Akker, J. (2006). Design research from a curriculum perspective. I Van den Akker, J., Gravemeijer, K, McKenney, S. & Nieveen, N. (Eds). (2006). Educational design research. London: Routledge, 62–90.Nieveen, N., McKenney, S., & van den Akker, J. (2006). Educational design research: The value of variety. In J. Van den Akker, K. Gravemeijer, S. McKenney & N. Nieveen (Eds.), Educational Design Research (pp. 151–158). London: Routledge.Nuthall, G. (2004). Relating classroom teaching to student learning: A critical analysis of why research has failed to bridge the theory-practice gap. Harvard Educational Review, 74(3), 273–306.Olteanu, L. (2016). Framg?ngsrik kommunikation i matematikklassrummet. Diss. Linnaeus University Press.Olteanu, L. (2014). Effective communication, critical aspects and compositionality in algebra. International Journal of Mathematical Education in Science and Technology, 45(7), 1021–1033.Olteanu, C., & Olteanu, L. (2013). Enhancing mathematics communication using critical aspects and dimensions of variation, International Journal of Mathematical Education in Science and Technology, 44(4), 513–522.Patton, M. Q. (2002). Qualitative evaluation and research methods (3rd ed.). Thousand Oaks, CA: Sage Publications, Inc.Ruthven, K. & Goodchild, S. (2008). Linking researching with teaching: towards synergy of scholarly and craft knowledge. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd. ed.) (pp. 561–588). Mahwah: Erlbaum. Stagoll, C. (2005). Force. In A. Parr (Ed.), The Deleuze Dictionary (pp. 106–108): Edinburgh University Press. Van den Akker, J. (1999). Principles and methods of development research. I J. Van den Akker, R. M. Branch, K. Gustafson, N. Nieveen, & T. Plomp (Eds.), Design approaches and tools in education and training. (pp. 1–14). Dordrecht: Kluwer Academic Publishers.Vanderlinde, R., & Van Braak, J. (2010). The gap between educational research and practice: Views of teachers, school leaders, intermediaries and researchers. British Educational Research Journal, 36(2), 299 – 316. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download