LEARNING AND TEACHING FUNCTIONS AND THE TRANSITION FROM LOWER SECONDARY TO UPPER SECONDARY SCHOOL

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1 LEARNING AND TEACHING FUNCTIONS AND THE TRANSITION FROM LOWER SECONDARY TO UPPER SECONDARY SCHOOL Hans Kristian Nilsen Sør-Trøndelag University College, Norway (PhD student) INTRODUCTION In Norway, the transition between different phases of schooling, particularly in relation to the learning and teaching of mathematics, is an area where little research has been done. The major part of the international research in this field concerns the transition from upper secondary school to university/university college (often denoted as the secondary-tertiary transition) (Gueudet, 2008; Guzmán et al., 1998). My own experiences as a student and a teacher, at both lower and upper secondary school levels have led me to believe that the traditions and beliefs in these institutions differ in ways which in turn might affect students learning. Kindergarden (1-5 years) Primary school (6-13) Lower Secondary school (14-16) Upper secondary vocational or general study program (17-19) University/univ ersity college (20- ) Figure 1: Transitions in the Norwegian school system. As a PhD student (in my third year), I have chosen this transition as the focus of my research. It is important to note that in Norway, upper secondary schooling is divided in two main programmes: the vocational programmes, which are orientated towards practical professions and the general study program, which aims to prepare students for higher education. The curriculum is different in these programmes and is considered to be more theoretical at the general study program. This is also the case for mathematics as a subject. Both of these programmes are included in this research. Further, I have chosen to focus on functions as this is an area highly relevant to both levels of schooling, and personally I find the development of students conceptual understanding of functions to be an interesting research area. It is also possible to expand this area of research, for example by taking the universities/university colleges into consideration, as the learning and teaching of functions is an important issue in several of these study programmes. RESEARCH QUESTIONS How do the students conceptions of functions develop from the 10th grade at lower secondary school to the 11th grade at upper secondary school? How do the students argue for their conception of functions at lower secondary and upper secondary school? How do the students argue for their conception of the slope of a function at lower secondary and upper secondary school? How do the students at upper secondary, general study programme, relate the slope of a function to the concept of differentiation? 1

2 How is the topic of functions mediated at lower secondary compared to upper secondary school? How is the concept of functions presented at lower secondary compared to upper secondary school? How is the slope of a function presented at lower secondary compared to upper secondary school, and in which way is this related to the concept of differentiation at upper secondary, general study programme? How do the teachers argue for the way that they are teaching functions? How do the students experience these two areas of learning (lower secondary and upper secondary) when it comes to the teaching and learning of mathematics, and functions in particular? THEORETICAL BACKGROUND I illustrate an overview of my theoretical framework by the use of the sketch below: Socio-cultural perspective Institutional perspective Brousseau, G. (1997); Chevallard, Y. (2005); Gueudet, G. (2008) Textbook and task analysis TEACHING Fucntions as a boundary object Star & Griesemer (1989) Historical development: Boyer (1959); Klein (1897); Kleiner (1989 LEARNING Sociocultural teories of learning Vygotsky (1978; 1981; 1987); Cole (1985); Lerman (2000) ; Pozzi et al (1998) Students conceptions: Vygotsky (1987); Pierce (1998); Presmeg (2005); Tall & Vinner (1981); Sfard (1991) Figure 2: An overview of my theoretical basis As indicated in the model above, I will use two perspectives for analyzing respectively what I call the teaching aspect and the learning aspect of the actual transition from lower to upper secondary school. In the teaching category, I will consider aspects which can be regarded as external to the individual student. Examples of this could be the teaching methods used and how the teachers approach the subject of functions. This information is mainly provided through observations and interviews. Another example could be the applied textbooks, and the different exercises given to the students. It will also be interesting to investigate upon whether the (later described) use- and/or exchange value is dominating in the mathematics teaching. Studying these issues, I find it useful to apply the institutional perspective as this provides me with an appropriate analytical tool for analyzing the actual teaching situations. I will also argue that the institutional perspective is coherent with the overarching socio-cultural perspective. When it comes to students learning I will be working within the frames of socio-cultural theory of learning. The core of this will be the students engagement in mathematical activities (provided by 2

3 conversations, interviews and handwritten material). Important here is the students concept formation and their conceptual understanding of the function concept. It is also of interest, as stated in my research questions in the previous section, how these conceptual understandings actually develop. The socio-cultural perspective serves as a fundamental basis in this research, overarching both teaching and learning. The socio-cultural perspective Acknowledging the fact that learning is a complex issue, which takes place in a certain social context within a given culture, this perspective to a great extent matches with my own assumptions and beliefs. In addition, it is evident that the concept of mediation is an essential part of this research. The following can serve as examples of psychological tools, and their complex systems: language; various systems for counting; mnemonic techniques; algebraic symbol systems; works of art; writing; schemes, diagrams, maps, and mechanical drawings; all sorts of conventional signs; and so on (Vygotsky, 1981c, p. 137) The quotation above contains some examples of what Vygotsky described as meditational means. Students hand-written materials, their work at computers, their answers and arguing during interviews and conversations, all related to the learning of mathematics (and in this case, functions) are all examples of such mediating means. Hence, in addition to the personal convictions mentioned above, this important role of mediation also brings in the pragmatic dimension in the construction of my theoretical platform Concept formation By basing my argumentation on the Vygotskian understanding of signs as mediating tools, I will approach concept formation from a perspective more in line with the socio-cultural way of thinking. Rooted in Pierce (1998), Presmeg (2005) describes signs through a triad, consisting of a representamen, an object and an interprentant. One can regard the representamen as the sign itself, for example the linear expression y = 2x 3. A classification of this expression (sign) in terms of being a function, an algebraic expression or a linear equation will relate to the object. Interpreting this sign, in terms of acting on it through different representations, for example to draw a straight line through a two-dimensional plane intersecting the y-axis at -3, making a value table or performing algebraic manipulations will all be acts of the interpretant. Figure 3: A representation of a nested chaining of three signs. (Presmeg, 2005, p. 107) 3

4 This interpretant involves meaning making: it is the result of trying to make sense of the relationship of the other two components, the object and the representamen. It is important to note that the entire first sign with its three components constitutes the second object, and the entire second sign constitutes the third object, which thus include both the first and the second signs. Each object may thus be thought of as the reification of the processes in the previous sign The role of students own interpretations in forming mathematical concepts is prominent most of Vygotskys work. Vygotsky separates between pseudoconcepts, concepts as we might use them in our everyday language and true concepts as they are defined and used for example within science and scientific research (Vygotsky, 1987). Working with students understanding of the function concept, it was apparent to me that some students maintained different pseudoconcepts, some of them pointing more in the direction of what function and functions mean in everyday life, than what it actually mean in a pure mathematical sense. In the possible transition from pseudoconcepts to true concepts, Vygotsky emphasises the importance of instruction: Conscious instruction of the pupil in new concepts (i.e. new forms of the word) is not only possible but may actually be the source for a higher form of development of the child s own concepts, particularly those that have developed prior to conscious instruction! (Vygotsky, 1987, p. 172) Functions as a boundary object I think it is an advantage that the focus of the mathematical content considered in such a comparative study as this, is regarded as relevant to both the parties involved. In accordance to the elaborations above, it is evident that functions are a major subject within school mathematics (as within mathematics as a whole). It is also evident (from the Norwegian curricula), that functions are prominent at both lower and upper secondary school. Boundary objects are objects that are both plastic enough to adapt to local needs and constraints of several parties employing them, yet robust enough to maintain a common identity across sites (Star & Griesemer, 1989, p. 46) The notion of boundary concept is used by Star & Griesemer (1989) as an analytical concept which both inhabit several intersecting worlds and satisfy the informational requirements of each of them (p. 393). It is my intention that conceiving of functions as a boundary object will justify the mathematical focus of this study as it (hopefully) creates a common ground for teachers and researchers interesting in developing mathematics teaching and the related transition between lower and upper secondary school. The institutional perspective I find the study of transitions, implying students shifting between two different institutions to be valuable not only for the sake of comparison, but also because it enables the researcher to study the issues of teaching and learning from different perspectives. An institutional perspective opens up for this, as it provides the researcher with the possibilities of analysing different cultural aspects of transition. As the cultural context are investigated and clearly defined to play a role in students learning, I find it evident that this perspective is in accordance with the underlying assumptions of the socio-cultural perspective. Questioning this change of cultures can lead researchers to consider precise mathematical content, and develop detailed transposition studies. It can also lead researchers to study more general institutional expectations (Gueudet, 2008, p. 245) 4

5 This institutional perspective is rooted in Brousseau s (1997) Theory of didactical situations and Chevallards anthropological theory of didactics. A key notion in Brousseau s theory is didactical transposition. Teachers isolate certain notions and properties, taking them away from the network of activities which provide their origin, meaning, motivation and use. They transpose them into a classroom context. (Brousseau, 1997, p. 21). Use- and exchange value Studying the transition between school and college in England, Hernandez-Martinez (2009) suggests that The Maths discourse at school is about exchange value, [as opposite to use value] which is influenced by the performativity system in which schools compete. Further he suggests that the Maths discourse at college is about use value. Students are asked for a certain level of abstraction and understanding of the mathematical concepts to be used, all in a relatively short period of time. The Marxist terms use value and exchange value are used to separate between the purposes of the mathematical discussions at the institutions. It would be of interest to see if similar findings may also apply for this study. METHODOLOGY As I believe in the multiple constructed nature of social phenomena, this makes me positioned in the ontology of constructivism (or constructionism). Constructionism is an ontological position (often also referred to as constructivism) that asserts that social phenomena and their meanings are continually being accomplished by social actors (Bryman, 2004, p. 17) Within this paradigm the methodology is qualitative, based on the hermeneutic tradition, where contextual factors are described (Mertens, 2005; Geertz, 1973). As a basis for my subsequent analysis I draw upon what Mertens (2005, p. 8) labels to the constructivists paradigm, the naturalistic paradigm of Lincoln & Guba (1985). The well-elaborated principles of this paradigm consist of five axioms: 1) Realities are multiple, constructed, and holistic. 2) Knower and known are interactive, inseparable. 3) Only time- and context bound working hypotheses (idiographic statements) are possible. 4) All entities are in a state of mutual simultaneous shaping, so that it is impossible to distinguish causes from effects. 5) Inquiry is value-bound. (Lincoln & Guba, 1985, p. 37) Samples and data collection Five different classes in five different lower secondary schools participated in this research. Two of these schools are private schools while the other three are public. The private schools were included in an attempt to seek some diversity in the sample, while the public schools were somewhat randomly selected, with the only criteria being that they, due to practical reasons, were located within a reasonable distance from my working place. As the Norwegian school system is quite homogenous I believe that these schools are representative to their area. The headmasters were contacted via telephone and their school was invited to participate. The number of students willing to participate from each class varied from three to ten. In total 33 students participated and I am currently conducting follow-up research on 12 of these as they entered upper secondary school. I have chosen the follow-up students on the basis of three criterions: equal gender distribution, students at both vocational and general study programmes, and variations of skills (on the basis of their marks). My purpose is to gain a rich material with some diversity. My data collection at lower secondary school mainly consisted of five phases : Observations of the teacher teaching, recorded conversations with the students engaging in mathematics in the classroom, interviews with the students, collection of students handwritten material and an interview with their teacher. This 5

6 provides me with a diverse material which allows me to study mathematics education from various perspectives. The data collection at upper secondary school is done in a similar way. My use of research instruments did vary somewhat from school to school, primarily due to the fact that some teachers imposed restrictions for example on my use of a video camera. I have mainly applied semistructured interviews (Kvale, 1997). The figure below shows how the distribution of the 12 participating students. Figure 4: Distribution of the 12 participating students. (The number in parenthesis indicates the number of students in each class. LS: Lower secondary, US: Upper secondary, VS: Vocational study programme and GS: General study program). ANALYSIS [These days (May, 2010) I am working with my data-analysis. As this work is conducted in this very moment it is hard to elaborate on my finding in this paper. Hopefully some of this work is ready for discussion at the summer school, and I hope to have the opportunity to discuss also this aspect with you in Palermo, even if no written material is provided at this point.] 6

7 REFERENCES Bos, H. (1980). Mathematics and Rational Mechanics. In Rousseau, G. S. & Porter R. (Eds.) Ferment of Knowledge (pp ). Cambridge University Press. Boyer, C. B. (1949). The history of the calculus and its conceptual development. Mineola, NY: Dover. Brousseau, G. (1997). Theory of Didactical Situations in Mathematics. Dordrecht: Kluwer Academic Publishers. Bryman, A. (2004). Social Research Methods. 2 nd ed. New York: Oxford University Press. Chevallard, Y. (2005). Steps Towards a New Epistemology in Mathematics Education. In Bosch, M. (Ed.) Proceeding of the fourth congress of the European Society for Research in Mathematics Education, CERME 4, Sant Feliu de Guíxols, Spain. Cole, M. (1985). The zone of proximal development: where culture and cognition create each other. In J. V. Wertsch (Ed.), Culture, Communication and Cognition: Vygotskian Perspectives. Cambridge: Cambridge University Press. pp Geertz, C. (1973). The Interpretation of Cultures. New York: Basic Books. Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), Guzmán, M., et al. (1998). Difficulties in the Passage from Secondary to Tertiary Education. Documenta Mathematica, 901(3), Hernandez-Martinez, P. (2009). Transition to post-compulsory education: the case of algebra as a boundary object between school and college. Presentation held at the European Conference on Educational Research: Vienna, Klein, F. (1897). Mathematical Theory of the Top. New York: C. Scribner s Sons. Kleiner, I. (1989). Evolution of the Function Concept: A Brief Survey. The College Mathematics Journal, 20(4), Kvale, S. (2007). Det kvalitative forkningsintervju. (Translation from S. Kvale, InterViews An Introduction to Qualitative Research Interviewing, 1997). Oslo: Gyldendal Norsk Forlag AS. Lerman, S. (2000). A case of interpretations of social: a response to Steffe and Thompson. Journal for Research in Mathematics Education, 31(2), Lincoln, Y.S. & Guba, E.G. (1985). Naturalistic Inquiry. California: Sage Publication. Mertens, D.M. (2005). Research and Evaluation in Education and Psychology. 2 nd ed. California: Sage Publications. Peirce, C. S. (1998). The essential Peirce. Selected philosophical writings. Vol. 2 ( ). Bloomington, IN: Indiana University Press. Pozzi, S., Noss, R. and Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in Mathematics 36(2), Presmeg, N. (2005). Metaphor and metonomy in processes of semiosis in mathematics education. I M. H. G. Hoffman, J. Lenhard, & F. Seeger (Red.), Activity and sign Grounding mathematics education (ss ). New York: Springer. Sfard, A. (1991). On the dual Nature of Mathematical Conceptions: Reflections on Processes And Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22(1), Star. S. L. & Griesemer, J. R. (1989). Institutional ecology, translations and boundary objects: Amateurs and professionals in Berkeley'sMuseum of Vertebrate Zoology, Social Stud. Sci. 19. Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 22(2), Vygotsky, L.S. (1978) Mind in Society: The Development of Higher Psychological Processes. Cambridge: Harvard University Press. Vygotsky, L.S. (1981) The instrumental Method in Psychology. In J.V. Wretsch (Ed. & 7

8 Trans.) The concept of activity in soviet psychology ( pp )Armonk, NY: M. E. Sharpe. (Russian original published in 1960) Vygotsky, L. S. (1987). Thinking and speech. In L. S. Vygotsky, The collected works of L. S. Vygotsky, Vol. 1, Problems of general psychology (pp ) (R. W. Rieber & A. S. Carton, Eds.; N. Minick, Trans.). New York: Plenum Press. (Original work published 1934). 8

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