http://www.math.unipa.it/~grim/YESS-5/PaperNilsen.pdf

LEARNING AND TEACHING FUNCTIONS AND THE TRANSITION FROMLOWER SECONDARY TO UPPER SECONDARY SCHOOLHans Kristian NilsenSr-Trndelag University College, Norway(PhD student)INTRODUCTIONIn Norway, the transition between different phases of schooling, particularly in relation to thelearning and teaching of mathematics, is an area where little research has been done. The major partof the international research in this field concerns the transition from upper secondary school touniversity/university college (often denoted as the secondary-tertiary transition) (Gueudet, 2008;Guzmn et al., 1998). My own experiences as a student and a teacher, at both lower and uppersecondary school levels have led me to believe that the traditions and beliefs in these institutionsdiffer in ways which in turn might affect students learning.Kindergarden(1-5 years)University/university college(20- )Uppersecondary vocational orgeneral studyprogram(17-19)LowerSecondaryschool(14-16)Primaryschool(6-13)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 isimportant 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 generalstudy program, which aims to prepare students for higher education. The curriculum is different inthese programmes and is considered to be more theoretical at the general study program. This isalso 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 ofschooling, and personally I find the development of students conceptual understanding of functionsto be an interesting research area. It is also possible to expand this area of research, for example bytaking the universities/university colleges into consideration, as the learning and teaching offunctions is an important issue in several of these study programmes.RESEARCH QUESTIONSHow do the students conceptions of functions develop from the 10th grade at lower secondaryschool to the 11th grade at upper secondary school? How do the students argue for their conception of functions at lower secondary andupper secondary school? How do the students argue for their conception of the slope of a function at lowersecondary and upper secondary school? How do the students at upper secondary, general study programme, relate the slopeof a function to the concept of differentiation? 2How 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 uppersecondary school? How is the slope of a function presented at lower secondary compared to uppersecondary school, and in which way is this related to the concept of differentiation atupper 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 andupper secondary) when it comes to the teaching and learning of mathematics, andfunctions in particular?THEORETICAL BACKGROUNDI illustrate an overview of my theoretical framework by the use of the sketch below:

Figure 2: An overview of my theoretical basisAs indicated in the model above, I will use two perspectives for analyzing respectively what I callthe teaching aspect and the learning aspect of the actual transition from lower to uppersecondary school. In the teaching category, I will consider aspects which can be regarded asexternal to the individual student. Examples of this could be the teaching methods used and how theteachers approach the subject of functions. This information is mainly provided throughobservations and interviews. Another example could be the applied textbooks, and the differentexercises given to the students. It will also be interesting to investigate upon whether the (laterdescribed) use- and/or exchange value is dominating in the mathematics teaching. Studying theseissues, I find it useful to apply the institutional perspective as this provides me with an appropriateanalytical tool for analyzing the actual teaching situations. I will also argue that the institutionalperspective 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 oflearning. The core of this will be the students engagement in mathematical activities (provided bySocio-cultural perspective Institutional perspectiveBrousseau, G. (1997); Chevallard, Y.(2005); Gueudet, G. (2008)Historical development:Boyer (1959); Klein (1897);Kleiner (1989Students conceptions:Vygotsky (1987); Pierce (1998);Presmeg (2005); Tall & Vinner(1981); Sfard (1991)Textbook and taskanalysisFucntions as aboundary objectStar & Griesemer(1989)Sociocultural teories oflearning Vygotsky (1978; 1981;1987); Cole (1985); Lerman(2000) ; Pozzi et al (1998) TEACHING LEARNING3conversations, interviews and handwritten material). Important here is the students conceptformation and their conceptual understanding of the function concept. It is also of interest, as statedin my research questions in the previous section, how these conceptual understandings actuallydevelop. The socio-cultural perspective serves as a fundamental basis in this research, overarchingboth teaching and learning.The socio-cultural perspectiveAcknowledging the fact that learning is a complex issue, which takes place in a certain socialcontext within a given culture, this perspective to a great extent matches with my own assumptionsand beliefs. In addition, it is evident that the concept of mediation is an essential part of thisresearch.

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 ofconventional 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 duringinterviews 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 convictionsmentioned above, this important role of mediation also brings in the pragmatic dimension in theconstruction of my theoretical platformConcept formationBy basing my argumentation on the Vygotskian understanding of signs as mediating tools, I willapproach concept formation from a perspective more in line with the socio-cultural way ofthinking. Rooted in Pierce (1998), Presmeg (2005) describes signs through a triad, consisting of arepresentamen, 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 ofbeing 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 todraw a straight line through a two-dimensional plane intersecting the y-axis at -3, making a valuetable 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) 4This interpretant involves meaning making: it is the result of trying to make sense of therelationship of the other two components, the object and the representamen. It is importantto note that the entire first sign with its three components constitutes the second object, andthe entire second sign constitutes the third object, which thus include both the first and thesecond signs. Each object may thus be thought of as the reification of the processes in theprevious signThe role of students own interpretations in forming mathematical concepts is prominent most ofVygotskys work. Vygotsky separates between pseudoconcepts, concepts as we might use them inour everyday language and true concepts as they are defined and used for example within scienceand scientific research (Vygotsky, 1987). Working with students understanding of the functionconcept, it was apparent to me that some students maintained different pseudoconcepts, some ofthem pointing more in the direction of what function and functions mean in everyday life, than whatit actually mean in a pure mathematical sense. In the possible transition from pseudoconcepts to trueconcepts, Vygotsky emphasises the importance of instruction:Conscious instruction of the pupil in new concepts (i.e. new forms of the word) is not onlypossible but may actually be the source for a higher form of development of the childs ownconcepts, particularly those that have developed prior to conscious instruction! (Vygotsky,1987, p. 172)Functions as a boundary objectI think it is an advantage that the focus of the mathematical content considered in such acomparative study as this, is regarded as relevant to both the parties involved. In accordance to theelaborations above, it is evident that functions are a major subject within school mathematics (aswithin mathematics as a whole). It is also evident (from the Norwegian curricula), that functions areprominent at both lower and upper secondary school.Boundary objects are objects that are both plastic enough to adapt to local needs andconstraints of several parties employing them, yet robust enough to maintain a commonidentity across sites (Star & Griesemer, 1989, p. 46)The notion of boundary concept is used by Star & Griesemer (1989) as an analytical conceptwhich both inhabit several intersecting worlds and satisfy the informational requirements of eachof them (p. 393). It is my intention that conceiving of functions as a boundary object will justifythe mathematical focus of this study as it (hopefully) creates a common ground for teachers andresearchers interesting in developing mathematics teaching and the related transition between lowerand upper secondary school.

The institutional perspectiveI find the study of transitions, implying students shifting between two different institutions to bevaluable not only for the sake of comparison, but also because it enables the researcher to study theissues of teaching and learning from different perspectives. An institutional perspective opens upfor this, as it provides the researcher with the possibilities of analysing different cultural aspects oftransition. As the cultural context are investigated and clearly defined to play a role in studentslearning, I find it evident that this perspective is in accordance with the underlying assumptions ofthe socio-cultural perspective.Questioning this change of cultures can lead researchers to consider precise mathematicalcontent, and develop detailed transposition studies. It can also lead researchers to study moregeneral institutional expectations (Gueudet, 2008, p. 245) 5This institutional perspective is rooted in Brousseaus (1997) Theory of didactical situations andChevallards anthropological theory of didactics. A key notion in Brousseaus theory is didacticaltransposition.Teachers isolate certain notions and properties, taking them away from the network ofactivities which provide their origin, meaning, motivation and use. They transpose them intoa classroom context. (Brousseau, 1997, p. 21).Use- and exchange valueStudying 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 usevalue] which is influenced by the performativity system in which schools compete. Further hesuggests that the Maths discourse at college is about use value. Students are asked for a certainlevel of abstraction and understanding of the mathematical concepts to be used, all in a relativelyshort period of time. The Marxist terms use value and exchange value are used to separatebetween the purposes of the mathematical discussions at the institutions. It would be of interest tosee if similar findings may also apply for this study.METHODOLOGYAs I believe in the multiple constructed nature of social phenomena, this makes me positioned inthe ontology of constructivism (or constructionism). Constructionism is an ontological position(often also referred to as constructivism) that asserts that social phenomena and their meanings arecontinually being accomplished by social actors (Bryman, 2004, p. 17)Within this paradigm the methodology is qualitative, based on the hermeneutic tradition, wherecontextual factors are described (Mertens, 2005; Geertz, 1973). As a basis for my subsequentanalysis I draw upon what Mertens (2005, p. 8) labels to the constructivists paradigm, thenaturalistic paradigm of Lincoln & Guba (1985). The well-elaborated principles of thisparadigm 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 todistinguish causes from effects.5) Inquiry is value-bound.(Lincoln & Guba, 1985, p. 37)Samples and data collectionFive different classes in five different lower secondary schools participated in this research. Two ofthese schools are private schools while the other three are public. The private schools were includedin an attempt to seek some diversity in the sample, while the public schools were somewhatrandomly selected, with the only criteria being that they, due to practical reasons, were locatedwithin a reasonable distance from my working place. As the Norwegian school system is quitehomogenous I believe that these schools are representative to their area. The headmasters werecontacted via telephone and their school was invited to participate. The number of students willingto participate from each class varied from three to ten. In total 33 students participated and I amcurrently conducting follow-up research on 12 of these as they entered upper secondary school. Ihave 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 oftheir marks). My purpose is to gain a rich material with some diversity. My data collection at lowersecondary school mainly consisted of five phases: Observations of the teacher teaching, recordedconversations with the students engaging in mathematics in the classroom, interviews with thestudents, collection of students handwritten material and an interview with their teacher. This 6provides me with a diverse material which allows me to study mathematics education from variousperspectives. The data collection at upper secondary school is done in a similar way. My use ofresearch instruments did vary somewhat from school to school, primarily due to the fact that someteachers imposed restrictions for example on my use of a video camera. I have mainly applied semistructuredinterviews (Kvale, 1997). The figure below shows how the distribution of the 12participating students.Figure 4: Distribution of the 12 participating students. (The number in parenthesis indicates the number ofstudents in each class. LS: Lower secondary, US: Upper secondary, VS: Vocational study programme andGS: General study program).ANALYSIS[These days (May, 2010) I am working with my data-analysis. As this work is conducted in thisvery moment it is hard to elaborate on my finding in this paper. Hopefully some of this work isready for discussion at the summer school, and I hope to have the opportunity to discuss also thisaspect with you in Palermo, even if no written material is provided at this point.] 7REFERENCESBos, H. (1980). Mathematics and Rational Mechanics. In Rousseau, G. S. & Porter R. (Eds.)Ferment of Knowledge (pp. 327-355). 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: KluwerAcademic Publishers.Bryman, A. (2004). Social Research Methods. 2nd ed. New York: Oxford University Press.Chevallard, Y. (2005). 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