Asimptota i asimptotsko ponašanje – didaktička transpozicija objekta znanja u gimnazijskom obrazovanju u Republici Hrvatskoj

Asimptota i asimptotsko ponašanje objekti su matematičkog znanja prisutni u različitim područjima matematike. Asimptotsko ponašanje istaknuto je svojstvo nekih funkcija i krivulja te važan alat za opisivanje i rješavanje raznovrsnih matematičkih i izvanmatematičkih problema. Različitost pristupa kojima se ostvaruje te širok spektar sadržaja i područja nastave matematike s kojima se može povezati, odnosno u kojima je relevantan objekt znanja, asimptotu čine zanimljivim objektom istraživanja. U ovom doktorskom radu istražena je i opisana didaktička transpozicija tog objekta znanja u gimnazijskom obrazovanju u Republici Hrvatskoj. Teorijski okvir unutar kojega je provedeno ovo istraživanje jest antropološka teorija didaktike (ATD), koju je razvio francuski matematičar Yves Chevallard početkom 1980-tih godina za potrebe istraživanja matematičkog obrazovanja. Pregled relevantne svjetske znanstvene literature ukazuje na to da su malobrojna istraživanja fokusirana na opisivanje asimptote kao objekta znanja. Osobito nedostaje onih u kojima se sagledavaju različiti aspekti tog matematičkog pojma te njihov međusobni utjecaj, odnosno različiti konteksti u kojima je taj pojam relevantan. Osim toga, ovo je prvo istraživanje u Republici Hrvatskoj kojim se tema iz gimnazijskog matematičkog obrazovanja teorijski utemeljeno adresira i povezuje s inicijalnim sveučilišnim obrazovanjem nastavnika matematike. Istraživanje je provedeno u skladu s odabranim teorijskim okvirom i obuhvaća tri faze: analizu udžbenika, upitnike sa studentima nastavničkog smjera matematike i intervju sa znanstvenicima. Izgrađen je referentni epistemološki model za objekt znanja asimptote koji je usklađen sa zahtjevima znanja za poučavanje te podržan akademskim znanjem i saznanjima epistemoloških istraživanja. Introduction: Asymptote and asymptotic behaviour are bodies of knowledge that have an important role in many traditional and modern mathematical disciplines and are supported by a rich and well-established abstract theory. Asymptotic behaviour is a significant property of some functions and curves and a tool for describing and solving various problems in theoretical and applied mathematics. Some basic aspects of these concepts must be utilized already in elementary mathematics as powerful tools in graphing and analyzing behaviour of elementary functions at infinity and near singularities and in graphing simple plane curves such as hyperbola. Therefore, this body of knowledge is a common part of upper secondary mathematics curricula worldwide. The notion of an asymptote appears as a theoretical concept, a part of a procedure or as a self-sufficient task in different contexts and different educational cycles. Hence, connection with a wide range of teaching contents and mathematical bodies of knowledge makes it an interesting object of research. This doctoral thesis investigates and elaborates the didactic transposition of asymptote as a body of knowledge in general secondary education in Croatia. The research is conducted within the theoretical framework of the anthropological theory of the didactic (ATD), developed by a French mathematician Y. Chevallard especially for research in mathematics education. Scientific literature review shows that studies focused on describing the body of knowledge asymptote are scarce within ATD and other theoretical frameworks. Corpus of scientific research especially lacks those that consider various aspects of this body of knowledge, their mutual influences and connections or various contexts in which this body of knowledge is relevant. Further, research conducted within this thesis pioneers in mathematics education research in Croatia since it addresses and connects, in a theoretically founded manner, initial university education of mathematics teachers and general secondary education. The main idea of ATD is to determine the relation \(R_I(p, O)\) between a body of knowledge \(O\) and a person that occupies position \(p\) in an institution \(I\). For this purpose mathematical knowledge and activities are described in terms of a praxeology \( [T, \tau, \theta, \Theta]\), where its practical component is represented with task \(T\) and technique \(\tau\) and discursive or theoretical component with technology \(\theta\) and theory \(\Theta\). ATD enables a researcher to develop a working model for a body of knowledge, that is reference epistemological model (REM). REM is a coherent structure consisted of complete and mutually connected praxeologies. It should be compliant, questioned and evaluated with respect to curricular and cultural demands, as well as academic and epistemological knowledge. Methodology : The comprehensive research conducted for this thesis is founded on a REM for the body of knowledge asymptote, \(A\). Asymptote is a body of knowledge consisted of a set of praxeologies for which it is a component of practical or theoretical block. Hence, the developed REM included praxeologies of different range and complexity that exist or can be constructed within curriculum and textbook themes connected to the notion of an asymptote. The relations \(R_B(p,A)\), \(R_S(p,A)\) and \(R_M(p,A)\) are analyzed and compared, where institutions considered are: \(B\) for the two Croatian mathematical gymnasium textbooks, \(S\) for the cohort of 40 the final, fifth year mathematics education students at the largest mathematical department in Croatia and \(M\) for the institution of academic mathematicians. The methodology of the implemented research provided information from all steps of the didactical transposition of the body of knowledge asymptote. It included (1) a praxeological analysis of the textbooks as representatives of the knowledge to be taught, (2) three questionnaires with open-ended questions for the prospective mathematics teachers to provide an insight in related knowledge available to students as a potential taught knowledge and (3) a semi-structured interview with two academics who represent scholarly knowledge. Results: Based on the results gained from three phases of the research, the proposed REM for the body of knowledge asymptote is verified and improved in order to develop an asymptote as a body of knowledge and to make suggestions for the teaching practice especially within the ongoing curricular reform in Croatia. In general, textbook organization is incoherent and contains mainly practical blocks, praxeologies are not connected, and available discourses are either not relevant or not connected to practical activities, except for algebraic manipulation with formulae and equations. Knowledge available to prospective math teachers is loaded with components of the secondary knowledge to be taught, techniques students chose are not the most efficient for the task in question and they do not utilize discursive components. For knowledge to be taught and knowledge available to students asymptote and asymptotic behaviour are available but not fully utilized as praxis or logos in relevant praxeologies. Even though praxeological organization of textbooks and students‟ praxeological equipment do not fully correspond to the proposed REM, we find that the implementation of such REM is possible with a proper support of the noosphere, i.e. the institutions responsible for the mathematics education. Considering this, it is our suggestion for the teaching practice especially within the ongoing curricular reform in Croatia that: all relevant properties of an object (e.g. asymptote, function, curve) should be emphasized and implemented in practical, applied and formal context; common properties of the object should be interrelated and diversely represented; simple, isolated and incomplete praxeologies should be organized into more coherent and complex praxeologies; and asymptotic behaviour should be more emphasized, adequately graphically represented and described by formal and informal mathematical discourse.