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Conceptual and Procedural Knowledge of Mathematical Functions dissertations | 34 | Pål Lauritzen| 34 | Pål | Pål Lauritzen Conceptual and Procedural Knowledge of Mathematical Functions Conceptual and Procedural Knowledge of Mathematical Functions of Conceptual Knowledge Procedural and Pål Lauritzen Function is one of the most important Conceptual and Procedural concepts and tools in mathematics. This thesis discusses it as a complex of Knowledge of Mathematical conceptual and procedural knowledge and suggests viable and sustainable Functions educational practices for mathematics teaching based on empirical research. Publications of the University of Eastern Finland Publications of the University of Eastern Finland Dissertations in Education, Humanities, and Theology Dissertations in Education, Humanities, and Theology No 34 isbn: 978-952-61-0892-6 (print) issnl: 1798-5625 issn: 1798-5625 isbn: 978-952-61-0893-3 (pdf) issn: 1798-5633 PÅL LAURITZEN Conceptual and Procedural Knowledge of Mathematical Functions Publications of the University of Eastern Finland Dissertations in Education, Humanities, and Theology No 34 University of Eastern Finland Joensuu 2012 Kopijyvä Oy Joensuu, 2012 Sarjan vastaava toimittaja: Jopi Nyman Myynti: Itä-Suomen yliopiston kirjasto ISBN: 978-952-61-0892-6 (print) ISBN: 978-952-61-0893-3 (PDF) ISSNL: 1798-5625 ISSN: 1798-5633 Lauritzen, Pål Conceptual and Procedural Knowledge of Mathematical Functions Joensuu: University of Eastern Finland, 2012, 158 pages Publications of the University of Eastern Finland Dissertations in Education, Humanities, and Theology ISBN: 978-952-61-0892-6 (print) ISBN: 978-952-61-0893-3 (PDF) ISSNL: 1798-5625 ISSN: 1798-5633 ABSTRACT Background. Function is one of the most important concepts and tools in mathematics. Its applicability depends on both conceptual and procedural knowledge. However, there are few studies of how these two knowledge types of function relate to each other and what could be an appropriate pedagogical implication. Even to find an instrument to measure the knowledge types independently from each other appears to be a hard task. Aims. The research explored how conceptual and procedural knowledge of functions can be measured, what is the relationship between them, and how the students’ ability to apply functions within economic and other mathematical tasks depends on the two types of knowledge. The outcome was related to the pedagogical philosophy applied to the study population at the upper secondary school. Methods. Data was collected at three different stages from 476 students in economics. Confirmatory factor analysis was applied to develop tasks to measure three components: ‘’procedural knowledge of functions’, ‘conceptual knowledge of functions’ and ‘the ability to apply functions’. A structural equation modelling technique allowed integrating factor analysis and regression analysis into one statistical model to study relationships. Even if causal relations could not be proven, the analysis was suitable to study whether the relations suggested in the model match the sample of data. Results. A large group of subjects showed good procedural knowledge but modest conceptual knowledge. Conceptual scores appeared even lower among those subjects who showed poor procedural knowledge. However, all students who scored high in conceptual tasks, scored also high in procedural tasks. Thus, the results support the genetic view that procedural knowledge is a necessary but not sufficient condition for conceptual knowledge. On the other hand, procedural knowledge alone seems to be insufficient for the student to be able to apply functions. The educational background of the subjects might have fostered this outcome. Interviews indicated that focus of the school teaching has been on simple procedures without links to abstract conceptual knowledge. Conclusions. The results refer to a quite polarized pedagogy concentrating on teaching simple procedures on one hand or giving lessons on abstract definitions without appropriate links to procedural knowledge on the other hand. This might reinforce the polarization among students to so-called conceptual learners and procedurally bounded learners. To develop practical pedagogical theories, it might be important to combine the systematic analysis of conceptual and procedural knowledge of functions with a theory of knowledge structures and scaffolding within constructivist views of teaching and learning in general. Key words: conceptual knowledge, function, genetic view, procedural knowledge. iii Lauritzen, Pål Matemaattisiin funktioihin liittyvä konseptuaalinen ja proseduraalinen tieto Joensuu: Itä-Suomen yliopisto, 2012, 158 sivua Publications of the University of Eastern Finland Dissertations in Education, Humanities, and Theology ISBN: 978-952-61-0892-6 (nid.) ISBN: 978-952-61-0893-3 (PDF) ISSNL: 1798-5625 ISSN: 1798-5633 TIIVISTELMÄ Tausta. Funktio on yksi matematiikan tärkeimpiä peruskäsitteitä ja työkaluja. Sen soveltaminen edellyttää sekä konseptuaalisen tiedon että proseduraalisen tiedon hallitsemista. On vähän tutkimuksia siitä, miten näitä kahta näkökohtaa tulisi painottaa. Jo pelkästään konseptuaalisen ja proseduraalisen tiedon mittaaminen on haasteellista. Tavoitteet. Tutkimuksessa selvitettiin, millaista konseptuaalista ja proseduraalista tietoa oppilaat liittävät funktioon käsitteeseen, miten näitä tiedon lajeja voidaan mitata, mikä on niiden keskinäinen suhde, ja miten ne vaikuttavat oppilaan kykyyn soveltaa funktioita taloustieteissä. Tulokset suhteutetaan kohdejoukon samaan lukio-opetukseen. Menetelmät. Kohdejoukkona oli 476 taloustieteen opiskelijaa, joita mitattiin kolmessa eri vaiheessa. Konfirmatorisen faktorianalyysin avulla kehitettiin kolme eri mittaria: funktioon liittyvän proseduraalisen tiedon mittaaminen, funktioon liittyvän konseptuaalisen tiedon mittaaminen, sekä kyky soveltaa funktioita. Näiden välisten suhteiden selvittämiseksi sovellettiin strukturaalisen mallintamisen tekniikkaa. Tulokset. Kohdejoukon enemmistöllä proseduraalinen osaaminen oli melko vahvaa, mutta konseptuaalinen osaaminen vaatimatonta. Proseduraalista tietoa huonosti hallinneilla konseptuaalisen tiedon hallinta oli vieläkin vaatimattomampaa. Sen sijaan kaikilla konseptuaalisen tiedon hallinneilla myös proseduraalisen tiedon hallinta oli korkealla tasolla. Tulokset tukevat geneettistä näkemystä, jonka mukaan proseduraalinen tieto on välttämätön, mutta ei riittävä ehto konseptuaalisen tiedon syntymiselle. Yksinomaan proseduraalinen tieto ei takaa myöskään sitä, että oppilas osaisi soveltaa funktiota, vaan hänen on hallittava myös konseptuaalinen tieto. Oppilaiden lukiossa saamalla matematiikan opetuksella oli ilmeinen vaikutus tuloksiin, sillä haastattelut paljastivat opetuksen keskittyneen yksinkertaisin proseduureihin vailla pyrkimyksiä linkittää niitä käsitteelliseen tietoon. Johtopäätökset. Tulokset viittaavat polarisoituneeseen matematiikan opetukseen, missä yhtäältä harjoitellaan yksinkertaisia proseduureja ja toisaalta käsitellään luentomaisesti abstrakteja asioita linkittämättä niitä käyttökelpoiseen proseduraaliseen tietoon. Tämä saa luultavasti aikaan polarisoitumisen myös oppilaiden oppimistyyleissä: ns. konseptuaaliset oppijat vs. proseduureihin sidotut oppijat. Edelliset pyrkivät ymmärtämään käsitteet vailla kiinnostusta niiden soveltamiseen, kun taas jälkimmäiset opettelevat ainoastaan yksinkertaisia proseduureja pyrkimättä ymmärtämään niiden pohjana olevia käsitteitä. Kestävien ja elinvoimaisten pedagogisten käyttöteorioiden kehittäminen edellyttää matematiikan konseptuaalis-proseduraalisten tietorakenteiden systemaattista analyysiä sekä samalla sen linkittämistä tieto- ja oppimisteorioihin. Avainsanat: geneettinen näkemys, funktio, konseptuaalinen tieto, proseduraalinen tieto. iv Acknowledgement The motivation for this dissertation developed during many years of employment as lecturer in mathematics at BI Norwegian Business School. It is written as my doctoral dissertation at the Philosophical Faculty at the University of Eastern Finland. I would like to thank Professor Lenni Haapasalo for encouragement and inspiration as my advisor. The discussions with him were always stimulating and constructive. I also thank Professor Bernd Zimmermann and Professor Jozef Hvorecky for reading the manuscript and giving valuable suggestions for improvement. Finally, I would like to express my gratitude to both my advisor and Professor Jopi Nyman, who worked a lot to help me improve the language of the thesis and its layout. Pål Lauritzen Oslo, August 2012 v Contents ABSTRACT ................................................................................................................... III TIIVISTELMÄ ............................................................................................................... IV ACKNOWLEDGEMENT ............................................................................................ V CONTENTS .................................................................................................................. VI 1 INTRODUCTION .................................................................................................... 1 2 BACKGROUND ....................................................................................................... 2 2.1 The duality of mathematical knowledge ...................................................................... 2 2.2 Terminology ...................................................................................................................... 9 2.3 Process and procept ......................................................................................................
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