INSTITUT FOR NATURFAGENES DIDAKTIK KØBENHAVNS UNIVERSITET Teaching infinitesimal calculus in high school with infinitesimals Mikkel Mathias Lindahl og Jonas Kyhnæb Kandidatspeciale September 2016 IND’s studenterserie nr. 53 INSTITUT FOR NATURFAGENES DIDAKTIK, www.ind.ku.dk Alle publikationer fra IND er tilgængelige via hjemmesiden. IND’s studenterserie 43. Lene Eriksen: Studie og forskningsforløb om modellering med variabelsammenhænge (2015) 44. Caroline Sofie Poulsen: Basic Algebra in the transition from lower secondary school to high school 45. Rasmus Olsen Svensson: Komparativ undersøgelse af deduktiv og induktiv matematikundervisning 46. Leonora Simony: Teaching authentic cutting-edge science to high school students(2016) 47. Lotte Nørtoft: The Trigonometric Functions - The transition from geometric tools to functions (2016) 48. Aske Henriksen: Pattern Analysis as Entrance to Algebraic Proof Situations at C-level (2016) 49. Maria Hørlyk Møller Kongshavn: Gymnasieelevers og Lærerstuderendes Viden Om Rationale Tal (2016) 50. Anne Kathrine Wellendorf Knudsen and Line Steckhahn Sørensen: The Themes of Trigonometry and Power Functions in Relation to the CAS Tool GeoGebra (2016) 51. Camilla Margrethe Mattson: A Study on Teacher Knowledge Employing Hypothetical Teacher Tasks - Based on the Principles of the Anthropological Theory of Didactics (2016) 52. Tanja Rosenberg Nielsen: Logical aspects of equations and equation solving - Upper secondary school students’ practices with equations (2016) 53. Mikkel Mathias Lindahl and Jonas Kyhnæb: Teaching infinitesimal calculus in high school - with infinitesimals (2016) Se tideligere serier på: www.ind.ku.dk/publikationer/studenterserien/ Abstract In high school students have their first encounter with the notion of infinitesimal calculus. A common problem encountered when introducing this subject is for the students to understand the theory behind the techniques they develop when being taught this. In this thesis the use of infinitesimals will be reinstated to teach this particular theme. They were used to teach infinitesimal calculus even after their abolishment, when the real numbers were constructed. With the dawn of the nonstandard analysis, through the construction of the hyperreal numbers, the infinitesimals were reinstalled as a mathematical sound quantity. This thesis will give reasons for or against the use of infinitesimals in the form of nonstandard analysis by employing the anthropological theory of didactics. The thesis will give a short introduction of how the hyperreal numbers are constructed and how the definition of the differential quotient and integral is equivalent statements to the definitions usually encountered in high school. A teaching course was done in order to survey if the didactical reasons suggested by the theories are noticeable when introducing infinitesimal calculus, with infinitesimals, in high school. As such a textbook material had to be developed in order for the students to prepare for the exam(s). This textbook material is void of the construction of the hyperreal numbers, as introduced in the thesis, but hinges on an intuitive construction of infinitesimal quantities to generate them. An analysis of the teaching employing the anthropological theory of didactics was done in order to determine if the merits of the infinitesimals are on par with the theoretical reasons. The thesis concludes that teaching based on infinitesimals are realistic and helps engage the students by letting their intuition guide them to a great degree.. IND’s studenterserie består af kandidatspecialer og bachelorprojekter skrevet ved eller i tilknytning til Institut for Naturfagenes Didaktik. Disse drejer sig ofte om uddannelsesfaglige problemstillinger, der har interesse også uden for universitetets mure. De publiceres derfor i elektronisk form, naturligvis under forudsætning af samtykke fra forfatterne. Det er tale om studenterarbejder, og ikke endelige forskningspublikationer. Se hele serien på: www.ind.ku.dk/publikationer/studenterserien/ Teaching infinitesimal calculus in high school with infinitesimals −1 −1 −1 1 1 1 − 1 + 1 −푑푥 0 푑푥 − 1 + 1 푑푥 푑푥 푑푥 푑푥 푑푥 푑푥 -2 -1 0 1 2 Master’s Thesis Mikkel Mathias Lindahl and Jonas Kyhnæb Supervisor: Carl Winsløw Department of Science Education Date of submission: 08.08.2016 Names: Mikkel Mathias Lindahl & Jonas Kyhnæb Title: Natsværmer Department: Department of Science Education Supervisor: Carl Winsløw Time frame: February – August, 2016 Abstract In high school students have their first encounter with the notion of infinitesimal calculus. A common problem encountered when introducing this subject is for the students to understand the theory behind the techniques they develop when being taught this. In this thesis the use of infinitesimals will be reinstated to teach this particular theme. They were used to teach infinitesimal calculus even after their abolishment, when the real numbers were constructed. With the dawn of the nonstandard analysis, through the construction of the hyperreal numbers, the infinitesimals were reinstalled as a mathematical sound quantity. This thesis will give reasons for or against the use of infinitesimals in the form of nonstandard analysis by employing the anthropological theory of didactics. The thesis will give a short introduction of how the hyperreal numbers are constructed and how the definition of the differential quotient and integral is equivalent statements to the definitions usually encountered in high school. A teaching course was done in order to survey if the didactical reasons suggested by the theories are noticeable when introducing infinitesimal calculus, with infinitesimals, in high school. As such a textbook material had to be developed in order for the students to prepare for the exam(s). This textbook material is void of the construction of the hyperreal numbers, as introduced in the thesis, but hinges on an intuitive construction of infinitesimal quantities to generate them. An analysis of the teaching employing the anthropological theory of didactics was done in order to determine if the merits of the infinitesimals are on par with the theoretical reasons. The thesis concludes that teaching based on infinitesimals are realistic and helps engage the students by letting their intuition guide them to a great degree. Page i of v Acknowledgements A great big thanks goes to Carl Winsløw for his help in shaping the thesis. The school, Roskilde Katedralskole, deserves a lot of credit for letting the study take place. An infinitely amount of gratitude and respect goes to the wonderfully sweet and awesome 1u, who took opened their arms wide and made the study all the more fun; a better class does not exist. Thanks for the constant surprises and for tolerating all the shenanigans. The biggest of thanks to family and girlfriend for being so patient and understanding. Thanks for the tremendous amount of support. The girls who sat in the other cubicle in the office for about 4 months, Anne and Line, thank you. To the new office, particularly Nine, thanks for making the final month or so bearable; from the breakfast table to the midday pokéhunt, all the way to the afterhours beer. 1 thanks to one another. How on Earth this thesis would have come to be without the lucrative cartalks, 푑푥 the mutual respect, The Trilogy, and the fantastic friendship that came with it all, is beyond infinity. The late nights paid off, let there be more of them. Page ii of v Contents 1 Introduction ......................................................................................................................................... 1 1.1 Motivation ................................................................................................................................... 1 1.2 Problematique.............................................................................................................................. 3 1.3 Research questions....................................................................................................................... 4 2 Theory ................................................................................................................................................. 5 2.1 Mathematical theory – NSA .......................................................................................................... 5 2.1.1 Real numbers ........................................................................................................................ 5 2.1.2 The hyperreal numbers ......................................................................................................... 6 2.1.3 Internal sets and functions, and the connection between ℝ and ∗ ℝ.................................... 9 2.1.4 Analysis using the hyperreal numbers ..................................................................................14 2.2 Didactical theory .........................................................................................................................26 2.2.1 Anthropological Theory of Didactics .....................................................................................26 2.2.2 What to be taught ...............................................................................................................29 2.2.3 Didactical reasons for or against using NSA ..........................................................................35 3 Concrete implementation of the knowledge to be taught ...................................................................38