Mathematics didactics

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Mathematics didactics in the context of their influencing variables

Mathematics didactics is a subject didactics for the subject of mathematics . As a science, it deals with teaching and learning mathematics for all ages and encompasses learning objectives , content and teaching methods in mathematics lessons as well as the learning behavior of students . As a university discipline, mathematics didactics is taught at colleges of education and universities and is an integral part of training for the teaching profession in mathematics.

The international society for mathematical didactics is the International Commission on Mathematical Instruction (ICMI) founded in 1908 , which organizes the International Congress on Mathematical Education (ICME) every four years .

Development of mathematics didactics

Geometry lesson, illustration based on a medieval edition of the elements of Euclid.

The beginnings of mathematics didactics are closely linked to the development of mathematics and the respective curricula for the subject. While in ancient times the didactics in Babylon and Egypt primarily on elementary mathematics concentrated, especially the Greeks joined Plato , Aristotle and Euclid Mathematics with philosophical questions. The latter wrote one of the first mathematical textbooks, "The Elements" . From the Middle Ages on , mathematical subjects such as arithmetic and geometry were taught as part of the Quadrivium . The arithmetic books by Adam Ries can be regarded as the first didactic works for a mathematics lesson. With the increase in the importance of trade , arithmetic and thus the teaching of mathematics in particular gained an important social position. Apprentices from skilled trades such as bricklayers, moneylenders and merchants learned job-related mathematical skills. First of all, subject-specific subject didactics developed for historical reasons: arithmetic, geometry, trigonometry , with a questioning and developing style of teaching predominating.

As a development and research area, mathematics didactics is a rather young area. It has developed particularly since the 19th century. It was shaped in the long term by the Humboldtian ideal of education , which goes hand in hand with the implementation of a general, state school system, the educational conception of new humanism and the goal of education for independence and personal development . The separation into different school sectors such as elementary school and high school as well as gender segregation in Higher daughters schools for girls and high schools for boys has led to the emergence of focus related mathematics didactics, particularly the relevant socio-cultural and anthropogenic conditions (see Berlin model ) considered. The discussion about set theory , which was essentially founded by Georg Cantor , and the introduction of the decimal system of measurement continued to have a special influence . Towards the end of the 19th century there was a change from general, experience-based mathematics didactics to an increasingly empirical science. In 1893 a chair for mathematics didactics was established at the Georg-August University in Göttingen . The Merano Conference in 1905 can be seen as a further milestone , at which Felix Klein in particular emphasized mathematics lessons as an important task in the education of functional thinking and spoke out in favor of including analysis in the curriculum. At the same time, intensive international cooperation began. In 1908, at the international congress of mathematicians in Rome, the "International Mathematical Commission" (IMUK) was founded with Klein as chairman.

In the time of National Socialism , mathematics and thus its didactics were subordinated to nationalist, racist and military goals and thematized in connection with applications of aerostatics , aerodynamics and ballistics . Representatives of such subject didactics were Erich Günther and Karl Hahn (both also exerted influence on school politics as educators). After the Second World War, Walter Lietzmann's methodology particularly influenced mathematics didactics. In the 1950s, discussions about teaching concepts such as action-oriented, emancipatory , genetic , application-oriented and problem-oriented mathematics lessons were reflected in works on mathematics didactics. The new mathematics that emerged in the 1970s and the mechanization that began in the 1980s, for example through the introduction of the pocket calculator , also found its way into this subject didactics. The latter was largely shaped by Hans-Georg Weigand . Since then, mathematics didactics has further established itself as a university discipline and contributes significantly to the qualification of mathematics teachers. In 1975, at the 9th Federal Congress of Mathematics Didactics, the " Society for Didactics of Mathematics " (GDM) was founded Countries to work together. In 1975 the first "World Congress of Mathematics Didactics" took place at the University of Karlsruhe (ICDM), which is planned for 2016 again in Germany at the University of Hamburg . In 1977 the “ Mathematik-Lehrs-Einheit-File ” (MUED), a self-administered organization of mathematics teachers from all types of schools, was set up with the idea of ​​developing, testing and exchanging application-oriented mathematics lessons . Furthermore, curricular developments such as the reform of the upper secondary level in the 1980s, the Saarbrücken framework agreement in the post-war years, in which mathematics was already listed as a compulsory subject , and the educational standards introduced in 2003 by the German Conference of Ministers of Education as a result of the PISA study had an impact on mathematics didactics. Similar standards have also been defined in other countries such as the United Kingdom and the United States.

More recent research in the field of this subject didactics ties in with theoretical analyzes as well as existing research results based on qualitative and quantitative studies and systematically uses empirical methods to clarify open theoretical questions. The research projects are very diverse. Examples include a. the area of ​​"mathematics didactics as design science" and, on a smaller scale, "error didactics".

Related Sciences

Mathematics didactics cannot be viewed in isolation, but applies methods and results from other scientific disciplines or takes their knowledge and experience into account. Relevant sciences are first of all mathematics itself, which is the subject of study, and then pedagogy , psychology , general didactics , social sciences , neurobiology , epistemology and educational research . In addition, in the area of ​​applications, for example, sciences such as art can be affected.

Areas of responsibility

Heinz Griesel (1931–2018) defined mathematics didactics as

"Science of the development of practicable courses for learning in the field of mathematics as well as the practical and empirical review of the courses including the considerations of the objective of the courses and the choice of material."

The main task of mathematics didactics is to research and improve the teaching and learning of mathematics with regard to its goals, conditions and methods. It has a normative as well as a descriptive and empirical character. Traditionally, the predominant part is the organization of learning processes in mathematics lessons as a practical task for children (primary school) and young people (secondary levels up to the Abitur). Mathematics didactics formulates prescriptive statements about which content and teaching methods are effective and meaningful, constructed and developed, etc. a. Curricula , teaching methods , learning materials and integratively connects and systematizes various dimensions of the field of activity of mathematics teachers. Following this point of view, mathematics didactics can be understood as the “professional discipline of the mathematics teacher”, which is developed within the framework of university education and through further education and training courses . For example, the project “ Teachers Teaching with Technology Germany” (T³ Germany for short) is active nationwide as an independent sponsor .

In addition to research in the field of general education, there are other job-related task areas such as vocational training , studying at colleges , adult education , in which special didactic aspects can be included. In addition to the general subject didactics, there are special didactic considerations for certain target groups such as elementary school students or sub-areas of mathematics such as didactics of analysis , geometry , algebra , arithmetic or stochastics .

Didactic principles

Mathematics didactics can be viewed as overriding in relation to its learning content , mathematics, as it reflects on ways of knowing the subject and incorporates didactic principles into its considerations. The latter focus on teaching methodology as a sub-area of ​​subject didactics, on the basis of which mathematics lessons can be planned and assessed. Didactic principles have a normative character. They are not necessarily math-specific to begin with, but some principles have special meaning for teaching and learning math. It should be noted here that social reference points are required for the normative statements and the decision about the learning content and the goals of education and upbringing is to be understood as politically and legally determined. A number of principles can be found in the mathematics didactic literature. The spiral principle , which goes back to Jerome Bruner , the genetic principle according to Martin Wagenschein and the operative principle, which was postulated by Jean Piaget and his student Hans Aebli , are considered fundamental . Further methodological concepts such as action-oriented , application-oriented or problem-oriented teaching can be derived from this.

Consequences for the math class

The question of legitimacy is one of the most important questions that mathematics didactics must ask, namely to what extent mathematics teaching is necessary at all and what arguments can be used to justify this necessity. We are looking for an answer as to whether mathematics lessons are really so important for individuals and society that every student must take part in them. Hans Werner Heymann answers this by orienting towards general education as part of general knowledge . According to Piaget, mathematics serves as an instrument for researching children's thought processes and the psychological and intellectual abilities of children. Dealing with mathematics can give students the opportunity to gain experience with scientific - in the sense of science propaedeutic - working methods and to deal with a mathematical description of reality at an early stage .

The national educational standards in mathematics formulated in the 2003 curricula aim to make mathematical knowledge verifiable and thus standardize it. They are to be understood as performance standards, but not as teaching standards, and should consciously allow creative freedom.

In the context of the discussion of mathematics didactics with the topic of how mathematics lessons can be better conveyed in schools and universities, in Germany in contrast to countries like France the question arises as to why fewer and fewer people want to take up the profession of mathematician despite good career prospects despite the application, there is apparently a socially even recognized disinterest in mathematics. Tasks play an essential role in math lessons. In the context of a task didactics, Timo Leuders , for example, names the following dimensions of task quality: authenticity , significance, relevance, openness and the nature of a challenge. Another decisive factor influencing the performance of students and their emotional relationship with mathematics is the teaching teacher , whose didactic competence is summarized in the following statement: "A good mathematics teacher is a teacher who passes on his knowledge and his love of mathematics to his students so that the students can also enjoy engaging with mathematics. ”Research into the subjective theories and behavioral patterns of mathematics teachers is also an area of ​​research in mathematics didactics.

literature

  • Heinz Jörg Claus: Introduction to the didactics of mathematics . Scientific Book Society, Darmstadt 1989, ISBN 3-534-08736-4 .
  • Timo Leuders (Ed.): Mathematics Didactics . Cornelsen Scriptor, Berlin 2007, ISBN 978-3-589-21695-6 .
  • Hans-Georg Steiner (Ed.): Didactics of Mathematics . Scientific Book Society, Darmstadt 1978, ISBN 3-534-06005-9 .
  • Friedrich Zech: Basic course in mathematics didactics. Theoretical and practical guides for teaching and learning math . Beltz, Weinheim / Basel 2002, ISBN 3-407-25216-1 .
  • Helge Lenné: Analysis of mathematics didactics in Germany . From the estate edited by Walter Jung, Klett 1969, 2nd edition 1975 (History of mathematics didactics in Germany until 1969)

Web links

Commons : Mathematics Didactics  - collection of pictures, videos and audio files

Individual evidence

  1. ↑ Bernd Hafenbrak: Introduction to Mathematics Didactics: Introduction. (PDF; 19 kB) (No longer available online.) Archived from the original on March 4, 2016 ; Retrieved May 24, 2013 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.didmath.ewf.uni-erlangen.de
  2. ↑ Marco Drönner: The historical development of mathematics didactics with the focus on the influence of didactics through Humboldt's educational ideal, the Meraner reform and the new mathematics . GRIN Verlag, Munich 2012, ISBN 978-3-656-25915-2 , p. 1 .
  3. ↑ Marco Drönner: The historical development of mathematics didactics with the focus on the influence of didactics through Humboldt's educational ideal, the Meraner reform and the new mathematics . GRIN Verlag, Munich 2012, ISBN 978-3-656-25915-2 , p. 2 f .
  4. ^ Heinz Jörg Claus: Introduction to the Didactics of Mathematics . Scientific Book Society, Darmstadt 1989, ISBN 3-534-08736-4 , pp. 5-7 .
  5. ^ Heinz Jörg Claus: Introduction to the Didactics of Mathematics . Scientific Book Society, Darmstadt 1989, ISBN 3-534-08736-4 , pp. 2 .
  6. ^ A b Bielefeld University: Ludger Huster: Documentation on the development of mathematics didactics in the 19th century: elementary school area . Bielefeld 1981, p. 9-13 .
  7. ↑ Hans-Georg Steiner (Ed.): Didaktik der Mathematik . Scientific Book Society, Darmstadt 1978, ISBN 3-534-06005-9 , pp. 9-21 .
  8. ↑ Marco Drönner: The historical development of mathematics didactics with the focus on the influence of didactics through Humboldt's educational ideal, the Meraner reform and the new mathematics . GRIN Verlag, Munich 2012, ISBN 978-3-656-25915-2 , p. 1-5 .
  9. ↑ Erich Günther: Handbook of Defense Physics. Frankfurt am Main 1936.
  10. ^ Karl Hahn: [Presentation on the physics class]. In: 25 years of Merano reform and the current situation in mathematics and science teaching. In: teaching sheets for mathematics and science. Volume 17, 1931, pp. 162-169.
  11. ↑ Jörg Willer: Didactics in the Third Reich using the example of physics. In: Medical historical messages. Journal for the history of science and specialist prose research. Volume 34, 2015, ISBN 978-3-86888-118-9 , pp. 105-121, passim.
  12. ^ Heinz Jörg Claus: Introduction to the Didactics of Mathematics . Scientific Book Society, Darmstadt 1989, ISBN 3-534-08736-4 , pp. 2-11 .
  13. ↑ About the GDM. Retrieved July 6, 2013 .
  14. ↑ World Congress of Mathematics Didactics. (No longer available online.) Formerly in the original ; Retrieved July 8, 2013 .  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.@1@ 2Template: Toter Link / www.braunschweig.ihk.de  
  15. ↑ The MUED: What we are - what we want. Retrieved July 7, 2013 .
  16. ^ Colloquium on Mathematics and its Didactics - SS 2013. (No longer available online.) Archived from the original on August 21, 2011 ; Retrieved July 9, 2013 .
  17. ^ The school curriculum. Retrieved July 9, 2013 .
  18. ^ Principles and Standards for School Mathematics. Retrieved July 7, 2013 .
  19. ↑ Category: Research Projects . Retrieved July 7, 2013 .
  20. ↑ Christoph Selter, Gerd Walther (Ed.): Mathematikdidaktik als design science . Klett Grundschulverlag, Leipzig 1999, ISBN 3-12-200060-1 .
  21. ↑ Learning from mistakes in math class. Retrieved July 9, 2013 .
  22. ↑ Reinhard Kahl: Teachers as enemies. The time 28/2007, accessed on July 9, 2013 .
  23. ↑ Katja Maaß: The field of tension between specialist science and specialist didactics. P. 28 , accessed on May 21, 2013 .
  24. ↑ Timo Leuders: Mathematics Didactics . Cornelsen Verlag, Berlin 2003, ISBN 3-589-21695-6 , pp. 11 .
  25. ^ Alfred Schreiber: Fundamentals of mathematics didactics. Retrieved May 11, 2013 .
  26. ↑ Heinz Griesel: The new mathematics for teachers and students: Vol. 1 . Schrödel Verlag, Hanover 1971, p. 296 .
  27. ↑ Erich Wittmann: Basic questions of mathematics lessons . 6th edition. Vieweg Verlag, Braunschweig 1981, p. 2 .
  28. ^ Alfred Schreiber: Fundamentals of mathematics didactics. Retrieved May 11, 2013 .
  29. ↑ Werner Blum, Günter Törner: Didaktik der Analysis . Vandenhoeck & Ruprecht, Göttingen 1983, ISBN 3-525-40545-6 .
  30. ↑ Hans Georg Weigand u. a .: Didactics of geometry for lower secondary level . Spektrum Verlag, Heidelberg 2009, ISBN 978-3-8274-1715-2 .
  31. ^ Hans Georg Weigand, Hans Joachim Vollrath: Algebra in the secondary level . 3. Edition. Spektrum Verlag, Heidelberg 2007, ISBN 978-3-8274-1803-6 .
  32. ^ Friedhelm Padberg: Didactics of arithmetic for teacher training and further training . Elsevier spectrum, Heidelberg 2009, ISBN 978-3-8274-0993-5 .
  33. ↑ Herbert Kütting: didactics of stochastics . BI Wissenschaftsverlag, Mannheim 1994, ISBN 3-411-16831-5 .
  34. ^ Alfred Schreiber: Fundamentals of mathematics didactics. Retrieved May 11, 2013 .
  35. ↑ Werner Blum, Günter Törner: Didaktik der Analysis . Vandenhoeck & Ruprecht, Göttingen 1983, ISBN 3-525-40545-6 , pp. 238-239 .
  36. ↑ Timo Leuders: Mathematics Didactics . Cornelsen Verlag, Berlin 2003, ISBN 3-589-21695-6 , pp. 11 .
  37. ↑ Hans Georg Weigand: Didactic principles. (PDF; 182 kB) p. 1 , accessed on July 10, 2013 .
  38. ^ Friedrich Zech: Basic course in mathematics didactics. Theoretical and practical guides for teaching and learning math . Beltz, Weinheim / Basel 2002, ISBN 3-407-25216-1 , p. 112-117 .
  39. ↑ Hans-Georg Bigalke: Sense and importance of mathematics didactics . In: Hans-Georg Steiner (Ed.) . Didactics of mathematics. Scientific Book Society, Darmstadt 1978, ISBN 3-534-06005-9 , pp. 115 .
  40. ↑ Hans Werner Heymann: General education as a task of the school and as a benchmark for subject teaching. (PDF; 60 kB) (No longer available online.) Archived from the original on December 24, 2012 ; Retrieved July 10, 2013 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.riemer-koeln.de
  41. ^ Emma Castelnuovo: Didactics of Mathematics . Akademische Verlagsgesellschaft, Frankfurt / Main 1968, p. 176-177 .
  42. ↑ Horst Jahner: methodology of mathematical teaching . 6th edition. Source and Meyer Verlag, Heidelberg / Wiesbaden 1985, ISBN 3-494-00977-5 , p. 10-11 .
  43. ↑ Werner Blum u. a .: Educational standards in mathematics: concrete . 5th edition. Cornelsen Scriptor, Berlin 2011, ISBN 978-3-589-22321-3 , pp. 15 .
  44. ↑ Math phobia is a German phenomenon. In: Berliner Zeitung. March 28, 2007, accessed July 10, 2013 .
  45. ↑ Timo Leuders: Quality in mathematics lessons at secondary level 1 and 2 . Cornelsen Scriptor, Berlin 2001, ISBN 3-589-21425-2 , pp. 94 .
  46. ↑ Timo Leuders: Quality in mathematics lessons at secondary level 1 and 2 . Cornelsen Scriptor, Berlin 2001, ISBN 3-589-21425-2 , pp. 99 .
  47. ↑ Karl Hehl: Why Mathematics. (PDF; 104 kB) (No longer available online.) P. 2 , archived from the original on March 4, 2016 ; Retrieved July 10, 2013 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.lehrer.uni-karlsruhe.de
  48. ^ Job-related cognitions, attitudes and subjective theories of math teachers. Retrieved July 10, 2013 .
  49. ↑ Patterns of behavior of teachers when using new media in the subjects of German, mathematics and computer science. Retrieved July 10, 2013 .