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Mathematical Modelling: Fighting Syllabusitis in Math Education

This talk discusses the approach to researching and developing mathematical modelling in mathematics education, addressing syllabusitis, competencies, and the structuring of curricula. Examples from Australia and Denmark are provided, along with a model for structuring teachers' work and development.

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Mathematical Modelling: Fighting Syllabusitis in Math Education

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  1. Mathematical modelling: Why not?On the fighting of syllabusitis in mathematicseducation

  2. The structure of the talk • Framing: My approach to researching and developingmathematicalmodelling as part of mathematicseducation. • Syllabusitis, competencies and the structuring of curricula: A framework and examplesfrom (Australia and) Denmark. • Structuringteachers’work and development: A model and a concretedevelopmentalexample. • Why not? Issues for discussion.

  3. Traditional curriculum structure (Niss & Højgaard, to appear) • The purposes of teaching. • A syllabus, i.e. an outline of the topics (concepts, procedures, results, events etc.) to be covered. • The instruments of assessment and testing. This traditionalstructurecan give rise to syllabusitis.

  4. Syllabusitis (Jensen, 1995, Højgaard, 2012) Syllabusitis is the name of an educationaldisease. It identifies the mastering of a subject with proficienciesrelated to a syllabus. Mastery of the syllabuscomes to define the essence of teaching, learning and assessment.

  5. Examples of Math education Reform approaches USA: Common Core State Standards in mathematics – practices and content. Australia: Mathematical proficiency and content strands, general capabilities, cross-curriculum priorities. Denmark: Mathematical competencies and contentareas, cross-curriculum priorities.

  6. Modelling the mathematicalmodellingprocess (Blomhøj & Jensen, 2007)

  7. How to put the reform ideasintopractice? A developmentalpremise From a learningperspective, reform objectivesonlymake a difference if they frame and give direction to teachersthinkingabouttheirteaching.

  8. Structuringteachers’work – A model from the project KOMPIS Two-dimensional structure of ambitions Planning by modules with associatedlearningobjectives Aug. Dec. June Arranging the role and activities of eachteaching session 2/9 19/8 23/8 26/8 30/8 16/8 Didacticalclassroommanage-mentbased on onesownlearningobjectives-guided arrangement of the teaching 9.00 8.15 9.50 (Højgaard, 2010, Højgaard et al. 2010, Sølberg et al., 2015)

  9. Syllabusitis and the curriculum tradition of ”bullets with comments” “reasoning” “problem solving” “understanding” “fluency” Textbooks Time Tests (Højgaard, 2012)

  10. Examples from currentmathematics curricula • Australia, grades p-10. • Queensland, grades p-10. • Denmark, grades 10-12, general stream (encl. 35).

  11. Competence and competencies I usecompetence to denotesomeone’sinsightfulreadiness to act in response to the challenges of a given situation. A (mathematical) competencyis then someone’s insightful readiness to act in response to a certain kind of (mathematical) challenge of a given situation. Competencies are • headed for action. • situational. • meaningful as learning objectives in their own right. (Blomhøj & Jensen, 2007)

  12. Syllabusitis ”vaccination” and the structuring of curricula Competency (Højgaard, 2012)

  13. Competencies in the KOM project Math. thinking competency: … carry out and have a critical attitude towards mathematical thinking. Problem handling competency: … formulate and solve both pure and applied mathematical problems and have a critical attitude towards such activities. Modelling competency: … carry out and have a critical attitude towards all parts of a mathematical modelling process. Reasoning competency: … carry out and have a critical attitude towards mathematical reasoning, comprising mathematical proofs. Representation competency: … use and have a critical attitude towards different representations of mathematical objects, phenomena, problems or situations. Symbols and formalism competency: … use and have a critical attitude towards mathematical symbols and formal systems. Communication competency: … communicate about mathematical matters and have a critical attitude towards such activities. Aids and tools competency: … use relevant aids and tools as part of mathematical activity and have a critical attitude towards the possibilities and limitations of such use. (Niss & Højgaard, to appear)

  14. Examples from currentmathematics curricula • Denmark, grades 8-9, adulteducation (encl. 28). • Denmark, grades 10-12, technicalstream: • Formal curriculum (encl. 21). • Official guidance for teachers. • Denmark, grades p-9, compulsoryeducation: • Common objectives. • Formal curriculum.

  15. Structuringteachers’work – A model from the project KOMPIS Two-dimensional structure of ambitions Planning by modules with associatedlearningobjectives Aug. Dec. June Arranging the role and activities of eachteaching session 2/9 19/8 23/8 26/8 30/8 16/8 Didacticalclassroommanage-mentbased on onesownlearningobjectives-guided arrangement of the teaching 9.00 8.15 9.50 (Højgaard, 2010, Højgaard et al. 2010, Sølberg et al., 2015)

  16. Why not? Issues for discussion • Systemicconservatism – ambitions measured by the width of the syllabus. • Traditions and the conservatism of teachers. • Teaching for competence is more meaningful, but also more difficult. The necessity of developingteachers’subjectspecificcompetencies – mathematically and didactically. • Assessingcompetencies is a complex, but necessarychallenge to face.

  17. References Blomhøj, M. & Jensen, T.H. (2007): What’s all the fuss about competencies? Experiences with using a competence perspective on mathematics education to develop the teaching of mathematical modelling. In: Blum, W. et al. (ed.), Modelling and applications in mathematics education – The 14th ICMI-study (pp. 45-56). Springer, New York, US. Højgaard, T. (2010). Communication: The Essential Difference Between Mathematical Modeling and Problem Solving, in R. Lesh et al. (eds), Modeling Students’ Mathematical Modeling Competencies, Springer, New York, USA, pp. 255–264. Højgaard, T., Bundsgaard, J., Sølberg, J. & Elmose, S. (2010). Kompetencemål i praksis – foranalysen bag projektet KOMPIS. MONA 3: 37-54. Højgaard, T. (2012). Competencies and The Fighting of Syllabusitis, in ICME 12 LOC (ed.), Pre-Proceedings of ICME 12, ICME 12 LOC, Seoul, South Korea, pp. 6412–6420. Available from the authors homepage. Jensen, J.H. (1995): Faglighed og pensumitis. Uddannelse 9, The Ministry of Education, Copenhagen, Denmark, pp. 464-468. National Curriculum Board (2009): Shape of the Australian Curriculum: Mathematics. Commonwealth of Australia, Australia. Niss, M. (2001). University Mathematics Based on Problem-Oriented Student Projects: 25 Years of Experience with the Roskilde Model. In: Holton, D. (ed.), The Teaching and Learning of mathematics at University Level: An ICMI Study (pp. 153–165). Kluwer, Dordrecht, The Netherlands. Niss, M. & Højgaard, T. (to appear): Mathematical Competencies in Mathematics Education: Past, Present and Future. Springer, New York, US. Sølberg, J., Bundsgaard, J. & Højgaard, T. (2015). Kompetencemål i praksis – hvad har vi lært af KOMPIS? MONA 2: 46-59.

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