Mathematics and science between skills , knowledge and process

1. Mathematics and science betweenskills, knowledge and process

2. Didactical TriangleI/didactical System Teaching Creating best possible conditions for learning – Planning and Interaction Teacher Student Learning Acquisition (cognition, individual) Participation (being part of social praxis – classroom set ups norms for (good) math activity) Subject Products (skills, knowledge) - Mastering Proces (develop algorithms and proofs, problemsolving) - understanding

3. Subject = Products and process Give examples of products and processes in math or science Discuss what could be the reasons for giving importance to products – to processes Have you experienced any development in the importance given to them? If so, what could have caused that development?

4. Subject = Products and process Introduction (M. Lampert in grade 5) • ___groups of 12 = 10 groups of 6 • 30 groups of 2 = ___groups of 4 • ___groups of 7 = ___groups of 21 • ___groups of 18 = ___groups of 21 • How many solutions can you to d) – can this be generalized? • How did you solve the problems? • What could be the reason for this problem in grade 5?

5. Products and process Products: Basic calculations, percent and fractions, basic equations, area, geometrical shapes, simple applications Key word: Mastering Impact of emphasizing products: Children will focus much more on REMEMBER HOW to do than SOUGHTING OUT WHAT to do

6. Products and process Background for the NCTM initiative • The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase (NCTM 2000)

7. Products and process NCTM 2000 Principle no 4: Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and previous knowledge. Standard 1:Number and Operations. ...deals with understanding numbers, developing meanings of operations, and computing fluently. Standard 6: Problem Solving. Solving problems is not only a goal of learning mathematics but also a major means of doing so. Standard 7: Reasoning and Proof. ....People who reason and think analytically tend to note patterns, structure, or regularities in both real-world and mathematical situations...... By exploring phenomena, justifying results, and using mathematical conjectures ..... students should see and expect that mathematics makes sense.

8. Products and process

9. Products and process The Danish Kom Project. The trend “Mathematics for all” was predominant since World War II but seemed during 90’s to fail in more aspects. - Lack of people with sufficient math education • Implementation problems due to teacher education and transition problems between the various levels of education system Summing up there were problems with the identity and coherence of mathematics as a subject across the levels. So the idea was to express mathematics in general terms – so called competencies - in order to overcome the stated problems

10. Products and process The Danish Kom Project. • To master mathematics means to posses mathematical competence. But then, what is that? • To posses a competence (to be competent) in some domain of personal, pro­fessional or social life is to master (to a fair degree, modulo the conditions and cir­cumstances) essential aspects of life in that domain. • Mathematical competence then means the ability to understand, judge, do, and use mathematics in a variety of in­tra- and extra-mathematical contexts and situations in which mathematics plays or could play a role. • Necessary, but certainly not sufficient, prerequisites for mathe­matical competence are lots of factual knowledge and technical skills, in the same way as vocabulary, orthography, and grammar are necessary but not sufficient pre­requisites for literacy. • A mathematical competency is a clearly recognisable and distinct, major constituent of mathematical competence.

11. Products and process Mathematicalcompetencies

12. Products and process • 5. Representing mathematical entities (DK)(objects and situations) such as • understanding and utilising (decoding, interpreting, distinguishing between) different sorts of representations of mathematical objects, phenomena and situations; • understanding and utilising the relations between different representations of the same entity, including knowing about their relative strengths and limitations; • choosing and switching between representations. Representations (NCTM 2000) Mathematical ideas can be represented in a variety of ways: pictures, concrete materials, tables, graphs, number and letter symbols, spreadsheet displays, and so on. The ways in which mathematical ideas are represented is fundamental to how people understand and use those ideas. Many of the representations we now take for granted are the result of a process of cultural refinement that took place over many years. When students gain access to mathematical representations and the ideas they express and when they can create representations to capture mathematical concepts or relationships, hey acquire a set of tools that significantly expand their capacity to model and interpret physical, social, and mathematical phenomena.

13. Products and process

14. Group work session wednesday • Discuss what could be the reasons for teaching math/science – if possible categorize the reasons. • Analyze the syllabus in question for (implicit) reasons/grounds for teaching the subject – are goals stated explicit/implicit? • Analyze the syllabus for scientific and every day knowledge • Analyze the syllabus for aspects of application • Discuss the pros and contras for tight curriculum/syllabus • Analyze the syllabus in question with respect to products, processes and competencies • Analyze your teaching reference for scientific and every day knowledge • Analyze your teaching reference for aspects of application • Analyze your teaching reference with respect to products, processes and competencies • If at all, what would be your top priority for a change in the syllabus – why?