1. Anna Jørgensen, DMUK 12-11-12 Mathematics teaching and learning in the Danish primary and secondary school (age 6 – 16) With a perspective on mathematical competences and working methods

2.  Bent Christiansen :  The matter meant  the matter taught  the matter learnt  Is it possible to close the gap? Anna Jørgensen, DMUK 12-11-12

3. Contents  Foreword  Common goals (FM 2009) History 1976 1995 2009 History 1976 1995 2009 Mathematical competences in Common Goals (FM 2009) Mathematical competences in Common Goals (FM 2009) Mathematical working methods in Common Goals (FM 2009) Mathematical working methods in Common Goals (FM 2009)  An example from grade 6  Is it possible for teachers in practice to teach in agreement with Common Goals? Anna Jørgensen, DMUK 12-11-12

4. From the Danish curriculum 1976 Goals concerning students working methods: ”It is considered to be a goal that the individual student gets an experimenting approach when gaining insight in new mathematical topics” Anna Jørgensen, DMUK 12-11-12

5. From the Danish curriculum 1995 From the Danish curriculum 1995 The students should be able to tackle mathematical problems, that are not routine problems, and they should be familiar with experimenting working methods Anna Jørgensen, DMUK 12-11-12

6. Common Goals 2009 Objective 1. The main goal is that students develop mathematical competences and achieve knowledge and skills so that they will be able to manage appropriately in mathematical situations concerning every day life and society. Anna Jørgensen, DMUK 12-11-12

7. Objective 2. Teaching is designed so that students independently and through dialog and cooperation with others can learn that working with mathematics requires and facilitates creative activities and that mathematics provides tools for problem solving, reasoning and communication Anna Jørgensen, DMUK 12-11-12

8. Mathematical competences  Mogens Niss: Mathematical competence 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. Anna Jørgensen, DMUK 12-11-12

9. Mathematical competences To ask in, with, about mathematics --------------------------------  Mathematical thinking competences  Problem tackling competences  Modelling competence  Reasoning competences To deal with mathematical language and tools ----------------------------------  Representing competences  Symbol and formalism competences  Communicating competences  Aids and tools competences Anna Jørgensen, DMUK 12-11-12

11. Common Goals 2009 Mathematical working methods (1) Goals after grade 9 The teaching should lead up to students getting knowledge and skills that enable them to...  take part in developing strategies and methods in connection with the mathematical topics  investigate, systematise, reason and generalise when working with mathematical problems Anna Jørgensen, DMUK 12-11-12

12. Common Goals 2009 Mathematical working methods (2) Goals after grade 9  read maths ”academic” texts and communicate about mathematics  Work individually and together with others on solving mathematical problems  Work with problem solving in a process, that builds on dialogue and on students different potentials Anna Jørgensen, DMUK 12-11-12

13. Mathematical Mathematicalworking methods Competences Take part in developing methods work inquiring work individually and collaborating collaborating Be in dialogue with other students Mathematical topics and concepts

14. Competence based description of mathematical knowledge In preparation, the teacher should focus on the mathematical topics and concepts, the mathematical competencies and the working methods at the same time. It is therefore important that the goals can ”work together”. E.g. a teaching sequence in grade 1-3 may, at the same time, focus on the pupils’ developing methods for addition, their problem solving and their ability to collaborate with others in solving maths problems Anna Jørgensen, DMUK 12-11-12

15. Congruence, similarity and scale in grade 6  Why and how can Common Goals 2009 support the teacher in his/her work? The idea is described in the article: ”Matematiklærer og Fælles Mål 2009” ( Mathematics teacher and common goals 2009) by Thomas Kaas The teacher is Heidi Kristiansen Anna Jørgensen, DMUK 12-11-12

16. Teacher’s reflection in every teaching situation  What am I going to teach?  Why am I going to teach it?  How am I going to teach it? Anna Jørgensen, DMUK 12-11-12

17. Mathematical Mathematicalworking methods Competences HOW WHY WHAT Mathematical topics and concepts

18. Example of activities Heidi Kristiansen and grade 6 at Lynghøjskolen, Roskilde  The teacher tells that congruente means somewhat like “identical” “the same”  Are the figures the same? 6 different sets of figures 6 different sets of figures (Mathematical thinking competence) Anna Jørgensen, DMUK 12-11-12

19. Congruent triangles? Congruent triangles?  What does it mean to be identical?  ”Can the figures be turned around?”  ”Can they be mirror images?” Anna Jørgensen, DMUK 12-11-12

20. What does it mean that two figures are identical?  The Book:  ”Two figures are congruent if they can be brought to cover each other point for point”  The class discusses: What does that mean?  One of the pupils says: ”One can cut them out and place them on top of each other. If they cover each other completely, then they are congruent, that is they are completely identical” Anna Jørgensen, DMUK 12-11-12

21. -and the discussion continues  Teacher:  ”Is it really necessary to cut them out?  Could we have measured them instead?”  -and the mathematical discussion continues Anna Jørgensen, DMUK 12-11-12

22. Similar figures  The teacher: ”Cut or draw similar figures. Write their scale. You decide what the scale should be. Present your work” Anna Jørgensen, DMUK 12-11-12

24. How does one write the scale in connection to the squares? Anna Jørgensen, DMUK 12-11-12

26. A pupil: ”Are they similar just because both are one bigger? Another pupil: ”Before, the scale was 4:7, but if we use the same trick here, there will be two different scales” A third pupil: ”They haven’t multiplied by the same number. Isn’t one supposed to do that?” Anna Jørgensen, DMUK 12-11-12

28. ”Oh Heidi! That’s not true. It’s not twice as big. It’s 4 times as big!” Anna Jørgensen, DMUK 12-11-12

29. The teacher uses the competences as indicators/pointers in preparing and teaching -Here mathematical thinking competence and reasoning competence -so that teaching goes beyond the mathematical topic -here it was congruence, similarity and scale Anna Jørgensen, DMUK 12-11-12

30.  Is it possible for teachers to teach in agreement with to teach in agreement with Common Goals 2009? Common Goals 2009?  Is it possible to descrease the gap between the matter meant, the matter taught and the matter learnt? Anna Jørgensen, DMUK 12-11-12