1. Researching how successful teachers structure the subject matter of mathematics Anne Watson BSRLM Nov 2008

2. CMTP project www.cmtp.co.uk Chronicle events for three departments Significant improvements at KS3 Looking at their lessons

3. Similarities Curriculum organised round big ideas Exploratory task and discussion Exposition and interaction Range of question-types Student participation of various kinds Accessible tasks, supportive input Wide variety of lesson structures … common features of good teaching

4. Differences Lesson structures Nature of mathematics within lessons Coherence and complexity (TIMSS) Interplay of mathematical ideas (patterns of participation; exchange systematicity) Structuring of mathematical ideas (lesson maps) Possibilities for conceptual understanding

5. Learning study Cycles of development that relate variation in tasks to variation in learning and hence identify ‘critical features’ from a learning perspective Lessons designed on these variation principles are then taught by non-study teachers – what happens?

6. Successful learning Lessons in which the dimensions of variation relevant for the critical features are opened up by teacher and/or pupils and available in the public arena

7. What is seen as learning in learning studies? Understanding the traditional canon of mathematical concepts Discernment of variation Inductive reasoning from these: classification, ordering etc. Distinguishing new ideas from old but similar ideas

8. How many kinds of mathematical knowledge? Kilpatrick: –Procedural fluency (rote, practice, memory, familiarity) –Conceptual understanding –Adaptive reasoning (extended tasks, problem- solving, exploratory tasks, reasoning etc.) –Strategic competency (ditto) –Productive disposition (ditto)

9. Learning concepts Piaget –cognitive conflict (e.g. Swan) Vygotsky –coordinating spontaneous and scientific concepts Cognitive approaches –structural complexity, information processing Mathematical approaches –generalising –covariation??

10. Learning more about how to teach for conceptual understanding Procedures/performance Instrumental/utility/ meaning-through-effect Relational/connected within mathematics Conceptual structures e.g.

11. An example: ideas and their representations Syntactic and semantic matching (sign: signifier and signified) –number line –grid multiplication –graphical representations of functions Matching variations –coordinating the relationship between changes in the sign and changes in the meaning, e.g. decimals represented on number-line –Understanding covariation

12. Other necessary shifts of perspective (add from earlier paper)

13. How to research? Bifurcation in teaching: methods and uses and mathematical enquiry – where is mathematical meaning? Some teachers are much better than others at helping students understand mathematical ideas Evidence is in adapting ideas for use in unfamiliar contexts; being able to build on them to learn composite ideas successfully; mathematical choices about domain of application; remembering and recognising structures Looking at tasks (coherence, complexity); variation (dimensions of variation); example use (inductive potential)