Researching how successful teachers structure the subject matter

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

CMTP project • www. cmtp. co. uk • Chronicle events for three departments • Significant improvements at KS 3 • Looking at their lessons

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

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

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?

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

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

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

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

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

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

Other necessary shifts of perspective • (add from earlier paper)

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)