parameters for practice and research in task design in mathematics education

1. parameters for practice and research in task design in mathematics education Anne Watson & Minoru OhtaniUniversities of Oxford, UK & Kanazawa, Japan ICME 13 TSG 36

2. ICME Study 22: Task Design in Mathematics Education • Emergent parameters for designers, users and researchers, from the Study

3. Nature and format of the Study • July 2013 ICMI Study 22 Task Design study conference • International IPC • 80 international authors • Collaboration researchers/designers/teachers/students • Syntheses as baselines and springboards (IPC) • Authorship in my paper: originator, synthesiser, me • Parameters: to be populated with theory-specific commitments in each design effort (whether design team, textbook authors, teachers …) • I claim that alltask design benefits from knowing where it stands in terms of each parameter

4. Grain size (Kieran, Doorman and Ohtani, 2015) • Grand theories • Intermediate frames • Domain-specific or local

5. Grand theories about: • the meaning and processes of learning inside/outside educational settings • the structure of mathematical concepts, their interrelationships and interdependencies

6. Intermediate frames: • guide practice across a variety of areas of mathematics • explain learning, and present complex interactions between task, teacher, teaching methods, educational environment, mathematical knowledge and learning • purposes and implications for task design are understood within the structure of practice

7. e.g. using conceptual change theory Van Dooren, Vamvakoussi, & Verschaffel, 2013

8. Domain-specific or local • particular aspects of the intermediate frames designed for a focused aim: e.g. (Sullivan, Knott and Yang 2015)

9. Necessary parameters for design ... • a general theory of learning • a commitment to a general structure of mathematical concepts • connections between task, teacher, teaching methods, educational environment, mathematical knowledge and learning • theory about domain or local focus Dependent on the broad purpose of education, the role of mathematics within education, curriculum and assessment regimes, aims for specific groups of learners within macro context

10. Task purposes conceptual/procedural understanding and fluency reasoning: conjecturing, persuading, proving application express and build connections using feedback/self-correcting language, vocabulary, symbols scope and sequence of an idea address a misconception memorisation practice or use something known revise knowledge develop mathematical habits of mind talking, writing and listening maths habits of enquiry problem solving capabilities develop dispositions towards mathematics relate mathematics to human values promote teamwork, discussion, literacy, self-confidence and other generic characteristics

11. Prediger et al. (2013)

12. Necessary design parameters ... • the meaning of ‘mathematical knowledge’ • the aims of mathematics teaching and learning • the local and specific learning goals for a task These combine to characterise the intention of the design

13. Intended mathematical activity • What are learners going to do? e.g. for algebra (Kieran 2004) • transforming:manipulations; • generating: representing and interpreting • integrating: coordinating manipulation and generation

15. Frames for thinking about activity, e.g. • mental activity (cognitive) • discourse/participation (sociocultural) • purposeful construction (constructionist)

16. Necessary design parameters ... • What activity? and how does it promote learning? • connect design intentions to the likely mental/symbolic/physical/symbolic actions of learners • explain how those mental/symbolic/physical/cognitive actions support progress towards the intended learning

17. Design of environment and pedagogy • advice about structuring learning environment (milieu) • advice about associated pedagogy (didactic contract) • general/specific • role of textbook or learning management system

18. Five dilemmas of task selection • Sullivan, Knott and Yang (2015), following Gimenez (2013): • the role of context • the relationship between language in the problem and in the solution; • structure and openness • how content appears • levels of interaction

19. Five dimensions of pedagogic decision • epistemic • cognitive • interactional • mediational • affective suitability (Gimenez et al. 2013) together these comprise “ecological suitability”

20. Example of parameters of task selection and use:

21. (Bartolini Bussi, Sun & Ramploud, 2013)

22. Pedagogic decision: China: 9 cases presented simultaneously; classify by similarities and differences. Italy: 9 cases split into parts; invent similar problems & discuss Role of intermediate frames: China: variation of representation and transformation Italy: learning as participation in a semiotic culture

23. Necessary design parameters ... • expectations about learners' point of view and likely responses • assumptions about available tools and materials to support activity • expectations of interaction, language and communication • local norms of teaching • the capacity to inform teacher decisions about typical dilemmas and suitability

24. Discrepancy potential • (Leung and Frant 2015) • tool use might model a mathematical idea in ways that do not match the concept closely, so that some discussion is necessary to understand outcomes. • this also happens with other representations in language, symbols and diagrams.

25. Necessary design parameter ... • the potential for uncertainty, predictable different responses, and need for contingent action

26. Summary of the necessary parameters:

27. Concluding remarks • Task designers, textbook authors and designers of learning management systems need to have a commitment to each of these parameters • Weighting may vary • Such a commitment would build on research about design, intention, implementation, and learning • Design teams and researchers should be articulate about these, and build on the tacit knowledge of teachers and other practitioners

28. What is an ICME study for? • This is where we are • Researchers, supervisors, investigators, reviewers, referees, examiners, editors have a responsibility to take heed and build on what has gone before anne.watson@education.ox.ac.uk