Content Matters: Towards a Symbiosis of General and Domain-Specific Theories of Learning and Instruction

1. Content Matters: Towards a Symbiosis of General and Domain-Specific Theories of Learning and Instruction Paul Cobb Vanderbilt University Nashville, USA

2. Top Down Relation Between Theory and Practice • The development of domain independent theory • The derivation of general principles for design from the theory • The translation of the principles into concrete designs in specific domains • The assessment of the concrete designs to test whether they work as anticipated

3. Bottom Up Relation: Methodological Orientation • Classroom design experiments • Responsible for a group of students’ learning • Teacher a member of the research team • From several weeks to a year or more in duration

4. Bottom Up Relation: Methodological Orientation • Sequences of instructional activities and associated tools • Analyses of the process of the students’ mathematical learning and the specific means by which that learning is supported and organized

5. Bottom Up Relation: Methodological Orientation • Means of support • Instructional tasks • Tools and resources • Organization of classroom activities • Classroom norms and discourse

6. Design Research Cycle

7. Instructional Design • Delineation of significant mathematical ideas within a domain -- problematize the content • Conjectured learning route or trajectory for students’ learning that culminates with these significant mathematical ideas • Conjectured means of organizing learning along the envisioned trajectory

8. Instructional Design • The conjectured learning route depends on the means by which learning is supported and organized • Attempting to “engineer” students’ development of particular forms of mathematical reasoning

9. Resources and Constraints for Design • General theories of learning and instruction • Students’ prior knowledge • Historical development of ideas and processes in the particular mathematical domain

10. Resources and Constraints for Design • Analyses of sequences of cognitive levels and of student misconceptions • Do not problematize the domain • Document the consequences of typical school instruction • Analyses of the process of students’ learning in innovative instructional environments

11. Overall Purpose • Not to demonstrate that the instructional design works • Not even to test whether the design works • Test, revise, and improve the conjectures inherent in the design

12. Domain Specific Instructional Theory • Substantiated learning trajectory that culminates with significant mathematical ideas • Demonstrated means of supporting and organizing learning along that trajectory

13. Design Heuristics • Abstract design heuristics from a number of domain specific instructional theories • Empirically grounded in a range concrete designs • Feeds forward to guide the development of other domain specific instructional theories • Open to continual refinement and improvement

14. Bottom Up Relation Between Theory and Practice • Instructional design serves as the context for the development of theory • If you want to understand something, try to change it • If you want to change something, try to understand it • Symbiotic relationship between psychology of mathematical learning and instructional design

15. Top Down Relation Between Theory and Practice • The development of domain independent theory • The derivation of principles for design from the theory • The translation of the principles into concrete designs • The assessment of the concrete designs to test whether they work as anticipated

16. Top Down Relation Between Theory and Practice • Rarely if ever occurs in practice • One-way chain of reasoning implied in research reports • Alive and well in the discourse of educational research

17. Top Down Relation Between Theory and Practice • Purpose: Test whether the design works • Experimental or quasi-experimental designs • Focus is typically on learning outcomes rather than learning processes • Limited guidance for improvement of designs

18. Top Down Relation Between Theory and Practice • Ill-suited for testing and revising theory • Weak feedback loop from design to theory • Failure of concrete designs rarely leads to revision of general theory • Attempt to demonstrate practical relevance of theory

19. Top Down Relation Between Theory and Practice • Assessed with respect to my concerns and interests as a mathematics educator • Development of general theories typically motivated by different sets of concerns and interests • Cognitive psychology • Developmental psychology • Sociocultural theory

20. Top Down Relation Between Theory and Practice • Emerged historically in response to different types of problems • Theories as conceptual tools developed for particular purposes • Ask different types of questions, attempt to gain insight into different types of phenomena, produce different forms of knowledge

21. Top Down Relation Between Theory and Practice • For what purposes are particular types of theories useful? • For who are particular types of theories useful? • General, domain independent theories are potentially useful to educational administrators

22. Design Research Cycle

23. Classroom-Based Analyses • Classrooms are complex and messy • Need interpretive frameworks that enable us to see pattern and order in seemingly ill-structured events • Key criterion: Should result in analyses that feed back to inform the improvement of instructional designs

24. Interpretive Frameworks

25. Social Norms • Students are obliged to: • Explain and justify their solutions • Make sense of others’ explanations • Indicate understanding and non-understanding • Ask clarifying questions or challenge alternatives when differences in interpretations have become apparent

26. Sociomathematical Norms • What counts as: • A different mathematical solution • A sophisticated mathematical solution • An efficient mathematical solution • An acceptable mathematical explanation

27. Classroom Mathematical Practices • Specific to particular mathematical ideas • Counting by one versus conceptualizing numbers as composed of tens and ones • How mathematical content is actually realized in the classroom

28. Analyses and Design • Accounts of the process of students’ mathematical learning that are tied to the means by which that learning was supported and organized • Develop testable conjectures about how the design might be improved

29. Classroom Learning Environment • Organization of classroom activities • Instructional tasks • Tools and resources • Classroom norms and discourse • Social norms • Sociomathematical norms • Classroom mathematical practices

30. Classroom Learning Environment • Instructional tasks as realized in the classroom are influenced by: • Organization of classroom activities • Tools and resources • Norms and discourse

31. Classroom Activity System • The teacher and students jointly constitute the classroom learning environment in the course of their ongoing interactions • Instructional tasks • Tools and resources • Organization of classroom activities • Norms and discourse

32. Top Down Perspective • Classroom learning environment composed of independent variables • Exists independently of teacher’s and students’ collective activity • Under the researcher’s control -- manipulable from the outside the classroom • Types of tasks • Types of tools • Social arrangements • Instructional strategies

33. Top Down Perspective • Investigate into how variations in the learning environment affect aggregate student performance • Design x works better or worse than design y • Conditions under which x works better or worse than y • Types of students for whom x works better or worse than y

34. Top Down Perspective • Limited value in improving designs • Teachers and domain specialists • Useful to educational administrators • Limited expertise in domain-specific processes of learning and teaching • Have to make a defend decisions that impact a large number of classrooms

35. Top Down Perspective • New cadre of educational administrators • Separate professional concerns from those of teachers • Both maintain a distance from and manage classroom instructional processes • Learning environment composed of manipulable variables

36. Top Down Perspective • Scientific research as the basis for the rationalization of educational systems • Folk beliefs about science rather than the reality of scientific practices • Quantification, experimentation, search for universal principles and laws

37. Domain Specific and General Theories • Hypothesis generating and hypothesis testing • Produce different forms of knowledge • Contrasting characterizations of learning environments • Contrasting characterizations of students • Useful to different groups of people

38. Division of Labor • Domain independent theories and experimental studies • Administrative perspective • Contribute to public policy discourse about education

39. Division of Labor • Domain specific theories and design experiments • Classroom perspective • Contribute to instructional design and teaching