Utrecht, 26-08-2013

1. Theory-enriched practical knowledgein mathematics teacher educationThe use of theory by student teachersin their reflections on practice Utrecht, 26-08-2013 Freudenthal Institute for Science and Mathematics Education Wil Oonk w.oonk@uu.nl

2. Programme • The Reason • The two foregoing studies • The two main studies • Method, especially the learning environment(s) for the st-teachers • Reflection Analysis Instrument: Nature and level of theory use • Results • Application • Discussion

3. The reason: Epistème, Phronèsis, both?

4. The Reasons • My continuous interest in the ‘Theory & Practice’-problem (e.g. 1960: my thesis subject TES: Dewey) • The Multimedia Interactive Learning Environment for PMTE, the MILE-project (Dolk et al., 1996-2002): two foregoing studies

5. First foregoing study: Pioneers in MILE 15 two-hour sessions: two student teachers and researcher Question: How develops the investigation process of student teachers in MILE and how do they construct knowledge Results: The investigation process was a cyclic process of planning, searching, observing, reflecting and evaluating Four levels of knowledge construction became manifest: assimilation (1) and accomodation (2) of knowledge, linking own practice (3) and theorizing (4)

6. Second foregoing study: Theory in actionTwo classes of 24 student teachers were followed during a ten meetings ‘MILE-course’ Question: What kind of connections do prospective teachers make between theory and the digital representation of actual practice? Results: 1. Fifteen ‘characteristics of theory use’, for example: • Theory explains situations • The theory generates new practical questions • Theory generates new questions about the student teachers' individual notions, ideas and opinions • Making connections between situations in MILE and own fieldwork experiences with the help of theory • Developing a personal theory to underpin own interpretations of a practical situation 2. Ideas for developing the next studies

7. What is Theory: Exploration of a phenomenon • The Theory of George Boole (Laws of Thought, 1854) • Gestalt Theory (Wertheimer, 1912; Koffka, 1935) • A local instruction Theory of Learning and Teaching to multiply (Freudenthal, 1984; Ter Heege, 1985; Treffers & De Moor, 1990)

8. Characteristics of theory, examples

9. Local instruction theory In this study (local instruction-) theory is considered as a collection of descriptive concepts that show cohesion, with that cohesion being supported by ‘reflection on practice.’ The character of the theory is determined by the extent to which intrinsic and extrinsic characteristics manifest themselves.

10. List of concepts for the theory of learning to teach multiplication

11. Name of student: Class: Name of teacher training college:

12. Intermezzo: Learning - to teach - the 5 times table“Which of the six concepts fit in most with this situation?” Structure Model Strategy Context Memorize Visualizing

13. By a narrative approach and theoretical enrichment of (own) practical knowledge to a coherent, cognitive network of concepts

14. The two Main StudiesMain research question: “In what way and to what extent do student teachers use theoretical concepts when they reflect on teaching practice in a learning environment that invites the use of theory and, how can this use of theory be described?”

15. Theoretical backgroundSocio-constructive vision, practical knowledge, reflection 1. The concept of practical knowledge (Elbaz, 1983; Verloop, 1991, 2001; Fenstermacher, 1994) Narrative knowing (McEwan & Egan, 1995) Knowledge construction and socio-constructivism (Kilpatrick, 1987; Schoenfeld, 1987; Cobb & McClain & Whitenack, 1997) The knowledge base of the (prospective) teacher (Shulman, 1986; Thiessen, 2000; Verloop & Van Driel & Meijer,2001) 2. Multimedia Learning Environments (Dolk & Faes & Goffree & Hermsen & Oonk, 1996; Goffree & Oonk, 2001; Goffree et al., 2003; Lampert & Ball, 1998; Lampert, 2001, 2010); Teaching adults (Tough, 1971) 3. Developmental Research Freudenthal Institute (Freudenthal, 1983; Goffree, 1979; Gravemeijer,1994, 1995; Treffers, 1978) 4. Reflective practice; reflective conversation (Dewey, 1904, 1933; Schön, 1983; Sparks-Langer & Colton & Pasch & Simmons & Starko, 1990; Korthagen, 2001, 2010) Structure and Insight. A theory of mathematics education (Van Hiele, 1986; Freudenthal, 1991)

16. Two studies Small scale study: 14 third-year students, one TES Large scale study: 269 student teachers (first, second and third-year students), eleven TES’s Course: “learning to teach multiplication” (6x3 hours; study load 80 hours)

17. Method • The primary data were collected from each individual student teacher • The data consisted of student teacher utterances, in which they used theory or notions of theory obtained from: • video-recorded observations during pre-service classroom discussions (small scale study) • Video-stimulated recall interviews which were held to get extra information about the data (small scale) • the reflective notes of the initial and final assessment (both studies)

18. Method-continuation Other instruments: • Written numeracy test (both studies) The student teachers' own numeracy served as an independent control variable in the study. A positive correlation was suspected between the ability of the student teachers to solve mathematical problems and their level of theory use. Example of a task: 0.25 x 2.5 x 48,000 = • Questionnaire (both studies) The 14 questions related to the evaluation of the course, particularly to how the students appreciated the theory as expressed in the course - a detailed manual for teacher educators (large study)

19. The Learning Environment: Requiring Enriched Practical Knowledge Ingredients of the Course “Learning to teach the tables of multiplying” • A cd-rom with 25 situations with 25 expert reflections • A list of 59 concepts (written and 2x on the cd-rom) • Initial / final assessment (reflection on practice situations) • Designing and formulating an investigating question • Activities: Concept game, ‘theorem’, hypothesizing students’ learning processes and justifying teacher choices, etc. • Whole group / small group discussions • Lecture about teaching strand of learning to multiply • Questionnaire

20. Reflection-Analysis-Instrument (RAI)The nature and level of theory use The nature of theory use points to the way students describe situations with the aid of theory. The four categories: factual description (A), interpretation (B), explanation (C) and responding to situations (D) form an inclusive relationship The level of theory use characterizes the level on which students use theory in their reflections

21. The nature of theory use • Factual description: the student teacher describes actual events only; no opinion is given, nor are any operations or expressions by either the teacher or the student teachers explained. • Interpretation: the student teacher relates what he or she thinks happens, without any supporting evidence or explanation (indicator words e.g. I think… in my opinion…). • Explanation: the student teacher explains why the teacher/student acts or thinks in a certain way. This concerns an unambiguous, "neutral" explanation on the basis of (previously mentioned) facts or observed events (indicator words e.g.: for this reason, because, as, as... if, probably, it could be possible that…). • Responding to situations: the student teacher relates or describes – for example in a design/preparation/evaluation – what could be done or thought (differently), what actions she as stand-in for a virtual teacher would take or want to take (indicator words e.g.: I expect, I predict, I would do, I make, I intend to, with the intention of…).

22. The level of theory use • No recognition and use of a theoretical concept • Recognizing theoretical concept(s). Correct description within a context; no network • Junctions (meaningful relationships) in a network of relations between concepts • Reasoning within the structure of a network of relations between concepts

23. Example of score D3

24. Example of score ??

26. Interrater reliability: for the nature = 0,80; for the level = 0,86; for the combination of nature and level  = 0,77.

27. Some Conclusions ‘Factual description’ (A) and ‘interpretation’ mostly occur on the first and second level , ‘explaining’ (C) and ‘responding to situations’ (D), mostly on the third level Two student teachers reacted on the fourth level (small scale; video-stimulated interview) Students use proportionally more general pedagogical concepts than pedagogical content concepts.

28. Continued There is a positive correlation between the level of numeracy and the use of theory (especially C3) Students with ‘Senior secondary vocational education without mathematics’ as prior education, show a low level of numeracy and score most A, B and level 1 Rises in level of theory use take place especially in interaction led by the teacher educator. The complaint voiced by many teacher educators that they do not have enough teaching hours for mathematics and didactics should therefore be taken very seriously

29. Applications: „Mathematics in Practice“ for PMTE 4 books and website MT K-2 (2010) MT Grade 3-6 (2010) MT Big ideas (2011) MT Differences in Class (2013)

30. Questions / Discussion