Theory-enriched practical knowledgein mathematics teacher education

The 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

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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

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The reason: Epistème, Phronèsis, both?

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

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 evaluatingFour levels of knowledge construction became manifest:

assimilation (1) and accomodation (2) of knowledge, linking own practice (3) and theorizing (4)

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 situation2. Ideas for developing the next studies

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)

Characteristics of theory, examples17 Characteristics of theory

Derived points of attention for using theory in TE

Intrinsic characteristics (10)1. Grounding: arguing, justifying,

proving; patterns2. The extent of formalization3. The presence of theory

charged examples.

ad.1 Underpin (intended) actions; patterns in what students do/thinkad.2 Levels of thinking and actingad.3 Narratives as a means for acquiring practical knowledge.

Extrinsic characteristics (7)1. The genesis and dynamics of

theory2. Theory in action; theory on

action3. The discourse in the

community of scientists

ad.1 The spark: intuition and creativityad.2 Theory as the basis for pedagogical reasoningad.3 The discourse (‘negotiating’) as the motor of constructing theoretical knowledge

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.

List of concepts for the theory of learning to teach multiplication

1. adaptive education2. algorithm (of multiplication)3. anchor points4. automate5. basic skills6. cognitive network7. commutative law8. context9. core objectives10. counting problem11. counting with jumps12. diagnose13. doubling (see halving)14. empty number line15. explain16. independent work17. factor18. grid structure19. halving20. informal procedures

21. Interaction22. learning environment23. learning line24. learning strand25. level26. main memory27. making concrete28. manipulatives29. memorize30. model31. multiplication sign32. multiply33. own construction34. own production35. put into words36. reconstruction pedagogy37. reflection38. reproduction39. rich problems40. rote (traditional) learning

41. schematize42. school climate43. solution strategy44. story related to sum45. strategy46. structure47. structured counting48. structured material49. the realistic strand50. the stages of a strand51. thinking model52. times table53. times table pedagogy54. to count by jumps55. to count rhythmically56. to exercise57. to give meaning58. understand59. visualizing

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

Concept

This concept has become more familiar to me. I can relate a teaching narrative in which this concept has meaning / has become clear.

Check below for yes, or leave it blank

The narrative is from:1=own teaching practice 2=the Guide3=magazine/book 4=college: instructions/ discuss. Circle (possibly more categories for each concept)

1. adaptive teaching 2. anchor points

(......)

59. rich problems

Intermezzo: Learning - to teach - the 5 times table“Which of the six concepts fit in most

with this situation?”

Structure

Model

Visualizing

Context

Strategy

Memorize

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By a narrative approach and theoretical enrichment of (own) practical knowledge

to a coherent, cognitive network of concepts

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?”

Theoretical background Socio-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)

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)

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)

Method-continuationOther 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)

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

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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

The nature of theory use

A. 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.

B. 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…).

C. 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…).

D. 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…).

The level of theory use

1. No recognition and use of a theoretical concept

2. Recognizing theoretical concept(s). Correct description within a context; no network

3. Junctions (meaningful relationships) in a network of relations between concepts

4. Reasoning within the structure of a network of relations between concepts

Example of score D3Example of meaningful unit

Clarification of the nature

Clarification of the level

Do the children really see the tens in the rectangle model? The teacher could have tested Fariet: “Fariet, how do you see the 10, 20…? Can you tell me or point it out, Fariet?”

The student teacher anticipates the situation in terms of a possible alternative to the teacher’s approach (D: Responding, gearing to).

The concepts “tens,” “really see” (notion of structure), “rectangle model” and “testing” are used coherently. The concepts are meaningfully related.(level 3)

Example of score ??Example of meaningful unit

Clarification of the nature

Clarification of the level

The class has already come up with 2 x 5 followed by 3 x 5. Because she visualizes the five times table for the children, they can also tell a story to accompany a problem. 1 x 5 will be possible to see as 1 tube times 5 balls. She also makes a connection between concrete material and a grid model. At one point Clayton is counting 10 x 5, the teacher confirms this for the class. In fact a transition is being made here from multiplication by counting to structured multiplication.

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Mean percentages categories A1 to D3 (large scale study)in the final assessment

AFactual

description

BInterpreta

tion

CExplaining

DResponding

Total

Level 1 12 5 12 7 36

Level 2 8 4 12 5 29

Level 3 5 3 18 9 35

Total 25 12 42 21 100

Correlation between nature and level of theory use (large scale study)

AFactual

description

BInterpreting

CExplaining

DRespond to

Level 1

A1Sig. 0,043Beta 0,129

B1Sig. 0,096Beta 0,106

C1Sig. 0,001Beta -0,214

D1Sig. 0,506Beta 0,043

Level 2

A2Sig. 0,020Beta 0,149

B2Sig. 0,015Beta 0,155

C2Sig. 0,105Beta -0,104

D2Sig. 0,007Beta -0,173

Level 3

A3Sig. 0,000Beta -0,230

B3Sig. 0,001Beta -0,212

C3Sig. 0,000Beta 0,282

D3Sig. 0,212Beta 0,080

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

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.

ContinuedThere 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

MT K-2 (2010)

Applications: „Mathematics in Practice“ for PMTE4 books and website

MT Grade 3-6 (2010)

MT Big ideas (2011)

MT Differences in Class (2013)

Questions / Discussion