Theory-enriched practical knowledge
in mathematics teacher education
The use of theory by student teachers
in their reflections on practice
Utrecht, 26-08-2013
Freudenthal Institute for Science and Mathematics Education Wil Oonk
w.oonk@uu.nl
Programme
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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
2
The reason: Epistème, Phronèsis, both?
3
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 evaluating
Four 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 action
Two 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
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, examples
17 Characteristics
of theory
Derived points of attention
for using theory in TE
Intrinsic characteristics (10)
1. Grounding: arguing, justifying,
proving; patterns
2. The extent of formalization
3. The presence of theory
charged examples.
ad.1 Underpin (intended) actions;
patterns in what students do/think
ad.2 Levels of thinking and acting
ad.3 Narratives as a means for
acquiring practical knowledge.
Extrinsic characteristics (7)
1. The genesis and dynamics of
theory
2. Theory in action; theory on
action
3. The discourse in the
community of scientists
ad.1 The spark: intuition and
creativity
ad.2 Theory as the basis for
pedagogical reasoning
ad.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.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
adaptive education
algorithm (of multiplication)
anchor points
automate
basic skills
cognitive network
commutative law
context
core objectives
counting problem
counting with jumps
diagnose
doubling (see halving)
empty number line
explain
independent work
factor
grid structure
halving
informal procedures
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
Interaction
learning environment
learning line
learning strand
level
main memory
making concrete
manipulatives
memorize
model
multiplication sign
multiply
own construction
own production
put into words
reconstruction pedagogy
reflection
reproduction
rich problems
rote (traditional) learning
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
schematize
school climate
solution strategy
story related to sum
strategy
structure
structured counting
structured material
the realistic strand
the stages of a strand
thinking model
times table
times table pedagogy
to count by jumps
to count rhythmically
to exercise
to give meaning
understand
visualizing
Name of student:
Class:
Name of teacher training college:
This concept has become more
familiar to me. I can relate a
teaching narrative in which this
concept has meaning / has
become clear.
Concept
1. adaptive
teaching
2. anchor points
(......)
59. rich
problems
Check below for yes, or leave it
blank
The narrative is from:
1=own teaching practice
2=the Guide
3=magazine/book
4=college: instructions/ discuss.
Circle (possibly more categories
for each concept)
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
By a narrative approach and theoretical
enrichment of (own) practical knowledge
to a coherent, cognitive network of concepts
13
The two Main Studies
Main 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
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)
1.
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-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)
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
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
20
The nature of theory use
A.
B.
C.
D.
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…).
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 D3
Example 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
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.
Clarification of
the nature
Clarification of
the level
Mean percentages categories A1 to D3 (large scale study)
in the final assessment
A
B
C
D
Factual
description
Interpreta
tion
Explaining
Responding
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
Total
25
Correlation between nature and level of theory use (large scale study)
A
Factual
description
Level 1
Level 2
Level 3
B
Interpreting
C
Explaining
D
Respond to
A1
Sig. 0,043
Beta 0,129
B1
Sig. 0,096
Beta 0,106
C1
Sig. 0,001
Beta -0,214
D1
Sig. 0,506
Beta 0,043
A2
Sig. 0,020
Beta 0,149
B2
Sig. 0,015
Beta 0,155
C2
Sig. 0,105
Beta -0,104
D2
Sig. 0,007
Beta -0,173
A3
Sig. 0,000
Beta -0,230
B3
Sig. 0,001
Beta -0,212
C3
Sig. 0,000
Beta 0,282
D3
Sig. 0,212
Beta 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.
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
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)
Questions / Discussion