1.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Educating Tomorrow’s Mathematics
Teachers
The Role of Classroom-Based Evidence
Ateng’ Ogwel
Centre for Mathematics, Science and Technology Education in Africa
Modeling in Mathematics Learning: Approaches for
P M
Classrooms of the Future
Makerere University 23–25 July, 2007 A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
2.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Outline
1 Background
Research Motivation
Theoretical Background: Steinbring
2 Epistemological Knowledge of Mathematics
Linear Equations: Confrey (1993)
Arithmetic Sequence
Similarity of Figures (Ogwel, 2007)
3 Classroom-Based Research
Role of Classroom-based Evidence
4 Concluding Remarks P M
Conclusions A
Implications
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
3.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Research Motivation
Problems of Mathematics Ed.: Motivation and
Attitude; Inadequate Rationale for School
Mathematics; Unsatisfactory exam performance
Solutions/ Interventions: Curricula Reviews;
Concretization of Instruction; Use of Real Life
Situations and Technology; Classroom
Organizations–group work; Alternative assessments
Students Roles: Active, creative, critical independent
and responsible participants
Teachers Roles: Design/ select tasks; Sequence
instruction–prior knowledge; Motivate learning; P M
Guide; Facilitate; Closely listen; Monitor progress;
A
Asses viable constructions
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
4.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Research Motivation
Reforms have been Reactionary: (Sputnik → New
math) Advocacy, fashion and urgency; Inadequate
teacher preparation; Decontextualized interventions;
Complexity of regular schools/teaching;
Inadequate opportunities for professional
development; Appropriate tools for analysis of
mathematics-speciﬁc discourse
Need for Professional Knowledge speciﬁc to
mathematics education, developed through
classroom-based research: Epistemological
P M
Knowledge for Mathematics Teaching
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
5.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Theoretical Background: Steinbring
ASSUMPTIONS OF EPISTEMOLOGICAL TRIANGLE
All knowledge is mediated: Object/Referent Contexts,
Sign/Symbols and Concepts
Old and New Mathematical knowledge: New
knowledge develops from but exceeds the old; is
subject to acceptance as new knowledge (social), but
must be theoretically consistent (epistemological)
Learning essentially involves some generalization
from particulars: representation or experiences
School mathematical knowledge, like scientiﬁc
mathematics develops subject to social and P M
epistemological constraints A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
6.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
EPISTEMOLOGICAL KNOWLEDGE OF MATHEMATICS
FOR TEACHING
A kind of PCK, a professional knowledge for
mathematics teaching
A conception of teaching and learning as
autonomous systems; (TR) provision of tasks,
monitoring of learning progress, variation of tasks,
and reﬂection; (ST) Subjective interpretation of tasks,
interactive reﬂection on individual reﬂections
Develops through theoretically informed analysis of
actual classroom learning episodes: Observation, P M
transcription, description interpretation and
A
classiﬁcation of students’ constructions
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
7.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Linear Equations: Confrey (1993)
Generalizing a Relation: Function Finder
X 1 2 3 4 5 a c
Y 3 5 7 9 11 b d
2 2
d −b
y −b = (x − a)
c −a
P M
(d − b)x + (bc − ad)
y = A
c −a
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
8.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Arithmetic Sequence
4, 7, 10, 13, ... (20th Term = ?)
Get difference btn 2 terms 20 times 3 = 60
Multiply it by 20 Plus 1 = 61
Then add first one Why?
Explain 4, 7, __, 13, ....
Common Diff = 3 4 X 3 = 12; 12+ 1 = 13
Times 20 = 60 Fifth term: 5 X 3 = 15; 15+1 = 16
Plus First term = 64 20th term: 20 X 3 = 60; 60+1 = 61
P M
Tn = a + (n − 1)d ⇔ Tn+1 = a − d + nd A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
9.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
The Task:
In Parallelogram ABCD, AF:FD = 3:4 and EF//BD. If area
of BCE = 10 Sq. units, ﬁnd area of BDF
A F D
E
I
H
P M
B C A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
10.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
A F D
E
I
H
B C
P M
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
11.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
A F D
E
B C
P M
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
12.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
A D
E
B C
P M
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
13.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
A F D
E
B C
P M
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
14.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
Interpretation of Student’s Solution
Object/reference Sign/Symbol
Context
A F D Parallel lines
Common base
E I Common height
B H F
D D
C
E E
B B C
P M
Concept
Equivalence A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
15.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
Problem-solving strategy: auxiliary line; reasoning
with structural properties of parallelogram and
parallel lines ⇒ Conception Similarity of ﬁgures as a
relation independent of position
Monitoring of students reasoning for theoretical
consistency, multiple representations
⇒epistemological knowledge of mathematics for
teaching
Verbalization of student responses: Accessibility of
the reasoning and time management
Elements of transitional demands of Grade 9–and
secondary mathematics education: conceptual, P M
ephemeral mathematical objects, mathematical A
connections, and minimized student speech
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
16.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Similarity of Figures (Ogwel, 2007)
Worthwhile Problems: Decision-making
If BC = 3; CD=6 and CP = 2. Find AP and BD
A
P D
B
P M
C
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
17.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Role of Classroom-based Evidence
Learning Need: Professional preparation and
development require evidence that challenge present
conceptions and practices, with documented
practices and students reasoning for reﬂection and
gradual improvement.
Provision of feedback on use of designed curricula,
instructional materials and theoretical interventions
Necessary for professional growth of mathematics
teachers and mathematics teacher educators
Demolishes illusions of successful innovations
Classroom-based research provides evidence for P M
reciprocal and mutual improvement in theory and
A
practice
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
18.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Conclusions
Conclusions
The developmental process of epistemological
knowledge of mathematics for teaching is consistent
with reform visions, but signiﬁcantly accounts for the
context of professional development and speciﬁcity of
mathematics education
The design and selection worthwhile tasks;
sequencing of instruction to achieve coherence,
linkage of school mathematics to real life situations
are complex tasks that teachers cannot effectively P M
manage on their own. A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
19.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Implications
Implications
Need for supporting teachers: Professional
preparation, practice and development;
Collaboration: Curriculum developers, task
designers, instructional material developers, teacher
educators, mathematicians, and teachers
Need for research: Technology, problem-solving and
modeling in the transition from secondary to higher
education
In Africa: focus on contextualizing research P M
models–through classroom-based research.
A
Challenges: Time, Funds, Logistics
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
20.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Implications
Confrey, J. (1993). Learning to see children’s mathematics:
Crucial challenges in constructivist reform. In K. Tobin (Ed.),
The practice of constructivism in science education (pp.
299–321). Hillsdale, NJ: Lawrence Erlbaum.
Good, T. L., Clark, S. N. & Clark, D. C. (1997). Reform efforts in
American schools: Will faddism continue to impede meaningful
change? In B. J. Biddle, T. L. Good & I. F. Goodson (Eds.),
International handbook of teachers and teaching (pp.
1387–1427). Dordrecht: Kluwer Academic Publishers.
Mason, J. (1998). Researching from the inside in mathematics
education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics
education as a research domain: A search for identity. An ICMI
Study (pp. 357–377). Dordrecht: Kluwer Academic Publishers.
Steinbring, H. (1998). Elements of epistemological knowledge P M
for mathematics teachers. Journal of Mathematics Teacher A
Education, 1(2), 157–189.
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
21.
Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks
Implications
Thanks you! Merci
Beacoup! Asanteni Sana!
P M
A
JCA OGWEL: Educating Tomorrow’s Mathematics Teachers
Modeling in Mathematics Learning: Makerere University: July 23–25, 2007
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