Educating Tomorrow's Mathematics Teachers

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Background Epistemological Knowledge of Mathematics Classroom-Based Research Concluding Remarks

Educating Tomorrow’s MathematicsTeachers

The Role of Classroom-Based Evidence

Ateng’ Ogwel

Centre for Mathematics, Science and Technology Education in Africa

Modeling in Mathematics Learning: Approaches forClassrooms of the Future

Makerere University 23–25 July, 2007

JCA OGWEL: Educating Tomorrow’s Mathematics Teachers

Modeling in Mathematics Learning: Makerere University: July 23–25, 2007

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Outline1 Background

Research MotivationTheoretical Background: Steinbring

2 Epistemological Knowledge of MathematicsLinear Equations: Confrey (1993)Arithmetic SequenceSimilarity of Figures (Ogwel, 2007)

3 Classroom-Based ResearchRole of Classroom-based Evidence

4 Concluding RemarksConclusionsImplications



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Research Motivation

Problems of Mathematics Ed.: Motivation andAttitude; Inadequate Rationale for SchoolMathematics; Unsatisfactory exam performance

Solutions/ Interventions: Curricula Reviews;Concretization of Instruction; Use of Real LifeSituations and Technology; ClassroomOrganizations–group work; Alternative assessments

Students Roles: Active, creative, critical independentand responsible participants

Teachers Roles: Design/ select tasks; Sequenceinstruction–prior knowledge; Motivate learning;Guide; Facilitate; Closely listen; Monitor progress;Asses viable constructions



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Research Motivation

Reforms have been Reactionary: (Sputnik → Newmath) Advocacy, fashion and urgency; Inadequateteacher preparation; Decontextualized interventions;Complexity of regular schools/teaching;

Inadequate opportunities for professionaldevelopment; Appropriate tools for analysis ofmathematics-specific discourse

Need for Professional Knowledge specific tomathematics education, developed throughclassroom-based research: EpistemologicalKnowledge for Mathematics Teaching



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Theoretical Background: Steinbring

ASSUMPTIONS OF EPISTEMOLOGICAL TRIANGLE

All knowledge is mediated: Object/Referent Contexts,Sign/Symbols and Concepts

Old and New Mathematical knowledge: Newknowledge develops from but exceeds the old; issubject to acceptance as new knowledge (social), butmust be theoretically consistent (epistemological)

Learning essentially involves some generalizationfrom particulars: representation or experiences

School mathematical knowledge, like scientificmathematics develops subject to social andepistemological constraints



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EPISTEMOLOGICAL KNOWLEDGE OF MATHEMATICSFOR TEACHING

A kind of PCK, a professional knowledge formathematics teaching

A conception of teaching and learning asautonomous 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 ofactual classroom learning episodes: Observation,transcription, description interpretation andclassification of students’ constructions



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Linear Equations: Confrey (1993)

Generalizing a Relation: Function Finder

X

Y

1 2 3 4 5 a c

3 5 7 9 11 b d

2 2

y − b =d − bc − a

(x − a)

y =(d − b)x + (bc − ad)

c − a



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Arithmetic Sequence

4, 7, 10, 13, ... (20th Term = ?)

Common Diff = 3Times 20 = 60Plus First term = 64

20 times 3 = 60Plus 1 = 61Why?

4, 7, __, 13, ....4 X 3 = 12; 12+ 1 = 13

Fifth term: 5 X 3 = 15; 15+1 = 1620th term: 20 X 3 = 60; 60+1 = 61

Get difference btn 2 termsMultiply it by 20Then add first one

Explain

Tn = a + (n − 1)d ⇔ Tn+1 = a − d + nd



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Similarity of Figures (Ogwel, 2007)

The Task:In Parallelogram ABCD, AF:FD = 3:4 and EF//BD. If areaof 4BCE = 10 Sq. units, find area of 4BDF

A F D

E

B C

H

I



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A F D

E

B C

H

I



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A F D

E

B C



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A D

E

B C



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A F D

E

B C



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Interpretation of Student’s Solution

Object/referenceContext

Sign/Symbol

Concept

Parallel lines

Equivalence

Common baseA

BC

D

E

F

HI Common height

B

D

E

F

B

D

E

C



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Problem-solving strategy: auxiliary line; reasoningwith structural properties of parallelogram andparallel lines ⇒ Conception Similarity of figures as arelation independent of positionMonitoring of students reasoning for theoreticalconsistency, multiple representations⇒epistemological knowledge of mathematics forteachingVerbalization of student responses: Accessibility ofthe reasoning and time managementElements of transitional demands of Grade 9–andsecondary mathematics education: conceptual,ephemeral mathematical objects, mathematicalconnections, and minimized student speech



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Worthwhile Problems: Decision-makingIf BC = 3; CD=6 and CP = 2. Find AP and BD

P

B

A

C

D



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Role of Classroom-based Evidence

Learning Need: Professional preparation anddevelopment require evidence that challenge presentconceptions and practices, with documentedpractices and students reasoning for reflection andgradual improvement.Provision of feedback on use of designed curricula,instructional materials and theoretical interventionsNecessary for professional growth of mathematicsteachers and mathematics teacher educatorsDemolishes illusions of successful innovationsClassroom-based research provides evidence forreciprocal and mutual improvement in theory andpractice



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Conclusions

Conclusions

The developmental process of epistemologicalknowledge of mathematics for teaching is consistentwith reform visions, but significantly accounts for thecontext of professional development and specificity ofmathematics education

The design and selection worthwhile tasks;sequencing of instruction to achieve coherence,linkage of school mathematics to real life situationsare complex tasks that teachers cannot effectivelymanage on their own.



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Implications

Implications

Need for supporting teachers: Professionalpreparation, practice and development;Collaboration: Curriculum developers, taskdesigners, instructional material developers, teachereducators, mathematicians, and teachersNeed for research: Technology, problem-solving andmodeling in the transition from secondary to highereducationIn Africa: focus on contextualizing researchmodels–through classroom-based research.Challenges: Time, Funds, Logistics



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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 inAmerican schools: Will faddism continue to impede meaningfulchange? 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 mathematicseducation. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematicseducation as a research domain: A search for identity. An ICMIStudy (pp. 357–377). Dordrecht: Kluwer Academic Publishers.Steinbring, H. (1998). Elements of epistemological knowledgefor mathematics teachers. Journal of Mathematics TeacherEducation, 1(2), 157–189.



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Implications

Thanks you! MerciBeacoup! Asanteni Sana!

