Educating Tomorrow's Mathematics Teachers

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  • 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