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FACULTY OF EDUCATION

FACULTY OF EDUCATION. The University of Auckland New Zeala1and. Pedagogy in the New Zealand Numeracy Projects Origins, the Present, the Future. A Shift in Normal Science.

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FACULTY OF EDUCATION

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  1. FACULTY OF EDUCATION The University of Auckland New Zeala1and

  2. Pedagogy in the New Zealand Numeracy Projects Origins, the Present, the Future

  3. A Shift in Normal Science Charles Smock at the University of Georgia was working to formulate a constructivist research and development program in mathematics education, including … an adaptation of Piaget's clinical interview. It was difficult, however, to overthrow the tyranny of the empiricist view of normal science in mathematics education.

  4. …It wasn't until 1983 that an article was published in the JRME with "constructivist" in the title (Cobb, & Steffe, 1983). There, it was argued that the constructivist researcher needed to be a teacher as well as a model builder.

  5. …As constructivist mathematics education researchers, we became oriented toward studying the construction of mathematical concepts and the operations by which children attend to and organize their experiences.

  6. In a teaching experiment, it is the mathematical actions and abstractions of children that are the source of understanding for the teacher-researcher. Steffe, L., Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711- 733

  7. The core of the numeracy project is derived from Children’s counting types: philosophy theory and application. Steffe, L., von Glasersfeld, E., Richards, J. & Cobb, P. (1983). New York: Paeder.

  8. Wright undertook PhD research at Georgia supervised by Leslie Steffe - based around Children’s Counting Types.

  9. Arithmetical Stages in Mathematics Recovery 0 Preperceptual Can’t count one-to one 1 Perceptual Can count visible collections 2 Figurative Can count screened collections from one

  10. Initial Number Sequence - Sequential Integrations Counts-on to solve additive and missing addend involving screened collections 4 Implicitly Nested number Sequence- Progressive Integrations -Sequential Integrations Uses counting -down-to solve subtractive tasks and can choose the more appropriate of counting-down-to and counting-down-from

  11. 5 Explicitly Nested Number Sequence- Part/whole Operations Uses a range of strategies which include procedures other than counting-by-ones such as compensation, using addition to work out subtraction, and using known fact such as doubles and sums which equal ten

  12. Wright constructed a slight variation for Count Me in Too which is used in the Diagnostic Assessment 0 Emergent Was Preperceptual 1 Perceptual No change 2 Figurative Counting Same

  13. 3 Counting-on Combines two stages 4 Facile Number Sequence Now Early Part-whole in NZ

  14. Limitations in the Framework Designed for Maths Recovery. It needed extension if it were to be useful for years 1 to 10.

  15. Counting-on 1999 Developmental sequences for understanding aspects of the numeration system Denvir & Brown, 1986 Fuson et al, 1997 Ross, 1986, 1989 Clark and Kamii 1996 Young-Loveridge 1999 Jones, G., Thornton, C., et al (1996)

  16. The Didactic Cut Arithmetic and Solution of Equations Divided into Two 2x + 4 = 5x - 11

  17. Similar Didactic Cut for whole numbers and decimals implicit in Jones, Thornton et al.

  18. New Zealand 2001 0-5 Extra Stage Inserted Some renaming for clarity for teachers The Blaxland Hotel 6 Advanced AdditiveMulti-digit addition/subtraction. Large jump from Early Part-whole

  19. 7 Advanced MultiplicativePart whole thinking in Mult and Div 8 Advanced ProportionalPart whole thinking in fractions, ratios, proportions

  20. The Strategy Framework and Pedagogy Is the strategy framework neo-piagetian?

  21. The teacher who understands where a child is on their conceptual development has a better change of promoting reflective abstraction than a teacher who just follows the curriculum Von Glasersfeld in Derry, S. (1996). Cognitive Schema theory in the constructivist debate. Educational Psychologist, 3 (3/4) 163-174. Lawrence Erlbaum Associates, Inc.

  22. Quality Teaching - The Teaching Model In the Count Me in Too trial in NZ in 2000 there was no model for encouraging more complex thinking as defined by the Strategy framework

  23. Quality Teaching - The Teaching Model In the Count Me in Too trial in NZ in 2000 there was no model for encouraging more complex thinking as defined by the Strategy framework

  24. The Problem of Material Use Even extensive experience with embodiments like base-ten blocks, and other place–value manipulatives does not appear to facilitate an understanding of place value… Ross (1989)

  25. "Bricks is bricks and sums is sums" Hart, 1989 NZAMT conference Hart noted the need for a bridge between “bricks” and “sums”

  26. Mathematics is the result of abstraction from operations on a level on which the sensory or motor material that provided the occasion for operating is disregarded. …. Such abstractions cannot be given to students, they have to be made by the students themselves.

  27. [Materials] can play an important role, but it would be naive to believe that the move from handling or perceiving objects to a mathematical abstraction is automatic. The sensory objects, no matter how ingenious they might be, merely offer an opportunity for actions from which the desired operative concepts may be abstracted; and one should never forget that the desired abstractions, no matter how trivial and obvious they might seem to the teacher, are never [obvious] to the novice. von Glasersfeld, E. (1992). ICME Montreal

  28. Bridging and Visualisation The use of concrete materials is important, but rather than moving directly from physical representations to the representations to the manipulation of abstract symbols … it is suggested that the emphasis be shifted to using visual imagery prior to the introduction of more formal procedures. (Bobis, J.)

  29. Primitive Knowing Image Making Image Having Property Noticing Formalising Observing Structuring Inventising Pirie-Kieren Learning Theory P-K theory comes out of constructivist teaching experiment

  30. Using Number Properties Using Imaging Using Materials New Knowledge & Strategies Existing Knowledge & Strategies Folding back is complex and not easy to reduce to a few simple rules for teachers to follow

  31. What are student thinking then they use imaging? How could we possibly know?

  32. An addition to the model and a comment on ability grouping When the teacher detects that the desired abstraction has been made the student is sent to independently to practice and extend their teaching Mix different stages together

  33. The Teaching Model as Tool The model is not P-K. It does not seek to explain student’s thinking Hopefully it is a tool for the teacher to make formative evaluations. With other tools the teacher reacts to the needs of the students and alters the lesson in real time.

  34. The Dark Side:The Teaching Model Ritualised Practice on Materials Practice Imaging Practice Number Properties (Abstraction)

  35. Quality Teaching and Pedagogical Content Knowledge PCK includes: an understanding of how particular topics, problems, or issues are organized, presented, and adapted to the diverse interests and abilities of learners, and presented for instruction. Shulman (1987)

  36. 9 6 10 5 What is 9 + 6?

  37. 11 12 13 14 15

  38. What might the pedagogical content issues be?

  39. •Fif means five • Teen is a dumb way to spell ten • For teen numbers the rule that the tens are said before the ones is broken

  40. 8 + 6 14

  41. Open Slather? 72 - 39

  42. Encouraging Algorithmic Thinking The danger is that repetition of similar problems just leads to another rule

  43. Avoiding Algorithmic Thinking Is the method suitable for 81 - 23?

  44. Generalising For ab - cd the method is efficient when • d is near ten and • b less than d This is algebraic thinking.

  45. Book 5 will get a rewrite to incorporate this change

  46. High End Objective Paul Cobb’s example: 62 x 45 • We should promotes use of the analysis of and adoption of efficient solution methods for all students 62 ÷ 2 =31, 45 x 2 = 90 31 x 90 = 2790

  47. Purposes of the Projects Provision of calculation and estimation skills for other subject users A way of thinking algebraically

  48. A (Mainly) Generic List of Quality Teaching Actions

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