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FACULTY OF EDUCATION The University of Auckland New Zeala1and

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

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A Shift in Normal Science It was difficult, however, to overthrow the tyranny of the empiricist view of normal science in mathematics education. 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.

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…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.

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…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.

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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

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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.

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

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

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3Initial Number Sequence - Sequential Integrations Counts-on to solve additive and missing addend involving screened collections 4Implicitly 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

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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

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

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

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

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

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

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

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

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7Advanced Multiplicative Part whole thinking in Mult and Div 8Advanced Proportional Part whole thinking in fractions, ratios, proportions

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

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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.

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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

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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

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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)

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

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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.

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[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

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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.) Bridging and Visualisation

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

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P-K Influences in the Teaching Model Using Imaging Using Number Properties Folding Back

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

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

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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

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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.

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

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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)

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96105 What is 9 + 6?

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12 13 14 15 11

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What might the pedagogical content issues be?

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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

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8 + 6 14

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Open Slather? 72 - 39

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

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

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

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Book 5 will get a rewrite to incorporate this change

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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

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

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Eliciting Facilitates students’ responding Elicits many solution methods for one problem from the entire class Waits for and listens to students’ descriptions of solution methods…. (10 categories) A (Mainly) Generic List of Quality Teaching Actions

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Supporting Supports describer’s thinking Reminds students of conceptually similar problem situations Provides background knowledge (13 categories)

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Extending Maintains high standards and expectations for all students Asks all students to attempt to solve difficult problems and to try various solution methods Encourages mathematical reflection Encourages students to analyse, compare, and generalise mathematical concepts… (10 categories)

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Journal for Research in Mathematics Education 1999, Vol. 30, No. 2,148-170 Judith L. Fraivillig, Rider University Lauren A. Murphy and Karen C. Fuson, Northwestern University Again the Constructivist Teaching Experiment

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Developments Post-Fraivillig: Manurewa Enhancement Initiative 1 Noticing: I have listened and observed for an appropriate time to students’ descriptions of their solution methods without intervening to correct errors.

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2 Understanding: I am satisfied I am competent at Noticing. I have understood the students' correct reasoning and the cause of any incorrect reasoning. If I did not I have listed the actions and words that I do not understand. I will discuss possible future actions with the observer in “I Focus on My Planning”, not now.

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3 Teaching: I am satisfied I am competent in Understanding student's thinking. I have taken appropriate teaching actions. The project books on the teacher’s knee

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Management I had routines for managing different student abilities. When a student's behaviour affected his/her or other's learning I consistently took appropriate action. My actions in dealing with inappropriate behaviour were immediate, consistent, predictable and fair. When working as a group or the whole class one person, including me, spoke at a time. The students knew what the next thing they had to do was and they did it. (12 categories) …. … Quality Classroom Management

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….. The students were supportive of each other when engaged in small groups or working independently. The thinking of each student was valued and respected by other students and me. In group and class situations students listened to others respectfully. When in groups students shared their thinking with everyone. (14 categories) … The students were comfortable in sharing their ideas even if their thinking was incorrect or only partially correct. … ( Quality Classroom Social Norms

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Some Notes About The Observer’s Role ( In training for the Citizens Advice Bureau advisors are asked to avoid "the tyranny of shoulds and musts". Advisers learn to assist a client to reach his/her own course of action rather than being directed.

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What If the Teacher Asks for Help During the Feedback? If the teacher asks “What should I do?” during the feedback, the observer will deflect this to the discussion in “Part D - I Focus on My Planning for Future Lessons”. This is important to help prevent the feedback session losing focus. (

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What If the Teacher Criticises Herself? Sometimes a teacher will say things critical of themselves. For example, “That lesson was rubbish wasn’t it?” or more simply, “that lesson was rubbish”. In such a situation it is important that the observer does not try to comfort the teacher by, for example, saying things like “You are being too hard on yourself” or “Actually I thought you did really well” and so on. (

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The observer will say something like “Give me examples of things you did that made you feel the lesson was rubbish” – the question is intended to focus the teacher back on evidence not feelings. (

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Pedagogical Shift. A Conjecture about Successful PD Reading Recovery trained teachers undergo a fundamental change in their teaching for the better. Central to this is learning to listen to and observe children, interpreting what they say and do, and then taking appropriate teaching actions

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If the shift is real it is permanent the teacher is not doing what they were told to do, rather they understand the pedagogy. If the shift is shallow the effect is transient and slips away as the external support is removed. But on-going support in Pedagogical Knowledge is needed even for the expert teacher

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Overlapping issues of The Future of the Numeracy Project Classroom Management Socio-culture norms Generic Pedagogy Pedagogical Content Knowledge

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The saddest lesson every official has to learn is that the teacher under pressure of instructions they have not understood or accepted have an infinite capacity for going on doing the same things under another name, so that only the shadow of progress can be achieved by regulations or exhortion. Dr Clarence Beeby

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Discovering, explicating, and codifying general teaching principles simplify the otherwise outrageously complex activity of teaching. The great danger occurs, however, when a general teaching principle is distorted into prescription, and when maxim becomes mandate. Lee Shulman A Caution about a Managerial Approach

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