Transcript Document

The University of Auckland
New Zeala1and
FACULTY OF
EDUCATION
Pedagogy in the New Zealand
Numeracy Projects
Origins, the Present, the Future
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.
…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.
…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.
In a teaching experiment, it is the mathematical
actions and abstractions of children that are the
source of understanding for the teacherresearcher.
Steffe, L., Kieren, T. (1994). Radical constructivism and
mathematics education. Journal for Research in Mathematics
Education, 25(6), 711- 733
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.
Wright undertook PhD research at Georgia
supervised by Leslie Steffe - based around
Children’s Counting Types.
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
3 Initial Number Sequence - Sequential
Integrations
Counts-on to solve additive and missing addend
involving screened collections
4 Implicitly Nested number SequenceProgressive Integrations -Sequential
Integrations
Uses counting -down-to solve subtractive tasks
and can choose the more appropriate of
counting-down-to and counting-down-from
5
Explicitly Nested Number SequencePart/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
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
3
Counting-on
Combines two stages
4
Facile Number Sequence
Now Early Part-whole in NZ
Limitations in the Framework
Designed for Maths Recovery.
It needed extension if it were to be useful for
years 1 to 10.
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)
The Didactic Cut
Arithmetic and Solution of Equations
Divided into Two
2x + 4 = 5x - 11
Similar Didactic Cut for whole numbers
and decimals implicit in Jones, Thornton
et al.
New Zealand 2001
0-5 Extra Stage Inserted
Some renaming for clarity for teachers
The Blaxland Hotel
6
Advanced Additive
Multi-digit addition/subtraction.
Large jump from Early Part-whole
7
Advanced Multiplicative
Part whole thinking in Mult and Div
8
Advanced Proportional
Part whole thinking in fractions,
ratios, proportions
The Strategy Framework and Pedagogy
Is the strategy framework neo-piagetian?
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.
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
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
The Problem of Material Use
Even extensive experience with embodiments like baseten blocks, and other place–value manipulatives does not
appear to facilitate an understanding of place value…
Ross (1989)
"Bricks is bricks and sums is sums"
Hart, 1989 NZAMT conference
Hart noted the need for a bridge between
“bricks” and “sums”
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.
[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
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.)
Pirie-Kieren Learning Theory
Primitive
Knowing
Image Making
Image Having
Property Noticing
Formalising
Observing
Structuring
Inventising
P-K theory comes out of constructivist teaching
experiment
Using Number Properties
Existing
Knowledge &
Strategies
Using Imaging
Using Materials
Folding back is complex and not
easy to reduce to a few simple
rules for teachers to follow
New Knowledge
& Strategies
What are student thinking then they
use imaging?
How could we possibly know?
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
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.
The Dark Side:The Teaching Model Ritualised
Practice on Materials
Practice Imaging
Practice Number Properties
(Abstraction)
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)
What is 9 + 6?
9
6
10
5
11
12
13
14
15
What might the pedagogical content issues be?
•
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
8+6
14
Open Slather?
72 - 39
40
+1
32 33
42
52
62
72
Encouraging Algorithmic Thinking
The danger is that repetition of
similar problems just leads to
another rule
Avoiding Algorithmic Thinking
Is the method suitable for 81 - 23?
Generalising
For ab - cd the method is efficient when
•
d is near ten and
•
b less than d
This is algebraic thinking.
Book 5 will get a rewrite to
incorporate this change
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
Purposes of the Projects
Provision of calculation and estimation skills
for other subject users
A way of thinking algebraically
A (Mainly) Generic List of Quality Teaching Actions
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)
Supporting
Supports describer’s thinking
Reminds students of conceptually similar
problem situations
Provides background knowledge
(13 categories)
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)
Again the Constructivist Teaching Experiment
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
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.
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.
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
Quality Classroom Management
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 Social Norms
…..
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)
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.
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.
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.
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.
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
If the shift is shallow the effect is transient and slips
away as the external support is removed.
If the shift is real it is permanent the teacher is not
doing what they were told to do, rather they understand
the pedagogy.
But on-going support in Pedagogical Knowledge is needed
even for the expert teacher
The Future of the Numeracy Project
Overlapping issues of
Socio-culture norms
Generic Pedagogy
Pedagogical Content Knowledge
Classroom Management
Dr Clarence Beeby
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.
A Caution about a Managerial Approach
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