Transcript Welcome to CS:Mathematics 1

Intervention in the number
learning of 8- to 10-year olds
A/Prof Bob Wright
David Ellemor-Collins
INTERVENTION IN NUMBER LEARNING
Acknowledgements
We gratefully acknowledge the support and contributions
from:
 Australian Research Council, grant LP0348932.
 Catholic Education Commission of Victoria.
 partner investigators Gerard Lewis and Cath Pearn.

participating teachers, students and schools.
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
1. NUMERACY INTERVENTION RESEARCH PROJECT
Joint project of
 Southern Cross University (SCU)
 Catholic Education Office, Melbourne (CEO)
Intervention with students- 8-10 years old (3rd and 4th grade)
 low-attaining in number learning
Design research methodology
 Three years in schools: 2004-2006
1. NUMERACY INTERVENTION RESEARCH PROJECT
Aim:
to develop pedagogical tools
for intervention.




Interview-based assessment schedules.
Learning framework.
Instructional framework.
Instructional procedures and sequences.
1. NUMERACY INTERVENTION RESEARCH PROJECT
Design research methodology.
 Three one-year design cycles.
 8 or 9 schools each year.
 8 intervention students each school.
 = total of 200 students in intervention.
Recorded on videotape:
 interview assessments
 instructional sessions
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
2. APPROACH TO INTERVENTION
The need for intervention in number
Significant proportion of students have
difficulties learning arithmetic.
(Mapping the territory, 2000)
Calls for an integrative approach to
develop intervention materials.
(e.g. Rivera, 1998)
2. APPROACH TO INTERVENTION
Intervention in number cont.
Intervention programs in early number.
(Dowker, 2004; Gervasoni, 2005; Pearn & Hunting, 1995;
Wright, Martland, Stafford, & Stanger, 2006)
NIRP extends to basic whole number arithmetic:
 Numbers in 100s and 1000s
 Multidigit addition and subtraction
 Early multiplication and division
2. APPROACH TO INTERVENTION
Organizing intervention by key aspects
We can describe number knowledge in terms of:
 “components” (Dowker, 2004).
 “domains” (Clarke, McDonough, & Sullivan, 2002).
 “aspects” (Wright, Martland et al., 2006).
NIRP uses an approach of organising key
aspects into a learning framework.
(Wright, Martland et al., 2006).
2. APPROACH TO INTERVENTION
Instructional design
NIRP design accords with the emergent models
approach. (e.g. Gravemeijer, Bowers, & Stephan, 2003)

Anticipate potential learning trajectory.

Devise instructional sequence of
instructional procedures.

Foster progressive mathematization.
2. APPROACH TO INTERVENTION
Instructional design cont.
Settings have an important role in instructional
sequences:

For initial context-dependent thinking, and

To become a model for more formal thinking.
(Gravemeijer, Cobb, Bowers, & Whitenack, 2000)
2. APPROACH TO INTERVENTION
Instructional design cont.
Instructional procedures incrementally:
 distance the materials.
 advance the complexity of the task.
 raise the sophistication of the student’s
thinking.
2. APPROACH TO INTERVENTION
Approach to number instruction
Detailed assessment of student’s knowledge
 Selection of instructional procedures.
On-going observational assessment
 Tuning instruction to cutting edge of learning.
Student engaged in sustained, independent
thinking on number tasks.
(Wright, Martland et al., 2006)
2. APPROACH TO INTERVENTION
Number instruction cont.
Build from students’ informal mental strategies.
Develop mathematically sophisticated
strategies.
Emphasize:
 Flexible, efficient computation.
 Strong numerical reasoning.
(e.g. Beishuizen & Anghileri, 1998; Gravemeijer, 1997;
McIntosh, Reys, & Reys, 1992; Yackel, 2001)
2. APPROACH TO INTERVENTION
Number instruction cont.
Low-attainers often:



Use inefficient count-by-ones strategies.
Use unreasoned rote procedures.
Depend on materials or fingers. (Gray & Tall, 1994)
Hence intervention instruction needs to:


develop students’ number knowledge to support noncount-by-ones strategies, and
move students to independence from materials.
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
ASPECT A: NUMBER WORDS AND NUMERALS
Low-attaining 8-10yo difficulties
NWSs
e.g. “52, 51, 40, 49, 48…”
“108, 109, 200, 201, 202…”
“108, 109, 1000, 1001…”
Tens off the decade
e.g. “24, 30, 34, 40…”
“24…34…44” counting-by-ones subvocally
Numerals: errors identifying and writing
e.g.
306, 6032, 3010, 1300, 1005
ASPECT A: NUMBER WORDS AND NUMERALS
Instruction
Facility is important, and requires explicit
attention for low-attainers. (Menne, 2001)
Reasoning with number word sequences and
numeral sequences.

Forwards and backwards
Bridging 10s, 100s, 1000s
By 10s and 100s, on and off the decade
By 2s, 3s, 5s

In range to 1000, and beyond (Wigley, 1997)
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

ASPECT A: NUMBER WORDS AND NUMERALS
Instruction: the numeral track
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See then say.
Say then see.
Work backwards.
Hop around.
ASPECT A: NUMBER WORDS AND NUMERALS
Instruction: the numeral track
Video clip—the numeral track.
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
ASPECT B: STRUCTURING NUMBERS 1 TO 20
Facile calculation 1-20
Initial strategies involve counting by ones.
(e.g. Fuson, 1988; Steffe & Cobb, 1988)
Facile strategies include:



Adding through ten (6+8=8+2+4)
Using fives (6+7=5+5+1+2)
Near-doubles (6+7=6+6+1)
Facile strategies build on knowledge of
combining and partitioning
(Bobis, 1996; Gravemeijer et al., 2000)
ASPECT B: STRUCTURING NUMBERS 1 TO 20
Facile calculation 1-20
Developing facile strategies is critical.

Reduces errors.

Reduces cognitive demand.
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Promotes number sense.
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Develops part-whole number concept.
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Prepares basis for later arithmetic.
(Steffe & Cobb, 1988; Treffers, 1991)
ASPECT B: STRUCTURING NUMBERS 1 TO 20
Low-attaining 8-10yo strategies
Typically use counting on and counting back.
17-15:
“17, 16,…2,1” 15 counts back, miscounted.
Two
numbers that add to 19:
“(6 second pause) 18 and (pause) 1”.
6+7:
Knows 6+6, but does not use doubles.
15-4:
Does not relate to 5-4 (ten structure of teen numbers).
ASPECT B: STRUCTURING NUMBERS 1 TO 20
Instruction: Arithmetic rack
Flexible patterning: pair-wise, 5-wise, 10-wise.
Phase 1: Making and reading numbers.
Phase 2: Addition of two numbers.
Phase 3: Subtraction, in various forms.
Use of screening and flashing
increasingly internalize the reasoning activity.
(Gravemeijer et al, 2000; Treffers, 1991; Wright, Stanger et al, 2006)
ASPECT B: STRUCTURING NUMBERS 1 TO 20
Instruction: Arithmetic rack
Video clip: arithmetic rack.
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
ASPECT C: CONCEPTUAL PLACE VALUE
Multidigit knowledge
Emphasis on mental strategies.
(Beishuizen & Anghileri, 1998; Fuson et al., 1997; Yackel, 2001)
Efficient, flexible strategies require network of
number structures. (Heirdsfeld, 2001; Threlfall, 2002)
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Additive place value (25 is 20 and 5).
Jumping by ten (40+20=60; 48+20=68).
Jumping through ten (68+5  68+2+3).
Relating to neighborhood (48+25  50+25-2).
ASPECT C: CONCEPTUAL PLACE VALUE
Multidigit knowledge
Where regular place value instruction is
intended to support the development of
standard written algorithms,
We propose conceptual place value as an
approach to support the development of
students’ intuitive arithmetical strategies.
ASPECT C: CONCEPTUAL PLACE VALUE
Low-attaining 8-10yo strategies
Not increment/decrement by ten off the decade.
48
+
10 10 5
“48, 49, 50…” Counting on dots.
“40+20=60, 8+5=13…?”
Attempt split, difficulty regrouping.
“485868, 69, 30, 31, 32, 33.”
Attempt jump, difficulty keeping
track of ones.
ASPECT C: CONCEPTUAL PLACE VALUE
Instruction
Base-ten settings: bundling sticks, dot-strips.
Flexibly incrementing and decrementing

by ones and tens.

later, by hundreds and thousands.
Use of screening.
Learning of place value is verbal, and additive.
ASPECT C: CONCEPTUAL PLACE VALUE
Instruction
Video clip: dot-strips and arrow cards.
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Facile 2-digit mental strategies
Foundation

for all further arithmetic.

for learning standard written algorithms.

for efficient use of calculators.
(Beishuizen & Anghileri, 1998; Treffers & Buys, 2001)
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Facile 2-digit mental strategies: 48+25
+10
58
+10
68
+2
70
+3

Jump: 48
73.

Split:

Various e.g. compensation: 50+25-2=73.
40+20=60; 8+5=13;  60+13=73.
All strategies involve
 jumping by ten.
 jumping through ten.
(e.g. Fuson et al., 1997; Klein, Beishuizen, & Treffers, 1998)
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Low-attaining 8-10yo strategies
Tend not to use jumping by ten, through ten.
(Menne, 2001)
Most successful students use jump/
most low-attainers use split.
Low-attainers using jump have more success.
(Foxman & Beishuizen, 2002; Klein, Beishuizen, & Treffers, 1998)
Split in subtraction has common procedural
difficulties (Fuson et al., 1997).
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Instruction
Jumping through ten: applying 1-digit knowledge
in higher decades requires instruction.
Setting: ten frame cards with full “bob” cards.

Adding and subtracting to and from a decuple:
68 +  = 70
54 -  = 50


70 + 3
50 - 4
Jumping through ten: 68 + 5
Tasks with two 2-digit numbers:
54 - 8
48 + 25
64-18
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Instruction
Use a notation system in conjunction with
mental strategies.
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Empty number line (ENL)—jump strategies.
Arrow notation—jump strategies.
Drop-down notation—split strategies.
Number sentences—jump and split.
(Gravemeijer et al., 2000; Klein et al., 1998)
ASPECT D: ADDITION AND SUBTRACTION 1 TO 100
Instruction
Video clip—Bob cards.
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Multiplicative thinking (MT)

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Coordinate two composite units.
Recognise multiplicative situations,
including equal groups and arrays.
Move beyond physical models toward
mental imagery.
MT builds in part on knowledge of skip-counting
and of addition/subtraction in range 1-100.
(Greer, 1992; Mulligan & Mitchelmore, 1997; Siemon et al, 2006;
Steffe, 1994; Sullivan et al, 2001; Wright, Martland & Stafford, 2006)
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Multiplicative thinking (MT)
Foundation

for number sense.

for learning standard written algorithms.

for efficient use of calculators.

for further arithmetic: fractions & decimals,
proportional reasoning, exponentials.
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Low-attaining 8-10yo strategies
Limited construction of composite units, tending
to count by ones.
Do not construct arrays in rows and columns.
Perceptual and figurative counting
e.g. Solves ‘Four 5-dot cards’ but not ‘4 times 5’.
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Low-attaining 8-10yo strategies
Limited knowledge of skip-counting NWS.
e.g. Counts by 2s and 5s, but not 3s or 4s.
Weak addition facility.
Repeated addition of 4s “8…12…16…21.”
Very limited knowledge of times tables facts.
e.g. Recalls a few 2s and 10s facts only.
e.g.
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Instruction
Multiplicative settings:

Equal groups dot cards, with 2-6 dots.

dot arrays, up to 10x10.
Multiplication, quotition, partition tasks.
Use of partial and full screening.
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Instruction
Promoting:

strategies using composite units.

mental imagery of equal groups and arrays.

familiarity with factor families in network of
number relations 1-100.

connection to formal written symbols.
Progress with other aspects is co-requisite:
skip-counting, structuring numbers 1-20,
conceptual place value, add/sub 1-100.
ASPECT E: EARLY MULTIPLICATION AND DIVISION
Instruction
Video clip—equal groups dot-cards.
3. EXPERIMENTAL LEARNING FRAMEWORK
Aspect A – Number Words and Numerals
Aspect B – Structuring Numbers 1 to 20
Aspect C – Conceptual Place Value
Aspect D – Addition and Subtraction 1 to 100
Aspect E – Early Multiplication and Division
3. EXPERIMENTAL LEARNING FRAMEWORK
Coherence of the Framework

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
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Approach to instructional design is
consistent.
Aspects overlap in assessment.
Aspects broadly concurrent in instruction.
Teacher makes connections between
aspects.
(Askew et al., 1997; Treffers, 1991)
3. EXPERIMENTAL LEARNING FRAMEWORK
Further lines of inquiry


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Analyze low-attainers’ learning in each
aspect.
Refine the instructional sequences in each
aspect.
Evaluate intervention programs based on
the framework.
Clarify the design research approach for
intervention.
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
4. CHILD-CENTRED TEACHING
Key features of intervention teaching
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
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9 Guiding principles for intervention
13 Key elements of instruction
9 Characteristics of children’s problemsolving
(Wright, Martland, Stafford & Stanger, 2006)
4. CHILD-CENTRED TEACHING
9 Guiding principles for intervention
1. Problem-based/inquiry-based teaching
2. Initial and on-going assessment
3. Teach just beyond the cutting edge (ZPD)
4. Select from a bank of teaching procedures
5. Engender more sophisticated strategies
6. Observe the child and fine-tune teaching
7. Incorporate symbolizing and notating
8. Sustained thinking and reflection
9. Child’s intrinsic satisfaction
4. CHILD-CENTRED TEACHING
13 Key elements of intervention instruction
1. Micro-adjusting
2. Scaffolding
3. Handling an impasse
4. Introducing a setting
5. Pre-formulating a task
6. Reformulating a task
7. Post-task wait-time
8. Within task setting change
…continued over…
4. CHILD-CENTRED TEACHING
13 Key elements of intervention instruction
…continued…
9. Screening, colour-coding, and flashing
10. Teacher reflection
11. Child checking
12. Affirmation
13. Behaviour-eliciting
4. CHILD-CENTRED TEACHING
9 Characteristics of children’s problem-solving
1. Cognitive reorganisation
2. Anticipation
3. Curtailment
4. Re-presentation
5. Spontaneity, robustness, and certitude
6. Asserting autonomy
7. Child engagement
8. Child reflection
9. Enjoying the challenge
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
5. RESEARCH METHODOLOGY
Design research
Design cycle:
1.
Devise pedagogical tools.
2.
Use tools in intervention program.
3.
Analyze learning and teaching in the
program.
4.
Refine tools.
Also on-going analysis and development.
(Cobb, 2003; Gravemeijer, 1994)
5. RESEARCH METHODOLOGY
Design research cont.
Analysis of teaching and learning informed by a
teaching experiment methodology.
(Steffe & Thompson, 2000)
Interview assessments and instructional
sessions videotaped for analysis.
5. RESEARCH METHODOLOGY
The Study
Intervention program in each school:
12 students identified as low-attaining.

term 2 - the 12 students interview-assessed.

term 3 - 8 students in teaching cycle.

term 4 - the 12 students assessed again.
Teaching cycle: 4 days/week x 10 weeks.
Classes: 2 singletons, 2 trios.
5. RESEARCH METHODOLOGY
The Study: totals
Years
3
Schools
25
Teachers
25
Students assessed (twice each) 300
Students taught individually
50
Students taught in trios
150
Students screening tested
2400 +
5. RESEARCH METHODOLOGY
Development of the Instructional Framework
Key aspects developed from 
Identifying areas of significance in lowattainers’ knowledge.

Profiling of assessments by teachers.

Focusing on key instructional sequences.

Clarifying a coherent framework for
instruction.
INTERVENTION IN NUMBER LEARNING
1.
2.
3.
NIRP: project overview
Approach to intervention
Experimental learning framework
Five key aspects: A, B, C, D, E
4.
5.
Child-centred teaching
Research methodology
Intervention in the number
learning of 8- to 10-year olds
A/Prof Bob Wright
David Ellemor-Collins
End.
A/Prof. Bob Wright
David Ellemor-Collins