Intervention in the number learning of 8- to 10-year olds

Intervention in the number learning of 8- to

10-year olds

A/Prof Bob WrightDavid Ellemor-Collins

INTERVENTION IN NUMBER

LEARNING

AcknowledgementsWe 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.

LEARNING

1. NIRP project overview2. Approach to intervention3. Experimental learning

frameworkFive key aspects: A, B, C, D, E

4. Child-centred teaching5. Research methodology

LEARNING

1. NIRP: project overview2. Approach to intervention3. Experimental learning

1. NUMERACY INTERVENTION RESEARCH PROJECT

Joint project of Southern Cross University (SCU) Catholic Education Office, Melbourne

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

Aim:to develop pedagogical tools for intervention.

Interview-based assessment schedules. Learning framework. Instructional framework. Instructional procedures and sequences.

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

LEARNING

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)

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

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).

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.

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)

Instructional design cont.

Instructional procedures incrementally: distance the materials. advance the complexity of the task. raise the sophistication of the student’s

thinking.

Approach to number instructionDetailed 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)

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)

Number instruction cont.

Low-attainers often: Use inefficient count-by-ones strategies. Use unreasoned rote procedures. Depend on materials or fingers. (Gray & Tall,

Hence intervention instruction needs to: develop students’ number knowledge to

support non-count-by-ones strategies, and move students to independence from materials.

LEARNING

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 difficultiesNWSs e.g. “52, 51, 40, 49, 48…”

“108, 109, 200, 201, 202…”“108, 109, 1000, 1001…”

Tens off the decadee.g. “24, 30, 34, 40…”

“24…34…44” counting-by-ones subvocally

Numerals: errors identifying and writinge.g. 306, 6032, 3010, 1300, 1005

InstructionFacility 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)

Instruction: the numeral track

See then say. Say then see. Work backwards. Hop around.

Instruction: the numeral track

Video clip—the numeral track.

ASPECT B: STRUCTURING NUMBERS 1 TO 20

Facile calculation 1-20Initial 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)

Facile calculation 1-20

Developing facile strategies is critical. Reduces errors. Reduces cognitive demand. Promotes number sense. Develops part-whole number concept. Prepares basis for later arithmetic.

(Steffe & Cobb, 1988; Treffers, 1991)

Low-attaining 8-10yo strategiesTypically 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).

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)

Instruction: Arithmetic rack

Video clip: arithmetic rack.

ASPECT C: CONCEPTUAL PLACE VALUE

Multidigit knowledge

Emphasis on mental strategies.(Beishuizen & Anghileri, 1998; Fuson et al., 1997; Yackel,

Efficient, flexible strategies require network of number structures. (Heirdsfeld, 2001; Threlfall, 2002)

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).

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.

Low-attaining 8-10yo strategies

Not increment/decrement by ten off the decade.

10 10 5

“48, 49, 50…” Counting on dots.

“40+20=60, 8+5=13…?” Attempt split, difficulty regrouping.

“485868, 69, 30, 31, 32, 33.”Attempt jump, difficulty keeping track of ones.

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.

Instruction

Video clip: dot-strips and arrow cards.

ASPECT D: ADDITION AND SUBTRACTION 1 TO

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)

Facile 2-digit mental strategies: 48+25 Jump:

Split: 40+20=60; 8+5=13; 60+13=73.

Various e.g. compensation: 50+25-2=73.

All strategies involve jumping by ten. jumping through ten.

(e.g. Fuson et al., 1997; Klein, Beishuizen, & Treffers, 1998)

48 58 68 70 73.

+10 +10 +2 +3

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).

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 70 + 3

54 - = 50 50 - 4

Jumping through ten: 68 + 5 54 - 8 Tasks with two 2-digit numbers: 48 + 25 64-18

Instruction

Use a notation system in conjunction with mental strategies.

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)

Instruction

Video clip—Bob cards.

ASPECT E: EARLY MULTIPLICATION AND DIVISION

Multiplicative thinking (MT)

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,

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.

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’.

Limited knowledge of skip-counting NWS. e.g. Counts by 2s and 5s, but not 3s or 4s.

Weak addition facility.

e.g. 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.

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.

InstructionPromoting: 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.

Instruction

Video clip—equal groups dot-cards.

Coherence of the Framework

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)

Further lines of inquiry

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.

LEARNING

4. CHILD-CENTRED TEACHING

Key features of intervention teaching

9 Guiding principles for intervention 13 Key elements of instruction 9 Characteristics of children’s problem-

solving(Wright, Martland, Stafford & Stanger, 2006)

9 Guiding principles for intervention1. Problem-based/inquiry-based teaching2. Initial and on-going assessment3. Teach just beyond the cutting edge (ZPD)4. Select from a bank of teaching procedures5. Engender more sophisticated strategies6. Observe the child and fine-tune teaching7. Incorporate symbolizing and notating8. Sustained thinking and reflection9. Child’s intrinsic satisfaction

13 Key elements of intervention instruction1. Micro-adjusting2. Scaffolding3. Handling an impasse4. Introducing a setting5. Pre-formulating a task6. Reformulating a task7. Post-task wait-time8. Within task setting change

…continued over…

13 Key elements of intervention instruction

…continued… 9. Screening, colour-coding, and flashing10. Teacher reflection11. Child checking12. Affirmation13. Behaviour-eliciting

9 Characteristics of children’s problem-solving1. Cognitive reorganisation2. Anticipation3. Curtailment4. Re-presentation5. Spontaneity, robustness, and certitude6. Asserting autonomy7. Child engagement8. Child reflection9. Enjoying the challenge

LEARNING

5. RESEARCH METHODOLOGY

Design researchDesign 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)

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.

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.

The Study: totals

Years 3Schools 25Teachers 25Students assessed (twice each) 300Students taught individually 50Students taught in trios 150Students screening tested 2400+

Development of the Instructional Framework

Key aspects developed from - Identifying areas of significance in low-

attainers’ knowledge. Profiling of assessments by teachers. Focusing on key instructional

sequences. Clarifying a coherent framework for

instruction.

LEARNING

Intervention in the number learning of 8- to

10-year olds

A/Prof Bob WrightDavid Ellemor-Collins

A/Prof. Bob WrightDavid Ellemor-Collins

Intervention in the number learning of 8- to 10-year olds

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