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  1. Intervention in the number learning of 8- to 10-year olds A/Prof Bob Wright David Ellemor-Collins

  2. 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.

  3. INTERVENTION IN NUMBER LEARNING NIRP project overview Approach to intervention Experimental learning framework Five key aspects: A, B, C, D, E Child-centred teaching Research methodology

  4. INTERVENTION IN NUMBER LEARNING NIRP: project overview Approach to intervention Experimental learning framework Five key aspects: A, B, C, D, E Child-centred teaching Research methodology

  5. 1. NUMERACYINTERVENTION RESEARCHPROJECT 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

  6. 1. NUMERACYINTERVENTION RESEARCHPROJECT Aim: to develop pedagogical tools for intervention. • Interview-based assessment schedules. • Learning framework. • Instructional framework. • Instructional procedures and sequences.

  7. 1. NUMERACYINTERVENTION RESEARCHPROJECT 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

  8. INTERVENTION IN NUMBER LEARNING NIRP: project overview Approach to intervention Experimental learning framework Five key aspects: A, B, C, D, E Child-centred teaching Research methodology

  9. 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)

  10. 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

  11. 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).

  12. 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.

  13. 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)

  14. 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.

  15. 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)

  16. 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)

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

  18. INTERVENTION IN NUMBER LEARNING NIRP: project overview Approach to intervention Experimental learning framework Five key aspects: A, B, C, D, E Child-centred teaching Research methodology

  19. 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

  20. 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

  21. 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

  22. 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)

  23. ASPECT A: NUMBER WORDS AND NUMERALS Instruction: the numeral track • See then say. • Say then see. • Work backwards. • Hop around.

  24. ASPECT A: NUMBER WORDS AND NUMERALS Instruction: the numeral track Video clip—the numeral track.

  25. 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

  26. 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)

  27. ASPECT B: STRUCTURING NUMBERS 1 TO 20 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)

  28. 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).

  29. 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)

  30. ASPECT B: STRUCTURING NUMBERS 1 TO 20 Instruction: Arithmetic rack Video clip: arithmetic rack.

  31. 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

  32. 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) • 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).

  33. 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.

  34. 10 10 5 ASPECT C: CONCEPTUAL PLACE VALUE Low-attaining 8-10yo strategies Not increment/decrement by ten off the decade. “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. 48 +

  35. 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.

  36. ASPECT C: CONCEPTUAL PLACE VALUE Instruction Video clip: dot-strips and arrow cards.

  37. 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

  38. 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)

  39. +10 +10 +2 +3 48 58 68 70 73. ASPECT D: ADDITION AND SUBTRACTION 1 TO 100 Facile 2-digit mental strategies: 48+25 • Jump: • Split:40+20=60; 8+5=13;  60+13=73. • Variouse.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)

  40. 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).

  41. 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 70 + 3 54 -  = 50 50 - 4 • Jumping through ten: 68 + 5 54 - 8 • Tasks with two 2-digit numbers: 48 + 25 64-18

  42. ASPECT D: ADDITION AND SUBTRACTION 1 TO 100 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)

  43. ASPECT D: ADDITION AND SUBTRACTION 1 TO 100 Instruction Video clip—Bob cards.

  44. 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

  45. 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, 2006)

  46. 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.

  47. 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’.

  48. 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. 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.

  49. 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.

  50. 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.