Developing Number Knowledge: Assessment, Teaching & Intervention with 7–11-Year-Olds
Publication Year: 2012
This fourth book in the Mathematics Recovery series equips teachers with detailed pedagogical knowledge and resources for teaching number to 7 to 11-year olds. Drawing on extensive programs of research, curriculum development, and teacher development, the book offers a coherent, up-to-date approach emphasizing computational fluency and the progressive development of students’ mathematical sophistication. The book is organized in key domains of number instruction, including structuring numbers 1 to 20, knowledge of number words and numerals, conceptual place value, mental computation, written computation methods, fractions, and early algebraic reasoning.
- Front Matter
- Back Matter
- Subject Index
- Part I
- Chapter 1: Professional Learning for Quality Instruction
- Chapter 2: Instruction in Arithmetic
- Part II
- Chapter 3: Number Words and Numerals
- Chapter 4: Structuring Numbers 1 to 20
- Chapter 5: Conceptual Place Value
- Chapter 6: Addition and Subtraction to 100
- Chapter 7: Multiplication and Division
- Chapter 8: Written Computation
- Part III
- Chapter 9: Early Algebraic Reasoning
- Chapter 10: Understanding Fractions
- Chapter 11: Connecting the Teaching and Learning of Fractions
Education at SAGE[Page ii]
SAGE is a leading international publisher of journals, books, and electronic media for academic, educational, and professional markets.
Our education publishing includes:
- accessible and comprehensive texts for aspiring education professionals and practitioners looking to further their careers through continuing professional development
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Find out more at: http://www.sagepub.co.uk/education
First published 2012
USMRC™ and Add+VantageMR™ are licensed trademarks belonging to U.S. Math Recovery Council, all rights are expressly reserved.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted in any form, or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.
All material on the accompanying CD-ROM can be printed off and photocopied by the purchaser/user of the book. The CD-ROM itself may not be reproduced in its entirety for use by others without prior written permission from SAGE. The CD-ROM may not be distributed or sold separately from the book without the prior written permission of SAGE. Should anyone wish to use the materials from the CD-ROM for conference purposes, they would require separate permission from SAGE. All material is © Robert J. Wright, David Ellemor-Collins, Pamela D. Tabor, 2012
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[Page v]To Patrick, Henry and Finn
Heather, Miriam and Jeremy
Ron, Jeremy and Zack[Page vi]
List of Boxes[Page xv]
- 3.1 Instruction in Number Word Sequences (NWSs) Using a Numeral Roll 32
- 4.1 Instructional Inquiry Encouraging Structuring 56
- 4.2 Arithmetic Rack: Subsets of Sums with Addends in the Range 1 to 10 58
- 4.3 Arithmetic Rack: A Variety of Task Types in the Range 1 to 20 60
- 5.1 (a) A Basic CPV Sequence, Using Bundling Sticks, (b) A More Complex CPV Sequence 79
- 6.1 Presenting Higher Decade Addition and Subtraction Tasks with The Ten-Frame Setting 113
- 7.1 Ways to Extend N-Tile Activities 151
List of Tables[Page xvi]
- 2.1 Inquiry mode and rehearsal mode 23
- 3.1 Examples of student difficulties with number word sequences (NWSs) by 1s 27
- 3.2 Examples of student difficulties with number word sequences (NWSs) by 10s 28
- 3.3 Examples of student difficulties with number word sequences (NWSs) by 3s 29
- 3.4 Examples of student difficulties with numerals 30
- 5.1 Distinguishing conceptual place value from conventional place value 83
- 6.1 Examples of mental strategies for addition 100
- 6.2 Examples of mental strategies for subtraction 101
- 6.3 Summary of seven mental strategies 103
- 6.4 Comparing transformation with other strategies 105
- 6.5 Complementary addition strategies for 53 − 39 or 39 + □ = 53 106
- 6.6 Arithmetical steps and foundational knowledge involved in a jump strategy for 37 + 25 108
- 6.7 Three foundations for mental addition and subtraction to 100 108
- 7.1 Examples of a strategy for a given multiplier 154
- 9.1 Factors and number of factors 209
- 9.2 Triangular numbers and square numbers 210
List of Figures[Page xvii]
- 2.1 A broad progression in the development of mental and written computation 12
- 3.1 Using a numeral roll and multi-lid screen 32
- 3.2 4056 built with arrow cards 33
- 3.3 Place value houses graphic organizer 40
- 3.4 A disappearing sequence for multiples of two 42
- 3.5 A numeral grid with chips marking beginning and ending of sequence 44
- 3.6 Blank numeral ladders 45
- 3.7 A Four Kings game underway, using a sequence of 3s 46
- 3.8 (a) Lovely Lucy card layout (b) Lovely Lucy game underway 48
- 4.1 Examples of ten-frames: five-wise patterns, pair-wise patterns, partitions of 10, combinations 55
- 4.2 Examples of standard patterns on the arithmetic rack 57
- 4.3 Table of additions marking subsets of sums 60
- 4.4 A ten-frame with students’ partitions notated 69
- 4.5 Example scoring tables for (a) Bulls-eye Dice, (b) Bulls-eye Combination and (c) Super Bulls-eye 76
- 5.1 Task board with screens, for Task Group A5.4. How many dots now, altogether? (33) 86
- 5.2 Three bundles of bundling sticks 89
- 5.3 Comparing cartons of ten with cartons of a dozen 91
- 5.4 Packaging a number of eggs in ten-egg cartons 91
- 5.5 Building 74 with arrow cards 93
- 5.6 Notating 50 + 20 on the empty number line 94
- 5.7 A sample crazy grid 96
- 5.8 Automatic Teller Machine (ATM) 97
- 6.1 Informal notations for mental strategies for 37 + 25 from Table 6.1 101
- 6.2 Informal notations for mental strategies for 53 – 19 from Table 6.2 102
- 6.3 Six types of tasks for higher decade addition and subtraction 111
- 6.4 Writing number sentences to notate mental strategies for 37 + 25 118
- 6.5 Jumping back to a decuple with ten-frames 10-dot cards for 64 − ? = 60 125
- 6.6 (a) Adding to 50 game board and (b) spinner 127
- 6.7 Jumping back from a decuple with ten-frames 10-dot cards for 70–3 128 [Page xviii]
- 6.8 Recording a series of number sentences for higher decade additions 130
- 6.9 A car got on the interstate at Exit 74 and drove 26 miles south 133
- 6.10 Notating different solution strategies for 74 − 26 133
- 7.1 15 ÷ 3 − Division in a grouping or quotitive sense 138
- 7.2 15 ÷ 3 − Division in a sharing or partitive sense 138
- 7.3 Key labels associated with the operation of multiplication 139
- 7.4 Key labels associated with the operation of division 140
- 7.5 Task involving the number of squares on a rectangular grid 149
- 7.6 Instructional settings for learning to structure numbers multiplicatively 151
- 7.7 An instructional setting for structuring numbers by threes 151
- 7.8 An instructional setting for multiplicative reasoning 153
- 7.9 Recording 3 × 8 using an empty number line 153
- 7.10 Recording multiplication with multi-digit factors 155
- 7.11 Multiplication with repeated equal groups 156
- 7.12 Multiplication with an array 158
- 7.13 Snack time recording sheet 163
- 7.14 Row of 3-pattern geometric dot tiles 164
- 7.15 100-bead arithmetic rack 165
- 7.16 Rainbow factoring for 24 166
- 7.17 Array bingo cards 167
- 7.18 ENLs for seven groups of five 169
- 7.19 Multiples of 6 game board 170
- 7.20 (a) Factoring game board, and (b) spinner 171
- 8.1 An example of a jotting, a semi-formal strategy and a formal algorithm for each of the four operations 174
- 8.2 16 × 43 solved using an open array diagram and partial products method 184
- 8.3 A written strategy to solve 680 ÷ 15, including a table of handy multiples of 15, and notation elaborated with words 185
- 8.4 Example of the equal additions technique in a subtraction algorithm 187
- 8.5 The school excursion task 188
- 8.6 The prompt for a semi-formal strategy task 190
- 8.7 The prompt for an algorithm error analysis task 191
- 8.8 Palindromizing 36 and 37 194
- 8.9 (a) Shortcut for 104 × 107, and (b) shortcut for 96 × 93 196
- 8.10 (a) Complete the addition using the digits 1–9, and (b) complete the subtraction using the digits 0–9 197
- 9.1 Function machine: multiply by 3 and then add 1 204
- 9.2 A repeating pattern and a growing pattern 205
- 9.3 Desk and chairs problem 206
- 9.4 The handshake problem 208
- 9.5 Triangular and square numbers shown graphically 210 [Page xix]
- 10.1 A Year 6 drawing showing an additive approach to representing larger denominators 215
- 10.2 An additive interpretation of fractions due to a lack of reference to the whole 216
- 10.3 A Year 5 student showing one-third as 3 equal parts and one-sixth as 6 equal parts 217
- 10.4 A Year 6 student's representation of fractions as a number of parts 218
- 10.5 Equidistant partitioning 218
- 10.6 Representing a number of parts rather than the area of the parts 220
- 10.7 Drawing four-thirds and showing coordination of units at three 222
- 10.8 People and pancakes drawing 222
- 10.9 People and pancakes table 223
- 11.1 What fraction is shaded? 226
- 11.2 Five-quarters recast as five-fifths 227
- 11.3 Coordinating units at three levels with proper fractions 227
- 11.4 Coordinating composition of partitioning and units at 3 levels 228
- 11.5 Comparing the size of 1/3 and 1/6 without referencing equal wholes 228
- 11.6 Folded strip of paper 229
- 11.7 Region with patterns blocks 230
- 11.8 Halving a discrete linear arrangement 231
- 11.9 Halving an equivalent linear arrangement 232
- 11.10 One-third of a bar 233
- 11.11 Folded paper: what fraction of the length is that?236
- 11.12 Three strips of paper 236
- 11.13 Strips of paper: variation 237
- 11.14 Strips of paper: more variations 237
- 11.15 A response showing correct partitioning and developing notation 240
- 11.16 A response showing accumulated three-quarters 240
- 11.17 The multiplicative relationship between fractional parts and the whole 241
- 11.18 Introducing mixed numbers 242
- 11.19 Economical partitioning 242
- 11.20 Partitioning involving quarters 243
- 11.21 Pancake and people recording table 245
- 11.22 Kit Kat bar prompt 246
List of Photographs[Page xx]
- 1.1 Using the book for professional development 6
- 1.2 Using the book for intervention teaching 8
- 1.3 Using the book for classroom teaching 9
- 2.1 A teacher's lesson plan organized using domains 14
- 2.2 Teacher and students discuss a solution to a task 20
- 2.3 Screening the setting to pose a task 22
- 2.4 Unscreening to verify the solution 22
- 3.1 Student writing numerals in a numeral ladder 33
- 3.2 Using arrow cards with digit cards 34
- 3.3 Reading numeral cards for numeral identification 35
- 3.4 Using the numeral roll with a window 43
- 4.1 Flashing double ten-frames 54
- 4.2 Organizing partitions ten-frames 55
- 4.3 Notating strategies in the context of the bead rack 61
- 4.4 Organizing expression cards 62
- 4.5 Making doubles on the arithmetic rack 71
- 4.6 Crackers the parrot puppet 74
- 5.1 Writing 10 more and 10 less 81
- 5.2 Using dot materials in conjunction with arrow cards 82
- 5.3 Screening bundling sticks to assess incrementing by tens off the decuple 85
- 6.1 Jump down from the decuple with 10-dot ten-frames 112
- 6.2 Solving an additive task in a setting of partly screened bundling sticks 114
- 6.3 Mentally solving bare number 2-digit addition tasks 117
- 6.4 Notating different strategies to facilitate discussion 118
- 7.1 Solving a task involving screened equal groups 148
- 7.2 Solving a task involving a partly screened array 148
- 7.3 (a) and (b) Posing a sequence of multiples of three using 3-tiles 152
- 7.4 Developing strategies to solve related bare number tasks 154
- 8.1 (a) and (b) Jotting to solve a division problem 175
- 8.2 (a) and (b) Using semi-formal written strategies 176
- 9.1 Connecting arithmetic to algebraic thinking 202
- 11.1 Finding half way 235
- 11.2 (a) and (b) Time series of locating three-eighths of the length of the whiteboard 239 [Page xxi]
- 11.3 (a) (b) (c) and (d) Four shots of dividing a circle into thirds 239
- 11.4 (a) and (b) Building towers activity: reasoning about fractions with linking cubes 243
- 11.5 Oops activity: reasoning about fractional length 244
Contents of Resource CD[Page xxii]
The files have been formatted into two styles: inch USA format and spellings; A4 International format and spellings. Each file is available in three file formats: MS Word, PDF and XPS.
- Chapter 3: Number Words and Numerals 25
4-Kings Deck by 3s
4-Kings Deck by 3s, 6s, 7s, 8s
4-Kings Deck by 10s off decuple
4-Kings Deck Template
4-Kings Playing Directions
Blank Number Grid
Disappearing Sequences Directions
Lovely Lucy Directions
Number Grid – 781–870
Number Grid – 941–1040
Numeral Ladder Graphic Organizer
Numeral Rolls 1–120, 101–220
Place Value Houses Graphic Organizer
- Chapter 4: Structuring Numbers 1 to 20 50
8 Plus Game
9 Plus Game
19 Plus Game
20 Minus Game
Addition Expression Cards
Addition Facts Grid
Bulls-Eye Dice Record Sheet
Decuple Minus Game
Making 6 Numeral Cards
Spinners 0–9, 1–9 Choose, 1–6, 4–9
Ten-Frames – Red and Green
Ten-Frames – Numbers 0 to 10
Ten-Frames – Parts of 10[Page xxiii]
Ten-Plus Dot Patterns
Two-Colour Double Ten-frames
- Chapter 5: Conceptual Place Value 77
Crazy Grid Master
Dot Strips – 1s and Instructions
Dot Strips – 10s and 100s
Egg Carton Recording Tables
Two-Colour Dot Strips and Squares
Uncovering Task A5-4
- Chapter 6: Addition and Subtraction to 100 99
20 Plus Game
32 Minus, 51 Minus Game
48 Plus, 67 Plus Game
70 Plus Game
Add or Subtract 12 Game
How Many More to Make 100? Game
How Many More to Make 60? Game
Jumping to 50 Game
Spinners 0–9, 1–9 Choose, 1–6, 4–9
- Chapter 7: Multiplication and Division 135
10 × 10 Array Graphic
Array Bingo Cards
Dot Tiles in Geometric Arrangements
Dot Tiles in Linear Arrangements
Factoring Game and Spinners
Multiples of 2 Game
Multiples of 3 Game
Multiples of 4 Game
Multiples of 5 Game
Multiples of 6 Game
Multiples of 7 Game
Multiples of 8 Game
Multiples of 9 Game
Perfect Squares Game
Snack Time Record Sheet
Spinners 0–9, 1–9 Choose, 1–6, 4–9
- Chapter 8: Written Computation 173
Alternative Algorithm Prompts
Error Analysis Prompts
Also included on the CD is each Instructional Activity in the book.
The three authors of this book share a consuming interest in advancing what is known about the teaching of number to children. Our collaborative work over the last eight years related to assessment and instruction in number, provides an appropriate back-drop for the writing of this book. In addition, the book is a major outcome of a multi-year research project for which the first and second authors were the lead and associate researchers respectively. The goal of this project was to develop pedagogical tools for intervention in the number learning of low-attaining 8–10-year-olds. During the course of the project, the research team worked with a teacher from each of 25 schools and a total of 200 low-attaining students.
The authors wish to express their gratitude and appreciation to the joint funders of this project – the Australian Research Council via the Linkage Grants Program under grant LP0348932, the Catholic Education Office Melbourne, and the Victorian Catholic Education Commission – in whose schools the project was conducted.
The authors also express their sincere gratitude and appreciation to the 25 schools involved in the project and to the participating students. We particularly wish to thank the 25 teachers, each of whom worked diligently over the course of a school year, to assess students, provide intensive instruction and actively participate in our on-going project workshops. Their hard work and collaboration in the development of new approaches to assessment and instruction have contributed significantly to the achievements and outcomes of the project.
We wish to acknowledge and give thanks to Andrea Dineen, Ann Stafford, Debra Meagher, Jackie Amato, Jeanna Gentile, Johti Kidd, Martin Gill, Rebecca Stewart and Ron Stump for contributions to the photographs, and Jeremy Tabor for contributions to the illustrations.
Our final thanks go to Mr Gerard Lewis, Team Leader in Mathematics for the School Services Staff Group of the Catholic Education Office, Melbourne, with whom we have worked closely for the duration of the planning and conducting of the project. We acknowledge and very much value his strong and on-going support, encouragement and affirmation for our endeavours on the project.
About the Authors and Contributor[Page xxv]Authors
Dr Robert J. (Bob) Wright holds Bachelor's and Master's degrees in mathematics from the University of Queensland (Australia) and a doctoral degree in mathematics education from the University of Georgia. His current position is adjunct professor in mathematics education at Southern Cross University in New South Wales. Bob is an internationally recognized leader in understanding and assessing children's numerical knowledge and strategies, publishing many articles and papers in this field. His work over the last 20 years has included the development of the Mathematics Recovery Programme which focuses on providing specialist training for teachers to advance the numeracy levels of young children assessed as low-attainers. In Australia and New Zealand, Ireland, the UK, the USA, Canada, Mexico and elsewhere, this programme has been implemented widely and applied extensively to classroom teaching and to average and able learners as well as low-attainers. He has conducted several research projects funded by the Australian Research Council including the most recent project focusing on assessment and intervention in the number learning of low-attaining 8–10-year-olds.
David Ellemor-Collins holds a Bachelor's degree with honours in mathematics and philosophy from Harvard University, and a Graduate Diploma in Education from the University of Melbourne. For the past seven years, he has worked as a researcher with an Australian Research Council-funded project focusing on assessment and intervention in the number learning of low-attaining 8–10-year-olds. David has worked as a mathematics teacher in primary schools, high schools and universities. He has contributed curriculum development to school mathematics programmes, provided professional development courses for schools, and published articles and papers for both researchers and practitioners in mathematics education.
Dr Pamela Tabor holds a Bachelor's degree in elementary education and Bible from Kentucky Christian University, a Master's degree in elementary education from East Tennessee State University, and a Doctor of Philosophy in mathematics education from Southern Cross University. She is a school-based elementary mathematics specialist and grant project manager with Harford County Public Schools, Harford County, Maryland. She has been a contracted researcher with several bodies researching fidelity of programme implementation, programme evaluation and professional development.[Page xxvi]
Dr Peter Gould holds Bachelor's and Master's degrees in science and mathematics from the University of Sydney (Australia). His doctoral dissertation examined the way that children come to develop a sense of fractions as numbers and the challenges they face along the way. Peter has been the Chief Education Officer in Mathematics with the New South Wales Department of Education and Training in Australia for 16 years. He taught mathematics classes for 13 years in schools serving disadvantaged communities as well as Technical and Further Education courses and University courses. Peter is well known for his work as a Mathematics Consultant and he readily acknowledges that his students and colleagues have taught him many useful things over the years.
Series Preface[Page xxvii]
If you ask educationalists and teachers whether numeracy intervention deserves equal attention with literacy intervention the overwhelming answer is ‘Yes, it should’. If you then ask whether this happens in their experience the answer is a resounding ‘No!’ What then are the reasons for this discrepancy? Research shows that teachers give more attention to addressing difficulties in literacy than difficulties in early numeracy. Teachers also state that there is a lack of suitable tools for assessing young children's numeracy skills and knowledge, and appropriate programmes available to address the deficits.
The four books in this series make a significant impact to redress the imbalance by providing practical help to enable schools and teachers to give equal status to early numeracy intervention. The books are:
- Early Numeracy: Assessment for Teaching and Intervention, 2nd edition, Robert J. Wright, Jim Martland and Ann K. Stafford, 2006
- Teaching Number: Advancing Children's Skills and Strategies, 2nd edition, Robert J. Wright, Jim Martland, Ann K. Stafford and Garry Stanger, 2006
- Teaching Number in the Classroom with 4–8 Year-olds, Robert J. Wright, Garry Stanger, Ann K. Stafford and Jim Martland, 2006
- Developing Number Knowledge: Assessment, Teaching and Intervention with 7–11-Year-olds, Robert J. Wright, David Ellemor-Collins and Pamela Tabor, 2012.
The authors are internationally recognized as leaders in the field of numeracy intervention. They draw on considerable practical experience of delivering training courses and materials on how to assess students’ mathematical knowledge, skills and strategies in addition, subtraction, multiplication and division. This is the focus of Early Numeracy. The revised edition contains six comprehensive diagnostic assessment tools to identify children's strengths and weaknesses and has a new chapter on how the assessment provides the direction and focus for intervention. Teaching Number sets out in detail nine principles which guide the teaching, together with 180 practical, exemplar teaching procedures to advance children to more sophisticated strategies for solving arithmetic problems. The third book, Teaching Number in the Classroom with 4–8 Year-olds, extends the work of assessment and intervention with individual and small groups to working with whole classes. In this text the authors have been assisted by expert, primary practitioners from Australia, the USA and the UK who have provided the best available Instructional Activities for each of eight major topics in early number learning. The fourth book, Developing Number Knowledge: Assessment, Teaching and Intervention with 7–11-Year-olds extends the focus of the series to include older students and more advanced learning and includes chapters on early algebraic reasoning and fractions.
[Page xxviii]The four books in this series provide a comprehensive package on :
- The identification, analysis and reporting of children's arithmetic knowledge, skills and strategies
- How to design, implement and evaluate a course of intervention
- How to incorporate both assessment and teaching in the daily numeracy programme in differing class organizations and contexts.
The series is distinct from others in the field because it draws on a substantial body of recent theoretical research supported by international, practical application. Because all the assessment and teaching activities portrayed have been empirically tested, the books have the additional, important distinction that they indicate to the practitioner, ranges of students’ responses and patterns of their behaviour.
The book series provides a package for professional growth and development and an invaluable, comprehensive resource for both the experienced teacher concerned with early numeracy intervention and for the primary teacher who has responsibility for teaching numeracy in kindergarten to upper primary levels. Primary numeracy consultants, mathematics advisors, special education teachers, teaching assistants and initial teacher trainees around the world will find much to enable them to put numeracy intervention on an equal standing with literacy. At a wider level the series will reveal many areas of interest to educational psychologists, researchers and academics.
- Abstract composite unit. A construct by which one can simultaneously regard an abstract number as both a composite and a unit. For example, ten is an abstract composite unit when it can be regarded as both ten ones and one ten. This can be distinguished from numerical composite which involves regarding an abstract number as a composite but not as a unit.
- Addend. A number to be added. In 7 + 4 = 11, 7 and 4 are addends, and 11 is the sum.
- Additive task. A generic label for tasks involving what adults would regard as addition. The label ‘additive task’ is used to emphasize that children will construe such tasks idiosyncratically that is, differently from each other and from the way adults will construe them.
- Algorithm. A step-wise procedure for carrying out a task. In arithmetic, a procedure for adding, subtracting, and so on. Also used to refer to the standard, written procedures for calculating with multi-digit numbers, for example, the division algorithm.
- Alternative algorithm. See Semi-formal written strategy.
- Arithmetic knowledge. A collective term for all the student knows about arithmetic (i.e. number and operations). The term ‘knowledge’ is sometimes juxtaposed with ‘strategies’ and in that case refers to knowledge not easily characterized as a strategy (for example, knowing the names of numerals).
- Arithmetic rack. See Appendix.
- Array. See Appendix.
- Arrow cards. See Appendix.
- Associative principle. The operation of addition is associative because, when any three numbers are added: 7 + 4 + 5, the order of performing the two addition operations does not affect the sum, for example, (7 + 4) + 5 = 7 + (4 + 5). Multiplication is also associative whereas subtraction and division are not associative.
- Automaticity. A capacity to quickly recall or figure out the answer to basic facts (e.g. 7 + 9, 4 × 8).
- Automatization. The development of automaticity.
- Backward number word sequence (BNWS). See Number word sequence.
- Bare number tasks. Arithmetic tasks presented in the absence of a setting or context, for example, 47 + 35, 86 × 3. [Page 248]
- Base ten. A characteristic of numeration systems and number naming systems whereby numbers are expressed in a form that involves grouping by tens, tens of tens, and larger powers of ten (1000; 10,000; etc.).
- Base-ten dot materials. See Appendix.
- Base-ten materials. A generic name for instructional settings consisting of materials organized into ones, tens, hundreds, and so on, such as bundling sticks and base-ten dot materials.
- Base-ten thinking. Thinking of numbers in terms of ones, tens, hundreds, and so on, for example thinking of 257 as consisting of 200 + 50 + 7.
- Basic facts. Combinations or number bonds of the form a + b = c (basic facts for addition) or a × b = c (basic facts for multiplication) where a and b are numbers in the range 0 to 10. Also, corresponding combinations involving subtraction or division.
- Bundling sticks. See Appendix.
- Centuple. A multiple of 100 (e.g. 100, 200, 300, 1300, 2500). This is distinguished from century which means a sequence of 100 numbers, for example from 267 to 366 or a period of 100 years.
- Century. See Centuple.
- Column computation. Arithmetical computation with multi-digit numbers that involves organizing numbers into columns, typically used before the formal algorithms are taught.
- Combinations. An alternative name for number bonds or basic facts, for example, 5 + 3 = 8, 8 – 2 = 6, 7 × 9 = 63, 48 ÷ 8 = 6.
- Combining. An arithmetical strategy involving combining (i.e. in a sense adding) two numbers whose sum is in the range one to 10, without counting, for example, 4 and 3, 7 and 2.
- Commutative principle. The operation of addition is commutative because, for any two numbers, their sum is unchanged when the numbers are commuted, for example, 7 + 4 = 4 + 7. Multiplication is also commutative whereas subtraction and division are not commutative.
- Compensation strategy. An arithmetical strategy that involves first changing one number to make an easier calculation and then compensating for the change, for example 17 + 38: calculate 17 + 40 and subtract 2.
- Complementary addition. A strategy for subtracting based on adding up, for example, 82 – 77 is solved as 77 + ? = 80, 80 + ? = 82, answer 5.
- Conceptual place value (CPV). An instructional topic focusing on developing students’ facility to increment and decrement flexibly by ones, tens, hundreds, and so on. CPV is distinguished from conventional place value, that is, the conventional instructional topic that is intended to provide a basis for learning formal written algorithms.
- Conventional place value. See Conceptual place value.
- Coordinating units. Conceiving of a unit fraction and a whole simultaneously. This involves knowing how to iterate a unit fraction such as one-third to form a whole. [Page 249]
- Counting by ones. A range of strategies used to solve arithmetical tasks. Some examples are, counting-on, counting-back and counting from one. Contrasted with non-counting (non-count-by-ones) strategies, that is, arithmetical strategies which do not involve counting by ones such as adding through ten, using a double, using a five-structure.
- Decade. See Decuple.
- Decrementing. See Incrementing.
- Decimalizing. Developing base-ten thinking, that is, approaches to arithmetic that exploit the decimal (base-ten) numeration system, such as using 10 as a unit, and organizing calculations into 1s, 10s and 100s.
- Decuple. A multiple of ten (e.g. 10, 20, 30, 180, 240). Distinguished from decade which means a sequence of 10 numbers, for example, from 27 to 36 or a period of 10 years.
- Difference. See Minuend.
- Digit. The digits are the ten basic symbols in the modern numeration system, that is ‘0’, ‘1’, … ‘9’.
- Digit cards. See Appendix.
- Distancing materials. An instructional technique involving progressively reducing the role of materials, for example, materials are unscreened, then flashed and screened, then screened without flashing and used only to check, and so on.
- Distributive principle. The principle that multiplication distributes over addition and subtraction as does division, for example, (7 – 5) × 3 = (7 × 3) – (5 × 3).
- Dividend. In a division equation such as 29 ÷ 4 = 7 r 1, 29 is the dividend, 4 is the divisor, 7 is the quotient and 1 is the remainder.
- Divisor. See Dividend.
- Domain. Used to refer to a broad area of arithmetical learning such as, ‘Number Words and Numerals’, ‘Conceptual Place Value’.
- Dot tiles. See Appendix.
- Doubles. The addition basic facts that involve adding a number to itself: 1 + 1, 2 + 2, …10 + 10.
- Drop-down notation. An informal notation for recording a split strategy for multi-digit addition and subtraction.
- Empty number line (ENL). See Appendix.
- Equidistant parallel partitioning. Drawing a set of equidistant parallel lines when attempting to partition a region into fractions of a specified size (e.g. fifths).
- Facile. Used in the sense of having good facility, that is, fluent or dexterous, for example, facile with a compensation strategy, or facile with the backward number word sequence. [Page 250]
- Factor. If a number F, when multiplied by a whole number gives a number M, we call Fafactor of M and Mamultiple of F. For example, 3 is a factor of 27 and 27 is a multiple of 3, because 3 × 9 = 27.
- Finger patterns. Arrangements of fingers used by students when calculating.
- Five-wise pattern. A spatial pattern on a ten-frame for a number in the range 1 to 10. Five-wise patterns are made by filling first one row, then the second. For example, a five-wise pattern for 4 has a row of 4 and a row of 0, a five-wise pattern for 7 has a row of 5 and a row of 2. This is contrasted with pair-wise patterns which are made by progressively filling the columns. For example, a pair-wise pattern for 4 has two pairs, a pair-wise pattern for 7 has three pairs and one single dot. On an arithmetic rack, these patterns can be made for numbers in the range 1 to 20. As well, ten-wise patterns can be made by first filling one row of the rack.
- Flashing. A technique which involves briefly displaying (typically for half a second) some part of an instructional setting. For example, a ten-frame with 8 red and 2 black dots is flashed.
- Formal algorithm. A standard written procedure for calculating with multi-digit numbers that relies on the conventions of formal place value; for example, the column-based procedures for adding, subtracting, and so on, contrasted with an informal strategy, for example, solving 58 + 25 by adding 58 and 20, and then 78 and 5.
- Formalizing. Developing an approach to arithmetic that involves more formal notation or more formal procedures. The term ‘formal’ is used to indicate adult-level, abstract mathematics.
- Forward number word sequence (FNWS). See Number word sequence.
- Generalizing. Reasoning that involves proceeding from a few cases to many cases.
- Groupable base-ten materials. Base-ten materials such as bundling sticks that can be aggregated into tens and disaggregated. This is contrasted with base-ten materials already grouped into tens (or hundreds and tens, etc.) which are referred to as pre-grouped base-ten materials.
- Higher decade addition. Typically used to refer to additions of a 2-digit number and a 1-digit number, for example 72 + 5, 47 + 6.
- Higher decade subtraction. Typically used to refer to subtraction involving a 2-digit number and a 1-digit number, for example 37 – 4, 52 – 7.
- Hurdle number. A number where students commonly have difficulty continuing a number sequence. For example, students may say ‘106, 107, 108, 109, 200’: there is a hurdle at 110.
- Incrementing. Increasing a number, typically by one or more ones, tens, hundreds or some combination of these. Similarly, decreasing a number in this way is called decrementing.
- Inquiry mode. A mode of working where students typically are investigating mathematical topics that are new to them and trying to solve tasks that are genuine problems for them. Contrasted with rehearsal mode which is a mode of working that involves repeating something with which the student is acquainted, with the intention of increasing familiarity and ease, and perhaps working towards automatization. [Page 251]
- Inquiry-based teaching. An approach to teaching that emphasizes the inquiry mode. Thus tasks are designed to be at the cutting-edge of students’ current levels of knowledge.
- Inverse relationship. Commencing with a number N, if another number, for example 6, is added to N and then subtracted from the sum obtained, then the result will be N. Thus addition and subtraction have an inverse relationship – each is the inverse of the other. Similarly, multiplication and division have an inverse relationship.
- Jump strategy. A category of mental strategies for 2-digit addition and subtraction. Strategies in this category involve starting from one number and incrementing or decrementing that number by first tens and then ones (or first ones then tens). Is also used with 3-digit numbers.
- Jump to the decuple. A variation of the jump strategy where the first step is to add up to the next decuple, for example, 37 + 25 as 37 + 3, 40 + 20, 60 + 2. Similarly for subtraction, for example, 73 – 35 as 73 – 3, 70 – 30, 40 – 2.
- Linking cubes. Instructional materials consisting of interlocking plastic cubes.
- Mathematization. See Progressive mathematization.
- Mental computation. Typically refers to doing whole number arithmetic with multi-digit numbers, and without any writing. Contrasted with written computation that is computation that involves writing.
- Minuend. The number from which another number is subtracted, for example in 12 – 3 = 9, 12 is the minuend, 3 is the subtrahend, that is the number subtracted, and 9 is the difference, that is the number obtained.
- Missing addend task. An arithmetical task where one addend and the sum are given, for example, 9 + ? = 13.
- Missing subtrahend task. An arithmetical task where the minuend and the difference are given, for example, 11 – ? = 8.
- Multi-digit Involving numbers with two or more digits.
- Multi-lid screen. See Appendix – Numeral roll and multi-lid screen.
- Multiple. See Factor.
- Multiplicand. The number multiplied, for example in 12 × 8 = 96 (interpreted as 12 multiplied by 8), 12 is the multiplicand, 8 is the multiplier and 96 is the product.
- Multiplier. See Multiplicand.
- Non-canonical. The number 64 can be expressed in the form of 50 + 14. This form is referred to as a non-canonical (non-standard) form of 64, whereas 60 + 4 is the canonical form of 64. Knowledge of non-canonical forms is useful in addition, subtraction, and so on.
- Non-counting strategy. See Counting by ones.
- Non-regrouping task. See Regrouping task. [Page 252]
- Notating. Purposeful writing in an arithmetical situation, for example, notating a jump strategy on an empty number line.
- Number. A number is the idea or concept associated with, for example, how many items in a collection. We distinguish among the number 24 – that is, the concept – the spoken or heard number word ‘twenty-four’, the numeral ‘24’ and also the read or written word ‘twenty-four’. These distinctions are important in understanding students’ numerical strategies.
- Number word. Number words are names or words for numbers. In most cases the term ‘number word’ refers to the spoken and heard names for numbers rather than the read or written names.
- Number word sequence (NWS). A regular sequence of number words, typically but not necessarily by ones, for example the NWS from 97 to 112, the NWS from 82 back to 75, the NWS by tens from 24, the NWS by threes to 30.
- Numeral. Numerals are symbols for numbers, for example, ‘5’, ‘27’, ‘307’.
- Numeral identification. Stating the name of a displayed numeral. The term is used similarly to the term ‘letter identification’ in early literacy. When assessing numeral identification, numerals are not displayed in numerical sequence.
- Numeral cards. See Appendix.
- Numeral roll. See Appendix.
- Numeral track. See Appendix.
- Over-jump strategy. A variation of a jump strategy that involves going beyond a given number and then adjusting, for example 53 – 19 as 53 – 20 and then 33 + 1.
- Pair-wise pattern. See Five-wise pattern.
- Part–whole construction of number. The ability to conceive simultaneously of a whole and two parts. For example conceiving of 10 and also of the parts 6 and 4. This is characteristic of students who have progressed beyond a reliance on counting-by-ones to add and subtract.
- Partitioned fraction. A fraction regarded as being in a context or setting, for example, half an apple, two-fifths of a rectangle. This is contrasted with quantity fraction, that is, a fraction regarded as a number (rational number) in abstract mathematics.
- Partitioning. An arithmetical strategy involving partitioning a number into two parts without counting, for example, when solving 8 + 5, 5 is partitioned into 2 and 3.
- Partitions of a number. The ways a number can be expressed as a sum of two numbers, for example, the partitions of 6 are 1 and 5, 2 and 4, 3 and 3, 4 and 2, and 5 and 1.
- Partitive division. A division equation such as 15 ÷ 3 is interpreted as distributing 15 items into three groups, that is, three partitions. This is contrasted with Quotitive division where 15 ÷ 3 is interpreted as distributing 15 items into groups of three, that is, groups with a quota of three.
- Pedagogical engineering. The pedagogical process of designing, trialling and refining instructional procedures. [Page 253]
- Period. A part of a numeral consisting of three decimal places. In the numeral 46,275,406, for example, 275 is the thousands period.
- Pre-grouped materials. See Groupable base-ten materials.
- Procedure. See Strategy.
- Product. See Multiplicand.
- Progressive mathematization. The development over time, of the mathematical sophistication of students’ knowledge and reasoning, with respect to a specific topic, for example, addition.
- Quantity fraction. See Partitioned fraction.
- Quotient. See Dividend.
- Quotitive division. See Partitive division.
- Regional model. A shape with shaded regions drawn to illustrate a fraction, for example, shading a circle to illustrate one-sixth.
- Regrouping task. In the case of addition of two 2-digit numbers, a task where the sum of the numbers in the ones column exceeds 9, for example, in 37 + 48, 7 + 8 exceeds 9. In the case of subtraction involving two 2-digit numbers, a task where, in the ones column, the subtrahend exceeds the minuend, for example in 95 – 48, 8 exceeds 5. This is similarly applied to addition and subtraction with numbers with three or more digits.
- Rehearsal mode. See Inquiry mode.
- Remainder. See Dividend.
- Scaffolding. Actions on the part of the teacher to provide support for students to reason about or solve a task beyond what they could manage on their own.
- Screening. A technique used in the presentation of instructional tasks which involves placing a small screen or cover over all or part of an instructional setting (for example, screening a collection of 6 bundles of sticks).
- Semi-formal written strategy. A well-organized, standardized, written strategy – less formal than a formal written algorithm – for performing an arithmetical calculation.
- Setting. Materials used as a standard context for posing arithmetical tasks, for example, Two-colour ten-frames, Numeral roll and multi-lid frame, Bundling sticks, Arrow cards. See Appendix for more examples.
- Split strategy. A category of mental strategies for 2-digit addition and subtraction. Strategies in this category involve splitting the numbers into tens and ones and working separately with the tens and ones before recombining them. Split strategies can also be used with 3-digit and larger numbers.
- Split-jump strategy. A hybrid strategy, for example, 47 + 25 as 40 + 20, 60 + 7, 67 + 5.
- Strategy. A generic label for a method by which a student solves an arithmetical task, for example, an add through ten strategy: 8 + 6 as 8 + 2, and 10 + 4. The term procedure is used with similar meaning. [Page 254]
- Structuring numbers. Coming to know numbers through organizing numbers in terms of networks of relationships, and applying that knowledge to computation. For example, thinking of 16 as 10 + 5 + 1 or as double 8.
- Subtrahend. See Minuend.
- Sum. See Addend.
- Symbolization. A process of symbolizing in the sense of developing and using symbols in a context of arithmetical reasoning.
- Task. A generic label for problems or questions presented to a student.
- Ten-frame. See Appendix.
- Ten-wise pattern. See Five-wise pattern.
- Transforming strategy. An arithmetical strategy that involves simultaneously changing two numbers to make an easier calculation, for example, 17 + 38 is transformed to 15 + 40, 83 – 17 is transformed to 80 – 14.
- Unit. A thing that is countable and therefore is regarded as a single item. For example, when one counts how many 3s in 18: one 3, two 3s, three 3s … six 3s; the 3s are regarded as units.
- Unitizing. A conceptual procedure that involves regarding a number larger than one, as a unit, for example, three is regarded as a unit of three rather than three ones, or 10 is regarded as a unit of 10. Unitizing enables students to focus on the unitary rather than the composite aspect of the number.
- Written computation. See Mental computation.
Appendix: Instructional Settings[Page 255]
Arithmetic rack. An instructional device consisting of two rows of ten beads which can be moved like beads on an abacus. In each row the beads appear in two groups of five, demarcated by colour. The rack is used to support students’ additive reasoning in the range 1 to 20. See Figure 4.2. Array. A rectangular grid of dots used as a setting for multiplication, for example, a 6 × 4 array has six rows and four columns. See Figure 7.12. Arrow cards. A set of 36 cards with a card for each of the following numerals: 1, 2, … 9; 10, 20, … 90 100, 200, … 900 1000, 2000, … 9000. The cards are used to build multi-digit numerals, and each card has an arrow on the right hand side to support students’ orienting and locating the cards. See Figure 3.2. Base-ten dot materials. Materials consisting of strips with 1 to 9 dots, strips with 10 dots and squares with 100 dots. Dots are grey or black in order to demarcate a 5 in the 6- to 9-dot strips, two 5s in the 10-dot strip and two 50s in the 100-dot square. Bundling sticks. Wooden sticks used to show 10s and 1s. Rubber bands are used to make bundles of 10 and groups of ten 10s. See Figure 5.2 [Page 256] Digit cards. A set of cards used to build numerals. Each card displays a digit (i.e. 0, 1, 2 … 9). A set includes several cards for each digit, in order to account for numerals with repeating digits (e.g. 464, 3333). Dot tiles. An instructional setting for multiplication consisting of tiles with a given number of dots. Hence 2-tile, 3-tile and so on. See Figure 7.6. Empty number line (ENL). A setting consisting of a simple arc or line which is used by students and teachers to record and explain mental strategies for adding, subtracting, multiplying and dividing. See Figure 5.6. Numeral cards. A set of cards with each card displaying a numeral. Numeral roll. An instructional setting consisting of a long strip of paper containing a relatively long sequence of numerals, increasing from left to right, for example, from 1 to 120 or 80 to 220. Numeral roll and window. A numeral roll threaded through a slotted card so that one numeral only is displayed. Numeral roll and multi-lid screen. A numeral roll and a screen with 10 or more lids enabling screening and unscreening of individual numerals. See Figure 3.1. Numeral track. An instructional setting consisting of a strip of cardboard containing a sequence of numerals and for each numeral, a hinged lid which can be used to screen or display the numeral. Ten-frame. An instructional setting consisting of a card with a 2 × 5 rectangular array used to support students’ additive reasoning in the range 1 to 10. See Figure 4.1. Ten-frames – 10-dot cards. Ten-frames with 10 dots of one colour. See Figure 6.7. Ten-frames – combinations. A set of ten-frames with the different combinations of (a) 0 to 5 dots in the upper row, and (b) 0 to 5 dots in the lower row, typically with the rows in differing colours. A total of 36 cards. See Figure 4.1. [Page 257] Ten-frames – five-wise. A set of 11 ten-frames showing the five-wise patterns for 0, 1, … 10. See Figure 4.1. Ten-frames – pair-wise. A set of 11 ten-frames showing the pair-wise patterns for 0, 1, … 10. See Figure 4.1. Ten-frames – partitions. A set of 11 ten-frames showing the partitions of 10 demarcated by colour (i.e. 10 & 0, 9 & 1, 8 & 2, … 0 & 10). Typically, a five-wise set and a pair-wise set. See Figure 4.1.
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