Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page i Mathematical Misconceptions Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page ii Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page iii Mathematical Misconceptions A Guide for Primary Teachers Edited by Anne D. Cockburn and Graham Littler Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page iv ©AnneD.CockburnandGrahamLittler2008 Firstpublished2008 Apartfromanyfairdealingforthepurposesof researchorprivatestudy,orcriticismorreview,as permittedundertheCopyright,DesignsandPatentsAct, 1988,thispublicationmaybereproduced,storedor transmittedinanyform,orbyanymeans,onlywith thepriorpermissioninwritingofthepublishers, orinthecaseofreprographicreproduction,inaccordance withthetermsoflicencesissuedbytheCopyrightLicensing Agency.Enquiriesconcerningreproductionoutsidethose termsshouldbesenttothepublishers. SAGEPublicationsLtd 1Oliver’sYard 55CityRoad LondonEC1Y1SP SAGEPublicationsInc. 2455TellerRoad ThousandOaks,California91320 SAGEPublicationsIndiaPvtLtd B1/I1MohanCooperativeIndustrialArea MathuraRoad NewDelhi110044 SAGEPublicationsAsia-PacificPteLtd 33PekinStreet#02-01 FarEastSquare Singapore048763 LibraryofCongressControlNumber2008924327 BritishLibraryCataloguinginPublicationdata AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN978-1-84787-440-5 ISBN978-1-84787-441-2(pbk) TypesetbyC&MDigitals(P)Ltd,Chennai,India PrintedinIndiabyReplikaPress Printedonpaperfromsustainableresources Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page v Contents List of figures vi List of pictures ix List of tables x Introduction 1 0 Zero: understanding an apparently paradoxical number 7 Anne D. Cockburn and Paul Parslow-Williams 1 Equality: getting the right balance 23 Paul Parslow-Williams and Anne D. Cockburn 2 Beginning to unravel misconceptions 39 Sara Hershkovitz, Dina Tirosh and Pessia Tsamir 3 Insights into children’s intuitions of addition, subtraction, 54 multiplication and division Dina Tirosh, Pessia Tsamir and Sara Hershkovitz 4 Right or wrong? Exploring misconceptions in division 71 Pessia Tsamir, Sarah Hershkovitz and Dina Tirosh 5 Developing an understanding of children’s acquisition of number concepts 86 Anne D. Cockburn 6 Highlighting the learning processes 101 Graham Littler and Darina Jirotková 7 Everyday numbers under a mathematical magnifying glass 123 Carlo Marchini and Paola Vighi Appendix 152 Index 159 Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page vi List of figures Chapter 0 7 Figure 0.1 Penny’s ‘47’ 15 Figure 0.2 Using a number line as part of the ‘squeeze game’ 17 Chapter 1 23 Figure 1.1 Extract from RobertRecorde’s The Whetstone of Witte, 1557 (Cajori, 1928: 165) 24 Figure 1.2 A series of equality problems based on addition 25 Figure 1.3 A series of equality problems based on subtraction 26 Figure 1.4 Examples of pupils’ recording strategies for solving equality problems 28 Figure 1.5 Equality problems represented in a visual form 32 Figure 1.6 Visual representation of a ‘missing number’ equality problem using a suspension balance 34 Figure 1.7 Picture cards used to represent objects in the equality relationship 2 = 3 – 1 35 Figure 1.8 An example of a pictorial model that could be used as a basis for discussion of number sentence structures 35 Chapter 2 39 Figure 2.1 Arranging four digits to create a range of calculations 41 Figure 2.2 Arranging four digits to create a calculation involving addition 41 Figure 2.3 Addition tasks to explore children’s understanding of place value 43 Figure 2.4 Addition task to deepen the pupils’ understanding 43 Figure 2.5 New task developed by teachers 44 Figure 2.6 Mathematical reasoning for the addition task 44 Figure 2.7 Arranging four digits to create a calculation involving subtraction 44 Figure 2.8 Subtraction tasks to explore children’s understanding of place value 47 Figure 2.9 Mathematical reasoning for the subtraction task 47 Figure 2.10 Arranging four digits to create a calculation involving division 48 Figure 2.11 Arranging four digits to create a calculation involving multiplication 49 Figure 2.12 Mathematical reasoning for the multiplication task 51 Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page vii LISTOFFIGURES vii Figure 2.13 Division and multiplication tasks to explore children’s understanding of place value 52 Figure 2.14 Arranging four digits to create a calculation involving multiplication – the case of impossible solution 52 Chapter 3 54 Figure 3.1 Debby’s calculation task 55 Figure 3.2 Debby’s trueor false statements 56 Figure 3.3 A sample of Ben’s response 56 Figure 3.4 Children’s correct and incorrect responses to ‘addition makes bigger’ 57 Figure 3.5 Debby’s subtraction task 58 Figure 3.6 Correct and incorrect responses to ‘subtraction makes smaller’ 59 Figure 3.7 The multiplication task the teachers were set 60 Figure 3.8 Debby’s division task for teachers 61 Figure 3.9 Common intuitive beliefs about division 62 Figure 3.10 Debby’s estimation task 63 Figure 3.11 Dividing decimals: algorithmic, formal and intuitive mistakes 64 Figure 3.12 Teaching by analogy: Dan’s example 65 Figure 3.13 The three steps of the cognitive conflict teaching method 66 Chapter 4 71 Figure 4.1 Right or wrong? 76 Figure 4.2 A division task with one divisor 80 Figure 4.3 A division task with several divisors 80 Figure 4.4 Tammy’s first subtraction task 83 Figure 4.5 Tammy’s second subtraction task 83 Figure 4.6 Tammy’ s final task 84 Chapter 5 86 Figure 5.1 Two buildings with the same number of rooms, illustrating the commutativity of multiplication 90 Figure 5.2 A wood illustrating the commutativity of multiplication 90 Figure 5.3 A child’s system to assist in the counting out 100 one-penny coins 93 Figure 5.4 Exchanging ten ones for a ten rod 96 Figure 5.5 Making connections 97 Chapter 6 101 Figure 6.1 Cubes lying down 104 Figure 6.2 Cubes as a tower 105 Figure 6.3 Three piles of matchsticks 106 Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page viii viii MATHEMATICALMISCONCEPTIONS Figure 6.4 Matchsticks in pattern 106 Figure 6.5 Predicted strategies for completing task B 115 Chapter 7 123 Figure 7.1 A first landscape of natural numbers 125 Figure 7.2 A landscape ofNwith addition 127 Figure 7.3 A landscape ofNwith addition and subtraction 128 Figure 7.4 A landscape ofNwith operations (addition, multiplication), and subtraction 129 Figure 7.5 A landscape ofNwith operations (addition, multiplication), and algorithms (subtraction, division) 132 Figure 7.6 A landscape ofNwith operations (addition, multiplication), algorithms (subtraction, division), and relations (equality, order) 135 Figure 7.7 The newspaper game 138 Figure 7.8 Possible solutions of the game (disregarding accompanying text conditions) 138 Figure 7.9 The (newspaper) solution of the game 139 Figure 7.10 A diagram illustrating the building up of theNandz number systems 140 Figure 7.11 Nandz, their propertiesand the connections between them 140 Figure 7.12 Carlo’s game 141 Figure 7.13 The number systemsN,zandq, including the absolute rational numbers,Q 144 a Cockburn-Prelims:Cockburn-Prelims 9/10/2008 11:04 AM Page ix List of pictures Introduction 1 ‘I want 3 as I’m 3’ 4 The honey collecting task 4 The 3 little pigs 5 Patsy’s bus stop task 5 ‘You are only nothing’ 6 Chapter 0 7 0 in the bag 9 A fist of zero 10 18 number fan 18 400 and 30 and 9 18 439 19 Chapter 2 39 A primary mathematics lesson in Israel 39 Chapter 3 54 Debby’s group of teachers 55 ‘4 ÷0.5 = 8’ or 8 half-apples 68