Key understandings in mathematics learning

1 Paper 2: understanding whole numbersBy Terezinha Nunes and Peter Bryant, University of Oxford Key understandings inmathematics learningA review commissioned by the Nuffield Foundation22 SUMMARY PAPER 2: understanding whole numbers2 Paper 2: understanding whole numbersIn 2007, the Nuffield Foundation commissioned ateam from the University of Oxford to review theavailable research literature on how children learnmathematics. The resulting review is presented in aseries of eight papers:Paper 1: OverviewPaper 2: understanding extensive quantities andwhole numbersPaper 3: understanding rational numbers andintensive quantitiesPaper 4: understanding relations and their graphicalrepresentationPaper 5: understanding space and its representationin mathematicsPaper 6: Algebraic reasoningPaper 7: Modelling, problem-solving and integratingconceptsPaper 8.

2 Methodological appendixPapers 2 to 5 focus mainly on mathematics relevantto primary schools (pupils to age 11 years), whilepapers 6 and 7 consider aspects of mathematics in secondary 1 includes a summary of the review, which has been published separately as Introduction andsummary of of papers 1-7 have been publishedtogether as Summary publications are available to download from our website, of paper 23 understanding extensive quantities and whole numbers7 References 34 About the authorsTerezinha Nunes is Professor of EducationalStudies at the University of Oxford.

3 Peter Bryant is Senior Research Fellow in theDepartment of Education, University of the Nuffield FoundationThe Nuffield Foundation is an endowedcharitable trust established in 1943 by WilliamMorris (Lord Nuffield), the founder of MorrisMotors, with the aim of advancing social wellbeing. We fund research and practicalexperiment and the development of capacity to undertake them; working across education,science, social science and social policy. Whilemost of the Foundation s expenditure is onresponsive grant programmes we also undertake our own this review 3 Key understandings in mathematics learningHeadlines Whole numbers are used in primary school torepresent quantities and relations.

4 It is crucial forchildren s success in learning mathematics inprimary school to establish clear connectionsbetween numbers, quantities and relations. Using different schemes of action, such as settingobjects in correspondence, children can judgewhether two quantities are equivalent, and if they arenot, make judgements about their order of insights are used in understanding the numbersystem beyond simply producing a string of numberwords in a fixed order. It takes children some time tomake links between their understanding of quantitiesand their knowledge of number. Children start school with varying levels of ability in using different action schemes to solvearithmetic problems in the context of stories.

5 Theydo not need to know arithmetic facts to solvethese problems: they count in different waysdepending on whether the problems they aresolving involve the ideas of addition, subtraction,multiplication or division. Individual differences in the use of action schemesto solve problems predict children s progress inlearning mathematics in school. Interventions that help children learn to use theiraction schemes to solve problems lead to betterlearning of mathematics in school. It is considerably more difficult for children to usenumbers to represent relations than to representquantities.

6 understanding relations is crucial for theirfurther development in mathematics in children s everyday lives and before they startschool, they have experiences of manipulating andcomparing quantities. For example, even at age four,many children can share sweets fairly between tworecipients by using correspondences: they share givingone-for-you, one-for-me, until there are no sweets do sometimes make mistakes but they knowthat, when the sharing is done fairly, the two peoplewill have the same amount of sweets at the end. Evenyounger children know some things about quantities:they know that if you add sweets to a group ofsweets, there will be more sweets there, and if youtake some away, there will be fewer.

7 However, theymight not know that if you add a certain number andtake away the same number, there will be just as manysweets as there were the same time that young children are developingthese ideas about quantities, they are often learningto count. They learn to say the sequence of numberwords in the right order, they know that each objectthat they are counting must be counted once andonly once, and that it does not matter if you count arow of sweets from left to right or from right to left,you should get to the same are thus amazing learners ofmathematics. But they lack one thing which is cruciallyimportant: they do not at first make connectionsbetween their understanding of quantities and theirknowledge of numbers.

8 So if you ask a four-year-old, who just shared some sweetsfairly between two dolls, to count the sweets thatone doll has and then tell you, without counting, howmany sweets the other doll has, the majority (about60%) will tell you that they do not know. Knowingthat the dolls have the same quantity is not sufficientSummary of paper 2: understanding whole numbers4 SUMMARY PAPER 2: understanding whole numbersto know that if one has 8 sweets, the other one has 8 sweets also, has the same and numbers are not the same thing. Wecan use numbers as measures of quantities, but wecan think about quantities without actually having ameasure for them.

9 Until children can understand theconnections between numbers and quantities, theycannot use their knowledge of quantities to supporttheir understanding of numbers and vice the connections between quantities andnumbers are many and varied, learning about theseconnections could take three to four years inprimary important link that children must make betweennumber and quantity is the link between the order of number words in the counting sequence and themagnitude of the quantity represented. How dochildren come to understand that the any number in the counting sequence is equal to the precedingnumber plus 1?

10 Different explanations have been proposed in theliterature. One is that they simply see that magnitudeincreases as they count. But this explanation doesnot work well: our perception of magnitude isapproximate and knowing that any number is equalto its predecessor plus 1 is a very precise piece ofknowledge. A second explanation is that children use perception, language and inferences together toreach this understanding . Young children discriminatewell, for example, one puppet from two puppets andtwo puppets from three puppets. Because theyknow these differences precisely, they put these two pieces of information together, and learn thattwo is one more than one, and three is one morethan two.