The relative importance of two different mathmatical abilities to mathematical achievement

The relative importance of two different mathematical abilities to mathematical achievement

Terezinha Nunes∗ , Peter Bryant, Rossana Barros and Kathy Sylva

Department of Education, University of Oxford, UK

Background.
     Two distinct abilities, mathematical reasoning and arithmetic skill, might make separate and specific contributions to mathematical achievement. However, there is little evidence to inform theory and educational practice on this matter.

Aims.
     The aims of this study were (1) to assess whether mathematical reasoning and arithmetic make independent contributions to the longitudinal prediction of mathematical achievement over 5 years and (2) to test the specificity of this prediction.

Sample.
     Data from Avon Longitudinal Study of Parents and Children (ALSPAC) were available on 2,579 participants for analyses of KS2 achievement and on 1,680 for the analyses of KS3 achievement.

Method.
     Hierarchical regression analyses were used to assess the independence and specificity of the contribution of mathematical reasoning and arithmetic skill to the prediction of achievement in KS2 and KS3 mathematics, science, and English. Age, intelligence, and working memory (WM) were controls in these analyses.

Results.
     Mathematical reasoning and arithmetic did make independent contributions to the prediction of mathematical achievement; mathematical reasoning was by far the stronger predictor of the two. These predictions were specific in so far as these measures were more strongly related to mathematics than to science or English. Intelligence and WM were non-specific predictors; intelligence contributed more to the prediction of science than of maths, and WM predicted maths and English equally well.

Conclusions.
There is clear justification for making a distinction between mathemat-

ical reasoning and arithmetic skills. The implication is that schools must plan explicitly

to improve mathematical reasoning as well as arithmetic skills.

The relative importance of two different mathematical abilities to mathematical

achievement

The mathematical problems that we give to children at school make two kinds of demand

on them. The most obvious of the two is that children have to be able make the necessary

calculation correctly. The other demand, though less obvious, is also important. It is to

Figure 1. An example of two additive reasoning problems that exemplify how the relations between

the quantities determine which calculation should be carried out.

analyse the quantitative relations involved in the problem in order to work out how to

manipulate the numbers (e.g., what calculation they have to make or how to count).

For example, the children’s task in both the problems in Figure 1 is to work out the

distance between the boy and the girl when they both stop walking. Children certainly

have to be able to calculate that 6–2 = 4 to solve the first of the two problems and that

5 + 3 = 8 to solve the second. However, they also need to reason that, because the two

children travelled in the same direction in the first problem but in opposite directions

in the second, the final gap between them is the shorter distance walked by the girl

subtracted from the longer distance walked by the boy in the first problem, and in the

second problem, the gap is the distance walked by the girl added to that walked by

the boy. This additive reasoning involves an understanding of spatial relations and also

of additive composition (e.g., that a 6-km length consists of a 2-km length plus a 4-km

length).

The distinction between reasoning about the relevant mathematical relations and

doing the calculation applies in the same way to other kinds of arithmetical problems,

and the actual quantitative relations that the child has to reason about vary with

the type of problem. In many problems, the main decision that has to be made is

about the nature of the relations, whether these are additive or multiplicative relations

(Vergnaud 1982; 1983). The quantitative relations that the child must consider in additive

reasoning problems are part-whole relations, and in multiplicative reasoning, problems

are one-to-many correspondences (Becker 1993; Nunes, Bryant, Evans, & Bell 2010) and

proportional relations (Vergnaud 1983). Figure 2 presents two problems, one additive

and one multiplicative, in which there are no verbal expressions that could give a clue to

the type of relation, such as ‘got x from’, ‘gave y to’, ‘each has z’. In such problems, which

include a reference to a practical situation, the oral presentation does not contribute to

the choice of operations.

Children who succeed in these problems often do not carry out computations but

count, and what they count and how they count depends on the relations they establish

between the different quantities in the problems (for a discussion of the use of counting

Figure 2. An additive reasoning (left) and a multiplicative reasoning (right) problem that children often

solve by reasoning and counting rather than calculating.

in both problems, see Nunes & Bryant 1996). Previous research (e.g., Brown 1981;

Cramer, Post, & Currier 1993; De Bock, Van Dooren, Janssens, & Verschaffel 2002;

Van Dooren, Bock, Hessels, Janssens, & Verschaffel 2004; Verschaffel, Greer, & de

Corte 2007) shows that children sometimes use additive reasoning when multiplicative

reasoning is required and that the opposite is also true. Even after apparently correctly

deciding that a problem is multiplicative, there are still decisions to be made, which

involve whether to multiply or divide and, in the latter case, which number is the

dividend and which is the divisor. These are is not always easy decisions for children,

who sometimes consider the irrelevant feature number size when making these decisions

rather than the relation between the quantities (e.g., De Corte, Verschaffel, & Van Coillie

1988; Greer 1988).

These examples do not exhaust the types of reasoning that children need to carry

out in mathematics in school: they clearly also need to reason about space relations in

order to learn geometry, about relations between numbers within an operation (e.g.,

for the same dividend, the bigger the divisor, the smaller the quotient; the order of

the addends in an addition does not affect the total), and about relations between

operations (addition and subtraction are inverse operations; multiplication and division

are inverse operations). Much interesting research has been carried out about children’s

understanding of these relations (e.g., see Siegler & Stern 1998, for the inverse relation

between addition and subtraction; see Baroody & Gannon 1984, for commutativity).

Some of this research shows that children understand such relations first in the context

of quantities and only later as number relations (e.g., Bryant, Christie & Rendu 1999)

and that the type of situation affects whether the number relations are understood or

not (e.g., Nunes & Bryant 1995; Squire & Bryant 2003).

The distinction between calculating and reasoning raises an interesting question

about how well children learn mathematics at school. It is whether two quite separable

abilities may play a part in children’s mathematical learning: these are the ability to reason

about the underlying quantitative relations in arithmetical problems and the ability to

calculate. These two abilities may make separate and independent contributions to

individual children’s progress in mathematics at school. If this were so, it would be

interesting and important to know about the relative strengths of these two determinants

of schoolchildren’s success in mathematical learning.

There is little direct evidence, at the moment, either on the question of the two

separate abilities or on the relative importance of the two abilities, in children’s success in

mathematics at school. There is also little evidence on whether these abilities, which are

easy to distinguish conceptually, can also be distinguished empirically, using quantitative

methods. There are longitudinal studies of how well children’s number knowledge and

computational skills predict their success in learning about mathematics (De Smedt,

Verschaffel, & Ghesqui`re 2009; Durand, Hulme, Larkin, & Snowling 2005; Jordan,

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Levine & Huttenlocher 1994; Jordan & Montani 1997; Krajewski & Schneider 2009), and

there is one study on how children’s mathematical reasoning at kindergarten predicts

their mathematics achievement in the first year in school (van de Rijt, van Luit, &

Pennings 1999). The results of this research, on the whole, have been positive. Individual

children’s success in tests of arithmetic and number skills and mathematical reasoning

do predict how well they learn mathematics at school later on. However, these are

separate studies and, thus, cannot provide information on the independence and the

relative contribution of mathematical reasoning and of arithmetic to the prediction of

mathematics achievement. Stern (1999) carried out an interesting and comprehensive

longitudinal study of children’s mathematical abilities between the ages of 4 and 12

years, which did involve both calculation and reasoning problems. However, her report

of the results of this study did not include measures of the children’s achievement in

mathematics at school.

To our knowledge, only one longitudinal study has included measures of both arith-

metic and mathematical reasoning and analysed their predictive value for mathematics

achievement at school (Nunes et al., 2007). Its results were quite positive and indicated

that mathematical reasoning and arithmetic make independent contributions to the

prediction of mathematics achievement. The prediction was specific, in the sense that

it could not be explained by general intelligence or working memory (WM), because

these were controlled for in the regression analyses.

It is often, and very plausibly, argued that children’s general ability to handle

information has an important effect on how well they learn mathematics. The aspect of

information processing that has received the most attention as a possible determinant

of children’s mathematical learning is WM. When children use a procedure, such as

addition, to solve a problem, they need to keep in mind the information in the problem

and the steps they must take to implement the solution, while monitoring what they

have done and what still needs to be done. WM should affect how well they can

keep this information in mind and, thus, their success with the procedure. Different

researchers have shown a connection between WM and arithmetic computations (e.g.,

Adams & Hitch 1997; Bull & Johnston 1997; D’Amico & Guarnera 2005; De Smedt,

Janssen, Bouwens, Verschaffel, Boets, & Ghesqui`re 2009; Hitch & McAuley 1991;

e

McLean & Hitch 1999; Siegel & Linder 1984; Siegel & Ryan 1989; Towse & Hitch

1995). The connection between WM and mathematical achievement is also supported

by experimental studies that show that disrupting WM interferes with arithmetic

performance and by evidence that children who have mathematical difficulties tend

to produce low scores in WM tasks (Barrouillet & L´pine 2005; Passolunghi & Siegel

e

2004). Because it is quite plausible that WM not only predicts children’s arithmetic skills

but also explains why arithmetic skills and mathematical reasoning predict mathematics

achievement (Geary & Brown 1991; Swanson & Beebe-Frankenberger 2004), WM should

be used as a control in predictive studies whose aim is to show a specific connection of

mathematical reasoning and arithmetic with mathematical achievement.

General intelligence is an ability defined more broadly than WM because it includes

crystallized intelligence as well, which might affect how well children do in mathematics.

So, predictive studies should control for WM and general intelligence in order to test

whether predictors such as mathematical reasoning and arithmetic have a specific

connection with mathematical achievement.

It is most unlikely that general intelligence and WM affect children’s mathematical

learning only or even that they affect their progress in mathematics more than in non-

mathematical subjects, since it is hard to think of any intellectual activity that does not

involve intelligence and WM. This raises a second issue related to specificity. WM and

intelligence measures predict children’s success in mathematics but would probably

be just as strongly related to non-mathematical subjects as well, whereas, according to

our hypothesis, mathematical reasoning and arithmetic should be specific predictors of

mathematical achievement and not very good predictors of non-mathematical school

subjects such as English.

Thus, research on the relationship between children’s abilities and their success in a

particular subject, such as mathematics, should deal with the question of specificity in

two senses. Mathematical reasoning and arithmetic seem on the face of it to be abilities

that will specifically affect children’s progress in mathematics but not in other, non-

mathematical subjects, such as English. This, however, needs to be checked empirically,

which can be done in a longitudinal study by putting in control outcome measures. The

issue of specificity is relevant to the choice of predictive, as well as of outcome measures.

One simply has to see how well the abilities in question predict children’s success in

non-mathematical subjects as well as in mathematics. If the role that the two abilities

play in children’s progress in school is specific to mathematics, the children’s scores on

measures of these abilities should predict their mathematical success much better than

their success in English or in some other non-mathematical outcome measure. This kind

of design is all too rare in developmental and educational research. It has occasionally

been adopted in longitudinal studies of children’s reading (Bradley & Bryant 1983), but

we know of no study of children’s mathematics that has included such controls, though

in our view, they should be regarded as an important part of longitudinal research on

any aspect of children’s learning.

To summarize, our knowledge of the abilities that influence children’s mathematical

learning at school is fragmentary. Measures of children’s mathematical reasoning, of

their calculation skills and of their intelligence and WM are related to their mathematical

knowledge, but there is little research to show to what extent these represent

separate abilities or whether each of them makes independent predictions of children’s

mathematical progress or what their relative importance is. Nor do we know how specific

the links are between these predictive measures and mathematics. Yet, the answers to

these unsolved questions are important. They will affect what children are taught in

mathematics and how this teaching is organized. To put this in the form of concrete

questions, should teachers emphasize mathematical reasoning more than they do now?

Would it be better to concentrate on strengthening the children’s calculation skills? Is

there a case for trying to improve children’s WM, and would such an improvement

influence their progress in non-mathematical as well as in mathematical skills?

The present study

We shall describe the results of longitudinal research over a period of 5 years with a large

number of children that provides some answers to these questions. We employed four

predictive measures, which were of children’s quantitative reasoning, of their calculation

skills, of their general intelligence, and their WM. Working with a sample from the Avon

Longitudinal Study of Parents and Children (ALSPAC), we investigated the links between

these measures and the children’s performance in three school subjects, Mathematics,

Science, and English over the subsequent 5 years.

The large body of data in the project includes information about the children’s

educational progress, including their progress in mathematics, and their performance

in psychological tests such as the Wechsler Intelligence Scale for Children (WISC-III;

Wechsler 1992) and a Test of Mathematical Reasoning; the latter two assessments were

given to the children when they were in their fourth year in school.

The WISC includes (1) a sub-test, backward digit recall, which is a measure of WM

that Gersten, Jordan and Flojo (2005) identified as a reliable predictor of mathematical

difficulties and (2) a sub-test, Arithmetic, designed to assess numerical skills.

The Test of Mathematical Reasoning was designed by Nunes and Bryant (see Nunes,

Campos, Magina, & Bryant 2001), drawing on the work of van den Heuvel-Panhuizen

(1990). The items in these tests require very simple arithmetic computations but make

clear demands on relational reasoning.

The children’s mean age when they were given the WISC was 8 years 6 months, and

their mean age when they took the mathematical reasoning test was 8 years 9 months.

The project also contains information about the participants’ results in two stan-

dardized tests of mathematical achievement, designed by the UK government and

administered by teachers, referred to as Key Stage Assessments. One assessment, Key

Stage 2 (KS2), was given to the children when they were in sixth grade; their mean age

at the time was 11 years and 2 months. The second assessment, Key Stage 3 (KS3), was

given to the children when they were in ninth grade; their mean age at the time was

14 years and 1 month. Both KS tests measure a variety of aspects of mathematics and

are seen as ecologically valid measures of mathematical achievement because of the role

that they play in the British educational system.

Method

Participants

ALSPAC is a longitudinal study of children born in Avon in the West of England in

1991–92. Golding, Pembey, Jones, and the ALSPAC team (2001) described the variety of

methods used to engage the interest of pregnant women in participating, which were

wide reaching in the community and engaged the cooperation of health professionals

working with pregnant women. At the time of recruitment, the ALSPAC team compared

the sample with that described in the British national sample of the Child Health and

Education study and found the ALSPAC sample to be comparable to the national sample

in many ways, including rural versus urban living, ethnic background, and prevalence of

different health indicators (for details, see Golding et al. 2001). The characteristics of the

sample originally recruited for ALSPAC are reflected in the samples analysed here. Thus,

there was not much selective attrition in the sample in terms of ethnic background or

Relative importance of different mathematical abilities to mathematical achievement

7

maternal occupation at KS2 or KS3 (for details on the sample, see Nunes, Bryant, Sylva,

& Barros 2009). In the sample analysed for KS2 results, 48% of the children are boys and

52% are girls; in the sample analysed for KS3 results, 47% of the children are boys and

53% are girls.

The children were recruited for participation in the measures considered here in two

ways. For the individually administered measure (the WISC), they were recruited by a

letter sent to the mother inviting her to attend a clinic. If no response was obtained, a

second letter was sent. The ALSPAC data base contains information on the IQ of 7,354

children.

The mathematical reasoning assessment was administered to the children by the

teachers. Teachers were invited to participate if there was a child in the class who was

included in the ALSPAC sample; 5,234 children were given the mathematical reasoning

measure.

KS2 and KS3 mathematics are also administered by teachers but these are not optional;

they are part of the information required by the government from the schools for

monitoring school performance. The ALSPAC database contains presently information

on 12,472 for KS2 and 8,519 children for KS3 results. The decrease in sample size from

KS2 to KS3 results from the fact that, when the data were analysed, KS3 data were not

available for the children born later, in 1992.

In total, full information regarding all the measures used in this study was available

on 2,579 participants for the analyses of KS2 mathematical achievement and for 1,680

for the analyses of KS3 achievement.

Measures

Predictors

Mathematical reasoning. The mathematical reasoning task included three types of

items, additive reasoning about quantities, additive reasoning about relations, and

multiplicative reasoning items. Additive reasoning items involve part-whole relations

(Carpenter, Ansell, Franke, Fennema, & Weisbeck 1993; Carpenter & Moser 1982;

Vergnaud 1982); the parts can be static (i.e., two parts form a whole) or involve change

(i.e., a quantity is added to or subtracted from an original one). Within the domain of

part-whole relations, one can ask questions about quantities or about relations. The latter

can involve, for example, comparisons (e.g., how much more does x have than y?) or

distance (how far is x from y?), as illustrated by the items in Figure 1 (developed from

an item used by Brown 1981). Examples of additive reasoning items are presented in

Figures 1 and 2, left.

Multiplicative reasoning items involve proportional relations between two quantities

(Vergnaud 1983). Figures 2, right, and 3 illustrate multiplicative reasoning items used in

this study; the example on the left, Figure 3, was adapted from van den Heuvel-Panhuizen

(1990).

All items are presented orally with the support of pictures, which reduces memory

demands and affords the use of a variety of strategies in finding the numerical answers: for

example, counting, addition, or multiplication might be used to solve the item presented

in Figure 3, left. The children’s booklets, where they are asked to write their answers,

contain no text, only drawings; the story is read by the teacher to the class.

The assessment contains a total of 17 items and it is not timed; administration usually

takes approximately 25–30 min. The child’s score is the number of correct answers.

Figure 3. Two multiplicative reasoning problems in which the arithmetic is quite simple, once the

student knows how to think about the relations between quantities.

Cronbach’s alpha inter-item reliability for this assessment was 0.74; thus, the assessment

had a good level of internal consistency (Kline 1999).

Arithmetic. The WISC Arithmetic sub-test was used to assess children’s computational

ability. It is a standardized measure of arithmetic knowledge, in which the questions

are presented as word problems that place little demand on relational reasoning.

The arithmetic required to solve the problems becomes progressively more difficult,

although, the relational reasoning demanded of the child does not increase. For example,

item 8 is: ’Joseph has 5 cakes. He gives 1 to Sam and 1 to Alice. How many does he have

left?’ and item 18 is: ’A shop had 25 cartons of milk and sold 14 of them. How many

cartons were left?’ When small numbers are used, pre-school children show high rates of

success in such problems (Becker 1993; Carpenter, Hiebert, & Moser, 1981; Carpenter &

Moser 1982; Carpenter et al. 1993); a result interpreted in the literature as demonstrating

that these problems require little relational reasoning (Vergnaud 1979; 1982). The child

is required to solve all the problems without the use of paper and pencil and the test is

interrupted after three consecutive failures. The child’s score is the number of correctly

answered questions. The split-half reliability for 8-year-olds is 0.78 (Wechsler 1992, p.

60); and the average correlation with the Wide Range Achievement Test Arithmetic

Score is .62 (Wechsler 1992, p. 76), which makes this a valid and economical assessment

of children’s arithmetic knowledge, thus, suited for large-scale studies such as this.

Controls

Working memory (WM). The most common measure of WM used in longitudinal

predictive research of mathematical achievement and difficulties is the Backward Digit

Recall, which is one of the sub-tests in the WISC-III (Wechsler 1992) and also a sub-test

in the Working Memory Battery for Children (Pickering & Gathercole 2001). In this

sub-test, children hear a series of digits and are asked to repeat them in the reverse

order. The number of digits to be recalled is increased by one over the trials until the

children can no longer recall the digits in the correct order. The child’s score is the

highest number of digits correctly recalled in the reverse order in four (of six) trials

of the same span. According to Pickering and Gathercole (2001), backward digit recall

has a high loading on the central executive factor of WM, which is a strong predictor

of mathematical achievement (Gathercole & Pickering 2000). The split-half reliability of

the WISC-III Digit Recall for age 8 is 0.84 (Wechsler 1992, p. 60).

As argued earlier on, it is important to control for WM to assess whether the relation

of mathematical reasoning and arithmetic to mathematical achievement is specific. This

measure will be used in regression analyses, each with one of the outcome measures,

where this subtest will be the only measure of information processing used as a

control.General intelligence. It was expected that children’s performance in a measure

of general intelligence correlates significantly with all three of the above predictors and

also with the children’s mathematical achievement. It was, therefore, desirable to include

a measure of general intelligence as a control in the regression analyses. The measure

used in this study was the WISC-III (Wechsler 1992). For the regression analyses, the

general IQ was estimated on the basis of 10 sub-tests; the sub-tests Arithmetic and Digit

Span (forward and backward recall) were excluded from the estimation, as these were

entered separately in the analyses.

Outcome measures

The outcome measures of mathematical achievement were standardized assessments

designed by the British government to measure school standards. They are administered

and scored by the teachers, and often used to make decisions about students’ placement

in mathematics attainment streams. Therefore, these are not only ecologically valid

measures but also high-stake tests. The tests are redesigned each year; the participants

in this study took the tests in different years as they are from different birth cohorts; the

majority took KS2 tests in 2003 and KS3 tests in 2006. The descriptions presented here

are taken from KS2 in 2003 and KS3 in 2006.

KS2 had three papers (see QCA 2003: http://www.emaths.co.uk/KS2SAT.htm);

students were allowed to use a calculator only in one of these. All papers were timed;

the mental arithmetic paper was timed by question and the other two were timed

as a whole. The papers assess a variety of aspects of mathematical knowledge that

children are taught about by the time they reach their sixth year in primary school:

for example, knowledge of decimals, arithmetic (calculation and problem solving),

geometric reasoning, measurement of space and time, identification of number patterns

of sequences of figures, graph reading (line and bar graphs).

The mental arithmetic paper includes mostly questions with no references to

quantities but simply to numbers (e.g., ‘divide ninety by three’; ‘subtract one point

nine from two point seven’; ‘when h has the value twelve, calculate five h minus two’),

but there are also questions that asses knowledge of scales of measurement and involve

calculation (e.g., ‘how many grams are in 12 kilograms’; ‘how much must I add to

four point ninety to make six pounds’) and questions related to geometry (e.g., ‘look

at the figures on the paper; put a ring around the figure which has only one line of

symmetry’ and ‘look at the clock; what angle is made by the hands of the clock when at

four o’clock’; ‘calculate the perimeter of a rectangle which is eleven meters long and four

meters wide’.) Only two of the 20 questions in this paper made reference to a practical

situation (e.g., ‘A yogurt costs forty-five pence. How many yogurts can be bought for five

pounds?’). This paper is more similar to the WISC arithmetic than to the mathematical

reasoning assessment.

The second paper to be answered without a calculator included 26 items, distributed

in different categories such as calculation either as a direct command (e.g., ’calculate

309 – 198’; ‘calculate 2307 × 8’) or in numerical expressions with one missing number

represented by a box (e.g., ‘600 × 4 = box’; ‘50 ÷ box = 2.5’). Item also referred to

measurement and money (e.g., ‘how many coins of only 1 p, only 10 p or only 20 p do you

need to have £1.60’), reading tables and graphs, rounding numbers, identifying patterns

in series of numbers and figures, naming geometrical figures, calculating percentages,

and perimeter. Six items referred to practical situations; five of these do not require

much relational reasoning (e.g., ‘Tom and Nadia have 16 cards each, Tom gives Nadia 12

of his cards, how many cards do Tom and Nadia each have now?’) and one does require

relational reasoning because it is about unequal division (‘30 children are going on a trip.

It costs £5 including lunch. Some children take their own packed lunch and pay only £3.

The 30 children pay a total of £110. How many children are taking their packed lunch?’)

The paper in which the children are allowed to use a calculator contains similar types

of questions but with larger numbers and fractions. There are seven (of 24) items that use

numbers without a reference to quantities; in general, the calculations are more difficult

in this paper than in the preceding one. For example, the missing number problems with

a box include questions such as ‘37 × box = 111’; ‘225 – box = 115’ and ‘box × box =

378’. Calculations also involve larger numbers such as ‘what is 3/8 of 980’. Number

relations are explored in questions such as ‘here are five digit cards [the digits are from

1 to 5]; fill in the boxes to make this sum correct [the children have to fill in three

addends, one with a single digit and two with two digits]; the result is 60’ and ‘Karen

makes a fraction using two number cards. She says, my fraction is equivalent to1/2. One

of the number cards is 6. What could Karen’s fraction be? Give both possible answers’; ‘ k

+ m + n = 1500; m is three times as big as n; k is twice as big as n; calculate the numbers

m, k and n’. Although these problems do not involve reference to practical situations,

they involve thinking about relations between numbers. Problems that refer to practical

situations used in this paper sometimes require the students to obtain information from

graphs or tables and sometimes present the information in words; seven of 24 items can

be described as word problem in this sense. An example of a simple word problem is

‘there are 5 balloons in a packet. There are 18 packets in a box. How many balloons

are there altogether in a box?’ An example of a more difficult word problem is ‘250 000

people visited a theme park in one year. 15% of the people visited the park in April and

40% visited the park in August. How many people visited the park in the rest of the

year?’). The remaining 10 items are about geometry (e.g., ‘draw two straight lines from

point A to divide the shaded shape into a square and a triangle’; ‘which of the diagrams

below shows is a reflection of the mirror line for this figure’), measurement [e.g., ‘Here

is a clock (a digital display shows 14:53). What time will the clock show in 20 minutes’;

‘write these lengths in order, starting with the shortest: 1/2 m; 3.5 cm; 25 mm; 20 cm’]

and probability (a square pinner is divided into unequal sections; the students are asked

to verify which statements about the probability of the spinner stopping at one number

are correct).

This detailed description shows that there is a wide range of items in the papers.

Although the mental calculation paper is mostly about numbers without reference to

quantities, and could be seen as giving greater weight to arithmetic than reasoning in

the KS tests, calculation and reasoning are for success in the three papers.

KS3 tests are administered 3 years after the KS2 tests. Similar to KS2, there are three

papers in KS3 tests, one of which is mental arithmetic, and calculators are allowed in only

one of the papers. KS3 papers are designed for four different levels of difficulty to avoid

giving the most difficult papers to students who would find them too frustrating. Thus,

there are 12 different papers for KS3; a detailed description of these tests is beyond the

scope of this paper. Suffice it to say here that they include questions designed to assess

the same topics included in KS2 at a higher level of difficulty. Three new topics appear,

proof, probability, and algebra; questions about fractions, calculations with decimals,

and missing numbers in expressions involve larger values. Some calculation questions

in the paper in which the students are not allowed to use a calculator explore the

students’ understanding of properties of operations. For example, students are asked in

one question: ‘part a: show that 9 × 28 is 252; part b: What is 27 × 28? You can use

part a to help you’.

The KS tests provide two types of score, one in attainment level, which varies between

1 and 9, and a points measure, which is a finer numerical scale. Our analyses showed

no difference in the pattern of results produced by the two scales. We report here the

analyses carried out with the finer score. The two scales do not vary in range across the

years in which the tests were given.

The children in the sample included two cohorts, as they were born on different

years, and thus, they took different tests. In each case, we ran the overall analyses with

both cohorts and also separately by cohort. The results replicated the patterns across

cohorts; so, we report here only the overall results, in which both samples are combined.

Results

Correlations

The correlations in Table 1 provide some preliminary information about the relationships

between predictors and outcome measures. The control measures, WM, and general

intelligence, and the two predictor variables, arithmetic and mathematical reasoning,

were correlated with each other as well as with each of the outcome measures.

The correlations between mathematical reasoning and the mathematics tests at

11 (.66) and 14 years (.68) were stronger than those between arithmetic and each

of these outcome measures. It is noteworthy that mathematical reasoning and general

intelligence show almost the same correlations with each of these two measures of

mathematics achievement. WM shows weaker correlations with the outcome measures

of mathematics than mathematical reasoning. Thus, the children’s ability to reason about

quantitative relations was a particularly good predictor of their progress in mathematics

over the next 5 years.

The children’s arithmetic scores also predicted their performance in the national tests

of mathematics well (0.57 with the 11-year and 0.58 with the 14-year national tests of

mathematics), though not as strongly as their mathematics reasoning scores had done.

The children’s general intelligence was more strongly related with their mathematics

achievement than arithmetic, but arithmetic scores had higher correlations with the

outcome measures than WM.

Table 1 also provides some preliminary evidence on the question of specificity. We

had argued that the children’s scores in the mathematical reasoning and arithmetic tasks

would predict their progress in mathematics more strongly than in science or English.

Table 1. Correlations between each of the predictors (mathematical reasoning and arithmetic), control

measures (IQ estimate and backward digit span) and outcome measures (KS2 and 3 Mathematics)

IQ estimate

without WM

and arithmetic

1

.29∗∗

.51∗∗

.50∗∗

.63∗∗

.68∗∗

.65∗∗

.69∗∗

.60∗∗

.57∗∗

Backwards

digit span

1

.28∗∗

.32∗∗

.33∗∗

.34∗∗

.26∗∗

.28∗∗

.31∗∗

.29∗∗

Mathematical

reasoning

Predictors

IQ estimate without WM and Arithmetic

Backwards digit span

Mathematical reasoning

Arithmetic

KS2 Math

KS3 Math

KS2 Science

KS3 Science

KS2 English

KS3 English

∗∗

∗

Arithmetic

1

.49∗∗

.66∗∗

.68∗∗

.55∗∗

.58∗∗

.48∗∗

.50∗∗

1

.57∗∗

.58∗∗

.45∗∗

.49∗∗

.44∗∗

.42∗∗

Correlation is significant at the .01 level (two-tailed).

Correlation is significant at the .05 level (two-tailed).

N = 2,413 for KS2 outcome analyses; N = 1,588 for KS3 outcome analyses

We expected that the two measures of general processing ability, WM and general

intelligence, would show similar correlations with mathematics, science, and English.

This hypothesis receives some support from the correlational analysis. Mathematical

reasoning and arithmetic both show stronger correlations with the mathematics tests

than with the science and English tests; the differences between the correlations

between the predictors with mathematics and with science or English are significant

statistically both at KS2 and KS3. This result is not surprising because the sample

size was rather large in both analyses, and N-3 is one of the terms in the numerator

of the formula for calculating the t value when comparing correlations in correlated

samples (Ferguson 1971). Therefore, this information becomes more important when

one considers whether the correlations between the cognitive measures of intelligence

and WM with KS2 and KS3 mathematics also differed significantly from those observed

for English and science KS tests.

First, one should note that the correlation between general intelligence and achieve-

ment in science is actually higher both at KS2 and KS3 than the correlation between

general intelligence and mathematical achievement. Although the difference in the

coefficients is small, one must conclude that general intelligence predicts science

achievement at least as well as it predicts mathematical achievement. It is noteworthy that

the opposite is true when it comes to English achievement: general intelligence predicts

mathematics (and science) better, and significantly so, than English achievement. At

KS2, the difference between the correlations of general intelligence with mathematical

achievement (r = .63) and with English achievement (r = .60), although small, still

reaches significance at .01 level (t = 3.37; p < .01).

In contrast, the children’s WM scores predicted the students’ performance in

mathematics and English better than it predicted their science achievement at both

KS tests. The correlation between WM and mathematical achievement did not differ

significantly from the correlation between WM and English achievement at KS2 (t =

1.17; ns), but this difference was significant at KS3 (t = 3.933; p < .01).

This close look at the correlations, therefore, suggests that both general intelligence

and WM cannot be seen as specific predictors of mathematics, as the first predicts science

achievement better than mathematics, whereas, the second predicted KS2 mathematics

and English achievement equally well, in spite of the effect of the very large sample size

on the statistical comparison between correlations.

Finally, we turn to the relations among the predictors and the controls. The correlation

between mathematical reasoning and arithmetic was quite high (r = .49) but far from

perfect. The reason for the strength of this correlation may be that the children could

use an arithmetical calculation in each of the mathematical reasoning items (although, as

pointed out, they could also solve many problems by reasoning and counting). Although

we did our best to make sure that the calculations did not tax the arithmetical skills,

they may still have done so for some of the children in the project. This would have

led to the correlation between mathematical reasoning and the arithmetic task, which

only measures ability to calculate correctly. We, therefore, had to control for the link

between the two predictors in any further examination of the relationship between

mathematical reasoning, and we shall report how we did this in multiple regressions in

the next section.

The correlations between these two tasks and the controls, WM and general intelli-

gence, were positive and significant; they were higher for general intelligence and lower

for WM. The existence of these correlations also emphasized the importance for us, when

considering the relationship between each of the predictors and the various outcome

measures, to control for the impact of the other two predictors. This is an important test

of the specificity of the relationship between the predictors, mathematical reasoning

and arithmetic, and the outcome measures, KS2 and KS3 mathematics attainment.

Multiple regressions: Prediction of mathematical achievement

The next question was whether the two predictors – mathematical reasoning and arith-

metic – made independent contributions to the prediction of mathematics achievement.

In order to test whether these contributions are specific to the measures, rather than

explained by more general abilities, we will in these analyses control for WM and general

intelligence. We used four hierarchical, fixed-order multiple regressions to answer this

question. The outcome measure in two analyses was the children’s performance in

mathematics tests at 11 years (KS2, see Table 2), and in the other two analyses, it was

their performance in mathematics tests at 14 years (KS3, see Table 3). The difference

between each pair of analyses was the predictor that was entered as the last step in the

equation. Mathematical reasoning was the last step in one analysis and arithmetic in the

other. This allowed us to see if each of the two variables accounted for a significant

amount of variance in the outcome measure after the effect of the other predictor had

been controlled.

The first point to note about the two pairs of regressions is that they accounted for

a highly satisfactory amount of the variance in the children’s performance in the two

mathematics assessments. The multiple regressions in which the children’s mathematical

achievement at 11 years (KS2, Table 2) was the outcome measure accounted for 58% of

the variance in that measure. The analyses accounted for 62% of the total variance in the

children’s achievement in mathematics at 14 years (KS3; Table3), which is a very high

level for a longitudinal predictive analysis over a period of 5 years.

Both tables show that each of the two predictors accounted for a significant amount

of additional variance in KS mathematics 2 and 3, after controlling for all the other

independent variables in the analysis. Thus, each predictor made an independent

contribution to predicting children’s mathematical achievement over the next 5 years.

Table 2. Prediction of achievement in KS2 mathematics. Two multiple regressions in which the first

three variables entered were the controls: (1) age at key stage assessment; (2) IQ; and (3) WM. The

fourth and fifth steps are changed across analyses to test whether the main predictors make independent

contributions to the prediction of KS2 attainment. The B and ␤ coefficients are those for the regression

when all the predictors have been entered (N = 2,413)

Step in regression

All regressions first step

Second step

Age at outcome

WISC IQ estimate

without arithmetic and

WM

WISC WM

WISC arithmetic

Maths reasoning task

Maths reasoning task

WISC arithmetic

R2 change

.033∗∗

.369∗∗

.031∗∗

.073∗∗

.076∗∗

.119∗∗

.030∗∗

␤ coefficient

.129∗∗

.316∗∗

.086∗∗

.211∗∗

.344∗∗

B

0.658

0.389

Standard

error of B

0.070

0.020

Third step

First regression

Fourth step

Fifth step

Second regression

Foutth step

Fifth step

∗∗

2.027

1.202

2.272

0.335

0.092

0.109

Significant at p Ͻ .001

This prediction is specific in the sense that it cannot be explained by general factors,

such as age, intelligence, and WM.

The tables also show that the ␤ value for mathematical reasoning was higher

than for all the other measures, including general intelligence, in the prediction of

children’s mathematical attainment at 11 years. The ␤ co-efficient was higher for general

intelligence than for mathematical reasoning in predicting the children’s mathematical

achievement at 14 years but was still far greater for mathematical reasoning than for

Table 3. Prediction of achievement in KS3 mathematics. Two multiple regressions in which the first

three variables entered were the controls: (1) age at key stage assessment; (2) IQ; and (3) WM. The

fourth and fifth steps are changed across analyses to test whether the main predictors make independent

contributions to the prediction of KS3 attainment. The B and ␤ coefficients are those for the regression

when all the predictors have been entered (N = 1,595)

Step in regression

All regressions first step

Second step

Age at outcome

WISC IQ estimate

without arithmetic and

WM

WISC WM

WISC arithmetic

Math reasoning task

Math reasoning task

WISC arithmetic

R2 change

.011∗∗

.463∗∗

.019∗∗

.059∗∗

.075∗∗

.111∗∗

.023∗∗

␤ coefficient

.085∗∗

.404∗∗

.063∗∗

.184∗∗

.340∗∗

B

0.028

0.030

Standard

error of B

0.005

0.001

Third step

First regression

Fourth step

Fifth step

Second regression

Fourth step

Fifth step

∗∗

0.092

0.065

0.138

0.024

0.007

0.008

Significant at p Ͻ .001

the other variables. There is a remarkable consistency in the results of the two analyses:

the order of importance of the two predictors is the same – mathematical reasoning is

a stronger predictor than arithmetic – and the ␤ values are also quite similar. This is

an impressive replication, considering that the two outcome measures of mathematics

achievement are different and were given to the children 3 years apart.

Multiple regressions: Specificity of prediction

The children were also given national assessments of science and of English at the same

time as they took the mathematics assessments. This allowed us to investigate how

specific to mathematics was the pattern of relationships that we have just described.

The children’s mathematical reasoning scores, and to a lesser extent their arithmetic

scores, predicted their mathematical achievement over the following 5 years relatively

well. Was this relative success in the predictive power of these two variables specific

to their mathematical achievement, or did they predict other outcome measures just

as well? We expected the first of these two alternative answers would be the right

one, because our hypothesis was that the relative success of these two predictors was

specifically due to mathematical factors. For example, we claimed that the outstanding

success of the mathematical reasoning task as a predictor was due to its success in testing

children’s ability to reason about quantities.

The second question was whether the children’s WM and intelligence scores would

predict their achievement in non-mathematical subjects as well as in mathematics. Our

hypothesis was that variation in children’s WM and intelligence would play as large a

part in the children’s achievement in subjects such as English as in their mathematics

because of the general information processing skills measured by these tests.

Our aim was to compare the strength of the relations among each of the two

predictors and each of the controls and each of these three kinds of outcome measures.

We had already done the appropriate analyses with achievement in mathematics as

the outcome measure (see Tables 2 and 3). We now carried out additional multiple

regressions with exactly the same five predictor variables as in the regressions that we

described in Tables 2 and 3, but with different outcome measures. There were four new

outcome measures: the national test scores in science at 11 and at 14 years and the

English scores at 11 and 14 years.

Table 4 presents the ␤ values for the relationship between the three main predictors

and the three kinds of outcome measure (achievement in mathematics, science and

English at 11 and 14 years). The figures in this table support our hypothesis about

the predictive power of the arithmetic and also about WM and general intelligence.

We had predicted that the arithmetic scores would be much more strongly related to

achievement in the national tests in mathematics than in science and English and that

WM and general intelligence scores, in contrast, would predict achievement in non-

mathematical as well as in mathematical subjects. Table 4 shows that the children’s

arithmetic scores had higher ␤ coefficients in the regressions with mathematics as the

outcome measure than in regressions with science and English as the outcomes. The

table also shows that the WM and intelligence scores were as strongly related to the

children’s achievement in English as in mathematics and science both in the 11- and

in the 14-year national achievement tests. As we expected, there was no evidence that

these measures of general processing are more important in learning about mathematics

than about either of the other two subjects.

Table 4 did provide one surprise. We had expected that the mathematical reasoning

scores would be far more strongly related to achievement in mathematics than either

.

Table 4. The relations between the specific predictors and the controls and the children’s achievement

in mathematics, science, and English

Outcome measures: Achievement tests at 11 years

Mathematics

Percent of total variance explained by all five steps in the regression

␤ coefficients

Mathematical reasoning

Arithmetic

WM

General Intelligence

58.0

0.34

0.21

0.09

0.32

Science

47.7

0.19

0.11

0.04

0.49

English

40.8

0.14

0.15

0.12

0.41

Outcome measures: Achievement tests at 14 –years

Mathematics

Percent of total variance explained by all five steps in the regression

␤ coefficients

Mathematical reasoning

Arithmetic

WM

General Intelligence

62.7

0.34

0.18

0.06

0.40

Science

54.8

0.21

0.11

0.03

0.54

English

38.3

0.14

0.13

0.07

0.42

in science or in English, and the results showed that this was strikingly the case.

The children’s mathematical reasoning scores were very strongly related to their

achievement in mathematics, as we have already noted, while the relations between this

predictor and science and English were much lower. So, these figures do demonstrate

the specificity in the relationship between mathematical reasoning and children’s

mathematical achievement that we had expected.

However, the children’s mathematical reasoning scores also predicted their achieve-

ment in science a great deal better than their achievement in English, as well as predicting

science achievement much better than arithmetic and WM did. What is the reason for

this relatively strong relationship between the children’s mathematical reasoning and

their achievement in science? One possibility is that it is due to the strong mathematical

element that undoubtedly is a part of the scientific curriculum. Scientific exercises

involve a great deal of measurement and calculation, of course, and many scientific

concepts, such as density and temperature, are intensive quantities (Howe, Nunes, &

Bryant 2010; Howe, Nunes, Bryant, Bell, & Desli 2010; Nunes & Bryant 2008; Nunes,

Desli, & Bell 2003) and their measurement is based on ratios; children, therefore, have

to be able to reason about proportions to understand several aspects of science. None

of this is true of English lessons. Therefore, the reason for the quite high connection

between children’s mathematical reasoning and their eventual achievement in science

may exist for specifically mathematical reasons. So, the pattern of predictions that we

have just described actually supports the idea of a highly specific connection between

mathematical reasoning and children’s understanding and use of mathematics.

It is also interesting that the children’s scores in arithmetic did not predict their

achievement in science any better than in English. This suggests that, in learning about

science, it is more important for children to be able to reason about quantities than to

manage to do the actual calculations correctly.

Discussion

As far as we know, ours is the first large-scale longitudinal study to have measured

the contribution of mathematical reasoning separately from mathematical calculation,

to children’s school mathematical achievement. Therefore, the finding of a particularly

strong link between children’s reasoning and their mathematical achievement in school

is a result of considerable importance. This result makes a significant contribution to

the debate on how much emphasis teachers should give to mathematical reasoning

and to knowledge of arithmetic in the classroom. Time is a precious resource in the

classroom, and our results suggest that greater investment in developing students’ math-

ematical reasoning should produce a higher pay-off in terms of students’ mathematical

achievements.

Arithmetic made a smaller, but nevertheless significant and independent contribution,

to the children’s achievement in mathematics. WM and general intelligence also made

independent contributions to the prediction of mathematical achievement. The contri-

bution of WM was relatively modest in size, but still consistently significant even after

controlling for general intelligence. The contribution of general intelligence was compa-

rable to that of mathematical reasoning at age 11 years but slightly higher at age 14 years.

Although mathematical reasoning and knowledge of arithmetic were significantly and

moderately correlated, their separate and independent contributions to the prediction

of mathematics achievement signify that they should be treated as distinct constructs.

For some time, researchers in psychology and mathematics education have made a

distinction between procedural and conceptual knowledge, most often using qualitative

descriptions of students’ problem solving activities, in which the students reveal that

they have some procedural knowledge without understanding or, alternatively, some

conceptual knowledge in the absence of related procedural skills (see, e.g., Hiebert

& Lefevre 1986; Rittle-Johnson & Siegler 1998). However, procedural and conceptual

knowledge are highly correlated (Hallett, Nunes, & Bryant 2010; Rittle-Johnson, Siegler,

& Alibali 2001), and this makes it difficult to justify their independence with quantitative

methods such as factor analyses, which only consider the relationship between measures

of procedural and conceptual knowledge. This study provides a clear empirical basis

for distinguishing mathematical reasoning as a form of conceptual knowledge and

knowledge of arithmetic as separate constructs.

An important next step will be to explore the boundary between these two forms of

knowledge. We are certain that all the items in our reasoning task were genuine tests

of quantitative reasoning: this reasoning was about part-whole relations and additive

composition, and one-to-many correspondence and proportions, and had an adequate

level of difficulty for this age level. Working with a younger cohort, Nunes et al. (2007)

used items that assessed reasoning about one-to-one and one-to-many correspondence,

additive composition and the inverse relation between addition and subtraction as

predictors of mathematics achievement for younger children. Future research on the

impact of children’s reasoning on their mathematical achievement at school could

explore other aspects of reasoning as well as different aspects of children’s knowledge

of arithmetic.

We are also confident that the task that we used to test children’s ability to calculate

was also valid. All the questions in the arithmetic task are quite explicit about the

calculation that is needed, and thus, the constraint on the children’s performance is

not how well they reason but how well they do the calculation that they are asked

to do. However, there are many other tasks that could be used in predictive studies,

and which could shed light on the relationship between arithmetic and mathematical

reasoning. Recently, for example, there has been a great deal of interest in the possibility

of children using an internal number line to judge which of two numbers is the larger.

The evidence for the existence of this form of representation is the ‘distance effect’ that

takes the form of discriminations between numbers that are further apart from each

other being easier than discriminations between numbers that are close to each other

(Butterworth 2005; Dehaene 1997; Durand et al. 2005; De Smedt et al. 2009). Because

the judgements in such comparison tasks are about the number system, they could be

relevant to children’s ability to use numbers to make calculations. However, it could

be argued that they are also about simple quantitative relations (larger, smaller) and

would, therefore, count as reasoning as well. Longitudinal research that includes such

tasks as well as the reasoning and arithmetic tasks that were included in our study could

contribute to a better understanding of the role that the abilities measured by such tasks

play in mathematical achievement.

The multiple regressions, in which the outcome measures were science and English,

demonstrated a strong specificity in the relations between mathematical reasoning and

arithmetic and the children’s progress in mathematics at school. Both variables predicted

children’s mathematical achievement much better than their achievement in English,

which is an entirely non-mathematical subject. This strongly suggests that these two vari-

ables predict mathematics because they are measures of specific mathematical abilities.

The relatively strong relation between the children’s mathematical reasoning scores

and their achievement in science suggests that mathematical reasoning also plays an

important part in children’s learning about science at school. We need to know the

reason for this undoubtedly important connection. Our suggestion is that it is at least

partly due to the mathematical nature of some basic scientific concepts. Temperature

and density, for example, are intensive quantities: this means that both variables are

based on ratios and, thus, make demands on children’s proportional reasoning. Thus,

density is the ratio of mass to volume, and children will only understand how to vary

density and what the effects of such variations will be if they grasp density’s proportional

nature. This idea, and other possible alternative ideas, about the reason for the strong

relationship between mathematical reasoning and children’s scientific achievement need

to be investigated. In general, the link between children’s mathematical knowledge and

their progress in learning about science is a neglected topic, but our results suggest that

it is an important one.

Our study provides strong evidence that mathematical reasoning should receive a

greater emphasis than calculation skills from the early years in primary school and

arguably to the end of secondary school. This innovation should produce gains in

students’ mathematical and scientific achievement in the future.

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Received 7 September 2010; revised version received 22 March 2011