1
Why does competence in basic calculation matter? Why do primary children differ in it?
1
2
1
Richard Cowan , Chris Donlan , Donna-Lynn Shepherd , and Rachel Cole-Fletcher
1
1
Institute of Education University of London
2
University College London
Paper presented at the British Educational Research Association Annual Conference, University of
Manchester, 2-5 September 2009
This research derives from a project funded by the ESRC and subsidized by the Institute of Education
and University College London. We are grateful for their support and the collaboration of the Royal
Borough of Windsor and Maidenhead schools, teachers, pupils, and parents. Correspondence
concerning this paper or the larger project may be addressed to Richard Cowan at the Department of
Psychology and Human Development, Institute of Education University of London, 20 Bedford Way,
London WC1H 0AL or via email to [email protected].
Introduction
Differences between children in mathematical progress in primary school have long been
acknowledged to be considerable (Cockcroft, 1982). Substantial curriculum reorganisation in recent
years does not appear to have reduced them (Brown, Askew, Millett, & Rhodes, 2003). Basic
calculation – the addition of single digit numbers and corresponding subtractions- is a very simple
component of integer arithmetic. Most children can do some basic calculations when they start school
but at school they develop further in knowledge of numerical principles and number facts and the
strategies they use to solve problems. Several studies indicate that basic calculation proficiency
covaries with more general mathematical proficiency (Durand, Hulme, Larkin, & Snowling, 2005;
Geary & Brown, 1991; Hecht, Torgesen, Wagner, & Rashotte, 2001; Siegler, 1988) and children
identified as making unusually poor progress in mathematics consistently show deficiencies in basic
calculation (Geary, Hoard, Byrd-Craven, & DeSoto, 2004; Jordan, Hanich, & Kaplan, 2003; Landerl,
Bevan, & Butterworth, 2004).
Why differences in basic calculation proficiency are related to differences in more general
arithmetical attainment is uncertain. It could be that this reflects the importance of basic calculation
competence in arithmetic: primary maths is dominated by arithmetic. It could simply reflect the
overrepresentation of calculation items in standardized maths tests. Another possibility is that the
same factors that cause children to differ in basic calculation also affect educational progress more
generally. Such factors include parental support and socio-emotional functioning (Sacker, Schoon, &
Bartley, 2002) as well as general cognitive factors such as language skills (Cowan, Donlan, Newton,
& Lloyd, 2005), speed of information processing (Bull & Johnston, 1997), and working memory
functioning (Gathercole & Pickering, 2000). There might also be specific numerical factors that explain
the connection between basic calculation and arithmetic. These include counting sequence
knowledge (Donlan, Cowan, Newton, & Lloyd, 2007), number sense (Griffin, Case, & Siegler, 1994),
as well as the capacity measures in a dyscalculia screener (Butterworth, 2003).
The aims of this study are to assess 1) the relations between components of basic calculation
proficiency, namely knowledge of number facts, grasp of arithmetical principles, and strategy
efficiency 2) the relations between basic calculation and both general factors (parental support, socioemotional functioning, language skills, processing speed, and working memory) and specific
numerical factors (counting number knowledge, number sense, capacity ) and 3) the relations
between basic calculation proficiency and general mathematical progress after controlling for the
general and specific numerical factors.
Method
Participants
2
Participants were 144 Year 3 children (76 male, 68 female) attending 5 state schools in
England whose parents had given consent. The schools served socially mixed catchment areas. The
children’s ages ranged from 7 years 6 months to 9 years 5 months (Mean 8 years 2 months, SD = 4
months).
General factors
Table 1: Descriptive Statistics
Measure
Language
TROG-E
BPVS II
BPVS II Standard scores
Maximum
M
SD
Range
20
168
13.74
88.25
104.90
03.34
13.75
11.94
4 - 19
9 - 118
50 - 136
54
42
27.31
20.26
04.03
03.33
12 - 40
12 - 30
54
42
23.15
13.72
04.39
06.94
1 - 35
0 - 29
36
36
12.06
10.90
03.58
03.45
0 - 23
0 - 19
Speed of information processing
WISC III Symbol matching
Woodcock –Johnson III Pair cancellation
45
69
18.80
39.26
03.97
09.47
6 - 28
15 - 68
Social and emotional functioning
Difficulties
40
05.44
05.51
0 - 26
18
24
11.25
18.47
.7079
.4075
04.03
03.72
.1361
.0382
1 - 18
5 - 24
0 - 9377
2596 - 5011
28
12
18
16
16
14.19
04.99
15.82
05.06
02.58
06.42
02.94
02.56
04.08
02.23
0 - 27
0 - 12
0 - 18
0 - 14
0 - 14
Working Memory (WM)
Phonological loop (WM: PL)
Digit recall
Word list recall
Visuo-spatial sketchpad (WM: VSSP)
Block recall
Mazes memory
Central executive (WM: CE)
Backwards digit recall
Listening recall
Specific number factors
Counting number knowledge
Number sense
Dot enumeration
Numeral comparison
Basic calculation
Number fact knowledge
Derived facts
Explain
Efficient strategies
Counting errors
Language. Two language measures were used: TROG-E, the electronic version of the Test
for Reception of Grammar (Bishop, 1983), and the British Picture Vocabulary Scale (BPVS II).
TROG-E is a computer-presented test of language comprehension used in identifying specific
language impairment. It consists of 20 blocks of four items and testing is discontinued if the child fails
one or more items in five consecutive blocks. All blocks, except the first three, assess comprehension
of oral statements. Each item requires identification of the picture, out of four, that matches the
utterance, e.g. 'the pencil is above the flower'. A child's score is the number of blocks for which they
answered every item correctly.
The BPVS II is a vocabulary test which, like the TROG-E, requires the child to pick an
appropriate picture from a set of four. Trials are administered in blocks of 12 and testing is
discontinued if the child fails 10 items within a block.
3
Table 1 shows the descriptive statistics for both tests. A composite measure was formed by
averaging the z scores on the two tests. Table 1 also shows the range and mean for the BPVS II
standardized scores. This indicates that the sample is slightly above average.
Processing Speed. Two measures of processing speed were used: the Symbol Matching
subtest of WISC III and the Pair Cancellation subtest of Woodcock-Johnson III. Symbol Matching
presents the child with rows of abstract geometric designs. In each row there are two target symbols
and a group of five symbols. The child has to decide whether either target is present in the set of five.
The child has two minutes to work through 45 rows. The score is the number of correct decisions less
the number of incorrect decisions.
In the Pair Cancellation subtest the child is shown an array consisting of pictures of dogs and
balls. The child has to circle instances where a dog is adjacent and to the right of the ball. The child
has three minutes to complete the task. The score is the number of correct identifications.
Table 1 shows the descriptives for each test. A composite speed score was formed by
averaging the z scores on the two tests.
Working Memory. Subtests of the Working Memory Test Battery for Children (Pickering &
Gathercole, 2001) were used to assess functioning: two phonological loop (PL) subtests (Digit Recall,
Word List Recall), two visuo-spatial sketchpad (VSSP) tests (Block Recall, Mazes Memory), and two
central executive (CE) tests (Backwards Digit Recall, Listening Recall). They were administered in
accordance with the manual with each child receiving the subtests in the same fixed order. Each
subtest involves trials at increasing span levels. Up to six trials are given at each span level. The child
progresses to the next level as soon as they have correctly answered four items. A subtest is
discontinued when a child fails three trials at the same span level. Table 1 shows the descriptives for
each subtest. Composites for each working memory component were derived by averaging the z
scores on the two relevant subtests.
Mathematical progress. Teachers were asked to indicate pupils’ current estimated curriculum
levels in the three National Curriculum Mathematics strands: Number; Shape, Space and Measures;
and Handling Data. Only 79 teachers complied. The estimated curriculum levels were highly
correlated (all rs > .9). The reliability of teachers in making these kinds of assessments was
established by Oliver et al. (2004).
Teachers were also asked to rate their pupils’ understanding of mathematical ideas and ability
to carry out arithmetical procedures on a five point scale where 1 was labelled ‘cause for concern’ , 2
was ‘below average’, 3 was ‘ average’, 4 was ‘above average’, and 5 ‘outstanding’. All teachers
complied. Table 2 shows the distributions of ratings. The correlations between estimated curriculum
levels and ratings of understanding and ability to carry out arithmetical procedures were also high (all
rs > .8 for the 79 pupils rated on both measures). A composite mathematical progress score was
formed by averaging the z scores on the two items concerning understanding and procedural ability.
Table 2: Distributions of Teachers’ Ratings
Item
Understanding of mathematical ideas
Ability to carry out arithmetical
procedures
Attendance in current year
Parental interest and contact
Cause for
concern
7
8
Below
average
28
24
Average
Outstanding
58
55
Above
average
47
49
1
0
2
9
33
47
68
51
40
37
4
8
Parental support. Teachers rated the child’s attendance and parental interest in educational
progress on the five point scale where 1 was labelled ‘cause for concern’ and 5 ‘outstanding’. Table 2
shows the distributions of ratings. A composite parental support score was formed by averaging the z
scores on the two items concerning attendance and interest.
Social and emotional functioning. Teachers completed Strengths and Difficulties
Questionnaires (Goodman, 1997) for each pupil in their classes. This comprises statements for which
the teacher has to judge for that child whether they are certainly true, somewhat true or not true. It
yields an overall difficulty score out of 40. Table 1 shows the descriptives.
Specific numerical factors
4
Counting number knowledge. Knowledge of the natural number sequence was assessed
orally and then with numerals. Both versions involved ascending and descending sequences. A warm
up oral item was given in which the child was asked to count up from 5 to 16. Children were give
support if necessary to enable them to recite the numbers by themselves. Following this they
attempted a set of ascending sequences (25 – 32, 194 – 210, 2,995 – 3,004) and then a set of
descending sequences (46 – 38, 325 – 317, 1006 – 997, 20, 005 – 19, 998).
Numeral sequences were presented in column grids with the first few items filled in. The
experimenter read out the numbers up to the continuation point and the child was asked to continue
by writing in the cells below what came next. The digits of each number were in separate cells of the
grid. Support was given with the first item if required to ensure that the child only wrote one digit in a
cell. The first item required the child to continue from 13 to 16 and the numbers from 5 to 12 were
printed above 13. Subsequent ascending sequences were 28 - 31, 899 - 901, 7,999 - 8,001, and
59,999 – 60, 001. The set of descending sequences were 11 – 9, 41 – 38, 601 – 599, 6,001 – 5,998,
and 70,001 - 69, 999.
Testing within a set was discontinued once a child had made errors on two sequences in a
set, or when they did not wish to attempt an item. Table 1 shows the descriptives and that internal
reliability was good.
Number sense. Items were derived from the version of the Levels of Number Knowledge test
in Griffin (unpub.). It consisted of four subtasks: Number sequence knowledge, Relative magnitude,
Numerical distance, Distances. Each subtask consisted of warm up items and six test items.
Numbers were shown on computer as well as being named by the experimenter.
Number sequence items required children to name the number in given positions in the
number sequence. The three warm up items were ‘What number comes right after 7?’, ‘What number
comes before 5?’, and ‘What number comes two numbers after 3?’. Correct answers were explained
if necessary. The test items were ‘two numbers after 7’, ‘right after 9’, ‘five numbers after 49’, ‘four
numbers before 60’, ‘ten numbers after 99’, and ‘nine numbers after 999’.
Relative magnitude items required children to identify the bigger of two numbers. Warm up
items asked children ‘Which is bigger: 5 or 4?’ and ‘Which is bigger: 6 or 7?’. The pairs of numbers in
the test items were 9 & 7, 13 & 14, 69 & 71, 32 & 28, 51 & 39, and 199 7 203.
Numerical distance items required children to identify which of two numbers was closer to a
target number. Warm up items were ‘Which number is closer to 3: 2 or 6?’ and ‘Which number is
closer to 4: 6 or 1?’. The triads of numbers in the test items were ‘7: 4 or 9?’, ‘13: 14 or 11?’, ’21: 25 or
18?’, ’49: 51 or 45?’, ‘28: 31 or 24?’, ‘102: 98 or 109?’.
Differences items asked children to identify which of two pairs of numbers had the greater
difference. In introducing the warm up item the experimenter ensured that the child understood what
was meant by difference. The warm up item asked ‘Which difference is bigger: the difference between
4 and 2 or the difference between 6 and 3?’. The test items featured the following contrasting pairs of
numbers: 10 & 5 vs. 10 & 7, 9 & 6 vs. 8 & 3, 6 & 2 vs. 8 & 5, 20 & 17 vs. 25 & 20, 25 & 11 vs. 99 & 92,
48 & 36 vs. 84 & 73.
Testing was discontinued within a subtask after the child had made three errors. The
differences items were derived from Level 3 which is the average 10-year-old level. Only children who
had not been discontinued on previous subtasks were invited to try it. They were told it was for older
children and that it was fine if they did not want to do it. Table 1 shows the descriptives for the
combination of the four subtasks into a single scale.
A composite number sense and knowledge score was formed by averaging the z scores on
the counting number knowledge and number sense tests.
Capacity measures. The two capacity tasks, dot enumeration and magnitude comparison,
from the computer-presented Dyscalculia Screener (Butterworth, 2003) were used. Dot enumeration
required children to judge whether the number of spots matched a given numeral. Numeral
comparison required children to identify the numerically greater of two numerals where sometimes the
physically larger numeral is numerically smaller. The screener yields scores for each task that are
adjusted for accuracy and simple response time. A composite capacity score was derived by
averaging the z scores for each task.
Basic calculation measures
5
Number facts. Knowledge of addition and subtraction facts was assessed using a forced
retrieval procedure (Jordan, Hanich, & Kaplan, 2003) and children were audio recorded for this task.
For the first warm up item, the experimenter showed a card with 4 -2 and asked the child to read it.
She used the way the child read it, whether ‘4 take away 2’ or ‘4 minus 2’, and said most people knew
the answer to this without having to work it out. She explained she was going to show some other
sums and that if the child knew the answer they should tell her as fast as possible. If they would have
to work it out then they should just say ‘work out’. The second warm up item was ‘3 – 2’. Then
followed 18 subtraction items in the same order: 10 – 5, 3 - 3, 10 - 9, 6 - 4, 15 - 10, 8 - 4, 13 - 5, 12 11, 7 - 7, 14 - 10, 8 - 0, 12 - 6, 13 - 9, 16 - 7, 15 – 8, 11 – 8. As the experimenter showed each card
she read out the problem on it.
Then the addition items were introduced with two warm up items: 2 + 2, and 3 + 2. Then 10
addition items were presented: 4 + 2, 10 + 8, 9 + 3, 6 + 6, 3 + 4, 6 + 10, 9 + 4, 3 + 8, 7 + 5, 6 + 7, 9 +
9, 7 + 9. Timings were derived from the audio recordings. Children were credited with knowing a
combination if they gave the correct answer within 3 seconds from when the experimenter read out
the problem.
Principles. Understanding of principles was assessed with two tasks: derived facts and
explaining patterns and principles. Derived facts assessed children’s understanding of calculation
principles and patterns in addition and subtraction. It drew on procedures used by Dowker (2005) and
Jordan et al. (2003). Children were presented with pairs of problems in which the given answer to the
first could be used to solve the second, i.e. the answer could be derived from the given fact. Each pair
of problems was presented on computer and read to the child. Audio recordings were made of each
child’s performance.
The first warm up item involved the experimenter explaining that the child would see a
problem with the answer and another which she wanted them to solve as fast as possible and that the
first problem might help them. The first problem was 32 + 19 = 51 and the second was 32 + 19 = ?. In
the second warm up item, the first problem was 20 – 5 = 15 and the second was 21 – 5 = ?.
Then followed twelve pairs of problems. There were two of each that were related by the
following principles and patterns: commutativity of addition ( 47 + 86 = 133, 86 + 47 = ?; 94 + 68 =
162, 68 + 94 = ?), subtrahend minus one ( 46 – 28 = 18, 46 – 27 = ?; 273 – 245 = 28, 273 – 244 = ?),
subtraction complement principle (153 – 19 = 134, 153 – 134 = ?; 84 – 27 = 57, 84 – 57 = ?), doubles
plus one pattern (37 + 37 = 74, 37 + 38 = ?; 64 + 64 = 128, 65 + 64 = ?), inverse relation between
addition and subtraction ( 27 + 69 = 96, 96 – 69 = ?; 36 + 98 = 134, 134 – 36 = ? ), subtrahend plus
one (64 – 36 = 28, 64 – 37 = ?; 157 – 92 = 165, 157 – 93 = ?).
Children were credited with using a principle or pattern if they answered correctly within 5
seconds of the second problem appearing.
Explaining patterns and principles. The aim of this task was to assess children’s knowledge
of numerical rules, patterns and principles and their beliefs in the generality of them. Audio recordings
were made of each child’s performance.
The experimenter introduced it as follows. ‘There are some patterns and rules for adding and
subtracting. Some you might know about already and some you might be learning about later. I’m
going to show sets of problems that have a pattern or rule in common.’ The warm up item involved
showing three n + 1 problems (4 + 1, 37 + 1, 125 + 1). The experimenter asked what they had in
common. If necessary, she explained the number after rule, i.e. that when one is added to a number
the answer is the next number. She asked if they knew that already. She then pointed out that it was
true for all numbers when you add one to them.
The six main items followed a similar pattern. First a set of three problems was shown on the
computer and the child asked to articulate the connection between them (spontaneous naming).
Subsequently the experimenter described the connection and the child was asked if they recognized it
(recognition). Finally the experimenter asked if it held for all problems with the connection
(generalization). The six patterns used were n – n, n + 10, n – 0, n - (n-1) = 1, n + 0, and 1 + n.
Strategy assessment: Addition and subtraction. Children’s strategies were assessed with a
set of 16 addition and subtraction problems. Audio recordings were made of each child’s
performance.
In two warm up items (2 + 5, 5 – 3) the experimenter explained that she was going to ask
them to do some adding problems and some take-aways, that they could solve them any way they
6
liked and they should try and get the right answer, and that after they had solved them she would ask
how they had solved it. The problems were presented on computer and read out by the experimenter.
They were 4 + 6, 17 – 9, 11 + 5, 10 – 4, 3 + 14, 7 – 6, 8 + 9, 7 - 7, 7 + 8, 12 – 7, 4 + 9, 16 – 8, 12 – 6,
15 + 3, 6 + 9, 15 – 8.
For each problem attempted, children’s strategies and accuracies were coded. Strategies
were classified as either retrieval (if they knew the answer without having to work it out), or efficient (if
they used a related fact or decomposed the numbers in the problem), or counting if they counted on
from one of the addends or up from the subtrahend or down from the minuend. Two variables were
derived: accurate efficient strategy use and counting errors.
Table 3: Correlations between basic calculation components, general mathematical progress, general
factors and specific numerical factors
Principles
Derived
Explain
facts
- .42
-
Derived facts
Explain
Efficient
Counting errors
Basic calculation
Strategies
Efficient
Counting
errors
.37
- .34
.43
- .43
- .53
-
Facts
- .55
- .62
- .59
- .63
Mathematical progress
- .55
- .57
-. 57
- .53
- .76
Language
Speed
WM:PL
WM: VSSP
WM: CE
- .27
- .41
- .22
- .32
- .28
- .39
- .36
- .32
- .33
- .39
- .29
- .38
- .26
- .20
- .33
- .20
- .35
- .24
- .30
- .31
- .38
- .52
- .31
- .27
- .41
Parental support
Difficulties
- .17
- .10
- . 11
- .25
- . 25
- .18
- .13
- .23
- .23
- .23
Number sense and knowledge
Capacity
- .48
- .29
- .58
- .40
- .52
- .32
- .64
- .44
- .75
- .42
Age
- .06
-.--.033
-.12
- .03
- .02
Correlations greater than .20 or less than - .20 have a probability of less than .05. Correlations greater
than .29 or less than - .29 have a probability of less than .0005.
Results and Discussion
All basic calculation measures correlated substantially with each other and with teachers’
ratings of overall mathematical progress as Table 3 shows. They also correlated with the general
factors (parental support, socio-emotional functioning, working memory, information processing
speed, and language skills) and the specific numerical factors. Significant correlations remained
between basic calculation components and overall progress even when the general and specific
factors were partialled out as Table 4 shows.
Table 4: Partial correlations between basic calculation components and general mathematical
progress controlling for age and general and specific factors
Derived facts
Explain
Efficient
Counting errors
Principles
Derived facts
Explain
- .20*
-
Efficient
.12*
.15*
-
Strategies
Counting errors
- .05**
- .07**
- .31**
-
Facts
- .31**
- .35**
- .29**
- .28**
7
Mathematical progress
- .34**
- .20*
-.24*
- .10**
- .41**
* p < .05. ** p < .0005.
As with any study there are limitations. No measure is perfect and some qualities may be only
roughly approximated with quantitative methods. For example the reliance on teacher ratings is
questionable though as Oliver et al. (2004) argued, teachers are best placed to judge pupils’
mathematical progress and within our study there were substantial correlations between teacher
ratings and estimated curriculum levels. Teacher ratings of children’s socio-emotional functioning on
the SDQ have been used in several studies. Using teacher assessments of parental support is novel
but they are surely in a good position to know about parental contact with the school and the child’s
attendance record. Ensuring their children attend is an indicator of parental commitment to children’s
education. Turning up to parent evenings and meetings is also an indicator of parental involvement.
Our sample is not demographically representative of the UK. Poorer families are less well
represented: eligibility for free school meals is approximately 7%, compared to 18% nationally. The
five participating schools all volunteered to take part. They included one first school. The four
participating primary schools have substantially above average success in KS2 results in maths (85 96 % of pupils achieving Level 4 or better, compared to 79% nationally and 82% for the LA).The
implications this may have for the generalizability of our findings is uncertain.
These results suggest competence in basic calculation matters because it is more intimately
connected with general mathematical competence than any of the other factors considered. Our study
does not establish causal direction or even causality: it is just a correlational study. Basic calculation
competence may reflect general mathematical competence as much as it influences it.
Secondly although knowledge of number facts is a major component of basic calculation and
a substantial correlate of general mathematical competence it is not the only component: there is
more to basic calculation competence than knowing facts. Understanding principles and making use
of these in solving problems is important too. While components of basic calculation substantially
covary there are limits to this. For example a child who shows a superior knowledge of facts does not
invariably show a good understanding of principles. This indicates the validity of the claim that there is
not a single mathematical ability (Dowker, 2005).
Finally the relations between basic calculation proficiency and the general and specific factors
suggest that there may be many reasons why children differ. The limitations in the extent of these
indicate that there may well be other causes of variation.
References
Bishop, D. V. M. (1983). The test for reception of grammar (TROG). Manchester, England: Age and
Cognitive Performance Research, University of Manchester.
Brown, M., Askew, M., Millett, A., & Rhodes, V. (2003). The key role of educational research in the
development and evaluation of the National Numeracy Strategy. British Educational Research
Journal, 29, 655-672.
Bull, R., & Johnston, R. S. (1997). Children's arithmetical difficulties: Contributions from processing
speed, item identification, and short-term memory. Journal of Experimental Child Psychology,
65, 1-24.
Butterworth, B. (2003). Dyscalculia Screener. London: Nelson.
Cockcroft, W. (1982). Mathematics counts. London: HMSO.
Cowan, R., Donlan, C., Newton, E. J., & Lloyd, D. (2005). Number skills and knowledge in children
with specific language impairment. Journal of Educational Psychology, 97, 732-744.
Donlan, C., Cowan, R., Newton, E. J., & Lloyd, D. (2007). The role of language in mathematical
development: Evidence from children with Specific Language Impairments. Cognition, 103,
23-33.
8
Dowker, A. (2005). Individual differences in arithmetic. Hove: Psychology Press.
Durand, M., Hulme, C., Larkin, R., & Snowling, M. (2005). The cognitive foundations of reading and
arithmetic skills in 7-to 10-year-olds. Journal of Experimental Child Psychology, 91, 113-136.
Gathercole, S. E., & Pickering, S. J. (2000). Working memory deficits in children with low attainments
in the national curriculum at 7 years of age. British Journal of Educational Psychology, 70,
177-194.
Geary, D. C., & Brown, S. C. (1991). Cognitive addition: Strategy choice, and speed-of-processing
differences in gifted, normal, and mathematically disabled children. Developmental
Psychology, 27, 398-406.
Geary, D. C., Hoard, M. K., Byrd-Craven, J., & DeSoto, M. C. (2004). Strategy choices in simple and
complex addition: Contributions of working memory and counting knowledge for children with
mathematical disability. Journal of Experimental Child Psychology, 88, 121-151.
Goodman, R. (1997). The strengths and difficulties questionnaire: A research note. Journal of Child
Psychology and Psychiatry, 38, 581-586.
Griffin, S. A., Case, R., & Siegler, R. S. (1994). Rightstart: providing the central conceptual
prerequisites for first formal learning of arithmetic to students at risk for school failure. In K.
McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp.
25-49). Cambridge, Mass: MIT.
Hecht, S. A., Torgesen, J. K., Wagner, R. K., & Rashotte, C. A. (2001). The relations between
phonological processing abilities and emerging individual differences in mathematical
computation skills: a longitudinal study from second to fifth grades. Journal of Experimental
Child Psychology, 79, 192-227.
Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies
in children with specific mathematics difficulties versus children with comorbid mathematics
and reading difficulties. Child Development, 74, 834-850.
Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical
capacities: a study of 8-9-year-old students. Cognition, 93, 99-125.
Oliver, B., Harlaar, N., Hayiou Thomas, M. E., Kovas, Y., Walker, S. O., Petrill, S. A., et al. (2004). A
twin study of teacher-reported mathematics performance and low performance in 7-year-olds.
Journal of Educational Psychology, 96, 504-517.
Pickering, S. J., & Gathercole, S. E. (2001). Working Memory Test Battery for Children (WMTB-C).
London: The Psychological Corporation.
Sacker, A., Schoon, I., & Bartley, M. (2002). Social inequality in educational achievement and
psychosocial adjustment throughout childhood: magnitude and mechanisms. Social Science
& Medicine, 55, 863-880.
Siegler, R. S. (1988). Individual differences in strategy choices: Good students, not-so-good students,
and perfectionists. Child Development, 59, 833-851.
This document was added to the Education-line collection on 25 August 2009