Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus: Posttest and long-term Effects of a two-year intervention
Michael Shayer,
King’s College, London
Mundher Adhami
King’s College, London
Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus: Posttest and long-term Effects of a two-year intervention
Background. In the context of the British Government's policy directed on improving
standards in schools, this paper presents research on the effects of a programme intended to
promote the cognitive development of children in the first two years of Primary school (Y1 &
2, aged 5 to 7 years). The programme is based on earlier work dealing with classroom-based
interventions with older children at both primary and secondary levels of schooling.
Aim. The hypothesis tested is that it is possible to increase the cognitive ability of children by
assisting teachers toward that aim in the context of mathematics. A corollary hypothesis is
that such an increase would result in an increase in long-term school achievement.
Sample. The participants were 8 teachers in one LEA and 10 teachers in another. Data was
analysed on 275 children present at Year 1 Pre-test in 2002 and at long-term Key Stage 2
Post-test in 2008.
Method. Two intervention methods were employed: a Y1 set of interactive activities designed
around Piagetian concrete operational schemata, and mathematics lessons in both Y1 and Y2
designed from a theory-base derived from both Piaget and Vygotsky.
Results. At post-test in 2004 the mean effect-sizes for cognitive development of the
children—assessed by the Piagetian test Spatial Relations— were 0.71SD in one LEA and
0.60 SD in the other. Five classes achieved a median increase of 1.3 SD. The mean gains over
Pre-test in 2002 for all children in Key Stage 1 English in 2004 were 0.51 SD, and at Key
Stage 2 English in 2008—the long-term effect—were 0.36 SD, an improvement of 14
percentile points.
Conclusions. The main hypothesis was supported by the data on cognitive development. The
corollary hypothesis is supported by the gains in English. The implications of this study are
that relative intelligence can be increased and is not fixed, and that children can be led into
collaborating with each other to the benefit of their own thinking, and that there does exist a
theory-based methodology for the improvement of teaching.
Introduction
This paper presents research on the effects of a programme intended to
promote the cognitive development of children in the first two years of Primary
school (Y1 & 2, aged 5 to 7 years). Detail of its relation to previous intervention work
with older children may be seen in Shayer (1999) and Shayer & Adhami (2007).
Earlier research on cognitive development of 10 to 16 year-olds had revealed
that on Piagetian measures the levels of 11/12 year-olds entering secondary education
spanned some twelve developmental years—i.e. from the average 6 year-old to the
above-average 16 year-old (Shayer, Küchemann & Wylam, 1976). On the other hand
research on 5 to 8 year-olds (Shayer, Demetriou & Pervez, 1988) had shown that 20%
of the 7 year-olds were already at the mature concrete level (2B), the average level of
12 year-olds. A possible interpretation of these data —consistent also with data on 16
year-olds reported in Shayer & Wylam, (1978) and the development charted in
Inhelder & Piaget, 1958—is that only 20% of the child population follow the
complete developmental path described by Piaget. Bearing in mind that in Britain in
1945 only 20% of the population were deemed suitable for Grammar school
education, it seemed desirable to test whether an intervention at the outset of Primary
education might increase this proportion by the age of 7, thus increasing the potential
for more children to benefit from the rest of their time in school. This intention was
reinforced, after the start of the research project, by the data reported in Shayer,
Ginsburg and Coe (2007) which showed that on the test of Piagetian conservations,
Volume&Heaviness, the average performance of the 11/12 year-olds had slipped by
2004 to the level of the 8 to 9 year-olds in 1976.
The grounding of the RCPCM1 work in the maths education research
literature
The detailed specifics, and how they were used, are found in Shayer (2003)
and Shayer & Adhami (2003). But from Cable (1997) and Vergnaud (1983) were
drawn the importance of concepts of measure in even 5 year-olds’ development of
their concept of number, neglected in much current teaching including the England
and Wales National Numeracy scheme. From the work of Butterworth (1999) and
Dehaene (1997)—and by implication from the early work of Dienes (1963)— come
the importance of multiple representations of number using both sides of the brain.
From Piaget & Szeminska (1952) comes the insight that even 5 and 6 year-olds have
primitive concepts of multiplicative relations between numbers as well as additive,
linking with measures, which are based on the ratio between the size of an object and
the measure unit. And from much of the work of Nunes (2002; with Bryant, 1996))
detailed empirical research of how to promote multiplicative thinking was drawn on
in the construction of TM lessons, with the intention, by the end of Year 2, of evening
up the balance in children’s experience of number between the additive and
multiplicative (ratio) aspects.
In Resnick, Bill & Lesgold (1992) one of the largest educational intervention
effect-sizes ever achieved —2 SDs on mathematics achievement—is reported. The
study describes work done in a single first year Primary class in arithmetic (6 to 7
year-olds): the authors comment
‘The children with whom we have worked come disproportionally from
among the least favoured of American families. Many of them are considered to be
educationally at risk…Yet these children learned effectively in a type of programme
1
Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus: research project
funded by the Economic and Social Research council at King’s College, London 2001-2004
that, if present in schools at all, has been reserved for children judged able and
talented–most often those from favoured social groups’.
This experiment, featuring a collaborative learning style for the children’s
mathematics work having much in common with that used in the RCPCM classrooms,
was also drawn on.
The Thinking Maths (TM) lessons
Insights into cognitive development from both Piaget and Vygotsky are used
in the conduct of a TM lesson (Adhami, Shayer & Robertson, 2004; Adhami, Shayer
& Twiss, 2005). Both believed in the virtues of collaborative learning. But Piaget
himself made a seminal contribution to the mathematics literature in his book on
Number (Piaget & Szeminska, 1952). From this and elsewhere (Shayer & Adey,
1981; Hart, 1981) it was possible to assess all the statements in the England and
Wales National Curriculum (NC) in terms of demand levels from early concrete
through to concrete generalisation. Similarly the levels of the different agendas in a
TM lesson were also assessed. Given the data in the monograph Shayer, Demetriou &
Pervez, 1988, the target aimed for in Year 2 (Y2) TM lessons, when the children
would be 7, was mature concrete (for the Year 1 lessons it had been middle concrete).
Each lesson started with an episode at a lower level, and the episodes—up to 3—were
designed so that children from a wide range of initial cognitive abilities could each
make some progress from their present level during each lesson.
Looking at the agenda of the TM lessons (and also the children) through
Piagetian eyes is thus an essential aspect of the art. But Piaget’s work tells us little
(except the importance of ‘cognitive conflict’) about how to teach the lessons. For this
we turn to a view of Vygotsky’s concept of the Zone of Proximal Development (ZPD)
that suggests that the mediating role of the teacher should be to maximize the
collaborative processes through which the children can mediate each other’s learning
(see the Appendix for an exposition of this view).
Each Episode of a TM lesson is a 3-Act play. The first, called Concrete
Preparation, introduces the context of the lesson agenda at a level where all in the
class can participate (5 to 8 minutes). Here the ‘argument within the collective’ is
begun. Several children are asked to describe their views of what the task ahead
involves, and to suggest possible strategies of solving it. In some cases the children
are asked quickly in pairs, while sitting on the carpet, to attempt the easier parts of the
task. Then different pairs are asked to offer their ideas, which others then discuss. In
the second act, Construction, children, in groups from two to 4 or 5 are asked to go to
their tables and attempt the first worksheet. In a few minutes time they will be asked
to present their ideas to the rest of the class, so they do not waste time in ‘neat work’:
they discuss vigorously with each other, using pencils on papers, until they feel they
have something worth saying or showing (typically, 7 to 12 minutes). Act 3,
Reflection, is again communal—often on the carpet in front of the whiteboard
again—where their different solutions are shown and shared with the whole class (8
minutes at least). Thus in Acts 1 and 2 the community of learning is created, in which
children offer different insights, which are made available for all to work with. But
Act 3, well managed by the teacher, gives maximum chances for the individual ZPDs
of the children to be completed as they witness the ideas produced by the other
groups. Typically Act 3 leads naturally to the next Episode with agenda set usually at
a higher level.
Method
The locations of the RCPCM research
The research team at Hammersmith and Fulham in 2001/2004: Hammersmith and
Fulham (H&F) is a Local Education Authority (LEA) in Inner London, with some
areas of social deprivation in which the experimental classes were located. The initial
task in 2001/2002 was to create the TM lessons for Y1 use in 2002/2003. Four
teachers, previously selected to assist in Professional Development (PD) for the
CASE @ KS1 Let’s Think2 lessons in H&F schools, were chosen as teacherresearchers. The research team consisted of the two authors from King’s College, an
Advisory Teacher for H&F, and the four teacher-researchers, and met for a whole
afternoon working session every 3 weeks. Initially the two authors would suggest
mathematical contexts for a possible lesson, and then the whole team would work on
how the lesson might be presented, and what worksheets etc. were needed. Then one
of the teacher-researchers would offer to trial the lesson with a Y1 class, with the two
authors in support. The first author would make detailed notes with a time line of the
trial lesson, which would be written up for the next team afternoon. From that
feedback the whole team would then revise the initial lesson plan, and one or the
other of the authors would produce all the materials needed for the project teachers.
Often an amended version of the lesson was re-trialled—sometimes more than once—
before the team were satisfied with it. Later some of the ideas for the TM lessons
came from the teacher-researchers. Twelve TM lessons were generated for 2002/2003
(Y1) and 19 for 2003/2004 (Y2). Thus the generation of lessons was based on the
same processes of collaborative learning, with the teacher-researchers, as the project
teachers would be asked to use with their children.
The research team at Bournemouth in 2001/2004: Bournemouth is a town on the
south coast of England with a school intake about the National average. The H&F
and King’s team had the responsibility of generating the TM lessons. But the
Bournemouth LEA on the south coast of England offered to run a parallel replication
2 CASE@KS1.H&F (1997-2000) Research project based at King’s College funded as part of a Single
Regeneration Budget granted to the Hammersmith and Fulham LEA by the DfES
of the RCPCM intervention in 2002/2004 in their schools—but taking the
collaborative learning aspect of the classroom practice as their focus.
The research design: 2002 to 2004
In a recent research project CASE@KS1.H&F3 , in the Local Education Authority
(LEA) of Hammersmith & Fulham, Adey, Robertson & Venville (2002) had reported
for a one-year intervention with Y1 children (5/6 years of age) Post-test effect-sizes in
relation to Control schools of 0.47 and 0.43 standard deviations (SDs) on Piagetian
tests. Children were given, every week, interactive and collaborative learning focused
on the major concrete operational schemata—classification, seriation, spatial
perception, time relations and causality— described by Piaget. Their lessons were
called Let’s Think (Adey, Robertson&Venville, 2001). Their interactive learning was
conducted for some 20 to 30 minutes each day with groups of six children around a
table with the teacher. Thus by the end of the week every child had experienced the
activity (given 30 children in the class) and the teacher had both been able to mediate
the children into collaborative learning, and also learnt much about their individual
learning difficulties. Yet the exercises in thinking had not been located within the
context of any specific part of the National Curriculum. This led to the design of the
RCPCM Project4 that the Hammersmith & Fulham LEA agreed to participate in.
In addition to experiencing the Let’s Think activities Year 1(Y1) children
would also receive whole-class Thinking Maths (TM) lessons in order to focus their
thinking skills within one major subject of the National Curriculum. Then in Y2, the
second year of full-time education, the Let’s Think work would have ceased but the
children, already used to the learning strategies practised in the Y1 TM lessons, would
now receive further whole class TM lessons at a rate of about one every 10 days. In
addition the teachers would be encouraged, where possible, to use the same teaching
3 CASE@KS1.H&F.(1997-2000) Research project based at King’s College funded as part of a Single
Regeneration Budget granted to the Hammersmith and Fulham LEA by the DfES.
4 Realising the Cognitive Potential of Children 5 to 7 with a Mathematics focus (2001/2004). Research project
funded by the Economic and Social Research Council at King’s College.
skills within the context of their ordinary Numeracy work, and also to establish a link
between the children’s TM insights and the National Curriculum learning objectives.
The RCPCM intervention featured 8 experimental classes in 8 H&F schools
and 10 experimental classes in 4 schools in Bournemouth (mean age at Y1 Pre-test 5
years, 7 months, SD 3.9 months). H&F provided 5 Control classes and Bournemouth
provided 11. Table 1 shows the research design.
Table 1: RCPCM Main Study schedule
Year
Sept.
2002-July
2003
Sept.
2003-July
2004
July 2004
Main Study Experimental Schools
Pre-tests: Piagetian Spatial Relations
test
Y1 teachers use Let’s Think each week
during the year
Y1 use 10 TM lessons during the year
Y2 teachers use 15 TM lessons during
the year and also ‘bridge’ to their
Numeracy work
Post-tests: Piagetian Spatial Relations
test and KS1 SATs in Maths &
English
Main Study Control Schools
Pre-tests: Piagetian Spatial
Relations test
Post-tests: Piagetian Spatial
Relations test and KS1 SATs
in Maths & English
The schools were selected by the LEA Advisory Teachers in consultation with
their Heads, on the assumption that the whole year group would be involved. The
H&F schools were all one-form entry: in Bournemouth all classes in the year group
were involved. The selection was not random, but the samples are believed to be
representative of the LEAs involved. In H&F the teachers received a two-day session
of professional development (PD) twice each term in Y1 and Y2 up to and including
the Spring term 2004. In Bournemouth there were four 1-day PD sessions from March
to May 2002 on the Let’s Think teaching, seven 1-days of PD during Y1 for the
Thinking Maths lessons, and two 1-day sessions in the first term of Y2.
Experimental and Control groups
Table 2 shows data used frequently in the UK as a measure of socio-economic
status of families: the percentage of children in the schools allowed free school meals
(%FSM). The control groups in each Local Education Authority (LEA) were chosen
as far as possible to cover the range of intakes of the experimental groups. For the
difference between the mean of all controls and all experimentals t = -1.8; p = 0.088
for %FSM. Thus although the socio-economic status of the intakes to the control
schools is higher (%Free school meals are lower) the difference is not so great as to
invalidate the study. It is convenient also to have control data that gives information
on the higher levels of performance at post-test. Details of all schools are shown in
Table 2 in order to display the wide variation in both experimental and control
schools’ intakes.
Table 2: Socio-Economic data on the children in this study
Group
H&F
Experimental group
1
2
3
4
5
6
7
8
Control group
9
10
11
12
13
Bournemouth
Experimental group
A
B
C
D
Control group
E
F
G
H
Total (Experimentals)
Total (Controls)
Mean (all Controls)
Mean (H&F Experimentals)
Mean (Bournemouth Exps)
N
%Free School meals
29
26
29
26
29
26
30
27
20.1
43.0
50.4
25.1
61.1
54.9
28.9
12.7
29
25
29
29
30
20.0
14.4
43.8
28.2
27.7
71 (3 classes)
55 (2 classes
106 (4 classes)
25
18.1
27.5
5.7
41.2
79 (3 classes)
3.2
19.9
14.9
9.2
88 (3 classes)
60 (2 classes)
80 (3 classes)
479
449
20.1
37.0
23.1
H&F had been given by the Government in 1998 a ‘Single Regeneration
Budget’—that is, a considerable sum of money in recognition of the social
disadvantage in the population supplying the LEA schools’ intake. Some of it was
spent in funding the previous CASE@KS1 project. The intention of the RCPCM
project was to enhance the cognition and learning ability of the H&F pupils to at least
the National average. At the same time it was thought desirable simultaneously to
replicate the intervention in an LEA with a quite different social composition. Table 3
compares the ethnic compositions of the two LEAs.
Table 3: Ethnic composition of groups in this study
Education Authority White%
H&F
Black%
Ethnic group
Mixed% Asian%
Chinese% Other%
Experimental
Control
Bournemouth
46.2
56.5
27.2
23.9
8.5
11.0
5.9
2.9
0.4
0.3
10.8
3.4
Experimental
Control
96.4
89.3
0.5
0.5
1.5
5.6
0.2
2.1
0.2
0.3
0.6
1.0
The details in both Table 2 and Table 3 are of the whole schools in 2002/2003,
so may not exactly describe the Y1 year-groups. The contrast between the two LEAs
can clearly be seen in Table 3, but inspection of Table 2 shows there is socioeconomic disadvantage comparable with that in H&F in the intake of at least two of
the Bournemouth experimental schools.
The tests used
The Piagetian Spatial Relations test (NFER, 1979) was used in the original
CSMS5 research covering the range early concrete (2A) through the concrete
generalisation (2B*) level. For the Y2 Post-test with the 7 year-olds it was found
possible to use it in its original form, as Piaget’s original research featured children of
this age. The two authors administered the test in H&F; in Bournemouth the first
author tested the classes in one school as a demonstration for those testing the other
schools. All the children’s responses are given in the form of their own drawings: no
reading or writing is involved. The first question involves them drawing a jam-jar
half-full of water at the various orientations the teacher shows them (upright, tilted at
5 CSMS: Concepts in Secondary Science and Mathematics. Research programme funded at Chelsea College by the SSRC
1974–1979.
30 degrees off vertical, and on its side: they have to imagine the water). The second
gives them a drawn hill and asks them to draw a house on one side, and three trees on
the other. The third repeats the agenda of question 1, but now with a jar containing a
plumb-line hanging from its lid. The fourth, permitting an assessment at the 2B* level
as well as all levels going down to Pre-operational, asks them to imagine themselves
standing in the middle of a road consisting of an avenue of trees, and to draw what
they would see, covering the near distance as well as afar. Assessment of the
children’s responses consists of seeing how many aspects of the situation, how many
relations, they can consider in the act of drawing. The scoring decisions are taken
from the drawings in Piaget & Inhelder, 1956. Thus with the hill, if they can handle
one relation at a time only (early concrete, 2A) then they relate the orientation of the
house just to the slope of the hill. But if they can handle two relations (mature
concrete, 2B) then they will also bear in mind earth’s horizontal, and they will draw
the houses and the trees upright. In effect the test measures the complexity of relations
the children can handle in the moment, and thus acts as a test of fluid intelligence
(Shayer, 2008). The two authors assessed all the children’s drawings from both LEAs.
The test reliability is 0.82. Items 1 and 3 have four scoring levels, ranging from <2A
(early concrete) to 2B (mature concrete). Item 2 has five scoring levels, ranging from
<2A to 2B* (concrete generalisation). Item 4 has six scoring levels, from <2A to the
2B*/3A (early formal) boundary. It is therefore equivalent to a 19 item test, and was
scored using the rating scale Rasch analysis to give an equal interval scale from 2.79
(just below 3 = 2A, early concrete) to >6.3 (higher range of concrete generalisation).
For the Pre-test at age 5 a modified version of the test designed by Adey for
the CASE@KS1 project was used (Adey, Robertson & Venville, 2002). Instead of
requiring the children to draw all their answers, the apparatus was drawn for them
(although still demonstrated by the teacher administering the test) so e.g., in the first
question they were simply asked to draw in the water for each orientation of the jam-
jar. Adey cited a reliability figure of 0.74, but this lower figure is due to the restriction
of range of the performance of the younger children.
KS1 SATs (Standard Assessment Tasks) are Government tests taken by all
children at the end of Y2, and used to assess schools’ performance and are not
published (KS1 is Key Stage 1 from 5 to 7 years, and Key Stage 2 is from 7 to 11).
The Key Stage 1 and 2 test data were supplied by the schools after any appeals had
been dealt with; the Y6 KS2 SAT results at the school level were published on April
1, 2009.
Results
Bearing in mind that long-term effects data were obtained for Key Stage 2 SATs in
both Maths and English the entire data-analysis is conducted on only those children
present at Pre-test in Y1 2002 and also in the 2008 SATs when they were 6½ years
older. In this way all the effects are analysed on the same children. Although this is a
43% loss of sample for the experimental groups it was thought this was an acceptable
price to pay for a cleaner analysis of the data. In both LEAs the mobility of children
in the experimental schools is high.
Effects on cognitive development
For Figures 1 to 5 the mode of data-analysis was first to obtain the residualised
gain scores (rgs) in relation to the initial Pre-test scores on the Spatial Relations test
on the 5 year-olds. For example, for Figure 2 the regression for the KS1 Maths class
mean scores on the class means for the Spatial Relations Pre-test was first computed
for the control classes. The Pre-test mean score for each experimental class was then
entered into the control’s regression equation. The difference between the class KS1
Maths mean score and the score predicted from the control regression line is the
residualised gain score (rgs) for the experimental class. The regression line averages
all the controls, and the rgs shows the difference from what would have been expected
had the experimental class been one of the controls. The correlation between pre and
post-test scores is, of course, not perfect, so one should imagine the control class
means lying scattered both above and below the regression line. In this way, in
Figures 1 to 5, in addition to the experimental effect-sizes, one sees also the extent to
which the control classes vary above and below their own regression line; their class
number weighted mean rgs sum to exactly zero. The rgs scores were then divided by
the standard deviation of all the control children to convert them into effect-sizes to
allow for comparison from test to test.
Figure 1 shows the effect-sizes at Post-test for all the classes for the Spatial
Relations test. An important variable in this study’s data is the variation between
teachers, both in the experimental and the control groups, so it was thought important
to display this in the Figures to follow.
Figure 1: Post-test Effects 2004 on the Spatial Relations test
The distribution of Experimentals’ effects is hardly normal. It can be read as trimodal, with one median about 1.3 SD showing what can be achieved by those who
grasp the whole model; a second median about 0.6 SD for those trying a method with
some success for the first time, and a third median around 0 SD. The average effectsizes are equivalent to raising a set of schools initially at the 50th percentile to the 70th
percentile.
In a similar way the Post-test results for Y2 KS1 Maths and English were
analysed in relation to children’s Y1 Piagetian Pre-test data. Figure 2 shows the
effect-sizes in 2004 for each class for KS1 Maths in relation to the original Pre-test
data. Figure 3 gives the KS1 English data.
Figure 2: Key Stage 1 Maths Effects 2004
Figure 3: Key Stage 1 English Effects 2004
A recent book—Hattie, 2008—summarizing all the meta-analytic evidence
there is available in educational research literature lists ‘prior achievement’ of
students as the major general factor affecting student achievement in schools. The
general effect of ‘teachers’ is about half as large. While Figure 1 shows data mainly in
terms of ‘prior achievement’ of students, effects of ‘teachers’ can already be seen as
well in the wider variation among the Control classes in Figure 2.
When the long-term effects for Key Stage 2 are inspected it needs to be born in
mind that in the four years between KS1 and KS2 children have usually experienced
learning with four different teachers, none of them using the teaching strategies which
are the subject of this paper, so an even greater source of variation in achievement is
seen. Figures 4 & 5 shows the 2008 KS2 Effects for Maths and English.
Figure 4: Key Stage 2 Maths Effects 2008
Figure 5: Key Stage 2 English Effects 2008
In both KS2 graphs it can be noticed that the H&F control classes have shown a
net improvement in relation to the Bournemouth controls.As a guide to the statistical
significance of these data, and also as a guide to LEAs that wish to judge whether an
investment in the PD required to deliver these effects would be justified, an
alternative data-analysis is presented in Table 4. Here the means of the data for all the
individual children in the experimental and control groups were first computed and
then converted into effect-sizes using the standard deviations (SDs) of the groups.
Then, using the formulae described in McNemar (1969, p90-98) for the difference
between the difference scores, the magnitudes of statistical significance required for
p≤ 0.05 and p≤ 0.01 were computed.
Table 4: Gains (E - C) in relation to the initial Pre-test in 2002
KS1 2002
Bournemouth
N Pre-test Post-test Maths English
Control mean
175
3.62
4.52
4.25
2.63
Experimental mean 148
3.28
4.58
3.99
2.69
SD difference
-0.50
KS2 2008
Maths English
4.64
4.64
4.38
4.52
0.10
-0.20
0.09
-0.28
-0.12
SD Gain over Pre-test
0.6
0.3
0.59
0.22
0.38
P≤0.05 for SD gain
P≤0.01 for SD gain
0.13
0.17
0.24
0.32
0.15
0.20
0.16
0.21
0.17
0.22
KS1
H&F
Control mean
Experimental
mean
N
94
127
Pre-test Post-test
3.33
4.23
KS2
Maths English Maths
3.74
3.76
4.71
English
4.72
3.13
4.48
4.15
3.93
4.65
4.75
-0.31
0.39
0.34
0.12
-0.07
0.03
SD Gain over Pre-test
0.71
0.65
0.43
0.24
0.34
P≤0.05 for SD gain
P≤0.01 for SD gain
0.19
0.24
0.30
0.39
0.39
0.51
0.21
0.27
0.19
0.25
SD difference
*Note: for Bournemouth KS1 English results were reported on a 3-point scale, NC 1 to 3. But
for all the other KS1 results the 5-point scale, 1, 2C, 2B, 2C, 3 was used.
The rows headed ‘SD difference’ give the experimental/control differences in SD
units. In the row headed ‘SD gain over Pre-test’ the value for KS2 Maths for
Bournemouth of 0.22 is the effect-size. Such a value is usually regarded as small, but
since it is greater than the 0.21 required for p≤ 0.01 it is still statistically significant.
Inspection of Table 4 shows that, e.g., the experimental group in Bournemouth
was initially 0.5 SD below the controls in terms of cognitive development. But by
post-test in 2004 they were actually 0.1 SD above the control group, an overall gain of
0.6 SD, compared with a figure of 0.17 SD required for significance at the 0.01 level.
In KS1 English in 2004 their relative gain of 0.59 SD was just as high. For KS1
Maths their relative gain was smaller (0.30 SD) but significant at the 0.05 level (0.24
required for p≤0.05). In this way the performance of all groups can be interpreted.
For Figures 6 and 7 the National average (obtained from published DfCFS6
statistics) was subtracted from the means, and the differences from the National
average shown in SD units.
Figure 6: H & F means in relation to National averages
Department for Children, Schools and Families: the Government Education department for
England and Wales.
6
Figure 7: Bournemouth means in relation to National averages
Here it can be seen that both H&F experimentals and controls were below the
National average at Pre-test, with controls 0.31 SD below experimentals. But for
Bournemouth the controls were about 0.3 SDs above National average at Pre-test, and
the Experimentals 0.2 SD below, a net difference of 0.5 SD as shown in Table 4.
Discussion
From Figure 7 it can be seen that for Bournemouth control schools the KS1 and
KS2 performance is relatively the same, with a mean of the maths and English results
0.17 SD above the National average at KS1 and 0.15 SD at KS2. The experimental
group did gain over the controls by 0.30 SD (gain significant at the p≤0.05 level,
Table 4) on KS1 Maths, but on KS1 English they had overcome an initial deficit of
0.5 SD. The mean of English and maths for the Controls at KS2 was 0.15 SD above
the National average compared with -0.05 SD for the experimentals. Hence the net
improvement compared with Pre-test of the experimentals versus the Controls was
0.30 SD (-0.50 compared to -0.20).
From Figure 6 it can be seen that H & F controls were about the National
average at KS1 for the mean of the maths and English in 2004, whereas in 2008 the
mean was 0.23 SD above average at KS2. The experimental group was well above
the controls on both maths and English at KS1, and at KS2 in 2008 was still giving
the same mean attainment as the control schools. Here there are two added-values to
be considered: in relation to the excellent performance of the control schools the
experimental schools give an mean increase of 0.30 SD, but in relation to National
statistics the added-value is around 0.65 SD.
Four years is a very long time—in children’s lives—between carrying out an
intervention in schools and looking for long-term effects in school achievement.
Remember that the organising hypothesis of the RCPCM research is that increasing
children’s cognitive level should give a permanent improvement to their learning
ability and hence later school achievement. With respect to the experimental classes
there is confirming evidence in Figures 6 & 7 from the KS2 English results. But other
factors can intervene subsequent to an initial intervention in schools, e.g. overall
school ethos; excellent (or otherwise) teaching, etc.
If Hattie’s (2008) ‘prior achievement’ is what can be assessed by a test of
intelligence or cognitive ability, then this has been raised by 0.71 SD for H&F and 0.6
SD for Bournemouth by 2004, with concomitant increases in achievement in English
at both KS1 and KS2. But the Bournemouth results for maths at KS2 require
comment and will be discussed below under Professional Development.
The process of Professional Development (PD)
Given the previous description of RCPCM classroom processes it follows that
simply ‘telling’ teachers how to do it was not an option. Just as each pupil—in
interaction with other children in their close environment—has to construct each
cognitive concept in their own mind, so too the PD of teachers needs a parallel
process. The teachers need that not only for the development of their own skills:
undergoing the same process themselves also gives them insight into managing their
pupils’ learning. In the H&F two-day PD sessions teachers were split into two groups
of 4: each group was given the teachers’ guides and materials for a new different TM
lesson. They were to discuss the lesson content and produce a lesson plan sufficiently
detailed so that they could then give and explain it to the others. After the groups had
shown each other the lesson plans, and answered questions arising, each group would
go to a different Primary school class accompanied by a member of the research team,
and team-teach one of their lessons. Later at the Teacher’s Centre each group would
report how their lesson plan had been realised, what difficulties they had encountered,
and further ideas for teaching the lesson. First thing next morning the two groups
would teach the other lesson, and then return to the Teachers’ Centre for feedback and
other PD agenda.
In addition each school was visited by one of the teacher-researchers to see
TM lessons taught and give the teacher personal feedback. Subsequent PD sessions
would begin with a detailed reporting by teachers of their experience of teaching the
lessons to their own pupils, and then followed by further work on new lessons as
before. Everything to do with the specifics of both the Piagetian and the Vygotskian
aspect of the art would be explored both in context and in terms of the teachers’ own
experience.
From what the two authors were able to see on visits to Bournemouth the PD
for Let’s Think in Y1 was well done: they did see the teaching in one school as well as
attending PD sessions. The teachers seemed to have learnt a lot from working with
groups of six children each day, and it may be that that the cognitive development of
the children was largely due to this, together with the information on children’s
learning difficulties that they may have applied to their conduct of other lessons. The
Bournemouth teachers found the mathematics more difficult to handle, and the two
authors had no time to assist in this in Bournemouth. The funding from ESRC did not
extend to paying for much of the PD in Bournemouth: thus less time was given to the
PD for the Thinking Maths lessons in Y1 & 2 than in H&F, and in particular the
collaborative learning between teachers planning and visiting each other’s lessons,
practiced in H&F, was not provided in Bournemouth. In addition, those attending the
PD sessions were the school Head and the lead teacher. Even had funding been
available it would still have been impossible for the 4 teachers from school D or the 3
teachers from school B (see Figure 1) to be absent together on the PD days. Even so
the experimentals did gain over the controls in KS1 maths: the subsequent four years
of teaching from Y3 to Y6 in the experimental schools appear to have favoured
learning in English rather than mathematics: this is probably Hattie’s ‘teacher effect’.
The research presented in this paper bears on two issues: the practice of PD and
also the practice of mathematics teaching. Evidence has been produced that
Government hopes for the improvement of standards in schools could be realized by
focus on and funding of the methods utilised in the research presented in this paper. In
the paper by Shayer, Coe & Ginsburg (2007) referred to earlier children’s grasp of
physical conservations at entry to secondary school—mainly related to their learning
experiences between the ages of 5 to 8—had deteriorated by an average of 0.79 SD in
comparison with 1976. The data in Table 4 and Figures 1 to 5 show the possibility
that this anti-Flynn effect could be reversed.
While it is true that the context of mathematics was used with the intention of
making the children better thinkers, as shown by the results for English for both H&F
and Bournemouth, it is claimed that the insights derived from the work both of Piaget
and Vygotsky into the learning difficulties of children is also a contribution to
mathematics education itself. But the contrast between the results in mathematics
between H&F and Bournemouth also shows that, without a simultaneous investment
in a collaborative style of PD, teachers are unlikely to construct, between each other
and for themselves, both the new insight into mathematics and also the teaching skills
they need to produce the better learning of their children.
Appendix
The interpretation of Vygotsky’s concept of the Zone of Proximal
Development (ZPD) used here is different from a view currently held widely, and
evident in Newman, Griffin & Cole (1989). They say (p59)
‘…the individual is not the most useful unit of analysis’
and that (p61)
‘We will use the more general interpretation of ZPD in these discussions,
referring to the zone as a location—in the teacher-child interaction—in which
new understandings can arise’
It is not denied that this approach usefully extends the Brunerian concept of
‘scaffolding’ in the direction of a general theory of instruction, but that it is a valid
interpretation of Vygotsky is questioned. Moreover it is inadequate to address the
deeper levels at which cognitive deficits are present in children. The individual child
is the only focus of the theory of ZPD, and unless this is understood Vygotsky’s
thinking on the social origins of much individual development cannot properly be
utilised.
The notion of the ZPD arose as Vygotsky developed a practice of dynamic
assessment of individual children intended to reveal far more of their current
spectrum of thinking that can be revealed in a static psychological test. ‘Mediation’
here means that, when the psychologist finds that on some aspect of thinking the child
does not succeed, he gives the child various signals intended to direct the child’s
perception to aspects of the task he needs to attend to before succeeding. The more
mediation that is required the less the child is on the path toward achievement, and yet
if success is attained a measurement of the child’s present position of partial
attainment has been recorded. By conducting this testing over important parts of the
psychological spectrum the child’s individual ZPD can be mapped in addition to what
he can do without mediation. Although Vygotsky left no detailed records one can see
ZPD specifics in all the published works of Piaget. By working with a sample of
children Piaget and his co-workers had recorded the many alternative steps through
which the ZPD of an individual child might move in the process of achieving a
concept. This process of dynamic assessment was extended by Feuerstein (1979) and
utilized in this country, in a replication of Feuerstein’s Instrumental Enrichment
programme, to assess the changes of children’s ZPDs over a two-year period (Beasley
& Shayer, 1990).
Thus, in addition to what can be revealed of children’s present competencies
by a psychometric test, the ZPD concept reminds us that each child has, in addition, a
spectrum of partially achieved competencies, some perhaps only a quarter others
three-quarters already completed. Vygotsky claimed that his ZPD measure better
predicted the next two years’ success of pupils than their psychometric test results
(Vygotsky, 1978, chapter 6). Only this view of the individual children enables us to
appreciate this widely quoted statement (Wertsch, 1979, p. 163):
We could formulate the general genetic law of cultural development as follows: Any
function in the child’s cultural development appears twice, or on two planes. First it
appears on the social plane, and then on the psychological plane. First it appears
between people as an interpsychological category, and then within the child as an
intrapsychological category. This is equally true with regard to voluntary attention,
logical memory, the formation of concepts, and the development of volition.
Vygotsky’ own gloss on this was:
Functions are first formed in the collective as relations among children and then
become mental functions for the individual. (Wertsch, 1979, p. 165).
and
Research shows that reflection [by the individual] is spawned from argument [in the
collective]. (Wertsch, 1979, p. 165, 5 lines later).
Thus in thinking about the social process of shared cognition that a group of
children can work within, while working on a problem, most if not all jumps in
children’s cognitive development are made when they witness a successful
performance in a child like themselves in their social milieu, and towards which they
may themselves have contributed. At that point they complete their ZPD by a process
of internalisation, and then perform as if they had always known that concept. The
conduct of the TM lessons is made so as to promote this process. The teacher is not a
psychologist and so cannot be asked to mediate in the style of a ZPD tester (even if
she were she could not possibly know all the pupils in this way). Instead she has a
different mediation role. Knowing that it is the acts and words of the other children in
the class that are closest to each child’s ZPD her mediating role is to maximise the
collaborative learning in the lessons—‘ reflection [by the individual] is spawned from
argument [in the collective]’— intervening only to give touches to the rudder as she
sees the children going off course or failing to address some important aspect of the
task.
6283 words to here
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