Jay and Betenson Mathematics at Your Fingertips

group where both sessions were delivered by the researcher. This A further set of measures were combined so that they could
additional input from the researcher was designed to remind the be administered to children in groups, as a series of pencil-and-
teachers of the activities and suitable mathematical language to paper tests, as follows:
use during the sessions, as had been practised during the training
days. Quantitative Skills Tests
In line with previous research, we expected the finger train- Number system knowledge was tested using an instrument
ing intervention to lead to improvement in finger gnosis scores based on the study by Gelman and Gallistel’s (1978) five implicit
(Gracia-Bafalluy and Noël, 2008), and the number games inter- principles of counting. The first page (count) required the pupil
vention to give rise to improvements in non-symbolic arithmetic to count numbers of objects which involved the one-to-one cor-
scores (Skwarchuk et al., 2014). We expected all groups bar the respondence principle where “one” and “two” are assigned to each
control group to show improvements in quantitative skills, with counted object, the abstraction principle which allows objects of
the highest level of improvement expected for those in the com- any kind to be collected together and counted and the cardinality
bined finger training and number games groups. principle where the final word tag defines the number of items
counted. The second page (dice) involved the recognition and
Participants the adding together of regular dot arrays displayed as dice. This
One hundred and thirty-seven children aged 6–7 years old took activity also involved the one-to one correspondence, abstraction,
part in the study. They were drawn from three primary schools in and cardinality principles. The third page (number) included the
a city in the South of England. Other than the age of the children, identification of the larger number of two, the ordering numbers
no specific criteria were used during recruitment other than an up to 102 and the placing of different numbers in the correct order
enthusiasm to take part in the research. The three schools are all on a number line. This was based on the stable order principle
larger than average sized primary schools, with pupils of similar, where the order of the word tags does not vary between sets of
diverse, ethnic backgrounds. The use of three schools allowed numbers. The fourth page (sequencing) involved the completion
for five different classes to be randomly allocated different ele- of number sequences going forwards and backwards in ones,
ments of the intervention program in order to reduce selection twos, fives, and tens. This also used the stable order principle. The
bias (see Table 1). final page (manipulation) involved splitting numbers into their
composite parts and creating a larger number from a combina-
tion of two or three smaller numbers. This incorporated the order
Instruments and Measures irrelevance principle where numbers can be put together in any
A set of measures were taken from all participants both before
sequence to make a given tagged number. The test pages used
and after the intervention sessions took place, in order to assess
both symbolic and non-symbolic representations of number and
different aspects of finger gnosis, symbolic number sense, and
were arranged on five different sheets of paper. One minute was
arithmetic fluency.
allowed for completion of each page. A maximum score of 126
Finger Gnosis Testing was possible on these five pages. Although the pages tested dif-
The finger gnosis test was administered on a one-to-one basis ferent aspects of counting, it was expected that there would be a
using a task adapted from the study by Gracia-Bafalluy and strong correlation between the scores for each of the five pages, as
Noël (2008). Each participant was asked to put their hand flat on each incorporated the five principles of counting. The tasks were
the table inside a box so that the participant could not see their all piloted and refined with children of the same age as the sample
fingers. The researcher then lightly pressed the pupil’s fingers in a of this study, and in collaboration with experienced teachers.
predetermined non-sequential order so that the pupil could say
Magnitude Comparison Test
the number corresponding to the finger which had been touched.
A 1 -min paper-and-pencil test of magnitude comparison was
Thirty trials were completed in the same manner. The same set of
used following a format from the study by Nosworthy et al.
trials was used for each pupil in order to ensure consistency and
(2013). Pupils were given 1 min to compare 20 pairs of dot arrays
to ensure no advantage was gained from a particular finger being
and to tick the larger set of dots each pair. The differences between
pressed more often than others. Participants’ responses were
the numbers of dots in each pair varied between one and six dots.
voice recorded so that they could be checked and collated after
The test was designed as a simple accuracy measure giving a
the testing had taken place. A correct answer was scored as 1 and
maximum total of 20 marks.
therefore a maximum total of 30 could be achieved on this test.
Procedure
TABLE 1 | Five experimental groups. All pupils completed the group-administered mathematics
achievement tests, which consisted of numeration and calculation
School 1 School 2 School 3
tests, and the magnitude comparison tests, prior to intervention
Class 1, 28 Class 2, Class 3, 27 Class 4, 28 Class 5, 27 in January 2014. These were delivered in their usual classroom
pupils 23 pupils pupils pupils pupils space with their teacher present to reduce any possible effects on
Teacher-led finger Control Teacher-led Teacher-led Researcher-led performance for children who found change in personnel or sur-
training and finger training number finger training and roundings distracting. The finger gnosis testing was administered
number games games number games individually in a quiet space outside the children’s classroom used

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Jay and Betenson Mathematics at Your Fingertips

for group work. This would reduce distraction from noises within revealed a significant effect of condition (F4, 128 = 16.71, p < 0.0005).
the classroom and yet to be in a space which was familiar to the Post hoc tests showed that both versions of the combined
students and therefore aid confidence. All tests were repeated after intervention (teacher-led and researcher-led) led to greater
the intervention sessions had been completed in March 2014. improvements in quantitative skills than the other three condi-
tions, which did not differ from one another (see Figure 1).
RESULTS The two versions of the combined intervention did not differ
significantly from one another in terms of improvement in
Analysis of Pretest Data quantitative skills.
The set of five tests used to measure quantitative skills showed In order to calculate an effect size for improvement in quantita-
a high level of reliability (five items; Cronbach’s α = 0.814). tive skills, the two combined intervention groups were combined,
Therefore, subsequent analysis used a composite “quantitative and compared with the three other groups combined; Hedges’
skills” score generated by adding scores from the five components. g = 1.32, a very large effect. See Table 2 for descriptive statistics
There was a significant positive correlation between finger gno- for the combined groups.
sis and quantitative skills at pretest (r = 0.33, n = 133, p < 0.0005), In order to determine whether the finger training aspect of
in line with previous research. There were also significant the intervention had been effective in improving finger gnosis
positive correlations between quantitative skills and magnitude scores, groups that received finger training were combined and
comparison (r = 0.37, n = 133, p < 0.0005), and between finger compared with those that did not. Descriptive statistics can be
gnosis and magnitude comparison (r = 0.171, n = 133, p = 0.05). seen in Table 3, and an independent t-test revealed a significant
However, the correlation between finger gnosis and magnitude difference in improvement in finger gnosis scores (t = 2.53,
comparison disappeared when quantitative skills were partialed df = 131, p = 0.013).
out (r = 0.058, n = 133, p = 0.511). In order to determine whether the number games aspect of the
intervention had an effect on non-symbolic magnitude compari-
Analysis of Posttest Data son, groups that received the number games intervention were
A one-way ANOVA was used to compare improvement in quan- combined and compared with those that did not. Descriptive sta-
titative skills between the five experimental conditions. This tistics can be seen in Table 4, and an independent t-test revealed a

FIGURE 1 | Improvement in numeracy score by experimental group.

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Jay and Betenson Mathematics at Your Fingertips

TABLE 2 | Improvement in quantitative skills—combined intervention Prior to the intervention taking place, the pretest data showed
compared with other groups.
a correlation between finger gnosis and number sense. This sup-
Condition N Mean SD SE ports previous findings from the study by Fayol et al. (1998) and
Andres et al. (2012). The findings do not fully corroborate the
Combined intervention (researcher-led and 55 19.80 9.02 1.22
teacher-led)
findings of Gracia-Bafalluy and Noël (2008), who found that an
Other groups combined (control, finger training 78 8.56 8.11 0.92 8-week finger training intervention alone improved quantifica-
only, number games only) tion skills in a sample of 6–7-year-old children (approximately the
same duration and age-range as in the current study). A possible
explanation for this difference is that Gracia-Bafalluy and Noël
TABLE 3 | Improvement in finger gnosis score—those that received finger selected participants who had poor finger gnosis at the outset of
training compared with other groups. their study, while the present study included participants with a
Condition N Mean SD SE full range of initial levels of finger gnosis. However, the findings
presented here do agree with the main findings of Gracia-Bafalluy
Groups that received finger training 82 1.90 4.26 0.47 and Noël, in that a finger training intervention—when combined
Other groups combined (control, number 51 0.16 3.13 0.44
games only)
with the number gains intervention—did lead to gain in finger
gnosis, and that these gains were accompanied by gains in quan-
titative skills.
The findings of the present study do not fully align with
TABLE 4 | Improvement in magnitude comparison score—those that
received the number games intervention compared with other groups.
those of the study by Siegler and Ramani (2008), Skwarchuk
et al. (2014), and others who have found that playing games
Condition N Mean SD SE involving symbolic representations of numbers can improve
Groups that received number games 83 3.08 4.30 0.47 children’s numerical knowledge. However, Siegler and Ramani
Other groups combined (control, finger training only) 50 1.3 4.03 0.57 (2008) worked with a sample of low-income children with low
levels of numerical knowledge at the outset of their experiment,
which again differed from the range of participants included in
significant difference in improvement in magnitude comparison the present study. Again, though, in combination with the finger
scores (t = 2.38, df = 131, p = 0.019). training, the number games intervention did lead to gains in
quantitative skills.
Summary The findings presented here suggest that for an interven-
The results show that the finger training aspect of the intervention tion to be successful in increasing children’s quantitative
was effective in improving participants’ finger gnosis scores, but on skills—when the children are starting within the normal range
its own was not effective in improving scores on the quantitative of ability—then the intervention should involve a combination
skills test. Similarly, the number games aspect of the intervention of number representations, rather than one particular set of
was effective in improving non-symbolic magnitude comparison representations.
scores, but on its own was not sufficient to improve quantitative Confidence in the above interpretation is added by the fact
skills. The two versions of the intervention that combined both that the finger training intervention (but not the number games
the finger training and number games aspects were successful in intervention) was shown to improve participants’ finger gnosis
improving participants’ quantitative skills relative to controls, and scores, and the number games intervention (but not the finger
with a large effect size. training intervention) was shown to improve non-symbolic
magnitude comparison scores. This supports the argument that
DISCUSSION AND CONCLUSION while both aspects of the intervention have a potential role to play
in supporting children’s learning, it is only in combination that
The findings show that the combined intervention, incorporating they can be shown to improve children’s quantitative skills. We
both finger training and symbolic number games, gave rise to argue that this provides evidence for the functional hypothesis,
significant improvements in participants’ numeration scores. regarding the relationship between finger gnosis and quantitative
Neither intervention alone had an effect on numeration scores. skills (Butterworth, 1999), whereby fingers act as a bridge, or
This is an important and original contribution to knowledge in mediator, between other representations of number.
this field, as this combination of interventions has not been tested
before, to our knowledge. This finding suggests that children’s Multiple Components of Numerical
developing number sense is best supported by experience of a Understanding
combination of representations of number—in this case includ- Further consideration is needed here, of possible mechanisms
ing fingers plus verbal, symbolic and non-symbolic representa- to explain the fact that the finger training and number games
tions—rather than by a particular set in isolation. Confidence in interventions led to significant increases in quantitative skills
the finding is added by the fact that the intervention led by a class’s when combined, but not in isolation. It will be useful to draw
usual teacher showed a similar increase in number sense as did on previous research relating to the complexity of numerical
the group led by the researcher. understanding, and its componential nature.

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Jay and Betenson Mathematics at Your Fingertips

One possibility is that the effects were additive, and only therefore not possible to know whether the gains in quantitative
reached significance in combination. This is somewhat unlikely skills demonstrated by the two groups participating in the com-
as each intervention in isolation led to levels of quantitative skills bined intervention would have persisted long enough to show
that were very close to those of the control group (Figure 1). an effect on a delayed test. If an intervention such as this is to
However, although differences were not significant, the mean be useful in a school context then it will be important to show
scores of the finger training-only group and the number games- both that gains persist and that they contribute to a more flexible
only group were higher than that of the control group, and so it foundation for further learning.
is not possible to discount this possibility without replication of Despite the limitations of the study, we argue that it has pro-
the current study. vided promising evidence that a combined finger training and
A second possibility is that the combined intervention led to number games intervention can contribute to young children’s
better results as it was more likely to match children’s particular quantitative skills and their developing mathematical under-
needs. Dowker (2005, 2008) has shown that there are several standing. We understand that further research will be needed in
components involved in children’s developing understanding order to fully determine the underlying mechanisms by which the
of number and arithmetic and that intervention is most effec- interventions leads to gains in skill, and to add confidence in the
tive when it addresses the particular component that child is effectiveness of the intervention, but argue that sufficient evidence
experiencing difficulty with. In the case of the current study, it is has been gained from this study to warrant such further work.
possible that some children may have needed more intervention
in finger gnosis and others in symbolic number manipulation or Conclusion
magnitude comparison. This study has shown that an intervention that combines finger
A third possibility is that the combined intervention helped training with number games can improve quantitative skills
children to make connections between representations of among 6–7-year-old children. It supports the findings of previous
number. This possibility follows from the functional hypothesis research arguing for a functional relationship between finger gno-
regarding the relationship between finger gnosis and quantitative sis and numeracy. We argue that this study provides evidence that
skills. Children generally need explicit exposure to relationships fingers represent a means for children to bridge between other
between numerical phenomena or relationships in order to (verbal, symbolic, and non-symbolic) representations of number
internalize them (Fuson, 1986). It may be that the combination of and that this contributes to children’s developing understanding.
activities in the intervention helped make relationships between The large effect size suggests that with further refinement and
different representations of number for participating children. replication, the combined finger training and number games
This hypothesis would place the current study within a growing intervention could be a useful tool for teachers to use to support
body of work showing that children often find it difficult to make children’s developing understanding of number.
connections between different aspects of number. Goffin and
Ansari (2016), for example, show that children’s judgments of ETHICS STATEMENT
cardinality and ordinality of number independently predict indi-
vidual differences in arithmetic fluency, and that there is a lack The study was approved by the ethics committee of the Graduate
of relationship between them. Similarly, De Smedt et al. (2013) School of Education, University of Bristol. Opt-out consent was
noted a lack of relationship between skills in processing symbolic gained from all parents of children who participated. Informed
and non-symbolic representations of number. Further work is consent was gained from head teachers and classroom teachers
needed in order to determine which relationships are necessary of all children who participated.
for the development of expertise with number, and how children
experience barriers and enablers for these in their learning. AUTHOR CONTRIBUTIONS
Limitations The study reported in this article was carried out as part of
Each of the five experimental groups comprised children who Dr. Betenson’s doctoral studies at the University of Bristol,
normally worked together as a class. This means that there may supervised by Dr. Jay. Dr. Jay adapted the article from
­
have been unobserved intra-cluster factors that affected learning Dr. Betenson’s thesis, carrying out a reanalysis of data and
and performance. For example, children in one class could con- ­additional literature review.
ceivably respond more positively or more flexibly to an interven-
tion than those in another class, with another teacher. A future FUNDING
fully randomized study could address this issue and provide more
convincing evidence for the effectiveness of the intervention. The research reported in this article was carried out as part of the
A second limitation relates to the fact that posttests were second author’s doctoral study, funded by the UK Economic and
carried out soon after the last session of the intervention. It is Social Research Council at the University of Bristol.

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Gracia-Bafalluy, M., and Noël, M. P. (2008). Does finger training increase Conflict of Interest Statement: The authors declare that the research was con-
young children’s numerical performance? Cortex 44, 368–375. doi:10.1016/j. ducted in the absence of any commercial or financial relationships that could be
cortex.2007.08.020 construed as a potential conflict of interest.
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competencies in children with specific mathematics difficulties versus children Copyright © 2017 Jay and Betenson. This is an open-access article distributed under
with comorbid mathematics and reading difficulties. Child Dev. 74, 834–850. the terms of the Creative Commons Attribution License (CC BY). The use, distribu-
doi:10.1111/1467-8624.00571 tion or reproduction in other forums is permitted, provided the original author(s)
LeFevre, J. A., Fast, L., Skwarchuk, S. L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., or licensor are credited and that the original publication in this journal is cited, in
et al. (2010). Pathways to mathematics: longitudinal predictors of performance. accordance with accepted academic practice. No use, distribution or reproduction is
Child Dev. 81, 1753–1767. doi:10.1111/j.1467-8624.2010.01508.x permitted which does not comply with these terms.

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