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Counting with your eyes
Can poor arithmetic students subitize more efficiently under
time pressure?
Author: Michal ter Kuile Project: Bachelor thesis Student number: 10729623 Supervisor: Ingmar Visser
Faculteit van Maatschappij- en Gedragswetenschappen (Faculty of Society and Behavioral Sciences)
Program group: Clinical developmental psychology Date: 29-04-2018
Word count: 5178
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Abstract
Subitizing is defined as the ability to recognize a number of items displayed for only a very brief time without counting. Subitizing ranges increase from two items at a young age to around four items in adulthood and the average speed increases as well. Differences in subitizing range and speed within a group of same aged children predict later arithmetic skills and explain variance in counting skills. This study tests the hypotheses that 1) subitizing range predicts arithmetic skills, 2) poor subitizers have a less effective subitizing strategy and 3) poor arithmetic students can subitize more efficiently under time pressure. Methods: 70 subjects between age 5 and 8 were tested on subitizing tasks. 57 subjects were tested with a mouse tracker, 13 subjects were tested with an eye tracker. The subjects were given a screen with a number of dots which either
disappeared after 300ms (time limited trials) or disappeared when the subject pressed the mouse (time unlimited trials). Arithmetic skills were calculated for each subject tested with the mouse tracker. Results: Subitizing range did not predict arithmetic skills. Poor subitizers had a less effective subitizing strategy measured as number of eye movements when time was not limited. Introducing a time limit helped all subjects to subitize larger numbers and poor subitizers showed the biggest improvement. Conclusions: hypothesis 1 had to be rejected. Hypothesis 2 was
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Index
1. Introduction 4 2. Methods 6 2.1. Participants 6 2.2. Inclusion/exclusion criteria 72.3. Measurements and procedure 7
3. Materials 8
3.1. Interface 8
3.2. Display 8
3.3. CITO and DLE 10
4. Results 10
4.1. Preliminary analyses 10
4.1.1. Different groups 10
4.1.2. Outliers 11
4.1.3. Correlation mouse/eye data 11
4.1.4. Descriptive statistics and assumptions 11
4.1.5. Figures 12 4.2. Hypothesis 1 16 4.3. Hypothesis 2 19 4.4. Hypothesis 3 22 4.5. Explorative analyses 23 5. Conclusions 27 6. Discussion 27 7. Literature 32 8. Appendix 36 I. counting test 36
II. Time limited trials stimuli 38
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1. Introduction
As the role of technology in our daily and professional lives increases, mathematical skills have become progressively more important on the work floor. Economically advanced countries like the US give a growing number of work permits to workers in the technical fields (Furner & Berman, 2005). In such an increasingly technology orientated world mathematical skills are a prized possession. Therefore, developing these skills benefits also those who do not have natural talents in this field. For example, many college students have math anxiety to some degree (Betz,1978) and between 3% and 8% of elementary school children tested in Belgium had mathematical learning disabilities (Desoete, Roeyers, & De Clercq, 2004). These data suggest that however wanted math skills are, for many they are not easy to acquire. Therefore it is crucial to understand the causes of difficulties with mathematical and arithmetic tasks and if they can be alleviated.
One predictor for mathematical skills is the ability to subitize, defined as the ability to swiftly identify small numbers of items without explicit counting. A pattern of low subitizing speed and small subitizing range has been found in practically all children with a diagnosis of dyscalculia (e.g. Schleifer, & Landerl, 2011; Landerl, 2013). This same pattern is also seen in students with poor arithmetic skills without diagnosed dyscalculia (Reeve, Reynolds,
Humberstone, & Butterworth, 2012; Moeller, Neuburger, Kaufmann, Landerl, & Nuerk, 2009). Subitizing has been the topic of multiple studies concerning mathematical abilities because speed of subitizing and subitizing range correlate significantly with early number sense and explains around 22% of variance in counting skills in five and six year olds (Kroesbergen, Van Luit, Van Lieshout, Van Loosbroek & Van de Rijt, 2009). Along this line, subitizing speed at age six predicts arithmetic abilities at ages 7, 8.5 and 9(Reeve, Reynolds, Humberstone, & Butterworth, 2012).
Adults subitize in around 50msec/item up to three/four items (subitizing range), with the reaction time (RT) going up substantially above three/four items to around 300 msec/item in the counting range (Chi & Klahr, 1975). Children show the same patterns of low RT while subitizing and RT going up at around three items. Children as young as two start to subitize and the normal patterns of development result in age related differences in speed of subitizing between younger and older children and adults (Svenson & Sjöberg,1983).
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dyscalculia show impaired subitizing and the fact that no single underlying neurological cause or mechanism has been found for dyscalculia (Rapin, 2016) indicates that mathematical problems are not a dichotomy but rather a scale. This is vital because it means that it is probable that certain problems of children with mathematical difficulties or dyscalculia might be alleviated by training as they are not dictated by a single unalterable cause.
As concluded above there are differences in subitizing skills between good and poor arithmetic students. However, there is no conclusive information on where these differences stem from.
Eye tracker data can reveal differences in how children solve subitizing problems and thereby help us understand some of the underlying mechanisms of arithmetic problems. Eye tracker data suggest that a close relationship exists between saccadic frequencies and RT on subitizing tasks. More saccadic movements lead to higher RT’s (Schleifer, & Landerl, 2011), indicating that children who use more saccadic movement are slower at subitizing. This suggests that children with poor arithmetic skills use a less effective subitizing strategy than good
arithmetic students.
If indeed poor arithmetic students are also poor subitizers, the question remains whether they can learn how to subitize more efficiently. Children with dyslexia can learn to read quicker when forced to read within a restricted time (Snellings, van der Leij, de Jong & Blok, 2009). In a similar vein Groffman (2009) shows that training children to subitize under time restriction resulted in better basic arithmetic skills. This would mean that not only can subitizing abilities be improved, but also that improving subitizing abilities helps with arithmetic skills. Groffman’s study however had only nine participants, all of them with severe arithmetic deficits. In our study we want to verify if poor but not (necessarily) dyscalculic students subitize better under time restriction and if this is due being forced to subitize more efficiently.
The question whether poor arithmetic students subitize more efficiently under time pressure shall be answered by the following (sub)questions:
Question 1) Does subitizing range predict arithmetic skills? Hypothesis 1) subitizing range predicts arithmetic skills
This question aims to verify if being a poor subitizer leads to being a poor arithmetic student, as research suggests.
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Question 2) Does number of fixations predict subitizing range? Hypothesis 2) poor subitizers have a less effective subitizing strategy
This question aims to verify whether more eye movements lead to lower subitizing range, thus showing that poor subitizers have a less effective subitizing strategy.
Question 3) do poor subitizers show higher subitizing ranges in time limited trials versus
unlimited time trials?
Hypothesis 3) poor subitizers will benefit from a time limit in subitizing
This question aims to explore the possible benefit of time limit and thereby the trainability of subitizing range.
2. Methods
2.1. Participants
Children from a single Dutch elementary school1 took part in the experiment for the mouse tracking data. The children were from kindergarten (N=17, 9 girls), first grade (N= 20, 11 girls) and second grade (N=20, 13 girls). The mean ages of the groups were M = 69.29 months (SD=4.67) for the kindergarteners, M = 82.20 months (SD=4.79) for the first graders and M = 92.25 (SD=4.08) for the second graders. For the eye tracking data children from multiple schools participated (N=9, 5 girls), with a mean age of 77.78 months (SD= 15.44. These children were asked to participate through sending emails to parents from different schools and asking parents known to the researchers.
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2.2. Inclusion/exclusion criteria
Inclusion criteria were an age between five and eight and for the mouse tracking students having a CITO score available (explained at materials). Exclusion criteria were not being able to do the counting check (counting 15 dots from a piece of paper, appendix 1).
2.3. Measurements and procedure
Eye tracker data and mouse tracker data were used to measure fixations. As Chen, Anderson, & Sohn (2001) have demonstrated that eye movement and mouse movements have a strong relationship, this enables the use of a mouse tracker to quantify eye movement. For this reason a mouse tracker was used during field measurements in schools and an eye tracker in the lab. Mouse tracking was also done in the eye tracker group to calibrate eye tracker and mouse tracker data.
In the school subitizing and counting tasks were completed in a quiet room. The
participants saw displays of dots on a laptop and had to say how many dots there were. The task started with time limited trials. In these trials different numbers of dots were displayed for 300ms, after which the child had to press the number of dots they thought they had seen on a screen that appeared after the dots. In time unlimited trials different numbers of dots were displayed on the screen, but this time the child had to press on the mouse when they knew the answer. Once a child pressed the mouse automatically the dots disappeared and a screen
appeared with different answer options. The child had to press what they thought was the correct number of dots they had just seen. After all the trials each child was asked to count dots of a page to see if they could count the maximum number of dots that were displayed during the measurement (15).
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Reaction time was measured on the unlimited time trials by measuring the time between the dots appearing on the screen and the moment that the child pressed the mouse. The
eye/mouse tracker was used to determine the number of fixations made per trial.
The procedure for the eye tracking group was the same except that the test was conducted in a dark and quiet room in the lab and the participants first had to look at a number of individual dots on the screen to calibrate eye movements.
3. Materials
3.1. Interface
The stimuli for the mouse tracker were presented on a Dell laptop (Latitude E5540) with a resolution of 1280x1024 pixels. The participant was holding the mouse while completing the tasks in both the eye tracker and mouse tracker condition.
3.2. Display
In both the mouse tracking and eye tracking conditions the dots were 2 mm with a
viewing distance of 60 cm and a minimal distance between dots of 1 cm with maximum distance of 7 cm. A 9x9cm light grey screen was presented in the middle of the screen, surrounded by a dark grey background.
There were 60 displays of different amount of dots. As this research was part of a larger study trials with up to 15 dots were used. This study only used the trials with 1 to 5 dots (30 trials, appendix 2). The dots were either randomized or in a pattern (no dice patterns were used). This study did not focus on the potential influence of the pattern of the dots, only on RT,
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Table 1
Number of trials per number of dots in time limited trials
Random Pattern 1 dot 1 0 2 dots 2 0 3 dots 3 0 4 dots 3 0 5 dots 1 0 Total 10 0 Table 2
Number of trials per number of dot in time unlimited trials
Random Pattern 1 dot 1 0 2 dots 3 2 3 dots 3 2 4 dots 3 2 5 dots 2 2 Total 12 8
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3.3. CITO and DLE
The ‘CITO rekenen’ test was used to quantify arithmetic skills. The CITO is a Dutch test that measures arithmetic skills from kindergarten to sixth grade. As CITO scores go up with age they were recalculated into DLE scores. These are age equivalent scores that state whether a child’s arithmetic ability is good, sufficient, poor or insufficient for their grade. There were no CITO/DLE scores available for the eye tracking group.
4. Results
4.1. Preliminary analyses
Because the eye tracking group only had nine participants, it was decided to analyze their data as explorative due to low power. Because of this descriptive statistics and figures were performed on the data from the mouse tracker group.
4.1.1. Different groups
Because children from different classrooms were tested (for each grade two clasrooms within the same school) an independent samples T-test was used to look at differences between the different kindergarten, first grade and second grade classrooms. There were no differences between different classrooms of the same grade in age, CITO scores, subitizing range limited time and unlimited time and average reaction time. In the kindergarteners and second grade no differences on DLE between classes was found. The two first grade classes differed significantly on DLE scores (t(18) = 2.36, p = .017). However, as we only tested 4 children from classroom A
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and 16 from class B, the differences might be due to chance. CITO scores, on which DLE is based on, did not differ between these two classrooms. Therefore we chose not to correct for this.
4.1.2. Outliers
In the eye tracking group the mouse tracking data of two participants could not be used because they didn’t operate the mouse themselves. These participants were excluded from all further analyses with mouse tracking data. No further outliers were removed.
4.1.3. Correlation mouse/eye data
For the correlation between mouse tracking data and eye tracking data a Pearson correlation was performed using number of fixations as measured by the eye tracker and the mouse tracker. Number of fixations over all trials was used, including the trials not used in this research and wrong answers.A correlation of 0.71 was found, p =.073. These results are not significant, but as there were only 7 participants who could be used to verify the correlation, a correlation of 0.71 was deemed enough.
4.1.4. Descriptive statistics and assumptions
Table 3 shows descriptive statistics for the reaction time (RT), DLE, and subitizing range for time limited and unlimited trials. Assumptions were checked and all assumptions were met. For RT we looked only at the time limited trials and took the mean of the RT of all correctly answered trials.
To calculate subitizing range (SR) for the time limited trials it was assumed that children stop subitizing if they make more than one mistakes with a specific number of dots. The
exception being five dots as there was only one trial of that number, so children had to succeed in that single trial to qualify for a subitizing range of five.
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calculated based only on correctly answered trials. It was assumed that children stop subitizing if the mean differences of reaction time between two consecutive dots was more than 1200ms. If a child started at one dot with a reaction time of 3200ms or higher the subitizing range was
counted as 0.
Mean fixations were calculated from all correctly answered trials.
Table 3
Means of all variables in different groups
Variables Reaction time DLE Subitizing range Subitizing range Fixations Fixations Condition Time unlimited NA Time limited Time unlimited Time limited Time unlimited Mean 4444ms 2.51 3.8 2.3 4.57 3.42 SD 1577ms 1.2 1.29 1.62 1.3 2.14 4.1.5. Figures
Figures were used to verify expected differences per group. As seen in figure 1, mean subitizing range goes up with age. Therefore we controlled for age in all analyses of hypotheses 1 and 2. Hypothesis 3 is based on the differences in subitizing range between the same students under different conditions, so age did not have to be controlled for in that case.
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Mean subitizing range per grade
Figure 2 shows a slight reduction of mean fixations per trials when subitizing range goes up.
Figure 2
Mean fixations per group of participants with a certain subitizing range
0,94 2,85 3 0 0,5 1 1,5 2 2,5 3 3,5
kindergarten(17) first grade(20) second grade(20)
m ea n su bi tiz in g r an ge
grade (number of participants in grade)
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As was to be expected from previous studies, reaction time in time unlimited trials goes up per dot (figure 3).
Figure 3
Mean reaction time per dot in time unlimited trials
4,8 3,9 4,3 3,8 3,4 3,4 0 1 2 3 4 5 6
zero(14) one(4) two(6) three(18) four(12) five(3)
mea n fix at oi ns p er tr ia l
subitzing range(amount of participants with this subitzing range)
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As seen in figure 4, CITO scores go up by grade. This is expected as CITO scores are not standardized per grade.
Figure 4
Mean CITO scores per grade
3008 3429 3755 5091 6912 0 1000 2000 3000 4000 5000 6000 7000 8000 1 2 3 4 5 mea n rea ct io n t ime number of dots
Reaction time
84,88 108,67 158,61 0 20 40 60 80 100 120 140 160 180kindergarten first grade second grade
mea n C IT O s co res grade
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DLE scores did not differ significantly per grade (figure 5). This is to be expected as DLE scores are standardized by correction for age.
Figure 5
Mean DLE per grade
All differences observed in the figures are expected and do not need to be controlled for.
4.2. Hypothesis 1: subitizing range predicts arithmetic skills
To establish whether subitizing range predicts arithmetic skills a hierarchical regression was calculated to try to predict DLE based on subitizing range (SR) and age in months. The prediction is that a lower subitizing range predicts lower DLE score. Two two-stage hierarchical regression analyses were conducted with DLE as the dependent variable. Age in months was
3 2,1 2,5 0 0,5 1 1,5 2 2,5 3 3,5
kindergarten first grade second grade
mea n DL E s co re grade
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entered at stage one of the regression to control for children being better at subitizing with
increasing age. SR for time (un)limited trials was entered at stage two. The results shown in table 4 show that subitizing range for neither limited nor unlimited time trials significantly predict DLE scores.
Table 4
Results of hierarchical regression predicting arithmetic skills based on subitizing range
F R2 p
Time limited trials Age in months F(1,55)=2.33 .04 .13
Subitizing range F(1,54)=1.33 .02 .25 Time unlimited trials Age in months
Subitizing range F(1,55)=2.33 F(1,54=.22 .04 .004 .13 .64
Figure 6 shows explained variance in DLE scores of age in months and subitizing range of (un)limited time trials.
Figure 6
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To visually distinguish the differences in DLE scores between poor and good subitizers
the group was divided in poor subitizers and good subitizers. This was done by using the mean subitizing range (SR) of unlimited time trials as a cut-off point. As the mean SR was 2.33 for time unlimited trials, pupils with a subitizing range of 0 to 2 were classified as poor subitizers, whereas pupils with a subitizing range of 3 to 5 were classified as good subitizers. As seen in figure 7, DLE scores for poor and good subitizers are practically the same, with a slightly higher score for poor subitizers.
Figure 7
Mean DLE scores for poor and good subitizers
4,1 4,1 2,3 0,4 0 1 2 3 4 5 6 7
limited time unltimited time
% ex pl ai ned v ar ia nc e of D LE sc or es
explained variance of DLE
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The hypothesis that subitizing range predicts arithmetic skills is to be rejected.
4.3. Hypothesis 2: poor subitizers have a less effective subitizing strategy
To establish whether poor subitizers have a less effective subitizing strategy, number of fixations was used to try to predict subitizing range. Previous research investigated saccadic frequency (e.g. Moeller et al., 2009) , but as saccades and fixations are practically the same, number of fixations will be used as a quantifier of saccadic frequency. The prediction is that a larger number of fixations will correspond to a lower subitizing range.
A hierarchical regression was calculated to predict subitizing range of time (un)limited trials based on number of fixations and age in months. Fixations were measured by the mouse tracker. The mean fixations across all correctly answered time (un)limited trials were used in the analyses. Two two-stage hierarchical regressions were conducted with subitizing range in (un)limited time trials as the dependent variable. Age in months was entered at stage one of the
2,7 2,4 0 0,5 1 1,5 2 2,5 3
poor subitizers(24) good subitizers(33)
mea n DL E s co res
group(number of participants in group)
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regression. Number of fixations in (un)limited time trials were entered at stage two. Mean
fixations of limited time trials were used to predict subitizing range of limited time trials and vice versa.
The number of fixations for the time limited trials does not significantly predict SR for the time limited trials (table 5). In contrast, number of fixations for the time unlimited trials did significantly predict subitizing range for the time unlimited trials. More fixations correspond to a lower subitizing range (figure 8).
Table 5
Results of hierarchical regression predicting subitizing range (un)limited time based on number of fixations in (un)limited time trials
F R2 p
Time limited trials Age in months F(1,55)=6.81 .01 .01
Fixations F(1,54)=.04 .001 .84
Time unlimited trials Age in months Fixations F(1,55)=16.57 F(1,54=.009 .23 .09 <.001 .009
Figure 8 shows explained variance of subitizing range of (un)limited time trials of age in months and number of fixations.
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Figure 8
Explained variance of subitizing range (un)limited time trials
The number of fixations was higher for poor subitizers (figure 9). This means that a larger number of fixations predicts a lower subitizing range in the time unlimited trials.
Figure 9
Mean fixations for poor and good subitizers
11 23,3 0,1 9,1 0 5 10 15 20 25 30 35
limited time unlimited time
% e xp la in ed v ar ia nce o f s ub itz in g r an ge
explained variance of subitizing range of
(un)limited time trials
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The hypothesis that poor subitizers have a less effective subitizing strategy is partially supported by the data.
4.4. Hypothesis 3: poor subitizers will benefit from a time limit in subitizing
To verify whether poor subitizers will benefit from a time limit the differences between subitizing range (SR) of the time unlimited and limited trials was calculated. The differences in difference of SR between poor and good subitizers was calculated with an independent T-test. The SR for the poor subitizers went up significantly more in the time limited range than the SR of good students goes up, t(55) = 5.22, p < .001 (calculated from the data of figure 10).
Figure 10
Mean subitizing range time (un)limited trials for poor and good subitizers
4,5 3,6 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5
poor subitizers good subitizers
mea
n fi
xa
tio
ns
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The hypothesis that poor subitizers will benefit from a time limit in subitizing is supported by the data.
4.5. Explorative analyses
Two two-stage hierarchical regressions with participants split into grades were conducted to see whether subitizing range predicts CITO scores. CITO scores were used as the dependent variable. Age in months was entered at stage one of the regression. Subitizing range in
(un)limited time trials were entered at stage two. Results in table 6 show that subitizing range of (un)limited time trials do not predict CITO scores. The percentage of explained variance is seen in figure 11. 0,67 3,5 3,3 4,2 0 0,5 1 1,5 2 2,53 3,54 4,5
poor subitizers good subitizers
m ea n su bi tiz in g r an ge group
subitizing range (un)limited time trials
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Results hierarchical regression predicting CITO scores based on subitizing range (un)limited time
R² F p
Kindergarten Explained variance of age .01 F(1,15)=.17 .69 Explained variance of subitizing
range limited time
.12 F (1,14)=1.94 .19
Explained variance of subitizing range unlimited time
.19 F (1,14)=3.34 .09
First grade Explained variance of age .06 F(1,16) = 1.0 .33 Explained variance of subitizing
range limited time
.08 F (1,15) = 1.4 .25
Explained variance of subitizing range unlimited time
.00 F (1,15) = .01 .94
Second grade Explained variance of age .03 F(1,16) = .51 .49 Explained variance of subitizing
range limited time
.01 F(1,15) = .07 .79
Explained variance of subitizing range unlimited time
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Percentage of explained variance of CITO scores
To see whether fixations as measured by eye tracker predict subitizing range, a
hierarchical regression was calculated. Analyses were only performed on eye tracking data, not on mouse tracking of the same participants, as the correlation between mouse tracking and eye tracking data was deemed high enough. Two two-stage hierarchical regressions were conducted to see whether number of fixations as measured by eye trackers predicts subitizing range. Subitizing range of (un)limited time trials was used as the dependent variable. Age in months was entered at stage one of the regression. Number of fixations of (un)limited trials were entered at stage two. Number of fixations of limited time trials was used to predict subitizing range of
1,1 6,1 3,1 12 8,1 0,5 19 0 0 0 5 10 15 20 25 30 35
kindergarten first grade second grade
% ex pl ai ned v ar ia nc e o f C IT O s co res grade
explained variance of CITO scores
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Number of fixations as measured by eye tracker data did not significantly predict subitizing range in unlimited time trials (table 7). However, for the limited time trials eye tracking data did predict subitizing range. Figure 12 shows explained variances.
Table 7
Results hierarchical regression predicting subitizing range time (un)limited trials based on number of eye fixations
R² F p Time limited trials Explained variance of age .001 F(1,7)=0.08 .79 Explained variance of fixations .49 F(1,6)=5.9 .05 Time unlimited trials Explained variance of age .17 F(1,7)=1.48 .26 Explained variance of fixations .10 F(1,6)=.84 .40 Figure 12
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5. Conclusions
This study has tried to verify if subitizing range could potentially be improved for poor arithmetic students by using a time limit. The results show that the subitizing range did not predict arithmetic skills. Furthermore when time was not limited the poor subitizing students showed a less effective subitizing strategy than the good subitizers. Imposing a time limit helped all students to reach a higher subitizing range. However the poor subitizers were most helped by the time constraint.
6. Discussion
This study focuses on three hypotheses. The first, that subitizing range predicts arithmetic skills, must be rejected, as bad subitizers turned out not necessarily to be bad arithmetic students.
1,1 17,5 49 10,1 0 10 20 30 40 50 60
time limited trials time unlimited trials
% e xp la in ed v ar ia nce o f s ub itz in g r an ge
explained variance of subitizing range
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This result contradicts previous research that has repeatedly found a relationship between subitizing skills and arithmetic skills (e.g. Kroesbergen et al., 2009; LeFevre et al., 2010; Reeve et al., 2012). When groups were analyzed per grade with CITO scores, subitizing range also did not predict arithmetic skills. Possibly DLE scores are not precise enough (going only from 1 to 4) and there were not enough participants per grade to yield significant results with CITO scores. However, in kindergartens a much larger part of variance in CITO was accounted for by the different subitizing ranges than in first and second grade. The explanation for this observation could be that subitizing range truly is a better predictor of arithmetic skills in young children or that the way arithmetic skills for kindergarteners is tested is substantially different and much closer to testing subitizing skills than the arithmetic tests for first grade and above are. This remains to be verified. In a similar note, observations on larger groups of participants per grade could establish whether DLE is not precise enough and CITO is or that subitizing range truly does not predict arithmetic skills. If subitizing skills actually do not predict arithmetic skills, this would implicate that improving subitizing skills of poor subitizers may not produce any further improvements in arithmetic skills. As these results are contradicted by extensive previous
research (e.g., Moeller et al., 2009; Landern, 2013) the relationship between subitizing skills and arithmetic skills must be verified.
The second hypotheses that bad subitizers have a less effective subitizing strategy as seen in number of fixations, was partially in agreement with the data. Poor subitizers had more
fixations (thus more eye movements) than good subitizers when time was not limited. This pattern did not show when time was limited. A cautionary note here is that in some of the unlimited time trials the dots were placed to form a pattern (other than a dice pattern). It is possible that good subitizers were helped by this pattern more than poor subitizers and that this
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has contributed to the differences. However since these were unfamiliar patterns, that could not yet be automated, and as Van Viersen, et al., (2013) show that poor and good arithmetic students differ in ability to automate, it is unlikely to have been the sole cause of the differences between poor and good subitizers. Eye tracking data suggested the exact opposite conclusion, but as the reliability of those analyses was questionable because of low power, these results were not incorporated in the overall analysis. Schleifer, & Landerl, 2011 already show that poor subitizers have more eye movements, which is in agreement with the data. However, their research does not provide evidence as to why this would not be the case under time pressure. The increased efficiency under time limitation that was documented in this study suggests that having a time limit helps poor subitizers to develop a more effective strategy. A cautionary note to the above conclusion is that because of the time limit, the time for eye/mouse movements was artificially reduced, which naturally results in fewer fixations. Because of this it cannot be stated with confidence that a reduction in the number of fixations points to a more effective subitizing strategy.
Hypothesis three, that poor subitizers will benefit from being forced to subitize faster by a time limit, was in agreement with the data. Similar observations were made by Groffman (2009). Subitizing range went up under time pressure for all students, although the poor
subitizers showed the biggest improvement. A possible explanation could be that due to a ceiling effect that applies to good subitizers these are simply not able to subitize higher numbers.
However, it is still useful to establish that a time limit leads to higher subitizing ranges, because improving subitizing skills can help to improve arithmetic skills (Groffman, 2009).
The reason that all participants, and especially poor subitizers, showed an improvement of subitizing range in time limited trials has not been addressed in this study. The possibility of
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experimental bias must be considered. On the one hand, it is possible that the time limit forced the participants to use a more efficient strategy. On the other hand an alternative explanation is that the jump in subitizing range that is seen from time unlimited to time limited range is simply because of the way the different subitizing ranges were calculated. For example in the time unlimited trials only 5.3% of students had a subitizing range of 5 while 42% of students had a subitizing range of 5 in the time limited trials. This is remarkable as the subitizing range goes up from 3 in early childhood to a maximum of 5 in adulthood (Starkey & Cooper, 1995). Other research is more conservative and puts the adult subitizing range around 4 (Mandler & Shebo, 1982). This means that it is highly unlikely that 42% of these children had an actual subitizing range of five. Even the 5.3% of children who had a subitizing range of 5 in time unlimited trials, seems unlikely considering the literature. Possibly the way the subitizing range was calculated in the unlimited time trials was not conservative enough.
An example of possible methodologically induced bias is that in time limited trials a lot of children who had a subitizing range of four did not see how many dots there were in the trial with five, but they did see it were more than they could count (more than four), so the correct answer options were 5 or 6. Hence, there is a 50% chance of a correct answer. For future studies it would be better to have more trials showing five dots with answer possibilities up to 7 to lower the chance of getting a subitizing range of five solely through guessing. Adding a few trials in both time conditions with 6 dots could help to clarify whether children are really subitizing to five or not, as we would definitely expect a jump at 6 dots, either in amount of mistakes or in reaction time. Another possible explanation is that all these children had an actual subitizing range of five and that this much higher range of subitizing while under time pressure will also been seen in adults. This would need to be verified.
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Groffman (2009) has already shown that arithmetic skills can be improved for very poor arithmetic students by training subitizing. Therefore it would be interesting to investigate whether training poor (but not dyscalculic) subitizers to subitize higher numbers in time limited trials also results in being able to subitize higher in time unlimited trials. If this should be the case, it can be examined if this training effect on subitizing also results in a training effect of arithmetic skills and thus improved arithmetic skills.
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8. Appendix
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