Who is better at Arithmetics?:
The visual number form - verbal word form association, symbolic
comparison skills, non-symbolic comparison skills and individual
differences in arithmetic skills
Fiona Beek
Master Thesis Clinical Developmental Psychology Student number: 10184163
Supervised by: Dr. Brenda Jansen Second Supervisor: Dr. Jurgen Tijms
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Contents
Abstract ... 3
Introduction ... 4
Assessment of learning a new VN-VW form association ... 7
Methods ... 8
Participants ... 8
Materials ... 8
Learning a new VN-VW form association ... 8
Assessing the strength of the VN-VW form association ... 9
Symbolic comparison task ... 10
Non-symbolic comparison task ... 11
Arithmetic skills ... 11
Procedure ... 11
Results ... 13
Discussion ... 13
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Abstract
In the present research it was studied whether the association between the visual number form and the verbal word form (VN-VW form association) could explain individual differences in arithmetic skills in children, in addition to symbolic and non-symbolic
comparison skills. 86 Fifth-graders participated in the study. A dynamic game was developed to teach the participants a new VN-VW form association. Also two tasks, A and Lexy-B, were created to measure the strength of the newly learned VN-VW form association. Non-symbolic comparison skills, Non-symbolic comparison skills and arithmetic skills were assessed. Results indicated that the Lexy tasks were not satisfying all psychometric conditions. Results further indicated that both symbolic and non-symbolic comparison skills accounted for variance in individual differences in arithmetic skills, but that symbolic comparison skills seemed to explain more variance. The strength of the VN-VW form association did not significantly account for variance in arithmetic skills. Results however did indicate that participants with lower scores on the Lexy tasks had significantly lower scores on the arithmetic skills test than participants who scored high on the Lexy tasks. For now it can be concluded that symbolic and non-symbolic comparison skills account for variance in
individual differences in arithmetic skills in children. Although the VN-VW form association does not significantly account for variance, there seems to be a connection between the VN-VW form association and arithmetic skills.
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Introduction
Numbers are very important in today's society. From birth on, children develop in an environment rich in numbers. They see numbers in the streets, on television and in books. Also, they hear their parents use numbers to count, when they use money or when they tell the time. Most people are eventually able to automate the number system. For some people however, this automation fails and they keep struggling with arithmetics. This has a lot of consequences. For example, Bynner and Parsons (1997) found that people with poor arithmetic skills left school early and had more difficulty getting and maintaining full-time jobs. Now-a-days, not much is known about what causes someone to be good or bad in arithmetics, which makes it difficult to develop effective treatments or learning methods for children with impaired arithmetic skills (Price & Ansari, 2013). Therefore, it is important to understand more about the cognitive impairments underlying arithmetic skills.
Dehaene and Cohen (1995; Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006) described a model about mental processes involved in arithmetic skills and number
processing, see Figure 1. This model, The Triple-Code model, assumes that there are three categories of mental representation in which numbers can be manipulated in the human brain. The first category is the visual Arabic number form, in which a number is represented as a visual symbol, for example 13. This representation is a form of symbolic number processing. Secondly there is the verbal word frame, in which numbers are represented as a syntactical string of words, such as /thirteen/. The final category is the analog magnitude representation. At this level, the numbers are represented in an analog, non-symbolic way and information about the quantity can be retrieved and compared to other numerical information. An example of an analog magnitude representation is ooooooooooooo. This representation is a form of non-symbolic number processing.
The visual Arabic number form, the verbal word frame and the analog magnitude representation are interconnected and play an important role in arithmetics. For instance, it is assumed that the visual Arabic representation plays a role in multi-digit operations, because it determines the exact numeric representation (Dehaene, 1992). The verbal representation is said to contain arithmetic facts and is mostly used for simple additions and for the tables of multiplication (Dehaene, 1992). The analog magnitude representation is said to play a role in approximations and number comparisons (Dehaene, 1992).
5 Figure 1
The Triple-Code Model from Dehaene & Cohen (Dehaene, 1992)
The Triple-Code Model is often used to study individual differences in arithmetic skills, focusing on the visual Arabic number form (symbolic number processing) and the analog magnitude representation (non-symbolic number processing) and their relation with arithmetic skills. An often used method in this kind of research is the Number Comparison
Task. However, there seems to be a difference between symbolic and non-symbolic
comparisons and their relation with arithmetic skills (DeSmedt, Noël, Gilmore & Ansari, 2013).
The Defective Number Sense Hypothesis states that there is a specific impairment in the innate ability to understand and represent quantities, which leads to difficulties in developing numerical and arithmetic skills (DeSmedt & Gilmore, 2011). In terms of the Triple-Code Model, this hypothesis states that people who have difficulties with arithmetics have impairments in their analog magnitude representation system. A number of researchers found a connection between the non-symbolic number processing and arithmetic skills (Bonny & Lorence, 2013; Halberda & Ferguson, 2008; Inglis, Attride, Batchelor & Gilmore,
6 2011). However, many other researchers did not find a connection between the two variables (DeSmedt & Gilmore, 2011; Price, Palmer, Battista, & Ansari, 2012; Rouselle & Noël, 2007). Results concerning The Defective Number Sense Hypothesis are thus far inconclusive and to date individual differences in arithmetic skills cannot be explained by impairments in non-symbolic number processing.
Many of the researchers whose results did not confirm the Defective Number Sense Hypothesis did find clues that arithmetic skills and access to the visual symbolic information of numbers are related. These results fit the Access Deficit Hypothesis. This hypothesis states that impairments in arithmetic skills are the result of limitations in the access to numeric information of symbols (DeSmedt & Gilmore, 2011). In terms of the Triple code model, children with difficulties in arithmetics do not have access to the visual Arabic number form. Consequently, the association between the symbolic and non-symbolic number form is disrupted (Rouselle & Noël, 2007), leading to impairments in the normal development of the analog number form representation (Wilson, Andrewes, Struthers, Rowe, Bogdanovic, & Waldie, 2015). This could also explain why results concerning impairments in the non-symbolic representation are inconsistent (Wilson et al., 2015) and results concerning impairment in the symbolic representation are more consistent. DeSmedt, Verschaffel and Ghesquière (2009) for example, found that skills in the area of symbolic magnitude
comparison can explain the later differences in arithmetic skills. Other researchers also found results supporting the Access Deficit Hypothesis (DeSmedt & Gilmore, 2011; Holloway & Ansari, 2009; Landerl & Kölle, 2009; Rouselle et al., 2007). This seems to suggest that individual differences in arithmetic skills might be explained by impairments in symbolic number processing.
Most studies focused on only two categories of the Triple-Code Model, that is the visual Arabic number form and the analog magnitude representation, respectively a form of symbolic and non-symbolic comparison skills, and their association with arithmetic skills. Earlier research seems to suggest that impairments in access to symbolic information underlie problems in arithmetic skills. Whether non-symbolic comparison skills are associated with arithmetic skills is not clear yet. Moreover, the third category, the verbal word form, and its role in arithmetics has not been studied.
Dehaene and Cohen (1995) suggest that there is a direct route that connects the visual Arabic number form and the verbal word form. Knowledge about the analog magnitude representation is not necessary for this. Furthermore, Piazza and colleagues (Piazza et al., 2010; Piazza, Pica, Izard, Spelke & Dehaene, 2012) maintain that learning new number words
7 leads to the development of a new numeric representation. This new numeric representation is very precise and has semantic content, which is based on ordinal information from the number line. This new numeric representation connects with the analog magnitude representation, hereby improving the precision of the analog magnitude representation (Piazza et al., 2010; Piazza et al., 2012).
Learning a new connection between the verbal word form and the visual Arabic number form hence seems to be important for developing a precise analog magnitude
representation and thus arithmetic skills. Therefore, it is interesting to examine whether there is a connection between the ability to learn a visual number form - verbal word form
association (VN-VW form association) and individual differences in arithmetic skills in children. It is also interesting to test how much variance this VN-VW form association accounts for in arithmetic skills, compared to the variance accounted for by the symbolic comparison skills and non-symbolic comparison skills.
Assessment of learning a new VN-VW form association
In this study, a new VN-VW form association will be taught to the participants
through a dynamic computer game. A dynamic test measures the ability to learn. The effect of hints, instruction and feedback on the learning potential of a child is examined (Resing, 2006). A benefit of dynamic testing is that the scores give a good impression of the learning potential (Resing, 2006). The game used in the present research offers feedback on accuracy of the VN-VW form association. Some children are assumed to be better in this game, and thus better in learning a new VN-VW form association, than others. The tests following this game will measure the strength of the newly learned VN-VW form association and in this way indicate the learning potential of the children.
However, it is not possible to assess the ability to learn a new Arabic VN-VW form association, because Dutch children are already familiar with Arabic number forms.
Therefore, a rather unknown visual number word form for Dutch children, the Thai number system, is used in the assessment of the VN-VW form association. This avoids the potential confounding effect of differences in prior exposure to visual number forms. The Thai number system has visual number forms that are not easy to relate to the Arabic number system. Only the Thai "zero" looks a lot like the Arabic "zero". Therefore, in order to make the participants truly learn a new VN-VW form association, only the numbers one through nine will be used in the dynamic game. The accuracy and response time of the assessment indicate the strength
8 of this newly learned VN-VW form association.
Thus, a dynamic game was developed to teach a new VN-VW form association. After playing the dynamic game, children had to complete two tasks (Lexy-A and Lexy-B) in which the strength of the newly learned VN-VW form association was tested. Children also
completed a symbolic comparison task and a non-symbolic comparison task. The relation between the strength of the newly learned VN-VW form association and individual
differences in arithmetic skills was studied, and it was investigated how much variance this VN-VW form association accounts for, compared to the symbolic and non-symbolic
comparison skills. It is expected that both the Lexy tasks are reliable instruments to measure the strength of the VN-VW association. Also, it is assumed that the newly learned VN-VW form association and symbolic comparison skills are both accurate predictors of individual differences in arithmetic skills in children and therefore both account for a great part of variance in arithmetic skills. Further, it is assumed that non-symbolic comparison skills are not an accurate predictor of individual differences in arithmetic skills in children and therefore account for little or none of the variance in arithmetic skills in children.
Methods
Participants
The sample in the present study consisted of 86 children, of which 34 males and 52 females. The average age was 9;1 years (SD = 5 months). The participants were recruited from four different elementary schools. Three schools were located in Amsterdam and one in Huizen. The children were all in fifth grade. Inclusion criterion was that the participants were not familiar with the Thai number system.
Materials
Learning a new VN-VW form association
First of all, a dynamic computer game was developed to teach the participants a new VN-VW form association, see Figure 2. Participants heard a number and had to shoot the corresponding Thai symbol. If the correct symbol was shot, the symbol disappeared and the playing field became emptier. If an incorrect symbol was shot, the symbol remained in its position. The game was played under time pressure: The symbols slowly went down. A red
9 line in the playing field indicated the demarcation point. If the symbols moved under this line, the game was lost and the participant had to restart the level. Thus, the game could only be won by shooting the correct symbols fast. In previous research a different version of this game has been used in order to teach a letter - sound association to children with dyslexia (Aravena, Snellings, Tijms & van der Molen, 2013). In the present study, the sounds and letters were converted to number words and Thai number symbols. This version of the game had not been used before in psychological research, therefore no psychometric characteristics are known.
Figure 2
Examples of the game in which the VN-VW form association is taught: Upper left: a correct symbol is shot and disappears. Upper right: An incorrect symbol is shot and it remains in its position. Lower left: A correct symbol, which is connected to multiple identical symbols, is shot and they disappear. Lower right: The symbols cross the red line and the game is lost.
Assessing the strength of the VN-VW form association
The strength of the newly learned VN-VW form association was tested by two computer tasks. In the first task (Lexy-A), participants heard a number between one and nine and had to choose the according Thai symbol as fast as possible, see Figure 3. In the second task (Lexy-B), participants saw two Thai symbols and had to choose which one had the highest magnitude, see Figure 3. Again participants had to choose as fast as possible. For both
10 Lexy tasks, the correct symbol could be chosen by either pressing the "D" or the "K" key on the laptop. Each of the nine symbols was combined with each of the other eight symbols, thus creating 72 items in total. Accuracy and response time were used as measures for an
indication of the strength of the VN-VW association.
Figure 3
Example of a trial of Lexy-A and Lexy-B, two tasks in which the participants have to choose the correct Thai symbol as fast as possible
Symbolic comparison task
To assess the symbolic comparison skills, a Symbolic Comparison Task was administered. Participants saw two Arabic symbols, for example a 3 on the left side of the laptop screen and a 7 on the right side of the screen, and had to choose which of the symbols had a higher magnitude. Participants could choose the correct symbol by either pressing the "D" or "K" key on the keyboard. Afterwards, the "Enter" key had to be pressed to continue to the next item. Again, participants had to choose as fast as possible. The Arabic symbols one through nine were used. Combining all the Arabic symbols with each other, the task consisted of 72 items. The Cronbach's Alpha of the symbolic comparison task used in the present study is 0.87, which suggests that the task has a good internal consistency (DeSmedt et al., 2013). Outcome values assessing the level of symbolic comparison skills were accuracy and response time.
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Non-Symbolic comparison task
Non-symbolic comparison skills were assessed with a Dot Comparison Task. For this task, the computer program Panamath (Halberda, Mazzocco & Feigenson, 2008) was
downloaded and installed on the laptops. In Panamath, the participants were presented with two clouds of dots on the screen and had to choose which one had the most dots. The according cloud could be chosen by either pressing the "D" or the "K" key on the laptop. Afterwards, the "Space Bar" had to be pressed to continue to the next item. Based on earlier research with children around age 10, the task consisted of 208 items with clouds consisting of five through 21 dots. The task lasted approximately 10 minutes. With these settings there was a split-half reliability of r = 0.70 (Halberda, Ly, Wilmer, Naiman, & Germine, 2012). Outcome values assessing the non-symbolic comparison skills were accuracy and response time.
Arithmetic Skills
A classical paper-and-pencil test, the TempoTest Automatiseren (TTA), was
administered to assess the level of arithmetic skills of the participants. The TTA measures the automations of different types of arithmetic skills. The TTA consisted of four pages, each containing a different type of math problems the participants had to solve mentally: additions, subtractions, multiplications and divisions. Each page consisted of 50 math problems of one type. Participants had exactly two minutes to solve as many problems on one page as possible. The total assessment of the TTA lasted about ten minutes. The level of arithmetic skills was indicated by calculating the total of all the correct answers, hereby creating an outcome value with a minimum of zero and a maximum of 200 for each participant. The TTA has
standardized scores based on a large normative sample (De Vos, 2010).
Procedure
Tests were administered by two researchers (master students), who first introduced themselves to the children in the classroom. They briefly explained why the research is
conducted and how the research would develop. It was emphasized that the participants would not receive grades for the different tasks, but that it was very important to do their very best. The tests consisted of an assessment in small groups and a classroom assessment.
The assessments in small groups were conducted with three participants at a time and one researcher present, or with four participants at a time and two researchers present. The
12 assessment in small groups was conducted in quiet spaces at the participants' schools. Each participant was seated behind a laptop. The volume was tested, after which the instruction of the VN-VW form association game was started. Participants played a practice trial before starting with the real game. The game lasted 20 minutes. Next, the Lexy tasks started.
Instructions for Lexy-A were: "Now you are going to hear a number and you have to pick the
according symbol. You can choose by pressing the "D" key if you think the left symbol is correct, or by pressing the "K" if you think the right symbol is correct. Try your best and choose as fast as possible". Lexy-A lasted approximately ten minutes. Thereafter, Lexy B was
started. The researcher explained to the participants that they had to choose the symbol with the highest magnitude. Again the use of the keys was explained and it was emphasized that participants had to try their best and choose as quick as possible. Lexy-B lasted approximately ten minutes. After finishing Lexy-B, Panamath was started. The instructions in Panamath were translated into Dutch. It was explained to the participants that the cloud with the largest number of dots had to be chosen. This could again be done by either pressing the "D" or "K" key. To continue to the next item, participants had to press the "Space Bar". It was
emphasized to choose as accurately and as fast as possible. Panamath lasted a maximum of ten minutes. The last task in the small group assessment was the Symbolic Comparison Task. Instructions on the screen were read aloud to the participants. Added was that they had to try their best and choose as fast as possible. It was also explained that they had to press the "D" or "K" keys to choose and use the "Enter" key to go to the next item. The total duration of the entire computer session was 45-55 minutes.
After all the participants in one class had taken the assessments on the laptop, the TTA was conducted in the classroom. Standard instructions were the following: "I would like to
know how fast you are in arithmetics. Before you is a form with math problems, which you later have to solve. You have to solve them mentally; you can only write the answers on the form in the spaces behind the math problems. You will have exactly two minutes to solve as many math problems as possible. I say 'as possible', because it is very hard to solve them all in under two minutes. When I say 'begin', you can start with the first math problem (name the first math problem, upper left). It is wise to try all the problems in the first block, before starting with the next, because the math problems will become more difficult. When I say 'Stop', you have to put your pencils down. Shortened versions of the instruction were repeated
for the subtraction, multiplication en division parts. Administration of the TTA lasted about ten minutes.
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Results
Breakdown of subjects. A lot of participants had response times well under 200ms.
These were considered too fast and were thus seen as invalid. The too fast response times only occurred in the two Lexy tasks. For Lexy-A, 73 participants responded too fast to one or more items. For Lexy-B, all 86 participants responded too fast to one or more items. On average, participants responded too fast to 22.35 items (SD = 26.81) in Lexy-A and to 22.17 items in Lexy-B (SD = 26.17). Next, the valid responses of the participants who had responded too fast to one or more items were further analyzed. The 73 participants with too fast responses on Lexy-A had an average of 45.33 valid responses (SD = 27.28) in Lexy-A. The 86 participants with too fast responses in B had on average 45.01 (SD = 26.73) valid responses in Lexy-B (See Figure 4).
Figure 4
Frequencies of the number of valid responses for Lexy-A and Lexy-B
As shown in Figure 4, in both Lexy tasks there seemed to be a gap around 40 valid responses. This suggested that there is a group of participants that has more than 40 valid items, and a group with less than 40 valid items. On average in the complete sample,
participants had 71.17 percent valid responses (SD = 18.28) on Lexy-A and 66.55 on Lexy-B (SD = 20.46). For the subsample with more than 40 valid responses, the average percentage correct was 78.08 (SD = 14.04) for Lexy-A and 73.47 (SD = 15.24) for Lexy-B. An
14 Independent T-Test was conducted (See Table 1) to test whether participants with more than 40 valid responses scored different on the TTA than participants with less than 40 valid responses. For Lexy-A, participants with less than 40 valid responses had on average a lower score (M = 84.89, SD = 33.40) on the TTA than participants with more than 40 valid
responses (M = 100.96, SD = 35.71). This difference was significant, t(83) = -1.99, p<.05, and represented a small effect, D = .46. For Lexy-B, participants with less than 40 valid responses on average also scored lower (M = 84.71, SD = 33.61) on the TTA than participants with more than 40 valid responses (M = 102.42, SD = 34.98). This difference was also significant
t(83) = -2.22, p<.05, which represented a medium effect, D = .52. This result indicated that
there were two different groups of participants and it was decided to run a two-track analysis: In the first track, analyses were performed for the complete sample. In the second track, analyses were performed only for the selection of participants who had more than 40 valid responses.
Table 1
Number of participants with more or less than 40 valid responses for Lexy-A and Lexy-B, their Mean score on the TTA (M) and the Standard Deviation (SD)
Reliability of the Lexy tasks. Assessed was whether the Lexy tasks were reliable.
Therefore, split-half reliability was calculated for response time (RT) and accuracy (ACC) for both the complete dataset and the subsample. In the complete sample, split-half reliability for Lexy-A-RT total was very low, r = .24. This was against expectations. Lexy-A-ACC had a good split-half reliability, r = .70. Lexy-B-RT and Lexy-B-ACC both had high split-half reliability, respectively r = .76 and r = .88. This was in line with expectations. Split-half reliabilities were also assessed for the subsample with more than 40 valid items. Lexy-A-RT had a relatively low split-half reliability, r = .66. Lexy-A-ACC, Lexy-B-RT and Lexy-B-ACC all had high split-half reliability, respectively r = .82, r = .76, and r = .85.
Number of valid responses N M SD
Lexy-A <40 28 84.89 33.40
>40 57 100.96 35.71
Lexy-B <40 28 84.71 33.61
15 Through a Kolmogorov-Smirnov Test, the assumption of normality for both Lexy tasks were assessed. The percentage of valid and correct responses in the complete sample for Lexy A, D(82) = 0.15, p<.01, and for Lexy-B, D(82) = .13, p<.01, were both significant and thus non-normal. This was against expectations. The distribution of the medians of the
response times in the complete sample for Lexy-A, D(82) = .09, p = .08, and Lexy-B, D(82) = .09, p = .07, were non-significant. The medians of the response times of both Lexy-A and B were thus normally distributed, as expected. Assumptions of normality were also assessed for the subsample with more than 40 valid responses. The percentage valid and correct responses for Lexy-A and Lexy-B in the subsample were both significant, D(55) = 0.15, p<.01, and thus non-normal. The distributions of the medians of the response times in the subsample for Lexy-A was non-significant, D(55) = 0.09, p = .07, and thus normally distributed. The medians of the response times in the subsample for Lexy-B were however significant,
D(55) = 0.12, p<.05, and thus non-normal. Because not all of the parametric characteristics
for both the complete and the subsample are satisfying, results concerning the Lexy tasks should be carefully interpreted.
Predictors of arithmetic skills. To check whether Lexy-A, Lexy-B, the non-symbolic
comparison task and the symbolic comparison task predicted individual differences in arithmetic skills, a hierarchical regression analysis was conducted. Because of the high number of too fast responses and the low reliability of Lexy-A-RT, only the percentage
correct items of Lexy-A and Lexy-B were included in the regression for the first track. For the non-symbolic comparison task and the symbolic comparison tasks both median RT and percentage correct were included in the regression analysis. The hierarchical regression analysis was conducted in two orders, for both the complete sample and the subsample with more than 40 valid responses. Results for the complete sample are described first.
In the first order, the following three steps were included in the analysis: In the first step, the median RT and percentage correct of the symbolic comparison task were included as predictors of the total TTA score. In the second step, the median RT and percentage correct of the non-symbolic comparison task were added as predictors. In the last step, the percentage correct of Lexy-A and Lexy-B were added. Results of these steps demonstrated that the symbolic comparison variables explained a significant amount of variance in arithmetic skills, ∆R2 = .27, F(2,80) = 14.39, p<.01, but both the non-symbolic comparison variables, ∆R2 =
.02, F(2,80) = 1.23, p = .30, and the Lexy variables, ∆R2 = .04, F(2,80) = 2.17, p = .12, did not account for variance in arithmetic skills. The Adjusted R-Square for the first step in this model was .24.
16 In the second order, the non-symbolic comparison variables were included in step one. In the second step the symbolic variables were added and in the third step the Lexy variables were added. The non-symbolic comparison variables explained a significant amount of variance in arithmetic skills, ∆R2 = .18, F(2,80) = 8.63, p<.001. Adding the symbolic comparison variables significantly improved the model, ∆R2 = .11, F(2,80) = 6.00, p<.01. Adding the Lexy variables however still did not significantly improve explained variance, ∆R2 = .04, F(2,80) = 2.17, p = .12. The corresponding Adjusted R-Square for the second step
in this model was .25. These results seemed to suggest that the non-symbolic comparison variables do account for some variance in arithmetic skills, but not more than the symbolic comparison variables. This was in line with the expectations. Against expectations the Lexy variables did not significantly account for variance in arithmetic skills. Because the non-symbolic comparison variables did account for some variation and the Adjusted R-Square was slightly bigger, the second model seems the most useful (see Table 2).
Next, the second track of the analysis was conducted, using only the subsample with more than 40 valid responses. Because this subsample has many valid responses, RT of Lexy-A and Lexy-B were this time included in the regression. Following the first order of inserting predictors, the symbolic comparison variables were inserted in the first step. In the second step the non-symbolic comparison variables were added and in the third step the Lexy-A and Lexy-B variables were added. Results seemed to indicate that only the symbolic comparison variables were significant predictors in the model, ∆R2 = .24, F(2,52) = 7.99, p<.001. Both the
non-symbolic comparison variables, ∆R2 = .04, F(2,50) = 1.24, p = .29, and the Lexy
variables, ∆R2 = .03, F(2,46) = 0.46, p = .76, did not significantly improve the model. The
corresponding Adjusted R-Square of the first step in this model was .20. Adding the Lexy variables seemed to decrease the predictive values of the model, Adjusted R-Square = .18.
Following the second order of inserting predictors, the non-symbolic comparison variables were inserted in the first step. In the second step, the symbolic variables were added and in the third step the Lexy-A and Lexy-B variables were added. Results of this model demonstrated that the non-symbolic comparison variables were significant predictors in the model, ∆R2 = .18, F(2,52) = 5.63, p<.01. Adding the symbolic variables significantly
improved the model, ∆R2 = .09, F(2,50) = 3.20, p<.05. However, adding the Lexy variables
did not significantly improve the model, ∆R2 = .03, F(2,46) = 0.46, p = .76. Again, adding the
Lexy Variables even seemed to decrease the predictive value of the model, Adjusted R-Square = .18 (See Table 3). The Adjusted R-R-Square of this model was 0.21 and thus seemed to account for slightly more variation than the first model. These results seemed to suggest
17 that both the non-symbolic comparison and symbolic variables were significant predictors in the model, but the Lexy variables were not. Also, the symbolic variables seemed to be stronger predictors than the non-symbolic comparison variables.
Table 2
The unstandardized regression coefficients (B), Standard Error (SE B) and the standardizes coefficient (β) for the three steps in the second model of the complete sample.
B SE B β Step 1 Constant 5.184 30.751 Non-Symbolic % 1.585 .404 .425*** Non-Symbolic Median RT -.044 .016 -.287** Step 2 Constant -15.102 48.309 Non-Symbolic % .714 .459 .191 Non-Symbolic Median RT -.010 .020 -.065 Symbolic % 1.431 .616 .272* Symbolic Median RT -.087 .027 -.423** Step 3 Constant -30.308 48.356 Non-Symbolic % .425 .473 .114 Non-Symbolic Median RT -.008 .020 -.055 Symbolic % 1.579 .612 .300** Symbolic Median RT -.097 .027 -.473*** Lexy-A % .404 .347 .187 Lexy-B % .056 .345 .026
Note: Adjusted R2 = .16 for step 1, Adjusted R2 = .25 for step 2 and Adjusted R2 = .27 for step 3. *p<.05, **p<.01 ***p<.001
18 Table 3
The unstandardized regression coefficients (B), its Standard Error (SE B) and the
standardized coefficient (β) for the second model of the second track regression analysis.
Note: Adjusted R2 = .15 for step 1, Adjusted R2 = .21 for step 2 and Adjusted R2 = .18 for step
3. *p<.05, **p<.01 B SE B β Step 1 Constant 5.517 44.018 Non-Symbolic % 1.667 .565 .394** Non-Symbolic Median RT -.045 .018 -.333* Step 2 Constant -64.298 66.299 Non-Symbolic % .944 .615 .223 Non-Symbolic Median RT -.023 .023 -.171 Symbolic % 1.734 .765 .315* Symbolic Median RT -.062 .034 -.308 Step 3 Constant -58.285 75.663 Non-Symbolic % .706 .671 .167 Non-Symbolic Median RT -.014 .025 -.106 Symbolic % 1.887 .812 .342* Symbolic Median RT -.069 .036 -.345 Lexy-A % .424 -.012 .492 .016 .171 -.126 Lexy-A RT Lexy-B % -.306 .004 .439 .009 -.134 .080 Lexy-B RT
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Discussion
In the present study it was investigated whether the strength of a newly learned VN-VW form association can explain individual differences in arithmetic skills and how much variance symbolic comparison skills, non-symbolic comparison skills and the VN-VW form association can account for. Participants were 86 fifth graders from different schools. To test the hypotheses, a dynamic game in which a new VN-VW form association was taught and two tasks measuring the strength of the VN-VW form association were developed. Further, symbolic comparison skills, non-symbolic comparison skills and arithmetic skills were assessed. Analysis revealed that there appeared to be a subsample in the tasks assessing the strength of the VN-VW form association: there was a group of the participants with more than 40 valid responses and a group with less than 40 valid responses. Also, some of the
psychometric characteristics of the tasks measuring the VN-VW form association were not satisfying conditions and results concerning these tasks should therefore be carefully interpreted. It was decided to run a two-track analysis: In the first track analysis were performed for the complete sample. In the second track analysis were performed for the subsample with more than 40 valid responses. Statistical analysis consisted of a reliability analysis and a hierarchical regression analysis with two different orders of predictors. Results indicated that symbolic and non-symbolic comparison skills both accounted for variance in arithmetic skills in children, although symbolic comparison skills seemed to account for more variance than the non-symbolic comparison skills. Although results indicated that the VN-VW form association does not significantly account for variance in individual differences in arithmetic skills when it is compared to symbolic and non-symbolic comparison skills, results do seem to imply that there is a relation between the strength of the VN-VW form association and the level of arithmetic skills. Thus, there seems to be some connection between the VN-VW form association and arithmetic skills, but the connection between non-symbolic and symbolic comparison skills and arithmetic skills is supposed to be much stronger.
Although the non-symbolic comparison skills account for less variance in individual differences in arithmetic skills than symbolic comparison skills, both seem to be related to arithmetics. The results in this study hence seem to support both the Defective Number Sense Hypothesis and the Access Deficit Hypothesis (DeSmedt & Gilmore, 2011). However, the Access Deficit Hypothesis states that the limited access to symbolic information disrupts the association between the symbolic and non-symbolic number form (Rouselle & Noël, 2007), consequently impairing the normal development of the non-symbolic number form
20 representation (Wilson et al., 2015). This could explain why results concerning impairments in the non-symbolic representation are inconsistent (Wilson et al., 2015). It also fits the results found in this study, clarifying why both types of comparison skills seem to be related to arithmetic skills in children and why the association with non-symbolic comparison skills and arithmetics seems to be weaker than the association with symbolic comparison skills and arithmetics.
An unexpected result was that the VN-VW form association does not seem to account for variance in individual differences in arithmetic skills when it is compared to the symbolic and non-symbolic comparison skills. This might be because symbolic and non-symbolic comparison skills are stronger related to arithmetic skills than the VN-VW form association. Thus, the VN-VW form association is related to arithmetics as the results in this study seemed to indicate, but its relation with arithmetics is less strong than those of the symbolic and non-symbolic comparison skills. It might also be possible that the strength of the VN-VW form association does not account for variance in arithmetic skills, because the psychometric characteristics of the task measuring the strength of the VN-VW form association were not all satisfying. This probably caused a lot of noise in the data, which may have resulted in the fact that the VN-VW form association does not account for variance in arithmetic skills in
children. An alternative explanation for the unexpected result could be that not everyone understood the game or liked the game equally, resulting in a weaker performance on the VN-VW form association task. Another alternative explanation could be that it takes children years to learn an Arabic VN-VW form association. Also, in reality, children are much more confronted with Arabic symbols than they were in the short time they were exposed to the Thai symbols. This could also clarify why the relation with symbolic comparison skills and non-symbolic comparison skills with aritmetics is stronger than the relation between the
VN-VW form association and arithmetics.
Another unexpected result was the fact that some of the psychometric characteristics of the tasks measuring the strength of the VN-VW form association were not satisfying conditions. A lot of participants had response times well under 200ms. This seems nearly impossible when one has to choose the correct symbol after hearing a number or pick the number with the highest magnitude, especially when those symbols are Thai and still
relatively unknown to participants. Also, on the symbolic and non-symbolic comparison tasks in which familiar symbols are used, none of the participants had response times this fast. The too fast response times thus seemed unreliable and therefore it was decided to remove
21 couple of responses left. Removing this part of the data probably caused the low reliability and non-normal distributions.
The fast response times may relate to a number of flaws of the dynamic game and the tasks concerning the VN-VW form association. During the game it sometimes took a while before the cannon appeared. Also, the game crashed occasionally and participants had to restart their level. These flaws cost playing time and therefore made it harder for the participant to win the level and provide less opportunity to learn the new VN-VW form association. Consequently, these flaws might have caused differences in performance between participants that are not related to the VN-VW form competences and/or hampered the
motivation of participants. Further, a lot of participants were able to respond in less than 200ms on the tasks measuring the strength of the VN-VW form association. However, these too fast response times did not occur in the symbolic comparison task or the non-symbolic comparison task. There are several differences between the comparison tasks and the tasks measuring the strength of the VN-VW association that can clarify why participants responded too fast in the VN-VW form association tasks, but not in the comparison tasks. Firstly,
participants may have perceived the tasks measuring the strength of the VN-VW form association as harder than the comparison tasks. Another explanation might be that, contrary to the symbolic comparison task and the non-symbolic comparison task, these too fast responses were not punished in the tasks measuring the strength of the VN-VW form association. In the comparison tasks, the programs gave a notification when the participant repeatedly responded too fast. Also, in the comparison tasks it was not possible to give the same response several times, because notifications were given when the participant responded with pressing the same key multiple times. These notifications were not given in the tasks measuring the strength of the VN-VW form association. Further, in the comparison tasks participants had to press a button to continue to the next item. This was not the case in the VN-VW form association tasks and it seems useful to implement this in the tasks.
For future studies, it is recommended to select an appropriate measure of arithmetic skills. One may comment that the TTA only measures a small part of arithmetic skills. The TTA only covers the ability to add, subtract, multiply and divide in limited time, but
arithmetic skills consist of much more than that. Dehaene and Cohen (1992) maintain that the Visual Arabic number form is related to multi-digit operations. Although the TTA covers multi-digit operation in the area of additions, subtractions, multiplication and divisions, there are many other areas of arithmetics in which multi-digit operations can be used. Also the verbal word form is related to counting, and the analog magnitude representation is often used
22 in comparisons and approximations (Dehaene & Cohen, 1992). These are all areas that the TTA does not cover, and it might be possible that stronger results are found when an instrument is used that covers more and different areas of arithmetic skills.
Despite these points of critique, for now it can be concluded that the present study has replicated the finding that the symbolic comparison skills seem to account for variance in individual differences in children's arithmetic skills (DeSmedt et al., 2009; DeSmedt & Gilmore, 2011; Holloway & Ansari, 2009; Landerl & Kölle, 2009; Rouselle et al., 2007) and that non-symbolic comparison skills relate to aritmetic skills among fifth graders as well (Bonny & Lorence, 2013; Halberda & Ferguson, 2008; Inglis, et al., 2011). Although the VN-VW form association does not seem to account for individual variance in arithmetic skills in children, there does seems to be a connection between a new VN-VW form association and the level of arithmetic skills of children. Further research concerning this subject therefore seems valuable. With additional research, it might be possible to create a new form of treatment for children with impaired arithmetic skills or design a new teaching method for arithmetics based on both symbolic comparison skills and non-symbolic comparison skills and possibly also the VN-VW form association skills of children.
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