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Skills Underlying Multiplication
The Neighbourhood Consistency and the Distance Effect and their Relation with
Inhibition in Child and Adult Multiplication
Vera van der Molen¹
Supervised by Camilla Gilmore² and Brenda Jansen¹
¹ Universiteit van Amsterdam
² Loughborough University
Master thesis
Research Master Psychology and Master Gezondheidszorgpsychologie
Student number: 5805643
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Abstract
This study looked at two effects in verification multiplication problems: the neighbourhood consistency effect, derived from the interacting neighbours model by Verguts and Fias (2005), and the distance effect. The neighbourhood consistency effect holds that when a multiplication problem is presented with an incorrect solution, it is more difficult to reject this solution when the solution is a solution to another multiplication problem that has one similar operand and one operand that only differs by one to the original problem (neighbour) and when this solution has the same decade as the correct solution (consistent). The distance effect holds that an incorrect solution is more difficult to reject when the numerical distance to the correct solution is small rather than large. When children and adults in this study were presented with multiplication problems with an incorrect solution, they indeed responded slower when the solution had a small rather than a large numerical distance to the correct solution. They also made more errors when the incorrect solution was a consistent neighbour to the correct solution than when this was an inconsistent non-neighbour, and this effect was even stronger when the numerical distance was small rather than large. These different effects for speed and accuracy show that they might have different underlying processes. Although our study did not show a relation between these effects and inhibition, inhibition cannot be ruled out as underlying mechanism yet, because different types of inhibition have to be taken into account. Finally, the same effects were found in children as well as in adults, which suggests that the same underlying
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Skills underlying multiplication
In making sense of the world around us, numbers are very important (Dehaene, 2011). Meanwhile, children consistently underachieve in mathematics, which is becoming a significant problem (Dowker, 2009). In order to improve children’s mathematics skills through effective teaching methods, it is important to understand the processes that underlie learning and performing mathematics. This study focused on the domain of multiplication and problem characteristics that affect performance in a multiplication verification task, as well as the potential role of inhibition as an underlying process for performance on this task.
There are four effects that have been robustly observed in multiplication performance in adults as well as children, namely the size effect, the tie effect, the size x tie interaction effect and the five effect (De Brauwer & Fias, 2009). First, the size effect holds that people respond slower to multiplication problems with large operands than small operands. Next, the tie effect holds that people find it easier to solve multiplication problems with two of the same operands than with two different operands. Third, the size x tie interaction effect holds that tie problems with large operands are found easier than would be expected based on their size. Last, the five effect holds that people respond faster to multiplication problems with five as an operand than with other numbers as operands.
Various researchers have designed models to explain the way people solve multiplication problems. All these models can explain the size effect and were designed to do this, but no model can effectively explain both the tie and the five effect, as well as the size x tie interaction effect (Verguts & Fias, 2005). The table search model (e.g. Geary, Widaman & Little, 1986, in Verguts & Fias, 2005) is one example. In this model the operands correspond to the rows and columns of a
multiplication table and the solution is retrieved by ‘walking’ to the appropriate row and column. Because this will take longer when operands are larger, the size effect occurs, but the five, tie and size x tie interaction effects cannot be explained. A second model is the Network Retrieval Model by Ashcraft (1987, in Verguts & Fias, 2005). In this model there are two input fields, one for each
4 operand, and one output field. The nodes in the input fields are connected to their solution nodes in the output fields: for example, 4 in the input field is connected to 4, 8, 12, 16 etc. in the output field. When a problem, for example 3 x 4, is presented, activation is spread to the neighbour problems, … x 3 and … x 4. Then, the most active node in the output field is selected as the response to the
problem. In this model, the size, tie and five effect are explained by their frequency of occurrence in textbooks: problems with large operands were said to occur less in textbooks whereas problems with five as an operand and problems with two equal operands were said to occur more frequently in textbooks. Whereas this is a valid explanation for the size effect, this is not the case for the five and tie effect (Verguts & Fias, 2005). A third model is the distribution of associations model (Siegler, 1988). This model states that multiplication problems are in general solved by retrieval. If no solution can be retrieved, children can use a sophisticated guessing approach to state any solution that comes to mind. If that doesn’t happen, the problem can be solved using a back-up strategy, like repeated addition. The final solution will become associated more with the problem, which will lead to the distribution of associations between problem and solution. This distribution will later be used when children try to answer the problem by retrieval. According to this model, the size effect occurs because problems with large operands are more difficult to solve using a backup strategy. The five effect occurs, because repeated addition is easier for fives than for other numbers (Verguts & Fias, 2005). However, how the tie effect and the size x tie interaction effect can be explained remains unclear. A fourth model is the connectionist model Mathnet by McCloskey and Lindemann (1992, in Verguts & Fias). This model consists of three layers with nodes: an input layer with the operands, an output layer with the solutions and a hidden layer in between with connections to the input and output layer. The connections are trained in the learning phase, where children are presented with the different multiplication problems. The size effect can be explained by the frequency of
occurrence of the different multiplication problems in the learning phase. The idea is that children are presented with multiplication problems with smaller operands more often, so the connections between these operands and their solutions are stronger than the connections between large
5 operands and their solutions. However, the model does not take into account the five and tie effect and as we have seen before, these effects cannot be explained based on their frequency of
occurrence in textbooks, or, in this case, the learning phase. The model also does not consider the size x tie interaction effect. The final model discussed here is the Network Interference Model by Campbell (1995, in Verguts & Fias, 2005). This model states that there are different problem nodes that correspond to a combination of an operand, an operation and a solution, for example 3, 4, x, 12. When a problem is presented, the nodes are activated depending on to what extent they match the nodes of the problem. So when the problem 3 x 4 is presented, the problem node 3, 4, x, 12 will be activated, but the problem node 2, 4, x, 8 will also be partially activated. Apart from this, a
magnitude code is calculated that approximates the size of the solution. If this magnitude code is large, it activates all large problems and when it is small, it activates all small problems. The
magnitude code is said to be able to explain the size effect: large magnitude codes will activate more large problems than small magnitude codes will activate small problems. Because more alternative problems need to be inhibited with a large problem, large problems are more difficult to answer. However, the exact reason of why according to this magnitude code more large than small problems are activated remains unclear. Campbell explains the tie and five effect by saying that these
problems are stored differently and that when a five problem is presented, non-five problems are only weakly activated and the same goes for tie problems. However, there is no empirical foundation for these different storages (Verguts & Fias, 2005), so this is not a valid explanation for these effects.
Because all models described above cannot properly explain all effects found in
multiplication, we have to look at another model that might be able to do so. Verguts and Fias (2005) introduced the interacting neighbours model describing the underlying mechanism of retrieval in multiplication, the most common strategy used by adults and children from grade 4 (Cooney, Swanson, & Ladd, 1988). The model holds that multiplication problems are stored in an associative network and explains the behavioural effects by the internal structure of the multiplication table. In this network, there are two input fields, one for each operand, and a semantic field in which all
6 solutions to multiplication problems are represented. This semantic field is presented in Figure 1. In this semantic field, problems with similar operands (for example 3 x 4 and 3 x 5) are stored closely together and of each commutative pair (e.g. 3 x 4 and 4 x 3) only one problem with its solution is represented. When a multiplication problem is presented, the neighbouring problems in the semantic field (up to a distance of two) will also be activated. This means that if problem 4 x 6 is presented, problems 4 x7, 4 x 8, 4 x 5, 4 x 4, 3 x 6, 2 x 6, 5 x 6 and 6 x 6 will also be activated. When the decades or units of the solutions to these problems match the decade or unit of the solution to the presented problem (consistent), the problem will be easier to solve through the process of cooperation. If this is not the case (inconsistent), competition will take place and it will be harder to solve the presented problem. For example, the problem 4 x 6 with its solution 24 has two decade consistent neighbours (20 and 28), and is therefore easier to solve than the problem 6 x 7, which has no decade consistent neighbours. Verguts and Fias state that all behavioural effects discussed above can be explained by the interacting neighbours model. First, the size effect occurs, because large problems have relatively more inconsistent than consistent neighbours. Secondly, there is a tie effect. The semantic field has a triangular structure due to the fact that only one problem of
commutative pairs is represented in the semantic field. Therefore, tie problems have less neighbours, and, since the majority of the neighbours is inconsistent, less inconsistent neighbours. This makes the tie easier to solve. Thirdly, according to Verguts and Fias (2005), there is a size x tie interaction effect because there is a positive correlation between size and inconsistent neighbours, but this correlation is smaller for the tie problems than for the non-tie problems. Finally, the five effect occurs, because all problems in the five times table have a neighbour with a distance of two that has a consistent unit. For example, 5 x 5 = 25 has the consistent neighbour 5 x 7 = 35 and 5 x 6 = 30 has the consistent neighbour 5 x 8 = 40.
7 x 2 3 4 5 6 7 8 9 2 4 3 6 9 4 8 12 16 5 10 15 20 25 6 12 18 24 30 36 7 14 21 28 35 42 49 8 16 24 32 40 48 56 64 9 18 27 36 45 54 63 72 81
Figure 1. Semantic field of the multiplication table.
Note that the original interacting neighbours model holds that neighbouring problems up to a distance of two are activated. However, in most studies concerning this model (e.g. Domahs et al., 2007; Campbell, Dowd, Frick, McCallum & Metcalfe, 2011) only direct neighbouring problems with a distance of one are considered to influence solving the multiplication problems. These studies do not explicitly clarify why they do this, but the reason could be that there are less consistent neighbouring problems with a distance of two and that these problems are less evenly distributed across the multiplication tables, see Table 1. This might make it more difficult to study the effect of having consistent neighbouring problems apart from the effects of the different tables, because studying the neighbourhood consistency effect will then be biased by an effect of the multiplication table. However, this is only one of perhaps many explanations and these need to be studied more in the future. Because that is beyond the scope of this study, we will for now also only focus on
neighbouring problems with a distance of one, which will make our study easier to fit in with earlier research.
8 Table 1
Number of Consistent Neighbours with a Distance of 1 and 2 for every Multiplication Table
Multiplication table
Number of consistent neighbours with a distance of 1 (percentage of total consistent neighbours with a distance of 1)
Number of consistent neighbours with a distance of 2 (percentage of total consistent neighbours with a distance of 2)
2 32 (16%) 22 (25%) 3 34 (17%) 8 (9%) 4 26 (13%) 8 (9%) 5 30 (15%) 24 (27%) 6 30 (15%) 6 (7%) 7 22 (11%) 12 (14%) 8 18 (9%) 4 (5%) 9 8 (4%) 4 (5%) Total 200 88
(Due to rounding to zero decimal places, the
sum of the above percentages is 101%)
The interacting neighbours model has above been explained using the production paradigm, but in this study we will use the verification paradigm. Because the model has different effects in the two paradigms, we will also explain the model in light of the verification paradigm. In the production paradigm participants are asked to produce the solution to a multiplication problem. In the
verification paradigm, participants are shown a multiplication problem with either a correct or incorrect solution and they have to decide whether the solution is correct or incorrect. According to Zbrodoff and Logan (1990), the macro processes involved in these two paradigms are different, but the paradigms use the same representations. However, the way we should look at the results of these paradigms is different: the factors that ease solving a multiplication problem in the production
9 paradigm complicate rejecting an incorrect solution in the verification paradigm. So whereas the interaction neighbours model states that problems with consistent neighbours are relatively easy to solve, in a verification paradigm it is more difficult to reject a consistent than an inconsistent solution to a problem. For example, the problem 3 x 4 is fairly easy to solve in the production paradigm because it has its decade in common with three of its neighbours (3 x 5 = 15, 3 x 6 = 18 and 4 x 4 = 16). However, in the verification paradigm it is difficult to reject 16 as the solution to 3 x 4, because 16 is a consistent neighbour of the correct solution 12.
Next, we will discuss evidence for the existence of the interacting neighbours model. First, studies show evidence for a neighbourhood effect: if a problem is presented, neighbouring problems, having an operand in common, will also be activated. Children produce more operand-related errors, i.e. solution that are part of the same multiplication table, than errors that include another incorrect solution from the multiplication tables (Campbell & Graham, 1985). The study by Domahs et al. (2007) confirms this result in a verification task. Adults had more trouble rejecting incorrect solutions that were operand-related to the correct solution than non-related solutions. Also, children in second and third grade reject incorrect solutions that are one step away in the multiplication table more slowly than solutions that are two, three or four steps away (De Brauwer & Fias, 2009). A second effect that the interacting neighbours model can account for is the consistency effect. Adults found it harder to reject incorrect solutions that were decade consistent with the correct solution than inconsistent incorrect solutions (Domahs et al.). A third effect is the combined neighbourhood consistency effect: when the time between presentation of problem and solution was short, consistent rather than inconsistent incorrect solutions were only harder to reject when the solution was also operand-related to the correct solution. Campbell, Dowd, Frick, McCallum and Metcalfe (2011) showed this effect by constructing two tables of arithmetic equations with a novel, made-up operation. In one table problems had consistent neighbours (same decades), in the other table problems had inconsistent neighbours (different decades). Producing solutions to the problems in
10 the set with consistent rather than inconsistent neighbours was easier for the adult participants in this study.
This neighbourhood consistency effect could be partly the same as another effect that has been found repeatedly in studies using the verification paradigm: the distance effect. This effect holds that it is easier to reject an incorrect solution with a large rather than a small numerical distance to the correct solution (De Rammelaere, Stuyven & Vandierendonck, 2001). This has also been found in addition (Ashcraft & Stazyk, 1981, Szűcs & Csépe, 2005). It is assumed that the correct solution is retrieved from memory and is compared to the presented incorrect solution (Ashcraft & Stazyk), since numerical distance between two numbers is the best predictor of the difficulty of comparing these numbers (Nuerk, Weger & Willmes, 2001), with larger distances resulting in faster comparisons.
To our knowledge, the neighbourhood consistency effect and the distance effect were never combined in a study, which is an important omission in the literature. Since neighbouring problems are problems with operands only differing by one, solutions to neighbouring problems generally have a small distance. Furthermore, decade-consistent solutions can differ by a maximum of nine, also resulting in a relatively small distance. This makes us question whether there are in fact two distinct effects, or whether the effects always co-occur. Because only measuring one effect means results can be confounded by the other, it is important to study these effects together in one study. Niedeggen and Rösler (1999) did look at the combined influence of the distance effect and the neighbourhood effect in adult participants and found that the effects didn’t interact. However, this study only concerned the neighbourhood effect and not the combined neighbourhood consistency effect. Moreover, this study, as many others (De Brauwer & Fias, 2009), concerned adult participants. Because any underlying processes are likely to be still developing within children, it is important to study these effects with child participants as well.
Apart from studying these effects, it is important to identify any underlying processes of the effects occurring during multiplication verification, one of which is inhibition. Inhibition could be
11 important for supressing irrelevant operational strategies in maths or intermediate solutions when solving a problem that requires multiple steps (Jansen, De Lange, & Van der Molen, 2013). Several studies indeed show a relation between inhibition and mathematical skills. Children with high rather than low mathematics ability suffer less of interference from irrelevant information while solving maths problems (Bull & Scerif, 2001) and inhibition and the end of year grades for mathematics are correlated (Oberle & Schonert-Reichl, 2013). Looking at the interacting neighbours model, it is likely that the role of inhibition in a verification task depends on which incorrect solution is presented with a multiplication problem. Inhibition could be required to reject consistent neighbour solutions, but this could be less when inconsistent non-neighbour solutions are presented. Inhibition could also be an underlying process of the distance effect, in the sense that incorrect solutions have to be inhibited actively. If the incorrect solution is very plausible because of its distance to the correct solution, more inhibition is needed which takes more effort, resulting in longer reaction times and more errors.
This study will explore the effects that influence performance on a multiplication verification task and the relation of these effects with inhibition. First, we will study whether children and adults show (interacting) neighbourhood consistency and distance effects by looking at their reaction time and errors. Are participants slower and do they make more errors in rejecting incorrect solutions that are consistent neighbours of the correct solution compared to incorrect solutions that are
inconsistent non-neighbours of the correct solution? Are participants slower and do they make more errors in rejecting incorrect solutions with a small rather than a larger distance to the correct
solution? Do these effects interact so that the distance effect is bigger for consistent neighbours than for inconsistent non-neighbours? Second, we will study whether one or both of these effects are driven by underlying processes of inhibition. Do children with higher levels of inhibitory control show reduced neighbourhood consistency and distance effects on reaction times and error rates compared to children with lower levels of inhibitory control?
In the research proposal of this study we first decided to only focus on children. For the practical reason that it took longer than expected to find schools to cooperate in this study, we
12 decided to also study the neighbourhood consistency and distance effects in adults to make use of the waiting time. Therefore, we will study these effects in adults and age differences in these effects in an explorative way.
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Method
Participants
Our sample consisted of 105 participants, of which 84 children and 21 adults. The child participants were recruited from two primary schools, one in a village near Loughborough (30 participants) and one in Nottingham (54 participants). The adult participants were recruited from staff and students of Loughborough University.
We have excluded 18 child participants after the study. Seventeen participants were excluded to make sure that all participants had sufficient multiplication ability. They were excluded because their score on the problems with correct solutions was not significantly above chance, meaning that they had less than 40 out of 64 trials correct. One participant was excluded because he wanted to quit during the experiment. For one child, the inhibition task was not administered correctly due to experimenter errors, so this participant is excluded from the analyses where inhibition scores are involved. From one child there were no maths scores available, because he was new at the school. This child is excluded from the analyses where maths scores are involved.
Therefore, our final sample consisted of a total of 87 participants, of which 21 adults (7 male) and 66 children (33 male) in Year 5 (31 children) and 6 (35 children). The mean age of the adult participants was 26 years and 2 months, ranging from 18 years and 9 months to 37 years and 7 months, and the mean age of the child participants was 10 years and 10 months, ranging from 9 years and 10 months to 11 years and 9 months.
For this study, we received ethical approval from the Loughborough University Ethics Approvals (Human Participants) Sub-Committee. Information letters were distributed amongst all parents of possible child participants. These letters contained an opt-out sheet parents could fill out if they didn’t want their child participating in the study.
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Material
The multiplication verification task consisted of 64 multiplication problems, each presented twice: with an incorrect solution and with the correct solution, resulting in a total number of 128 trials. At each trial, participants had to indicate whether the given solution to the problem was correct or incorrect. The problems were presented in a pseudo-random order in eight blocks of 16 problems, to prevent consecutive presentation of the same problem. The problems used in the multiplication task can be found in Appendix A. For every category of incorrect solutions, i.e. consistent neighbours with a small or large distance and inconsistent non-neighbours with a small or large distance, eight distinct problems were chosen from the multiplication table ranging from 2 x 2 to 9 x 9. The problems and their solutions were chosen such that they met three criteria. First, the ratio of the given incorrect and the actual correct solution to a problem was between 1.05 and 1.20 for the small distance sets and between 1.30 and 1.70 for the large distance sets. This resulted in mean distance ratios of 1.16 (consistent neighbours small distance), 1.46 (consistent neighbours large distance), 1.11 (inconsistent non-neighbours small distance) and 1.44 (inconsistent non-neighbours large distance). Secondly, in each category half of the incorrect solutions were larger and half were smaller than the correct solution. Thirdly, the parity of half of the incorrect solutions in each category was the same as the parity of the correct solution and the parity of the other half was different. For each problem, the commutative non-ties were placed in the same category and ties were placed in the same category twice, resulting in sixteen problems per category. From this task, a measure of multiplication ability was calculated by determining the accuracy (number of correct responses) on problems with a correct solution.
The multiplication task was implemented in PsychoPy and performed on a 15.1 inch Windows laptop. The problems were presented in white letters (font type Arial) on a grey
background. The distance between the children and the laptop could was set in such a way that the children could comfortably operate the computer.
15 For the inhibition measure, the inhibition subtest of the NEPSY (Korkman, Kirk & Kemp, 2007) was used. The NEPSY is a test battery that aims to measure several executive functions. The
inhibition subtest can be used to measure the ability to inhibit automatic responses in favour of novel responses in children aged 5 to 16 years (Korkman, Kirk & Kemp, 2007). The subtest consists of two naming tasks and two inhibition tasks. First, children complete a naming task, in which they see black and white circles and squares and have to name them (‘circle’ or ‘square’) as fast as possible. Secondly, children complete an inhibition task, in which they see the same black and white circles and squares, but now they have to give the opposite name for each shape, so ‘circle’ when they see a square and ‘square’ when they see a circle. Thirdly, children complete the second naming task, in which they see black and white up and down arrows and have to name them (‘up’ or ‘down’) as fast as possible. Finally, children complete the second inhibition task, in which they see the same black and white up and down arrows, but now they have to give the opposite name for each shape, so ‘up’ when they see a down arrow and ‘down’ when they see an up arrow. A naming combined scaled score is calculated based on the normed scores for completion time (maximum 360 seconds) and number of uncorrected and self-corrected errors (maximum 80 errors) on the two naming tasks. The score ranges from 1 to 19, with higher scores indicating better naming skills. An inhibition combined scaled score is calculated based on the normed scores for completion time (maximum 480 seconds) and number of uncorrected and self-corrected errors (maximum 80 errors) on the two inhibition tasks. The score ranges from 1 to 19 with higher scores indicating better inhibition skills. The naming-inhibition contrast scaled score is then calculated based on the naming combined scaled score and inhibition combined scaled score. The score ranges from 1 to 19 with higher scores indicating higher than expected inhibition skills considering the naming skills. The internal reliability for the inhibition subtest for our age group is r = .8 and the test-retest reliability is r = .66 for 9-10 year olds and r = .8 for 11-12 year olds. There is a large amount of validity evidence (Brooks, Sherman & Strauss, 2010).
The maths scores were directly retrieved from the schools. Due to availability issues in the schools, we used scores from two different tests. For all Year 5’s and the Year 6’s of the first school,
16 we used the summer midterm maths test administered at these schools. The scores of this test range from level 1 to 8, with every level subdivided into three levels. Level 4 for example consists of 4a (slightly above average level 4), 4b (average level 4) and 4c (slightly below average level 4). To be able to use these scores in our analyses, we constructed new maths scores ranging from 1 to 9, for which we changed the lowest maths score 3c into 1, the second lowest maths score 3b into 2, and so further until the highest maths score 5a, which was changed into 9. For the Year 6’s of the second school, we used the maths scores of the key stage 2 National Curriculum Tests, better known as SATs. The scores of this test also range from level 1 to 8, but with no subdivision within the levels. To be able to use these scores in our analyses, we constructed new maths scores ranging from 1 to 4, for which we changed the lowest maths score 3 into 1, the second lowest maths score 4 into 2, the second highest maths score 5 into 3 and the highest maths score 6 into 4.
Procedure
The adults were tested individually in the quiet lab in the Mathematics Education Centre at Loughborough University and received £2 for their participation. They were presented with some information about the study and signed an informed consent. Then, they performed the
multiplication task.
The children from the first school were tested individually in the library, which was a small area at the back of the class. The children from the second school were tested in the hallway of the school. Two children were tested at the same time by different experimenters. The children were given a certificate for their participation. They were asked to come to the testing area and were given an information sheet about the study, which they read together with the experimenter. This
information sheet gave some information about the tasks and stated that they could stop at any time without giving a reason to do so. Then the children performed the multiplication task and the
inhibition task, the order of which was counterbalanced. The order in which each experimenter executed the tasks was also counterbalanced.
17 At the beginning of the multiplication task, information about how the task had to be carried out was presented on the screen. The adult participants read this information by themselves,
whereas the children read the information together with the experimenter. One subtraction trial was answered together with the experimenter, to see whether the purpose of the task was clear,
followed by 20 practice subtraction trials that the participants carried out on their own. If necessary, extra help was given to make sure the task was carried out appropriately. After the practice trials and possible questions, the test trials were started. Each trial started with a ready signal (‘Ready?’). Participants had to press the space bar for appearance of the multiplication problem on the screen in the standard from a x b = c for 3000 msec. Only during this time could participants indicate whether the solution was correct or incorrect by pressing a green (correct) or red (incorrect) key on the keyboard (‘x’ or ‘,’) with the right or left index finger, which was counterbalanced. If participants didn’t respond within this given time, a warning was given to respond faster next time, which was the only feedback. Every 16 trials, a screen stating that the participant could take a break if necessary was presented. The order of the screens presented in the multiplication verification task is also presented in Figure 2.
Figure 2. Order of the screens presented in the multiplication verification task.
The inhibition task was presented according to the standard procedure described in the manual (Korkman, Kirk & Kemp, 2007).
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Results
Below we will first present the results on whether there are significant differences between children of the different years and between children of the different schools. Next, we will present the results on the neighbourhood consistency effects and the distance effects. The final section will go into inhibition as an underlying process of the distance and the neighbourhood consistency effect.
We decided to analyse the data from the adult and child participants together at first. If there turned out to be any interaction effects with age group (child or adult), we could later conduct the analyses by age group.
In the proposal we planned to test whether multiplication ability is correlated with
performance on problems with an incorrect solution and if so, to include multiplication ability as a covariate in the analyses. However, below we will only present results without multiplication ability as a covariate. The results with multiplication ability as a covariate as well as an explanation for not including these results in the main results section can be found in appendix B.
Differences between years and schools
First, we tested whether there were significant differences between the responses from children in Year 5 and children in Year 6. Therefore we conducted two independent samples t-tests to compare RTs (in seconds) and error rates (in number of errors) on problems with an incorrect solution for children in Year 5 and Year 6. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 2 = .025. For RTs, there were no significant differences between Year 5 (M = 1.88, SD = 0.29) and Year 6 (M = 1.77, SD = 0.28), t(64) = 1.50, p = .14. Because the error rates were not normally distributed in Year 6 (S-W = 0.81, df = 35, p < .001), we calculated the Mann-Whitney test to compare error rates in Year 5 and Year 6. There was no significant difference between the error rates in Year 5 (Mdn = 13) and Year 6 (Mdn = 9), U = 394.5, z = -1.91, p = .06.
19 Secondly, we tested whether there were significant differences between the responses from the children of the two different schools. Therefore, we conducted two independent samples t-tests to compare RTs and error rates on problems with an incorrect solution for children from the two different schools. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 2 = .025. For RTs, there were no significant differences between school 1 (M = 1.84, SD = 0.33) and school 2 (M = 1.81, SD = 0.26), t(64) = 0.35, p = .73. Because the error rates were not normally distributed for school 1 (S-W = 0.85, df = 24, p = .003) and school 2 (S-W = 0.94, df = 42, p = .029), we calculated the Mann-Whitney test to compare error rates in school 1 and school 2. There were no significant differences between school 1 (Mdn = 8.50) and school 2 (Mdn = 11.50), U = 462, z = -0.56, p = .58.
The above shows that there were no differences in the responses from the children from different years or different schools. Therefore, we collapsed the data from all children.
Neighbourhood consistency and distance effects
To test the neighbourhood consistency and distance effects, we conducted a 2 (consistent neighbour vs. inconsistent non-neighbour) x 2 (small vs. large distance) x 2 (adult vs. child) mixed design ANOVA on RTs of all problems presented with an incorrect solution, either correctly or incorrectly answered. The analysis showed a main effect of age group, F(1, 85) = 52.16, p < .001, with longer reaction times for children than for adults. There was also a significant main effect for distance, F(1, 85) = 12.43, p = .001, with longer reaction times for small distances than for large distances. The neighbourhood consistency effect was non-significant, F(1, 85) = 3.02, p = .08, as well as the distance x
neighbourhood interaction effect, F(1, 85) = 0.02, p = .88. All interactions with age group were non-significant, meaning that the effects were the same for adults as for children. The mean RTs are shown in Figure 3.
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Figure 3. Mean RT’s on problems presented with an incorrect solution. The error bars represent the
standard errors of the means.
We also conducted a 2 (consistent neighbour vs. inconsistent non-neighbour) x 2 (small vs. large distance) x 2 (adult vs. child) mixed design ANOVA on error rates of problems presented with an incorrect solution. The analysis showed a main effect of age group, F(1, 85) = 9.92, p = .002, with higher error rates for children than for adults. There was also a significant neighbourhood consistency effect, F(1, 85) = 46.00, p < .001, with more errors for consistent neighbours than for inconsistent non-neighbours. The distance effect was non-significant, F(1,85) = 3.84, p = .05. More importantly there was a significant neighbourhood x distance interaction effect, F(1, 85) = 4.49, p =
21 .04, showing a stronger neighbourhood consistency effect for small distances than for large
distances. To interpret this effect, we performed four paired samples t-tests between the error rates of the consistent neighbours with a small and a large distance and the inconsistent non-neighbours with a small and a large distance, for all children and adults together. The means and standard deviations are given in Table 2. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 4 = .0125. There was a significant difference between error rates of consistent neighbours with a small distance and consistent neighbours with a large distance, t(86) = 2.96, p = .004, but no significant difference between inconsistent non-neighbours with a small distance and inconsistent non-neighbours with a large distance, t(86) = -0.58, p = .567. This difference causes the interaction effect: there is a distance effect for consistent neighbours, with higher error rates for small distances than for large distances, but not for inconsistent non-neighbours. There were significant differences between error rates of consistent neighbours and inconsistent non-neighbours with a small distance, t(86) = 7.69, p < .001 and between consistent neighbours with a large distance and inconsistent non-neighbours with a large distance, t(86) = 4.15,
p < .001. This means that there was a neighbourhood consistency effect both for small distances as
well as large distances. All interactions with age group were non-significant, meaning that the effects were the same for adults as for children. The mean error rates are shown in Figure 4.
Table 2
Mean Error Rates and Standard Deviations (between Brackets) for Consistent Neighbours with a
Small and a Large Distance and Inconsistent Non-neighbours with a Small and a Large distance
Consistent neighbours Inconsistent non-neighbours
Small distance 3.90 (2.91) 1.99 (2.29)
22
Figure 4. Mean error rates on problems presented with an incorrect solution. The error bars
represent the standard errors of the means.
Inhibition
To see whether it would be useful to perform ANCOVA’s with inhibition as a covariate, we calculated the correlation between inhibition and RTs of problems presented with incorrect solutions and inhibition and error rates of problems with incorrect solutions. To do this, we used the naming-inhibition contrast score from the NEPSY naming-inhibition subtest. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 2 = .025. The correlation between inhibition and RTs is r = .04, p = .76 and the correlation between inhibition and error rates is r = -.05, p = .72.
23 This means there is no significant relation between inhibition and performance on problems
presented with an incorrect solution. Since both correlations are very small and non-significant, it is not useful to include inhibition as a covariate in the analyses. Because this is against our
expectations, we will look at some alternative explanations for these small correlations in an explorative way.
The small correlation could be due to the fact that there is too little variation in the inhibition measure. However, as can be seen in Figure 5, there is quite a lot of variation in the
naming-inhibition contrast score. Therefore, it doesn’t seem plausible that this is an explanation for the small correlation between inhibition and RTs or inhibition and error rates.
24 However, what can be noticed in Figure 5, is that there seem to be two groups of scores within the naming-inhibition contrast score: one ranging from 2 to 8 or 9 and one ranging from 9 or 10 to 16. It could be that different participants scored very differently on this variable and that no correlation can be found between inhibition and reaction times or error rates when all scores are taken into account. First, it is important to find out why these two groups of inhibition scores seem to exist. Three possibilities are that children in different years, from different ages or from different sexes score differently on the naming-inhibition contrast score. Therefore, Table 3 shows the percentage of children per year, age and sex scoring in a certain range of the naming-inhibition contrast score.
Table 3
Cumulative Percentages of Persons scoring in certain Ranges of the Naming-inhibition Contrast Score,
per Year, Age and Sex.
Naming-inhibition contrast score Year Age Sex
5 (n=31) 6 (n=34) 9 (n=6) 10 (n=30) 11 (n=29) male (n=33) female (n=32) 2-8 45.2 50.0 33.3 43.3 55.2 45.5 50.0 9-16 54.8 50.0 66.7 56.7 44.8 54.5 50.0 2-9 45.2 55.9 33.3 50.0 55.2 48.5 53.1 10-16 54.8 44.1 66.7 50.0 44.8 51.5 46.9
It can be concluded from the table that children in different years, from different ages or from different sexes do not score very differently on the naming-inhibition contrast score. Every time, about 50% of the children score in either group. The only time this percentage is very different from
25 50% is the larger amount of 9 year olds scoring highly on the inhibition measure, but this result seems distorted because of the small amount of 9 year old participants.
Apart from the fact that the reason for these two groups scoring differently on the inhibition measure is not easy to find, it could still be that correlations between the naming-inhibition contrast score and reaction times or error rates can only be found within one group of low or high scoring children. Therefore, we also calculated correlations between the naming-inhibition contrast score and reaction times and error rates per range of the naming-inhibition contrast score. The results can be seen in Table 4. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 8 = .006
Table 4
Correlations and their p-values between the Naming-inhibition Contrast Score and Reaction Times
and Error Rates, for certain Ranges of the Naming-Inhibition Contrast Score
Naming-inhibition contrast score Reaction times Error rates r p r p All .04 .76 -.05 .72 2-8 -.05 .80 .13 .49 9-16 .01 .95 -.32 .06 2-9 -.07 .70 .18 .30 10-16 -.03 .86 -.31 .09
Also after breaking down the naming-inhibition contrast score into two groups in two different ways, no significant correlations can be found between the naming-inhibition contrast score and reaction times or error rates. The correlations between the naming-inhibition contrast score are somewhat
26 higher in the groups where a higher naming-inhibition contrast score was obtained, with higher naming-inhibition contrast scores corresponding to lower error rates, but this correlation is not high enough to be significant.
Another explanation for the low correlation is the type of inhibition measure that is used. If the naming-inhibition contrast score is used, someone with a very low naming as well as a very low inhibition score will have the same naming-inhibition contrast score as someone who scores very high on both naming and inhibition. So when using the naming-inhibition contrast score, all information about absolute performance is lost. Therefore, as an alternative one can use the
inhibition score, while controlling for the naming score. We have calculated correlations between the inhibition score and RTs of problems presented with incorrect solutions and the inhibition score and error rates of problems presented with incorrect solutions. For this we used partial correlations in which we controlled for the naming score. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 2 = .025. While controlling for naming, the correlation between inhibition and RTs is r = .004, p = .98 and the correlation between inhibition and error rates is r = -0.135, p = .29. It can be concluded that using the inhibition score while controlling for the naming score does not alter the results obtained using the inhibition-naming scaled contrast score.
Although the NEPSY is a widely used instrument to measure inhibition, it could be that the naming-inhibition contrast score is just not a good inhibition score. Since it has been shown that inhibition and maths ability are related (Bul & Scerif, 2001; Oberle & Schonert-Reichl, 2013), we tested this by measuring correlations between inhibition and multiplication ability and between inhibition and maths ability. Because the maths ability scores are interval data, we will report the Spearman’s rank order correlation. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 3 = .017. The correlation between inhibition and multiplication ability is r = .005, p = .97, the correlation between inhibition and summer midterm maths is r = .004, p = .98 and the correlation between inhibition and SATs is r = .11, p = .66. This shows that all possible maths ability measures have very low and non-significant correlations with
27 the naming-inhibition contrast score of the NEPSY. These low correlations could indicate that the naming-inhibition contrast score is not a good inhibition score and so this could be an explanation for the low correlations we found between inhibition and performance on the problems with an
28
Discussion
In this study we looked at whether children and adults show neighbourhood consistency and distance effects when solving problems in a multiplication verification task. We also looked at whether one or both of these effects could be driven by underlying processes of inhibition. In this section we will discuss the found neighbourhood consistency and distance effects and how these relate to previous studies. We will also go into explanations of why our study did not show a relation between inhibition and performance on the multiplication task. Finally, we will discuss some
limitations of our study and the interacting neighbours model as well as give some recommendations for further research.
From our results it is evident that children were slower in responding to the multiplication problems and made more errors in their responses than adults. All other effects were the same for children and adults, which implies that children aged nine to eleven already solve multiplication verification problems in the same way as adults, only a bit slower and with more errors. Next, it was shown that there are different effects for reaction time and errors. Solutions with a small distance to the correct solution were answered more slowly than solutions with a large distance (distance effect). On the other hand, it was shown that more errors were made when the solution was a consistent neighbour to the correct solution than an inconsistent non-neighbour (neighbourhood consistency effect). This effect was stronger when the solution had a small distance to the correct solution than when it had a large distance.
Interestingly, the neighbourhood consistency effect with respect to reaction times, shown in previous studies, has disappeared when the distance from the incorrect solution to the correct solution is taken into account. It could be that this distance is actually more important than the neighbourhood consistency and that previous studies only found a neighbourhood consistency effect because they didn’t take into account the distance. This could be the case, because solutions that are neighbours and decade consistent with the correct solution, also have a small distance to the correct
29 solution. Another explanation is that the effects are simply different for adults than for children and since we used mainly children as our participants, we found a different effect than studies that only used adults as participants. However, this is not a likely explanation, since we found that the effects were the same for our child participants as for our adult participants.
A second way in which our study differs from previous studies is that we did find neighbourhood consistency and distance effects on the error rates, whereas previous studies generally didn’t. It could be that differences in methodology have caused these different effects. Some studies for example let participants decide for themselves how long they wanted to think about a problem before they responded, whereas our study limited this to three seconds. It could be that errors are limited when participants have unlimited time, masking a possible effect on error rates. To test this, future research should use different time limits to compare these time limits directly and see whether they are the cause for the appearance of any effects on error rates.
The fact that we found different effects on reaction times and on error rates suggests that there are different underlying mechanisms at work for speed and accuracy. Based on the results, it could be the case that when deciding whether a solution is correct or incorrect, it is mainly important whether the solution is a consistent neighbour or an inconsistent neighbour, but deciding this is easier for large distance solutions than for small distance solutions. The speed with which this decision is made is mainly determined by the distance of the solution to the correct solution, with faster decisions for large distance solutions than for small distance solutions.
Apart from the neighbourhood consistency and distance effects themselves, we also studied inhibition as a possible underlying process of these effects. Although there is an established
relationship between mathematics and inhibition, our study contradicted the idea that inhibition is the underlying mechanism for the neighbourhood consistency and the distance effect. An alternative explanation lies in the type of inhibition task used. It has been shown that mathematical ability is more strongly related to inhibition when the inhibition task contains numerical information than when the task measures inhibition skills in a more general way (Bull & Scerif, 2001). Bull and Scerif
30 administered two versions of the Stroop task. In the non-numerical task participants had to name the colour in which a word (e.g. the word BLUE) was printed, with the ink (e.g. green) not corresponding to the meaning of the word. In the numerical task participants had to name the quantity of a string of printed numbers (e.g. 777), with the quantity (3) not corresponding to the meaning of the printed numbers. There appeared to be a moderate to large significant correlation between the numerical inhibition task and a measure of mathematics, whereas the correlation between the non-numerical inhibition task and the measure of mathematics was non-significant. Gilmore, Keeble, Richardson and Cragg (2015) also found a stronger relationship between mathematical ability and performance on a numerical inhibition task than between mathematical ability and a non-numerical inhibition task. They found that being able to ignore irrelevant numerical information is particularly important for mathematical performance. It could be that mainly inhibition measured with a numerical inhibition task will be related to mathematical performance. Because we did not only find that inhibition wasn’t an underlying mechanism of our effects, but also found that performance on our non-numerical inhibition task was not related to multiplication ability, we think the nature of our inhibition task could very well be the explanation for the fact that we didn’t find inhibition to be an underlying mechanism of the found effects. However, not all studies with a non-numerical inhibition task found no correlation with mathematics ability. Oberle and Schonert-Reichl (2013) administered a non-numerical inhibition task in which a heart or a flower was presented on the left or right screen of a screen. Participants had to press a key on the same side if the stimulus was a heart (e.g. press left if the heart appeared left) and a key on the opposite side if the stimulus was a flower (e.g. press left if the flower appeared right), with the latter measuring Inhibition. Inhibition still appeared to be correlated with end of year grades for mathematics, even though a non-numerical inhibition task was used. The exact relation between the involvement of numerical information in an inhibition task and its relationship with mathematics is therefore still unclear. This stresses the importance of future studies looking into the different types of inhibition measured with different tasks and their relation with mathematical or more specifically multiplication ability.
31 One very important limitation of our study lies in the limitations of the multiplication table: when using the real multiplication table (which most studies use from 2 x 2 to 9 x 9), one is bound by the characteristics of this table and when constructing items for a multiplication task, confounds are inevitable. An example is that not every table has the same amount of consistent neighbours. Future studies should focus on solutions to this problem. One solution is making more use of made up multiplication tables, in which all factors can be controlled for, like in the study of Campbell, Dowd, Frick, McCallum and Metcalfe (2011). However, it can be questioned whether results from these made up multiplication tables can simply be generalized to the real multiplication tables, so it could be important to compare and combine results from real multiplication tables and made up
multiplication tables.
A limitation of the interacting neighbours model is that it speaks of consistent neighbours in terms of consistent with respect to their decades as well as their units. The five effect often found in multiplication can only be explained by taking into account the consistency in the units. However, all previous studies as well as this current study have only focused on the consistency of the decades. Because there are only a couple of multiplication problems with a consistent unit neighbour and all of these problems have five as an operand, it was not possible to include this when constructing our item set: the influence of having a consistent unit could then not be separated from having five as an operand. Although other studies have also only included decade consistent neighbours, we have not found a study explaining why the unit consistent neighbours were ignored. Because the unit
consistent neighbours are also part of the model of Verguts and Fias (2005), we think it is very important that future research looks at the contribution of the consistent units to the model. This could for example be studied by making use of made up multiplication tables, as mentioned earlier.
To our knowledge, our study has been the first to investigate both the neighbourhood consistency effect as well as the distance effect. We found different effects for speed than for accuracy, which shows that it is very important to combine both effects, because only studying one means the results can be confounded by the other. It also implies that speed and accuracy in
32 responding to a multiplication verification problem may have different underlying mechanisms. Although our study showed no relation between multiplication and inhibition, inhibition cannot be ruled out as an underlying mechanism, but it is very important to study different types of inhibition and identify whether some of these are an underlying mechanism of multiplication. For now this study has tried to shed more light on the processes that underlie learning and performing
multiplication, which will hopefully make it easier to improve children’s multiplication skills through effective teaching methods in the future.
33
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36
Appendix A: Problem set
Cooperative neighbours small distance
Problem Correct solution
Incorrect solution
Direction* Ratio Parity
2 x 9 18 16 - 1.13 Same 3 x 8 24 21 - 1.14 Different 4 x 6 24 28 + 1.17 Same 4 x 9 36 32 - 1.13 Same 5 x 6 30 35 + 1.17 Different 5 x 8 40 48 + 1.20 Same 6 x 7 42 49 + 1.17 Different 7 x 7 49 42 - 1.17 Different
Cooperative neighbours large distance
Problem Correct solution
Incorrect solution
Direction* Ratio Parity
2 x 2 4 6 + 1.50 Same 2 x 3 6 4 - 1.50 Same 2 x 5 10 15 + 1.50 Different 2 x 6 12 18 + 1.50 Same 3 x 3 9 6 - 1.50 Different 3 x 4 12 16 + 1.33 Same 3 x 5 15 10 - 1.50 Different 4 x 7 28 21 - 1.33 Different Competitive non-neighbours small distance Problem Correct solution Incorrect solution
Direction* Ratio Parity
2 x 4 8 9 + 1.13 Different 2 x 7 14 15 + 1.07 Different 3 x 9 27 25 - 1.08 Same 4 x 8 32 27 - 1.19 Different 5 x 5 25 21 - 1.19 Same 5 x 7 35 36 + 1.03 Different 5 x 9 45 49 + 1.09 Same 6 x 6 36 32 - 1.13 Same Competitive non-neighbours small distance Problem Correct solution Incorrect solution
Direction* Ratio Parity
2 x 8 16 21 + 1.31 Different
3 x 6 18 25 + 1.39 Different
3 x 7 21 32 + 1.52 Different
4 x 4 16 10 - 1.60 Same
37
6 x 8 48 63 + 1.31 Different
6 x 9 54 40 - 1.35 Same
7 x 8 56 42 - 1.33 Same
* If direction is +, the incorrect solution is larger than the correct solution. If direction is -, the incorrect solution is smaller than the correct solution.
38
Appendix B: Results with multiplication ability as covariate
In the research proposal of this study we planned to test whether accuracy on problems with the correct solution, reflecting multiplication ability, is correlated with accuracy on problems with an incorrect solution. If this would be the case, we would include accuracy on problems with the correct solution as a covariate in our analyses. However, after writing the research proposal we learned that when people are not randomly assigned to different groups, as is the case in our study where the groups are formed by children and adults, it is very likely that the covariate and the grouping variable are related and this should not be the case when including a covariate in the analysis (Miller & Chapman, 2001). Looking at the data, we see that there is indeed a significant difference between children (Mdn = 49) and adults (Mdn = 59) in accuracy on problems with the correct solution, U = 316.5, z = -4.50, p < .001. This means we actually should not include accuracy on problems with the correct solution as a covariate. However, because the analyses mentioned above were planned in the research proposal, we will present them below.
We tested whether performance on problems with the correct solution is correlated with RT and accuracy on problems with an incorrect solution. To minimize chances on a type I error, we conducted the Bonferroni correction and used α = .05 / 2 = .025. Because the assumption of
normality was not met for performance on problems with the correct solution (S-W = 0.95, df = 87, p = .002, negatively skewed) and for performance on problems with an incorrect solution (S-W = 0.94,
df = 87, p < .001, negatively skewed), we calculated Spearman’s correlation coefficients. There was a
significant positive correlation between multiplication ability and accuracy on problems with an incorrect solution, r = .83, p < .001. This means that participants with higher multiplication ability also had higher accuracy on problems with an incorrect solution. There was a significant negative
correlation between multiplication ability and RT on problems with an incorrect solution, r = -.46, p < .001. This means that participants with higher multiplication ability responded faster on problems with an incorrect solution.
39 The above shows that, as expected, multiplication ability is related to RT and accuracy on problems with an incorrect solution. Therefore, we will include multiplication ability as covariate in our analyses and present these results below.
We conducted a 2 (consistent neighbour vs. inconsistent non-neighbour) x 2 (small vs. large distance) x 2 (children vs. adults) mixed design ANOVA on RTs of problems with an incorrect solution with multiplication ability as covariate. The analysis showed a significant effect of age group, F(1, 84) = 30.72 p < .001, with longer reaction times for children than for adults . There was also a significant effect of multiplication ability, F(1, 84) = 33.86, p < .001, with longer reaction times for participants with lower multiplication ability than for participants with higher multiplication ability. All other effects were non-significant, all p’s > 0.05.
Next, we conducted a 2 (consistent neighbour vs. inconsistent non-neighbour) x 2 (small vs. large distance) x 2 (children vs. adults) mixed design ANOVA on error rates of problems with an incorrect solution with multiplication ability as covariate. The analysis showed that the effect of age group was non-significant, F(1, 84) = 0.55, p = .460, with equal error rates for children and adults. There was a significant effect of multiplication ability, F(1, 84) = 59.33, p < .001, with higher error rates for participants with lower multiplication ability than for participants with higher multiplication ability. Furthermore, there was a significant neighbourhood consistency effect, F(1, 84) = 16.41, p < .001, with more errors for consistent neighbours than for inconsistent non-neighbours. There was also a significant neighbourhood consistency x multiplication ability interaction effect, F(1, 84) = 10.15, p = .002. To interpret this effect, we calculated a variable for the neighbourhood consistency effect for every participant, reflecting the difference in error rates on problems with a consistent neighbour as solution and problems with an inconsistent non-neighbour as solution. The variable was calculated in the following way: neighbourhood consistency effect = mean error rate consistent neighbours (small and large distance) – mean error rate inconsistent non-neighbours (small and large distance). The relationship between the neighbourhood consistency effect and multiplication ability is shown in Figure 6. There is a significant negative correlation between neighbourhood consistency
40 effect and multiplication ability, r = -0.37, p < .001. This means that the higher a person’s
multiplication ability, the lower the neighbourhood consistency effect on error rates.
Figure 6. The relationship between the neighbourhood consistency effect and multiplication ability
on error rates.
The analysis on error rates also showed a significant distance effect, F(1, 84) = 5.86, p = .02, with more errors for small distances than for large distances. Furthermore, there was a significant distance x multiplication ability interaction effect, F(1, 84) = 4.79, p = .03. To interpret this effect, we calculated a variable for the distance effect for every participant, reflecting the difference in error rates on problems with a small distance solution and problems with a large distance solution. The
41 variable was calculated in the following way: distance effect = mean error rate small distances
(consistent neighbours and inconsistent non-neighbours) – mean error rate large distances (consistent neighbours and inconsistent non-neighbours). The relationship between the distance effect and multiplication ability is shown in Figure 7. There is a small to medium, but non-significant correlation between distance effect and multiplication ability, r = -0.19, p = .07. This means that the higher a person’s multiplication ability, the lower the distance effect on error rates. However, this needs to be interpreted carefully, because the correlation is non-significant. Other interaction effects were not significant, all p’s > 0.05.
