Introduction
Children with mathematical learning difficulties (MLD) have problems in estimating the positions of numbers on a number line; their estimations deviate more from the requested number as in typically developing children (e.g. Geary, Hoard, Byrd-Craven, Nugent, & Numtee,
2007; Geary, Hoard, Nugent, & Byrd-Craven,
2008; Van’t Noordende & Kolkman,
2013). However, the underlying causes of their estimation difficulties remain unclear. It is possible that they have problems in using estimation strategies, as the use of estimation strategies is related to number line performance (Ashcraft & Moore,
2012; Newman & Berger,
1984). In the last decade, there has been an increasing interest in the use of eye tracking to measure number processing (Hartmann,
2015; Mock, Huber, Klein, & Moeller, this issue). The aim of the current study is to unravel possible differences in number line estimation strategies between children with MLD and children with typical mathematical development using eye tracking.
Most of the previous research on number line estimation assumed that the estimations reflect an internal, mental representation of a number line. Participants’ estimations are modeled along a linear or logarithmic regression line, which leads to the assumption that magnitudes are represented logarithmic or linearly (e.g. Booth & Siegler,
2006; Siegler & Booth,
2004). Recently, researchers have criticized this assumption and developed new models like the two-linear model (Ebersbach, Luwel, Frick, Onghena, & Verschaffel,
2008; Moeller, Pixner, Kaufmann, & Nuerk,
2009) and the proportional judgment models (e.g. Barth & Paladino,
2011). These models are based on the view that children’s actual scores on an estimation task are influenced by the strategies they use and therefore do not allow for direct inferences on their mental representation. As such, a logarithmic estimation pattern does not necessarily imply an underlying logarithmic magnitude representation, but could also be caused by the inability to use an adequate estimation strategy (Sullivan & Barner,
2014). Thus, more research is needed to give insights into the actual strategies that children use during an estimation task.
The proportional judgment models propose the use of reference points (beginning, mid, and end) to estimate the target number on a line. These models have been tested with cyclic power models. It has been found that cyclic power models fit number line estimations better than linear and logarithmic models (Barth & Paladino,
2011; Friso-van den Bos et al.,
2015; Huber, Moeller, & Nuerk,
2014; Rouder & Geary,
2014; Slusser, Santiago, & Barth,
2013), suggesting reference points are indeed used for number line estimation. This has been confirmed by eye tracking studies (Schneider et al.,
2008; Sullivan, Juhasz, Slattery, & Barth,
2011). These studies examined at which aspects of the number line people fixate, and thus attend to before giving a response. It was found that the amount of fixations peaks around the beginning, mid- and endpoints of the line both in adults (Sullivan et al.,
2011) and in children (Schneider et al.,
2008), indicating that these points are used as reference points.
Developmental trends in number line estimation strategy use have been found during the first years of primary school. The use of the beginning, mid-, and endpoints seems to gradually develop from grade 1 onwards, starting with the use of the beginning point, than the beginning and endpoints and finally the use of all three points (Ashcraft & Moore,
2012; Friso-van den Bos et al.,
2015; Rouder & Geary,
2014; Schneider et al.,
2008; White & Szűcs,
2012). The use of all three reference points seems to appear earlier in development and seems to be more stable for smaller number ranges than for larger number ranges (Ashcraft & Moore,
2012). Moreover, Newman and Berger (
1984) found that younger children mainly reported using the beginning point of the line, whereas grade 3 children used the reference points more flexibly, according to their self-reports. The older children adapted their estimation strategy to the specific number to be estimated, i.e. they used the reference point closest to the target number.
To summarize, the described studies indicate that people make use of reference points when estimating numbers on a number line and an increasing use of different reference points becomes visible with increasing age and numerical experience. These estimation strategies are related to performance on number line tasks. For example, Newman and Berger (
1984) found that children who report using the reference points on the number line adaptively are more accurate in their estimations than children with a less flexible strategy use. Likewise, Sullivan and Barner (
2014) suggest that children who have problems with proportional reasoning will score low on a number line estimation task, because of problems with using adequate estimation strategies. This implies that the seemingly less linear—or more logarithmic—number line estimation patterns of children with MLD (e.g. Geary et al.,
2007,
2008) could actually be the reflection of inabilities to make use of efficient estimation strategies. This would be in line with other domains of mathematics, in which children with MLD also have shown to display difficulties in strategy use (Torbeyns, Verschaffel, & Ghesquière,
2004,
2006). They are likely to experience such problems on number line tasks as well, because they lag behind in mathematical skills needed to use estimation strategies. For example, the use of reference points on the number line is related to arithmetic procedures (Link, Nuerk, & Moeller,
2014) and children need to be aware that the midpoint of the line corresponds to the midpoint of the number range to correctly use it as a reference point (Ashcraft & Moore,
2012; White & Szűcs,
2012). This suggests that children who lag behind in mathematical abilities should have problems using reference points on number line tasks. Moreover, children with MLD often have difficulties in spatial cognition (Swanson & Jerman,
2006), a skill that is also needed to make use of proportional estimation strategies.
A recent study indeed showed differences in estimation strategies between children with MLD and a control group without MLD. As expected, children with MLD made less use of reference points compared to children without MLD. Surprisingly, however, the children with MLD looked
more at the midpoint than the control group (Van’t Noordende & Kolkman,
2013). Van’t Noordende and Kolkman (
2013) suggested that children with MLD know they can use the midpoint as a reference point, but do not adapt their estimation strategy to the number that has to be estimated. This hypothesis could not be confirmed, since strategy use was examined on task level (measuring strategy use across all estimated numbers) instead of item level (measuring strategy use per estimated number). Two other studies did assess differences in functionality of strategy use between children with and without developmental dyscalculia (Schot, Van Viersen, Van’t Noordende, Slot, & Kroesbergen,
2015; Van Viersen, Slot, Kroesbergen, Van’t Noordende, & Leseman,
2013). They defined the functionality of an estimation strategy by the proximity of the reference point to the number that had to be estimated, for example using the beginning point on a 0–100 number line to estimate the number 18. A dysfunctional estimation strategy was defined as using a reference point far away from the number that had to be estimated, for example using the endpoint on a 0–100 number line to estimate the number 18. A case study on a 9-year-old girl with developmental dyscalculia showed that the reference point used by this girl was dysfunctional in 26 % of the trials, whereas only 8 % of the estimation strategies used by the control group was dysfunctional (Van Viersen et al.,
2013). Schot et al. (
2015) included two children with developmental dyscalculia and plotted fixation patterns on the number line with respect to both the numbers that had to be estimated and the number that was estimated by the children. They found that the fixations of the children with developmental dyscalculia were more scattered across the number line and farther away from both the target number and the response than in the control group, indicating no—or at least a weaker—relation between the target number or the response and looking behavior. Together, these results suggest that children with MLD have problems in using functional estimation strategies in number line estimation. However, these studies were case studies and did not statistically test differences in functionality of strategy use. Therefore, in the current study strategy use on number lines 0–100 and 0–1000 will be tested with eye tracking in a larger group of children with MLD. The goal is to assess whether children with MLD differ in strategy use from children without MLD and more specifically, whether children with MLD use less functional estimation strategies than children without MLD. This will help us to understand the specific difficulties of children with MLD on number line estimation.
Method
Participants
A group of 14 children (2 boys and 12 girls;
M age = 11.09, SD = 1.10 years) with mathematical learning difficulties (MLD) participated in this study.
^{1} These children were recruited via the ambulatory service of Utrecht University specialized in dyscalculia to which they were referred because of problems with mathematical learning in school. All children in the specified age range of 10–12 years and whose parents gave permission to use test results for research purposes were included in the study. On average they lag behind 19 months in automatization in mathematics compared to typically developing children. All children met the criteria for dyscalculia used in the centre: they scored below the 10th percentile on standardized math tests [both a timed test with basic facts (TempoToets Automatiseren) and a standard national criterion-based math test (CITO) that is administered twice a year in almost every classroom in the Netherlands]. The CITO mathematics test consists of grade-appropriate mathematics problems, primarily word problems that cover a wide range of mathematics domains such as measurement, time, and proportions. Scores are converted into five categories: 0–10, 10–25, 25–50, 50–75, and 75–100 %. All MLD children scored in the lowest category on at least two assessments.
The age-matched control group consisted of 14 children (3 boys and 11 girls; M age = 10.71, SD = 0.89 years) without MLD, selected from primary schools. Their teachers did not report any known mathematical difficulties and all children scored at or above mean level on the CITO mathematics test (6 children scored between 75 and 100 %, 5 children scored between 50–75 % and 3 children scored between 25 and 50 %).
Procedure
All children were tested on a computer with Tobii T60 eye tracker in the Pedagogics lab at Utrecht University. The temporal resolution of the Tobii T60 is 60 Hz. The spatial resolution is 0.2°. A nine-point calibration was used. For all children, the 0–100 number line was administered first and the 0–1000 number line second.
Instruments
Two number line tasks were used to measure number line estimation: (1) a 0–100 number line task, and (2) a 0–1000 number line task. An empty number line was presented on the computer screen with numbers only at the beginning and endpoints (i.e. 0 and 100, or 0 and 1000, respectively). Then the number that had to be estimated was presented beneath the number line. Children were asked to read the number aloud and then estimate its position on the number line by placing the mouse cursor on the line. To make sure the numbers that had to be estimated were more or less equally distributed over the number line, the number line was divided into 33 equal sections and one number from each section was randomly selected to be used in the task. For the 0–100 number line task, the used numbers were: 3, 5, 9, 10, 14, 18, 19, 24, 27, 28, 32, 34, 37, 41, 43, 46, 49, 53, 57, 60, 61, 64, 66, 72, 74, 78, 80, 83, 87, 89, 91, 96, 99; for the 0–1000 number line task, the used numbers were: 4, 36, 68, 104, 135, 153, 201, 230, 261, 277, 308, 354, 385, 398, 422, 469, 510, 528, 542, 594, 613, 636, 684, 697, 723, 763, 804, 844, 862, 880, 919, 958, 996. The same numbers were presented to each child but in a different random order.
Data analysis
To quantify performance on the number line, the absolute error was calculated and expressed as a proportion of the range of the number line (called the percentage absolute error) using the following formula: (response − target number)/range number line (100 or 1000) × 100 (Siegler & Booth,
2004). Furthermore, for each participant the conventional linear and logarithmic model fit was computed by conducting a regression with the estimated number (response) as dependent variable and the target number (equaling the correct answer) as independent variable. Finally, a two-cycle power model was fitted to the individual data as an index of beginning point, midpoint and endpoint use (Slusser et al.,
2013). For all models,
R
^{2} was used as an index of model fit. A MANOVA was used to test for possible group differences in absolute error and model fit on each number line.
Eye movements
Eye movements were analyzed using Matlab (Mathworks Inc). They were classified as fixations when the absolute speed of the eyes was lower than 3 m/s for at least three consecutive samples (50 ms). Fixations were pooled if they were within 0.5 cm
^{2} of each other. Only fixations that fell within 3.5 cm above or below the number line and occurred between the start of the stimulus presentations and the participants’ response were included in the analyses. The number that had to be estimated was presented more than 3.5 cm under the number line, and fixations on this number were thus excluded from the analyses (Schot et al.,
2015).
Estimation strategies
To gain insights into the estimation strategies that the children used, we assessed whether children made use of the reference points (beginning point, midpoint, or endpoint). Fixations within a margin of 5 % from the beginning, mid-, or endpoint were classified as fixations on these respective reference points. When the fixations on a particular trial were confined to just one of these reference points, the estimation strategy for this trial was classified as a beginning, mid-, or endpoint estimation strategy depending on the reference point the child used. When there were fixations on multiple references the estimation strategy was classified as such (i.e. begin and mid, begin and end, mid and end, or all references). When there were no fixations on the references, but all fixations were within 5 % of the correct answer, and the given answer was within 5 % around the correct answer, the estimation strategy was classified as automatized. In trials with no fixations on the references and no fixations around the correct answer (within 5 %), the estimation strategy was classified as guess when all fixations were within 10 % of the given answer and classified as no references (NoRefs) when fixations were scattered over the number line. Trials in which no eye movement data were available (for example due to movement of the child) were excluded from the analyses. In total, 1.84 % of the trials was lost due to this constraint. We calculated the percentage of trials in which the children used each of the estimation strategies in both tasks. A MANOVA was used to test for possible group differences in strategy use on each number line. To avoid problems with dependency because the strategies sum up to 100 %, the ‘no references’ category was not included in the MANOVA.
Functionality of fixations
To examine the functionality of the eye-fixation behavior, the horizontal position of the fixations was plotted against the target number and against the response for each participant separately. The number line was then divided into three equal segments (0–33/0–333, 34–66/334–666 and 67–100/667–1000) to examine whether the fixations were near or farther away from the target number and the response over trials. For each trial, the percentage of fixations that fell in the same segment as the target number (near), in the segment next to the correct segment (in between), and two segments away
^{2} from the correct segment (far) was calculated per participant separately for each task. The same percentages were calculated relative to the participants’ response. For the control group, the mean of these percentages was calculated and plotted. Because of the large variance in the MLD group, results were plotted for each participant separately.
Adaptive strategy use
Finally, to assess whether strategy use was adaptive (i.e. related to the number that had to be estimated), we calculated the percentages of the estimation strategies used for both the MLD group and the control group for each trial separately. Based on the pattern of estimation strategy uses across the number range in the control group (see Online Resource 1), the number lines were then again divided into three equal sections (0–33, 34–66, 67–100; 0–333, 334–666, 667–1000) and the percentages of the use of beginning point, midpoint, and endpoint used within each section were compared between the MLD group and the control group.
Use of estimation strategies per number section
A repeated measures MANOVA was used with group (MLD, control) as independent variable, estimation strategy (beginning point, midpoint, and endpoint) as within subjects factor, and section of the number line as measure, to examine if there were differences between the MLD and control group in which estimation strategy they used most in each number section. Whenever the assumption of sphericity was violated, the Greenhouse-Geisser correction coefficient (\(\hat{\varepsilon }\)) is reported together with the uncorrected degrees of freedom and the corrected p value.
Use of most adaptive strategy per number section
A MANOVA was used to test for differences between the groups in use of the (theoretically) most adaptive strategy for each number section, with group as independent variable and the use of beginning point in the low number section, the use of midpoint in the medium number section and the use of endpoint in the high number section as dependent variables (since these are theoretically the most adaptive strategy for each number section).
For statistical analysis,
α = .05 was used. Effect sizes were classified according to the criteria of Cohen (
1988):
η
^{2} ≥ .01 is small,
η
^{2} ≥ .06 is medium,
η
^{2} ≥ .13 is large.