Children’s neural activity during number line estimations assessed by functional near-infrared spectroscopy (fNIRS)

Abstract

Number line estimation (NLE) is an educational task in which children estimate the location of a value (e.g., 25) on a blank line that represents a numerical range (e.g., 0–100). NLE performance is a strong predictor of success in mathematics, and error patterns on this task help provide a glimpse into how children may represent number internally. However, a missing and fundamental element of this puzzle is the identification of neural correlates of NLE in children. That is, understanding possible neural signatures related to NLE performance will provide valuable insight into the cognitive processes that underlie children’s development of NLE ability. Using functional near-infrared spectroscopy (fNIRS), we provide the first investigation of concurrent behavioral and cortical signatures of NLE performance in children. Specifically, our results highlight significant fronto-parietal changes in cortical activation in response to increases in NLE scale (e.g., 0–100 vs. 0–100,000). Furthermore, our results demonstrate that NLE performance feedback (auditory, visual, or audiovisual), as well as children’s grade (2^nd vs. 3^rd) influence cortical responding during an NLE task.

1.0. Introduction

The number line estimation (NLE) task is a common educational activity wherein children estimate the location of a numerical value (e.g., 25) on a line that represents a numerical range (e.g., 0–100). In order to perform accurately on an NLE task, children must correctly map the numerical ratio (i.e., 25:100) to the visual proportion of the line (i.e., 25% of the total line length) that correctly identifies the point of the line on which the target value lies relative to the numerical range. Far from trivial, NLE performance is a strong predictor of success in mathematics (Booth and Siegler, 2006; Siegler and Booth, 2004), and as a result much effort has been made to understand the development of NLE ability (Barth and Paladino, 2011; Thompson and Opfer, 2010).

A common behavioral signature of NLE in children is the tendency to inaccurately place the location of a target value within a less familiar numerical scale, despite accurate placement on known numerical scales whose target-to-scale ratio is identical. For example, a second grade child may accurately identify the point at which ‘25’ lies on a scale of ‘100’, but may fail to correctly identify where ‘250’ lies on a scale of ‘1,000’ (see Fig. 1). When plotted in Cartesian space with the true target-to-scale position along the x-axis and the child’s perceived correct location along the y-axis, these two examples result in highly linear and logarithmic functions, respectively. As children develop and are exposed to larger numerical scales throughout their education, estimation performance in progressively larger numerical scales may become linear (Barth and Paladino, 2011; Opfer and Siegler, 2007; Sella et al., 2015; Siegler and Booth, 2004; Siegler and Opfer, 2003; Slusser et al., 2013).

Figure 1.

Linear and logarithmic number line estimation functions

Perfect estimation performance results in a linear response function with a slope of 1 and an intercept of 0. Conversely, overestimation of estimation values on the low end of a scale result in a logarithmic function. In this example, the dot at ‘150’ indicates the correct location along the x-axis, whereas the square indicates a common overestimation for values at the low end of unknown scales. The data in this example are a hypothetical illustration.

The transition from logarithmic to linear NLE behavioral performance is an important milestone in children’s educational development. As such, much effort has been made to develop training paradigms that facilitate linear NLE performance. Multiple efforts have been made to explicitly highlight to children the relationship between known small numerical scales and progressively larger unknown scales. For example, progressive alignment allows children to make similarity comparisons over concrete, perceptual similarities such as monotonic increases in size across differently shaped stimuli. This process is thought to facilitate children’s ability to notice higher order relational commonalities across stimuli that possess fewer surface-level features in common (Kotovsky and Gentner, 1996). Similarly, analogies (e.g., loaf of bread : single slice of bread :: lemon : ________?) have been used to highlight consistencies across disparate stimuli with the hope that the connections between dissimilar objects will generalize to tasks such as number line estimations across scales (Abdellatif et al., 2008). More recently, researchers have shown that first-grade children’s NLE performance is significantly enhanced following embodied estimation training. For example, children who were allowed to make their estimations by walking along a large number line placed on the classroom floor showed more pronounced training effects than children in classical pen-and-paper based NLE training (Link et al., 2013).

The enhanced NLE performance that results from the training paradigms described above are commonly attributed to a log-to-linear “shift” in children’s numerical representation of number along the mental number line (Siegler and Booth, 2004). Alternatively, proportional judgment accounts of NLE enhancement posit that training enhances children’s ability to estimate proportions of the bounded number line (Link et al., 2014). That is, providing children with perceptual “landmarks” that evenly divide the number line into equal parts (e.g., quartile breakup), allows them to readjust their estimations relative to the landmark and thus brings them closer to linearity (Barth and Paladino, 2011; Slusser et al., 2013). Thus, the proportional judgment account suggests that the common log-to-linear developmental pattern occurs without a fundamental shift in internal number representation.

Despite its strong relationship with success in mathematics, very little is understood regarding the neural signatures of real-world number line estimation behavior. A wealth of evidence from neuroimaging studies implicates overlapping regions of the parietal and prefrontal cortices in underlying both numerosity and spatial cognitive processing (Arsalidou et al., 2018; Harvey et al., 2015; Peters & De Smedt, 2018; Soltanlou et al., 2018a; Soltanlou et al., 2018b), both of which are integral to NLE performance (Arsalidou and Taylor, 2011; Hubbard et al., 2005; Vogel et al., 2013). For example, the bilateral intraparietal sulci (IPS) have been shown to be specifically responsive to number-related processes including mental arithmetic (Burbaud et al., 1999; Chochon et al., 1999; Dehaene et al., 1999; Lee, 2000; Menon et al., 2000; Pesenti et al., 2000; Simon et al., 2002), number comparisons (Chochon et al., 1999; Cohen and Dehaene, 1996; Dehaene, 1996; Langdon and Warrington, 1997; Le Clec’H et al., 2000; Pesenti et al., 2000; Pinel et al., 2001; Rosselli and Ardila, 1989; Seymour et al., 1994), category specific representation and processing of number (Dehaene, 1995; Le Clec’H et al., 2000; Pesenti et al., 2000), parametric modulation of number (Dehaene, 1996; Piazza et al., 2002a; 2002b; Pinel et al., 2001; Stanescu-Cosson et al., 2000), and unconscious quantity processing (Dehaene, 2011; 1992; Dehaene et al., 1998; Dehaene and Marques, 2002; Naccache and Dehaene, 2001). Simultaneously, the lateral intraparietal cortex is involved in processing of events in space, and is active during saccadic eye movements believed to play a role in actively attending to peripheral targets (Astafiev et al., 2003). Furthermore, the ventral intraparietal responds selectively to motion in space (Bremmer et al., 2001), and the anterior IPS has been implicated in fine motor movements and grasping in space (Binkofski et al., 1998; Culham et al., 2003; Shikata et al., 2003). This overlap of neural structures related to processing of both number and space is thought to account for the behavioral interactions between representations of number and space (Hubbard et al., 2005; Walsh, 2003).

It is important to note, however, that the brains of children and adults have been shown to differ in their response profile to number and mathematics (Arsalidou et al., 2018; Peters & De Smedt, 2018). That is, children and adults recruit similar fronto-parietal regions when performing mathematics operations, although the activation magnitude is smaller in children compared to adults. Furthermore, observed patterns of a frontal-to-parietal shift indicate that children recruit greater prefrontal regions during mathematics, and that activation migrates to parietal regions throughout development. It is thought that this shift may reflect neural changes that are accompanied with an increased reliance on fact retrieval or automatization of arithmetic facts. Interestingly, such developmental shifts have been observed among children differing as little as one year in development (Arsalidou & Taylor, 2011).

Moreover, and of particular importance for the current study, many findings highlight significant overlap between the parietal and prefrontal regions underlying numerical cognition and those facilitating multisensory integration. For example, the IPS has been identified as a region of the brain most highly implicated in sensory convergence (Calvert, 2001). Specifically, the IPS appears to be specialized for synthesizing crossmodal spatial coordinate cues and mediating crossmodal links in attention (Banati, 2000; Bushara et al., 2001; 1999; Callan et al., 2001; Calvert, 2001; Eimer, 1999; Lewis et al., 2000; Macaluso et al., 2000). The involvement of the frontal cortex in multisensory integration is less understood, but it seems to be involved in integrating newly acquired crossmodal associations, such that frontal areas may be recruited when associations between crossmodal cues are essentially arbitrary (Banati, 2000; Bushara et al., 2001; Callan et al., 2001; Calvert, 2001; Calvert et al., 2000; Giard and Peronnet, 1999; Gonzalo et al., 2000; Lewis et al., 2000; Raij et al., 2000). These findings serve to underscore the claim that the brain is highly equipped to process multisensory information, and that processing of this information is distributed among brain regions that include the parietal and prefrontal cortices. Taken together, these findings further implicate the intraparietal and prefrontal regions as being directly involved in the processing of number, space, and multisensory integration.

An intriguing possibility is that providing multisensory feedback about estimation accuracy may influence NLE performance in children. The intersensory redundancy hypothesis suggests that representation of amodal properties such as number can be enhanced following the presentation of synchronous and redundant sources of information about the property (Bahrick et al., 2002; Bahrick and Lickliter, 2000; Baker and Jordan, 2014; Baker et al., 2014; Flom and Bahrick, 2007; Jordan and Baker, 2011; Lewkowicz and Lickliter, 2013). That is, multiple sources of sensory information (e.g., visual and auditory) that occur concurrently improves perception of the amodal property (e.g., number). The enhancing effects described by the intersensory redundancy hypothesis are thought to arise from an inherent attraction to multisensory stimuli that contributes in critical ways to perceptual development (Lewkowicz and Ghazanfar, 2009). While it remains to be tested, providing multisensory information about number throughout the NLE task may enhance children’s ability to generalize known to unknown number-to-scale relationships, thereby effectively enhancing NLE performance.

Here, we provide the first assessment of the neural and behavioral signatures of real-world number line estimations. Using functional near-infrared spectroscopy (fNIRS), we address the primary hypothesis that NLE scale (e.g., one hundred vs. one thousand) affects cortical activity within the bilateral parietal and prefrontal cortices of second- and third-grade children. In short, fNIRS uses light projected into the brain to assess hemodynamic changes in response to stimuli over time. Given its high tolerance to movement and portability (Baker et al., 2017), fNIRS has emerged as an optimal tool for studies of educational neuroscience, including those investigating educational training (Soltanlou et al., 2018a; Soltanlou et al., 2018b). Furthermore, we also aim to address the secondary hypothesEs that NLE performance feedback (e.g., visual, auditory, or audiovisual) and children’s grade (2^nd vs 3^rd) similarly influences neural and behavioral signatures of NLE performance.

2.0. Method

2.1. Participants.

Twenty-three second- and third-grade students (second grade = 9, third grade = 14; female = 16; mean age = 7.73yrs, range = 6 – 9yrs) varying in age between seven and nine years participated in the study. Sample size was based on power analysis of the main effect of scale on estimation accuracy. This analysis was conducted using G*Power software and assumed a Cohen’s d effect size of 0.611, a power of 0.8, and an alpha of 0.05. Two additional second grade children were recruited but did not finish the session due to illness and an unwillingness to wear the NIRS cap, respectively, and were therefore not included. Participants received an ice cream coupon in compensation for participation. All participants were typically developing, of normal intelligence¹, had normal hearing and normal or corrected to normal vision and were native English speakers.

2.2. Task design.

A mixed block and event-related design was employed. All participants experienced three distinct experimental blocks, each consisting of 30 NLE trials in a pseudorandomized order. In order to assess the efficacy of feedback on NLE performance, the task was broken into three blocks herein referred to as “pre-feedback”, “feedback”, and “post-feedback”. Each participant was randomly assigned to a visual, auditory, or audiovisual feedback condition (see below). Furthermore, half of all NLE trials were conducted on a “small” numerical scale (i.e., 0 – 100), while the other half were conducted on a “large” numerical scale (i.e., 0 – 100,000) The NLE scales were pseudorandomized prior to data collection.

The task (see Fig. 2 for task structure) began and ended with a 30 second rest period in which each participant was asked to close their eyes and clear their mind. Similarly, each experimental block was separated by identical 30-second rest periods. Within each experimental block, the participants completed 15 trials of each estimation scale (i.e., hundred and hundred thousand), totaling 90 trials (15 trials × 2 scales = 30 trials per block × 3 blocks = 90 total trials). Within blocks one and three (i.e., pre/post feedback), the duration of each trial was seven seconds. This duration increased to nine seconds in block two to include feedback during the final two seconds of each trial.

Figure 2.

Number line estimation task structure

Each trial began with a jittered inter-trial interval between 2 and 8 seconds long. Next, an estimation statement reading “Out of ____, show me ____” was presented for 3 seconds. The estimation statement was then replaced with a blank response line, which remained present for up to 3 seconds or until the participant responded. Responses were made by clicking the location on the blank line with the mouse on the subjectively correct location. On each trial a response display was then presented for at least 1 second, which displayed the participant’s estimated location. During feedback, condition-relevant feedback was then presented for 2 seconds.

Each trial contained four distinct sections: First, a jittered ITI between 2 sec and 8 seconds was presented. Next, an estimation statement (e.g., “Out of [scale], show me [value]”) was presented for 3 seconds. The “scale” and “value” within the estimation statement were presented as words, rather than Arabic numerals (e.g., “Out of one hundred, show me twenty-five”). In order to maintain consistency across scales and blocks, the same numerical values were used throughout each block of the task, and only differed by the magnitude of their scale (e.g., “sixty-three” and “sixty-three thousand” appeared in each block). In this manner, it was possible to assess estimation inaccuracies that arose strictly because of an increase in numerical scale. After three seconds, the estimation statement disappeared and was replaced by the response line, which was 1,000 pixels in length and was flagged on each end with short vertical lines that denoted its end points. The response line always appeared in the center of the computer screen and was the only object on the screen throughout the participant’s estimation. Each participant was given up to three seconds to respond on the response line. Immediately upon the participant’s response, the response display (i.e., a blue vertical “hash mark” and/or tone) appeared on the location of the participant’s mouse click. This response display remained visible for at least one second; however, its total presentation duration was allowed to vary depending on the speed of the participant’s response, to maintain a total trial duration of seven seconds. Participants were not able to adjust their estimates after their initial estimation. Within the feedback block, an extra two seconds of condition relevant feedback (i.e., visual, auditory, audiovisual) was presented immediately after the 7-second trial length. In this way, all trials within the feedback block lasted a total of 9 seconds. Each feedback type is described below.

Visual Feedback:

Visual feedback was provided by a vertical red “hash mark” placed on the trial’s correct value-to-scale location on the response line. As the purpose of the feedback was to provide the participants with information that allowed them to compare their own estimation location relative to the correct location on the line, their own blue response display remained visible throughout the feedback portion of each trial. Therefore, their own estimation location and the correct location could be observed simultaneously throughout the two seconds of feedback within each trial. On trials in which the participant provided no response, the feedback display appeared alone on the response line for two seconds.

Auditory Feedback:

Auditory feedback was provided as a pair of tones that varied in loudness depending on the spatial relationship on the response line between the participant’s estimation response (tone 1: participant’s response tone), and the correct value-to-scale location (tone 2: correct response tone). Throughout the response display block of this condition, the participant’s response display was accompanied by a pure tone (3000Hz) whose volume was position dependent. Specifically, a quiet-to-loud volume orientation was mapped onto the left-to-right physical orientation of the response line. Thus, a point on the left side of the response line coincided with a tone that was quieter than a tone that coincided with a point on the right side of the line. Throughout the 2-second feedback portion of each trial, a second tone was presented whose volume represented the correct value-to-scale location on the line². Therefore, participants were able to compare the accuracy of their estimations based on the magnitude of volume difference between their response tones and the correct location tones.

Audiovisual Feedback:

The audiovisual feedback condition provided a combination of both the visual and auditory feedback simultaneously. That is, similar to the auditory feedback condition, a position-dependent response tone (e.g., tone 1) accompanied the participants’ response display on the response line. Next, similar to the visual feedback condition, a red “hash mark” was presented in the correct value-to-scale location. However, in the audiovisual feedback condition, the red “hash mark” was also accompanied by a correct response tone (e.g., tone 2) whose volume also indicated the correct value-to-scale location. Therefore, participants received simultaneous audiovisual information about their estimation locations as well as the correct locations of the estimation values, and could thus use this combined sensory information to inform future estimations.

2.3. Practice Trials:

Immediately prior to beginning the task, each participant engaged in a practice session that introduced him or her to the task, as well as to his or her condition-relevant feedback. Participants were first shown an example estimation statement, in which a target and scale that were not used throughout the task were displayed (e.g., “Out of ten, show me two”). It was explained to the participant that he or she would have exactly three seconds to read this sentence before making his or her estimation. Next, participants were shown a blank estimation response line. The experimenter highlighted for the participant the end points of the line and described that, “the line equals the numerical scale from the statement you just read. That means, because the statement read ‘out of ten, show me two’, the length of this line equals zero to ten”. Next, the participant was instructed to click on the line in the location that represented “two out of ten”. This resulted in the blue response display being shown at the location they clicked. As this section of the practice was completely experimenter driven (i.e., not automatically timed, as throughout the task), it was possible to ensure that the participant understood how each trial would progress. Next, each participant underwent five practice trials that were structured and timed exactly like the trials they experienced throughout the task, except that the scale magnitudes used were smaller than those used throughout the task. The experimenter observed these trials to ensure that each participant was responding in the correct time limit. Moreover, general comprehension of the task (i.e., responding in the generally correct location on the line) was visually assessed. If additional practice trails were requested by the participant, or needed based on observed performance, the entire practice procedure was repeated. These practice trials allowed the participant to become familiar with the pacing of each trial, while not being exposed to the scale magnitudes that were to be used throughout the task.

Next, each participant was introduced to his or her condition-relevant feedback. First, a pre-programmed visual demonstration of an entire feedback trial was provided. These trial demonstrations were identical across each feedback condition, and only varied insofar as the type of feedback they produced. As with the initial practice section above, this section was experimenter driven so that the feedback could be described in detail, and could be repeated if necessary. If the participant expressed confusion, the feedback instructions were repeated. Following this introduction, each participant underwent an additional five feedback trials that progressed in the same manner as throughout the task. Following these practice trials, the NIRS cap was placed on the participant’s head, and the task moved forward as programmed.

2.4. Setting and Apparatus:

All experimental sessions were conducted in the Utah State University NIRS Laboratory, housed within the Emma Eccles Early Childhood Educational Research Center. All trials occurred on a PC desktop computer running Windows XP®, and were presented by E-Prime Stimulus Presentation Software (Schneider et al., 2002a; 2002b). The screen resolution was set to 1020 × 1280. Cortical activation data were recorded with a continuous wave fNIRS system (ETG 4000, Hitachi Medical Co., Japan) (Plichta et al., 2006) using two 3×5 optode probe sets, each with 22 data-recording channels (7 photo detectors and 8 light emitters). The interoptode distance was 30mm and the sampling rate was set to 10Hz. The probe sets were attached to participants’ heads by a simple elastic-band cap (see Fig. 3) that allowed for firm yet comfortable placement of each optode.

Figure 3.

Probe set positioning

Two 3×5 fNIRS optode patches were used to cover the bilateral prefrontal and bilateral parietal cortices. The International 10–20 system was used to guide the placement of each optode patch. Specifically, the middle optode on the bottom row of the frontal patch was placed on the midsaggital plane directly above the brow, and the bottom edge of the patch was made level with the brow. The middle optode of the second row of the parietal patch was placed directly on top of the 10/20 location ‘Pz’.

2.5. Placement Localization of fNIRS Probe Sets and Functional Regions of Interest:

Probe set placement localization was determined prior to each scan based on physical measurements of each participant’s head and corresponding international 10–20 system locations (Okamoto et al., 2004). Use of this system allowed for consistent placement of optodes despite changes in participant head size. For probe set 1, the middle optode on the first row of the set was placed directly over Fpz (i.e., 10% of nasion-to-inion distance above the nasion), directly on the midline of the head, which marks the frontopolar midpoint of the pre-frontal cortex. For probe set 2, the middle optode in the second row of the set was placed directly over Pz (i.e., 70% of nasion-to-inion distance from nasion), which marks the midpoint of the parietal cortex.

All functional regions of interest (fROI) used throughout the analyses were identified a priori based on previous fMRI studies of math processing in children and adults (Arsalidou and Taylor, 2011; Berteletti et al., 2015; Dehaene et al., 2004; Delazer et al., 2005; Hung et al., 2015; Kucian et al., 2011; Rosenberg-Lee et al., 2011; Vogel et al., 2013; 2015a; 2015b) and were localized for each participant based on their correspondence to the Montreal Neurological Institute (MNI) standardized neurological coordinate system (Okamoto et al., 2004). MNI coordinates of NIRS observation channels were obtained immediately after each scan session through measurements made by a 3D magnetic space digitizer (FASTRAK, Polhemus, Cochester, VT), which provides an accurate measurement of the NIRS optode positions and thus each observation channel, within a real-world coordinate system (Singh et al., 2005). Anatomical labels of the neural regions convolved within each NIRS observation channel were obtained and were used to identify the NIRS observation channels that constitute each fROI. In the prefrontal cortex, fROIs included the medial PFC (channels 7, 11, 12, 16), and the left dorsolateral PFC (channels: 6, 10, 11, 15, 19, and 20) (Delazer et al., 2005; Rosenberg-Lee et al., 2011). In the parietal cortex, fROIs included the left intraparietal sulcus (channels 28, 32, 33, and 37), the left angular gyrus (channels 32 and 36), the right intraparietal sulcus (channels 30, 34, 35, and 39), and the right angular gyrus (channels: 13 and 18) (Arsalidou and Taylor, 2011; Delazer et al., 2005; Rosenberg-Lee et al., 2011).

3.0. Results

3.1. Behavioral Data Analysis and Outcomes:

In order to examine participants’ NLE accuracy across blocks, we first converted each estimation point to a numerical value by multiplying the estimated proportion of the line (i.e., linear distance from “0” mark to the estimation point, divided by the total length of the line) by the scale value. Next, the percent absolute error (PAE) of each estimate (0–100%) was calculated by dividing the mean absolute difference between each estimated value and the actual value, divided by the total scale. Finally, accuracy scores were computed by subtracting percent absolute error from 100% (Thompson and Opfer, 2010) (see Table 1 for accuracy means and standard deviations). These accuracy scores were then submitted to repeated measures analysis of variance (ANOVA) as described below. Huynh-Feldt (H-F) corrections for violation of sphericity was applied to each ANOVA outcome.

Table 1.

Children’s Estimation Accuracy Means and Standard Deviations.

		Prefeedback		Feedback		Postfeedback
Variables		Mean	SD	Mean	SD	Mean	SD
Second grade
Auditory	One hundred	91.605	6.431	91.373	5.331	93.315	4.063
	One hundred thousand	79.250	22.093	88.778	17.164	92.500	4.647
Visual	One hundred	90.337	8.176	90.176	9.938	91.143	9.133
	One hundred thousand	78.410	21.112	86.064	19.936	88.718	12.098
Audiovisual	One hundred	87.920	11.434	88.106	13.388	85.051	16.006
	One hundred thousand	82.167	15.589	86.050	14.742	76.906	21.466
Third grade
Auditory	One hundred	88.300	10.540	89.506	12.536	90.239	8.818
	One hundred thousand	79.129	22.695	85.079	18.072	82.946	21.021
Visual	One hundred	93.297	5.310	93.710	9.818	94.634	4.500
	One hundred thousand	83.660	18.029	89.347	14.833	89.030	13.839
Audiovisual	One hundred	91.506	7.721	91.919	7.742	92.165	6.756
	One hundred thousand	81.826	16.468	90.792	8.915	90.431	11.087
Combined	One hundred	90.494	8.269	90.798	9.792	91.091	8.213
	One hundred thousand	80.740	19.331	87.685	15.610	86.755	14.026

First, a 3 (block) × 2 (scale) × 3 (feedback condition) × 2 (grade) repeated measures ANOVA was used to assess number line estimation accuracy. This analysis revealed a significant effect of block on number line estimations (F(2, 34) = 11.711, MSE = 151.769, H-F p < .001, partial η² = .408), which resulted from an increase in estimation accuracy as the task progressed (see Fig. 4). Follow-up pairwise comparisons with Bonferroni corrections indicated that the prefeedback block accuracy was significantly lower than the feedback block (p < .001), as well as the postfeedback block (p < .05). The feedback and postfeedback blocks did not differ in accuracy (p > .05). Moreover, repeated measures ANOVA revealed a significant effect of scale (F(1, 34) = 7.20, MSE = 968.229, H-F p < .05, partial η² = .298), driven by significantly lower accuracy on large compared to small estimation scales. Furthermore, a significant block × scale interaction was identified (F(2, 34) = 11.379, MSE = 112.127, H-F p < .001, partial η² = .401), which resulted from particularly poor estimation performance on large scales in the prefeedback block. No other effects were significant.

Figure 4.

Numberline estimation accuracy across blocks and scales

Children’s NLE performance was best fit by a linear compared to logarithmic function in each condition. However, NLE performance was significantly poorer in large compared to small scale estimations, and specifically in the pre-feedback condition. The x-axis of each subplot provides the true percentage of the number line presented on each trial (e.g., 25 out of 100 = 25% of number line), and the y-axis provides the estimated percent provided by the participants. The dotted trend line with a slope of 1 and intercept of 0 represents perfect NLE performance. The red trend line provides the smoothed mean performance, and the shaded area around the line is the 95% confidence interval. Performance above the dotted line indicates over estimation of the percent of the line that actually represents a given value, whereas performance below the dotted line indicates under estimation. Participants in our task tended to under estimated the location of numbers on the high end of each scale. Moreover, on large scale estimations, our participants tended to underestimate values on the low end of the scale.

See Table 2 for children’s response time means and standard deviations. An identical 3 × 2 × 3 × 2 repeated measures ANOVA of response times revealed a significant effect of block (F(2, 34) = 3.544, MSE = 269506.273, H-F p < .05, partial η² = .173), driven by an overall decrease in reaction times as the task progressed (see Fig. 5). Next, a significant main effect of scale was also identified (F(1, 17) = 8.812, MSE = 478162.616, H-F p < .05, partial η² = .341), which resulted from significantly longer response times for large scale estimations compared to small scale estimations. No other effects were significant.

Table 2.

Children’s Estimation Response Time Means and Standard Deviations.

		Prefeedback		Feedback		Postfeedback
Variables		Mean	SD	Mean	SD	Mean	SD
Second grade
Auditory	One hundred	1822.200	703.775	1752.421	660.942	1782.842	659.007
	One hundred thousand	1863.600	745.027	1801.000	676.168	1934.750	575.573
Visual	One hundred	1882.550	491.142	1872.447	622.9827	2016.077	699.984
	One hundred thousand	2079.950	569.2129	2223.513	625.3426	1975.368	754.700
Audiovisual	One hundred	2002.500	880.769	1872.567	779.226	1739.414	819.456
	One hundred thousand	2119.857	789.354	1970.300	861.534	1779.448	731.036
Third grade
Auditory	One hundred	1938.040	589.406	1871.460	665.426	1717.812	618.704
	One hundred thousand	2052.182	714.131	1875.125	541.399	1817.653	693.492
Visual	One hundred	2479.051	619.219	2123.351	622.980	2150.429	584.510
	One hundred thousand	2379.868	667.5627	2306.083	599.1616	2285.718	703.599
Audiovisual	One hundred	2221.717	582.444	2294.787	566.259	1957.714	567.778
	One hundred thousand	2481.587	603.630	2442.310	671.134	2291.104	611.692
Combined	One hundred	2057.676	644.459	1964.506	652.969	1894.048	658.240
	One hundred thousand	2162.841	681.486	2103.055	662.457	2014.007	678.349

Figure 5.

Numberline estimation response time across blocks and scales

Inter-quartile range of children’s NLE response time across blocks and scales. The top and bottom lines of each box represent the 75^th and 25^th percentile, respectively. The bold line in each box represents the median (i.e., 50^th percentile) response time. The length of each whisker represents the extent of the lowest and highest datum that lie within +/− 1.5 times the inter-quartile range. Data points outside of this range are represented by a dot. Our participants responded significantly faster to small compared to large scale estimations within each block of the task. Furthermore, response speed decreased significantly across task blocks.

In order to identify whether a linear or logarithmic function best fit the relationship between estimation accuracy and scale across each experimental condition and grade level, individual linear and non-linear regression analyses were conducted for each scale × block × grade possibility. Specifically, both models were identical in each case, although the non-linear model contained a exponent that allowed it to adjust its fit in the case on non-linear estimation trajectories. Consistent with previous studies (Siegler and Opfer, 2003; Thompson and Opfer, 2010; 2008), median values were used to reduce impact of outliers at group level. A linear function best fit each scale × block pair for both second grade and third grade children (see Table 3 for regression R² outcomes). These findings indicate that children in our study did not exhibit logarithmic functions on large scale estimation trials, despite significantly poorer performance.

Table 3.

Linear vs. Logarithmic Regression R² Outcomes.

Grade	Scale	Block	Function
			Linear	Logarithmic
Second	Small Scale	Pre-	0.878	0.880
		Feedback	0.957	0.823
		Post-	0.989	0.711
	Large Scale	Pre-	0.071	0.017
		Feedback	0.948	0.797
		Post-	0.951	0.598
Third	Small Scale	Pre-	0.963	0.688
		Feedback	0.965	0.881
		Post-	0.991	0.708
	Large Scale	Pre-	0.602	0.282
		Feedback	0.972	0.858
		Post-	0.977	0.654

3.2. fNIRS Data Analysis and Outcomes:

All treatments of NIRS data reported below were conducted with NIRS-SPM NIRS analysis package (Ye et al., 2009) for Matlab®. Initially, the modified Beers-Lambert law was used to convert optical density data recorded by an ETG 4000 (Hitachi), which provide an indicator of oxygenated (O₂Hb) and deoxygenated (O₂Hb) hemoglobin concentration levels within the blood of the cortex at each region of interest. Previous research has found oxygenated hemoglobin to be a more informative marker of cognitive processing compared to deoxygenated hemoglobin (Baird, 2002; Bortfeld et al., 2009; Grossmann et al., 2008; Minagawa Kawai et al., 2008; Peña et al., 2003; Wilcox et al., 2009; 2008); therefore, only oxygenated hemoglobin data were used throughout the following analyses. Next, because several physiological processes (e.g., respiration, blood-pressure changes, heartbeat, etc.) are known to produce structured “noise” within the data (i.e., autocorrelation), a precoloring treatment was performed, through which such temporal correlations are “swamped” by an imposed 4s Gaussian temporal correlation structure and are thus effectively reduced (Worsley and Friston, 1995). Furthermore, O₂Hb data were lowpass- and highpass-filtered with a cutoff of .5 and .01Hz respectively (Ye et al., 2009).

The processed data were analyzed using a general linear model (GLM) procedure developed by Plichta and colleagues (Plichta et al., 2007). Specifically, the start of each trial was entered into the GLM procedure as condition-specific onsets, and the entire trial duration prior to the onset of the inter-trial interval was modeled. All inter-trial intervals were pseudorandomly jittered between two and eight seconds. Thus, a standardized beta weight was estimated for each block, scale, and feedback condition. This procedure was first conducted on a channel-wise basis. Next, the largest beta outcome for all channels within a single ROI was selected and entered into the group-level analyses reported below. This functional localization procedure is identical to that used in previous reports (Sagiv et al., 2019; Baker et al., 2016; Baker et al., 2018; Bruno et al., 2018) and allows for person-to-person variation in the specific channel within each ROI that responds greatest to the experimental stimuli.

Prefrontal ROI Analyses:

First, a 3 (block) × 2 (scale) × 3 (feedback condition) × 2 (grade) repeated measures ANOVA for beta values corresponding to the left dorsolateral PFC identified a significant effect of block, indicating that cortical activation in this region differed between blocks (See Table 4 for a breakdown of left dorsolateral outcomes). Follow-up pairwise comparisons indicate that cortical activation was significantly greater during feedback compared to the pre-feedback block (p < .05). No other effects were identified within the prefrontal ROIs (see Table 5 for breakdown of medial prefrontal outcomes).

Table 4.

Breakdown of Children’s Left Dorsolateral Prefrontal Outcomes.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	151.134	75.567	3.594	.038a	0.175
Block × condition	4	268.283	67.071	3.190	.025b	0.273
Block × grade	2	21.909	10.955	0.521	0.599	0.030
Block × condition × grade	4	206.825	51.706	2.459	0.064	0.224
Scale	1	5.391	5.391	1.780	0.200	0.095
Scale × condition	2	1.950	0.975	0.322	0.729	0.037
Scale × grade	1	0.135	0.135	0.045	0.835	0.003
Scale × condition × grade	2	2.513	1.257	0.415	0.667	0.047
Block × scale	2	2.832	1.416	0.630	0.538	0.036
Block × scale × condition	4	1.815	0.454	0.202	0.936	0.023
Block × scale × grade	2	9.650	4.825	2.148	0.132	0.112
Block × scale × condition × grade	4	20.357	5.089	2.265	0.082	0.210
Condition	2	32.450	16.225	0.462	0.637	0.052
Grade	1	0.032	0.032	0.001	0.976	0.000
Condition × grade	2	119.304	59.652	1.700	0.212	0.167

^aActivation was greatest during the feedback block of the task.

^bIncreased activation during feedback was greatest in the audiovisual feedback condition.

Table 5.

Breakdown of Children’s Medial Prefrontal Outcomes.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	73.441	36.720	0.990	0.382	0.055
Block × condition	4	313.108	78.277	2.110	0.101	0.199
Block × grade	2	64.288	32.144	0.866	0.430	0.048
Block × condition × grade	4	501.445	125.361	3.379	,020a	0.284
Scale	1	9.763	9.763	2.917	0.106	0.146
Scale × condition	2	4.817	2.408	0.720	0.501	0.078
Scale × grade	1	3.613	3.613	1.080	0.313	0.060
Scale × condition × grade	2	13.478	6.739	2.014	0.164	0.192
Block × scale	2	16.910	8.455	4.373	.020b	0.205
Block × scale × condition	4	9.930	2.482	1.284	0.296	0.131
Block × scale × grade	2	22.289	11.144	5.763	.007c	0.253
Block × scale × condition × grade	4	18.998	4.750	2.456	0.064	0.224
Condition	2	1.872	0.936	0.026	0.974	0.003
Grade	1	2.699	2.699	0.076	0.786	0.004
Condition × grade	2	43.626	21.813	0.615	0.552	0.067

^aIncreased activation during feedback was greatest in second-grade children within the auditory feedback condition.

^bDriven by significantly greater activation during large- compared to small-scale estimations during the prefeedback condition.

^cThe block × scale interaction above was exacerbated within second- compared to third-grade children.

Parietal ROI Analyses:

Within the left IPS, identical repeated measures ANOVA of O₂Hb beta weights revealed a significant main effect of scale, driven by significantly higher activation during large scale estimations compared to small scale estimations (see Table 6 for a breakdown of left IPS outcomes). A significant main effect of grade was also identified, driven by significantly greater activation within second compared to third-grade children. Within the left angular gyrus, significant main effects of scale and grade were driven by greater activation in second compared to third-grade children (see Table 7 for a breakdown of left angular gyrus outcomes). No interaction between grade and condition was identified.

Table 6.

Breakdown of Children’s Left Intraparietal Sulcus Outcomes.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	83.638	41.819	0.612	0.548	0.035
Block × condition	4	358.585	89.646	1.311	0.285	0.134
Block × grade	2	19.241	9.620	0.141	0.869	0.008
Block × condition × grade	4	119.612	29.903	0.437	0.781	0.049
Scale	1	31.798	31.798	5.939	.026a	0.259
Scale × condition	2	3.584	1.792	0.335	0.720	0.038
Scale × grade	1	2.324	2.324	0.434	0.519	0.025
Scale × condition × grade	2	3.683	1.841	0.344	0.714	0.039
Block × scale	2	5.988	2.994	0.459	0.636	0.026
Block × scale × condition	4	13.547	3.387	0.519	0.722	0.058
Block × scale × grade	2	1.436	0.718	0.110	0.896	0.006
Block × scale × condition × grade	4	8.821	2.205	0.338	0.851	0.038
Condition	2	68.589	34.295	0.683	0.519	0.074
Grade	1	248.907	248.907	4.955	.040b	0.226
Condition × grade	2	646.086	323.043	6.430	.008c	0.431

^aActivation was greater in large- compared to small-scale estimations.

^bDriven by greater activation within second- compared to third-grade children.

^cDriven by significantly greater activation within second- compared to third-grade children within the auditory and audiovisual feedback conditions, as well as by significantly greater activation within third- compared to second-grade children within the visual feedback condition.

Table 7.

Breakdown of Children’s Left Angular Gyrus Beta Value Comparisons.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	143.235	71.617	0.802	0.457	0.045
Block × condition	4	371.080	92.770	1.039	0.405	0.109
Block × grade	2	137.613	68.807	0.770	0.471	0.043
Block × condition × grade	4	143.342	35.835	0.401	0.806	0.045
Scale	1	67.330	67.330	12.475	.003a	0.423
Scale × condition	2	1.706	0.853	0.158	0.855	0.018
Scale × grade	1	1.664	1.664	0.308	0.586	0.018
Scale × condition × grade	2	4.825	2.412	0.447	0.647	0.050
Block × scale	2	1.354	0.677	0.089	0.915	0.005
Block × scale × condition	4	16.315	4.079	0.536	0.710	0.059
Block × scale × grade	2	8.021	4.010	0.527	0.595	0.030
Block × scale × condition × grade	4	7.886	1.972	0.259	0.902	0.030
Condition	2	416.129	208.065	2.332	0.127	0.215
Grade	1	1349.480	1349.480	15.126	.001b	0.471
Condition × grade	2	547.473	279.739	3.068	0.073	2.65

^aActivation was greater in large- compared to small-scale estimations.

^bDriven by greater activation within second- compared to third-grade children.

While the left and right IPS were not compared statistically, activation patterns in the right IPS were different from that of the left parietal cortex (see Table 8 for a breakdown of right IPS outcomes). The effect of scale was present, although all other effects identified within the left parietal cortex were not. Next, an effect of block was identified, which was caused by a significant increase in activation during the feedback block of the task. Pairwise comparisons with Bonferroni corrections indicated that activation levels in the feedback block were significantly greater than both the pre-feedback and post-feedback conditions. An effect of scale was also significant within the right angular gyrus (see Table 9 for a breakdown of right angular gyrus outcomes). Similarly, results within the right angular gyrus highlighted a significant effect of block, the causes of which were identical to the right IPS effects reported above. No other effects were significant.

Table 8.

Breakdown of Children’s Right Intraparietal Sulcus Outcomes.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	1190.865	595.432	8.390	.001a	0.330
Block × condition	4	1360.796	340.199	4.794	.004b	0.361
Block × grade	2	499.919	249.960	3.522	.041c	0.172
Block × condition × grade	4	835.730	208.932	2.944	.034d	0.257
Scale	1	85.807	85.807	17.769	.001e	0.511
Scale × condition	2	11.196	5.598	1.159	0.337	0.120
Scale × grade	1	29.185	29.185	6.043	.025f	0.262
Scale × condition × grade	2	8.790	4.395	0.910	0.421	0.097
Block × scale	2	6.584	3.292	0.443	0.646	0.025
Block × scale × condition	4	27.253	6.813	0.917	0.465	0.097
Block × scale × grade	2	14.119	7.059	0.951	0.397	0.053
Block × scale × condition × grade	4	24.593	6.148	0.828	0.517	0.089
Condition	2	276.385	138.193	1.435	0.266	0.144
Grade	1	1.36.456	136.456	1.417	0.250	0.077
Condition × grade	2	137.704	68.852	0.715	0.503	0.078

^aActivation was significantly greater in the feedback block of the task.

^bIncreased activation during feedback was significantly greater in the auditory compared to visual and audiovisual feedback conditions.

^CIncreased activation during feedback was greater in second- compared to third-grade children.

^dIncreased activation during auditory feedback was exacerbated in second- compared to third-grade children.

^eActivation was significantly greater for large- compared to small-scale estimations.

^fIncreased activation for large- compared to small-scale estimations was exacerbated in second- compared to third-grade children.

Table 9.

Breakdown of Children’s Right Angular Gyrus Outcomes.

Variable	df	Sum of squares	Mean square	F ratio	p value	Partial eta-squared
Block	2	1030.593	515.296	4.924	.013a	0.225
Block × condition	4	1181.869	295.467	2.823	.040b	0.249
Block × grade	2	547.362	273.681	2.615	0.088	0.133
Block × condition × grade	4	844.517	211.129	2.017	0.114	0.192
Scale	1	188.822	188.822	17.615	.001c	0.509
Scale × condition	2	38.439	19.219	1.793	0.197	0.174
Scale × grade	1	33.261	33.261	3.103	0.096	0.154
Scale × condition × grade	2	30.247	15.123	1.411	0.271	0.142
Block × scale	2	5.570	2.785	0.300	0.743	0.017
Block × scale × condition	4	53.513	13.378	1.441	0.242	0.145
Block × scale × grade	2	13.325	6.663	0.718	0.495	0.041
Block × scale × condition × grade	4	12.285	3.071	0.331	0.855	0.037
Condition	2	641.444	320.722	2.761	0.092	0.245
Grade	1	284.258	284.258	2.447	0.136	0.126
Condition × grade	2	568.575	284.288	2.447	0.116	0.224

^aActivation was significantly greater in the feedback block of the task.

^bIncreased activation during feedback was significantly greater in the auditory compared to visual and audiovisual feedback conditions.

^cActivation was significantly greater for large- compared to small-scale estimations.

4. Discussion

Our results indicate that, as hypothesized, the numerical scale of a child’s NLE task significantly influences task performance. That is, second and third grade children performed similarly worse on large (i.e., hundred thousand) compared to small (i.e., hundred) numerical scales. As is apparent within Figure 4, children tended to overestimate the location of values on the small end of each scale, while underestimating the location on the large end. This tendency is greater in the large-scale estimation, and is particularly noticeable within the large-scale pre-feedback block of the task. In contrast with previous studies demonstrating a similar effect of scale on children’s estimation performance in response to increases in scale (Opfer and Siegler, 2007; Siegler and Booth, 2004; Thompson and Opfer, 2010), this decrease in accuracy was not accompanied by a logarithmic rather than linear fit of children’s data. Instead, a linear function was shown to be a better fit for all estimations, despite scale or grade level or block of the task. It is thus possible that children in both grades were already familiar with the large numerical scale used, and thus performed more linearly even prior to feedback. In order to more closely scrutinize the concurrent behavioral and neural signatures of logarithmic versus linear NLE performance, future studies should either target a younger student population (i.e., 1^st graders) who would ostensibly be less familiar with the ‘hundred thousand’ scale, or adjust the large estimation scale to more appropriately capture the logarithmic estimation function in second and third grade children.

Children’s estimation speed decreased throughout the task, but was significantly slower for large- compared to small-scale estimations. As seen in Figure 5, children’s median response times were consistently faster for small compared to large-scale estimations. Furthermore, children demonstrated less variance in response times for small-scale estimations compared to large-scale estimations. Taken together, these data indicate that children were both faster and more accurate on small-scale estimations. Although children in this study were slower to respond to large-scale estimations, their accuracy for large-scale estimations did improve as a function of feedback during the task. Second and third-grade children had similar response times regardless of the change in scale, although this relationship was not tested statistically.

A primary strength of the current study is our use of fNIRS to identify concurrent signatures of cortical activation during the NLE task. Notably, our data are some of the first to demonstrate significant information with this imaging modality regarding how children process this real-world math learning activity (Soltanlou et al., 2018a; Soltanlou et al., 2018b). Previous research has implicated the right IPS and the right angular gyrus in numerical processes such as numerical deviance discrimination (Ansari, 2007; Arsalidou and Taylor, 2011) (Arsalidou et al., 2018; Peters & De Smedt, 2018), such that greater activation in these regions is reported when greater numerical distance exists between two numerical stimuli. In the current task, as estimation scales increased in magnitude, so too did the numerical distance between the target value and the scale endpoints. Thus, greater right IPS activity within children that resulted from an increase in scale magnitude may be indicative of increased numerical deviance discrimination processes that occur naturally in healthy children. However, these data do not rule out the possibility that this increase in activation is also driven by other concurrent processes such as attention and visuospatial reasoning that occur in overlapping regions of the brain. Indeed, the feedback block of our task significantly increased activation in the right IPS and angular gyrus. It is possible that our task feedback elicited increased cognitive processes related to visuospatial navigation of the mental number line, thus increasing activation in these regions.

Despite similar behavioral performance between second and third-grade children, fNIRS data recorded within the left IPS and left angular gyrus highlight significant grade-related differences in children’s cortical response patterns throughout the task. These results may be expected, given the significant involvement of these ROI’s in number processing and calculation (Arsalidou and Taylor, 2011; Dehaene et al., 2003; Hung et al., 2015). However, overall activation in both areas was greater in second compared to third-grade children, indicating that young children may be engaging in greater amounts of effortful verbal numerical processing during our NLE task compared to older children. Finally, the effect of scale on cortical activation in the right AG may provide additional credence to the claim that more ventral areas than previously thought are relevant to mathematical calculations (Menon, 2015). Interestingly, this increased activation does not affect young children’s behavioral performance.

Cortical activation within the left dorsolateral PFC was particularly influenced by the feedback block of our task. The left dorsolateral PFC has been shown to be involved in calculation tasks, particularly when externally generated information about number needs to be monitored and manipulated (Arsalidou and Taylor, 2011; Christoff and Gabrieli, 2000; Curtis and D’esposito, 2003; Owen et al., 2005). In order for our feedback to effectively enhance subsequent NLE trials, participants must keep the relationship between their previous estimations and their respective feedback stimuli in working memory, calibrate the spatial layout of their mental number line, and use that updated representation on each subsequent trial. Within the auditory feedback condition, though, the response tone must be kept in working memory, and must then be compared to the feedback tone that follows it. Thus, greater amounts of working memory may be needed to assess differences in feedback within the auditory condition. This increase in working memory processing may alternatively be driving the increased dorsolateral PFC activation identified here. Further, the dorsolateral PFC has also been implicated in the integration of crossmodal stimuli, especially when such information is essentially arbitrary (Banati, 2000; Bushara et al., 2001; Callan et al., 2001; Calvert, 2001; Calvert et al., 2000; Giard and Peronnet, 1999; Gonzalo et al., 2000; Lewis et al., 2000; Raij et al., 2000). In the current experiment, the auditory feedback tone could have been difficult to interpret if it was similar to the magnitude, in volume, of the participant’s estimation tone. Thus, the coincident increase in right IPS and dorsolateral PFC activation in response to auditory feedback may indicate concurrent numerical deviance discrimination and perception of essentially arbitrary crossmodal stimuli. Our failure to identify a significant block × feedback condition interaction may have resulted from use of too few trials within the feedback block, thereby reducing our ability to identify a true effect. Future research is necessary to elucidate this issue further.

Regarding behavioral effects of NLE performance feedback, as hypothesized, feedback during our task significantly improved children’s estimation accuracy. However, each type of feedback benefitted children’s estimation performance equally. Thus, in the current experimental paradigm, redundant multisensory information about NLE estimation accuracy did not improve children’s NLE performance over and above either unisensory feedback type. The underlying cause of this result could stem from many factors. For example, children may have attended to only one of the two concurrently presented modalities of feedback and disregarded the other. Another possibility is that this audiovisual feedback did not sufficiently convey redundant multisensory information about the spatial position of a number along a bounded number line. Visual feedback is inherently bound to the physical dimensions of the number line, but the perceptual limitations of the volume of the tone feedback are unbounded and may thus have conveyed non-redundant perceptual information. Alternatively, multisensory information often benefits perception most when the information to be processed is difficult or ambiguous, an effect that has been observed in children’s representation of number (Jordan and Baker, 2011). Because children’s representations of number were already linear in the current study, it may be that audiovisual feedback would still benefit children’s number line estimations if representations were logarithmic. In sum, while both visual and auditory feedback may be used to improve a participant’s NLE performance, the cognitive processes by which both forms of feedback operate may differ.

5. Limitations and Future Directions

While our study is the first effort to elucidate the influence of numerical scale on behavioral and neural signatures of NLE performance, there are multiple shortcomings that may limit to impact of our results. First, our study was statistically powered to identify neural and behavioral effects of estimation scale only. This was done for the sake of replicating past results that focused on behavioral estimation. However, because our study was the first of its kind, all factors of interest were included in our analyses so that effect size estimations may be made. These estimations may, in turn. assist in sample size analyses for future research studies. Nevertheless, our study was likely underpowered to identify effects in all factors. Moreover, inclusion of all factors may have inadvertently increased the probability of committing a Type I (false positive) errors. Future research may overcome these drawbacks by basing the sample size needed to identify these effects from the effect size statistics reported in Table 1, as well as employing p-value correction methods on all main effect outcomes.

Second, in our study, the ability for children to map auditory magnitude (tone) to spatial magnitude (line position) was determined by the experimenter that was presenting the information to the children. Because a more objective approach was not used, it remains possible that some children may not have been capable of making this mapping appropriately. It will be important for future research to improve on this drawback. For example, future researchers may employ a pre-scan calibration task, in which the range of tone volume is based on each participant’s own psychophysical sensitivity (Baker et al., 2018). Furthermore, future research may require an accuracy criteria before progressing to the scan.

Next, our study does not include a control condition, in which no feedback is given. Inclusion of this condition would allow for comparison of both neural and behavioral signatures of NLE performance against a group that performed the same number of trials but did not receive feedback. Inclusion of a proper control condition is important and should be employed in future research. Moreover, while our study focused solely on oxygenated hemoglobing, multiple recent studies indicate that the use of deoxygenated hemoglobin or a combination of the two may provide additional useful information regarding cognitive function. Thus, future studies should consider the use of deoxygenated and combined hemoglobin metrics.

Our study failed to identify a logarithmic estimation function on large scale estimation trials. As discussed above, this may be due to the fact that the children in our study could have already been familiar with both numerical scales, or due to the use of other idiosyncratic strategies. In any case, future research should take care in identifying appropriate numerical estimation scales for similarly aged populations or recruit younger participants who are not familiar with the scales we used. Furthermore, future research should interview each child after the task to obtain information on the strategies they used. Finally, future research may consider investigating alternative approaches to providing redundant multisensory stimulation during number line tasks to further examine the possible beneficial effects this may have on NLE performance. As stated above, this should include the addition of a no-training group. As computer-based math teaching tools become increasingly more common, integration of redundant multisensory information into the tools themselves may provide a powerful method to enhance math learning.

6. Conclusion

Our results are the first to report concurrent behavioral and neural signatures of number line estimations in children. Our results corroborate previous behavioral findings indicating that increases in the scale of a number line estimation negatively impacts estimation performance. However, our results also offer unique insight into the role of the fronto-parietal cortex during this common math learning activity and help extend the growing body of literature aimed at elucidating the cognitive components underlying real world mathematics. Moreover, our study demonstrates the utility of fNIRS as a tool that is optimally suited for studies in Educational Neuroscience.

Research Highlights.

• Numerical scale influences children’s neural and behavioral response to number line estimations.

• Increases in numerical scale are associated with increased activation in the bilaterial parietal cortices

• Increases in numerical scale are associated with decreased estimation performance.