How number line estimation skills relate to neural activations in single digit subtraction problems
Introduction
Mathematical skill predicts later academic achievement more strongly than early reading and socio-emotional skills (Duncan et al., 2007, National Mathematics Advisory Panel, 2008). Therefore understanding individual differences has become a central research question for educational policies. Several studies suggest that differences in arithmetical performance are related to neural activations specific for numerical processing (De Smedt et al., 2011, De Smedt et al., 2013, Price et al., 2007).
A widely used numerical task that has shown to be predictive of future mathematical skill is the Number Line (NL) task (Booth and Siegler, 2006, Cowan and Powell, 2014, Siegler and Booth, 2004, Siegler and Opfer, 2003). This task requires both numerical magnitude and spatial processing since participants are asked to estimate the position of a given number (e.g., 21) onto a black horizontal line with the left and right ends labeled as 0 and 100 (or 1000), respectively. Younger children who have a less precise representation of numerical values overestimate small numbers and underestimate larger ones. Linearity is initially acquired for smaller ranges and progressively, with age and increasing number knowledge, linearity is extended to larger ranges (Berteletti et al., 2010, Siegler and Booth, 2004, Siegler et al., 2009). Performance on the NL task correlates with other estimation tasks (Booth and Siegler, 2006, Laski and Siegler, 2007), improves following interventions on children's linear and cardinal understanding of the numerical sequence (Ramani and Siegler, 2008, Siegler and Ramani, 2008), and is correlated with and is predictive of arithmetic learning and mathematical achievement (Booth and Siegler, 2008, Link et al., 2014, Linsen et al., 2014, Ostergren and Träff, 2013, Sasanguie et al., 2013, Schneider et al., 2009). Performance is also impaired or delayed in children with math learning difficulty (Geary et al., 2008, Landerl, 2013). Importantly, improving children's performance on the NL task enhances performance in numerical magnitude processing tasks as well as facilitates learning of multi-digit addition problems (Booth and Siegler, 2008, Kucian et al., 2011). Because the NL task has been shown to be a strong and unique predictor of arithmetical ability in grades 1 and 2 over and above other non-symbolic and symbolic tasks (Lyons et al., 2014), and because the NL tasks were correlated to mental multi-digit subtraction performance (Linsen et al., 2014), it can be argued that the NL task is specifically calling upon numerical processes that are crucial for later acquisition of mathematical competences. Two types of functional processes involved in the Number Line task might explain the relationship with mathematical performance and in particular with subtraction problems. First in both tasks, the symbolic numbers need to elicit the numerical magnitude representation. This allows for a comparison between the symbolic numbers: in the Subtraction task, the two digits need to be compared to determine the result; in the Number Line task, the number to position has to be compared to the numerical boundaries of the interval to determine it's relative numerical magnitude. Second, both tasks call upon visuo-spatial processes. For the Number Line task, the symbolic number that has to be estimated has to be translated into a visual segment. To do this, children need to focus first on the entire interval and then move their attention along the line to estimate the position. Subtraction tasks have also been shown to rely on visuo-spatial processes (Dehaene et al., 2003, Knops and Willmes, 2014, Knops et al., 2009, Rotzer et al., 2009), and visuo-spatial proficiency has been shown to predict arithmetic performance in children (De Smedt et al., 2009, Rotzer et al., 2009). This result may be explained by the observation that efficient strategies rely on mechanisms that involve shifts of spatial attention (Knops et al., 2009). Indeed, studies have shown that subtraction and addition problems lead to under- and over-estimation and this Operational Momentum effect has been explained in terms of attentional shifts on a spatially organized mental representation of numbers (Knops et al., 2009, Knops et al., 2013, Knops et al., 2014) as if quantities were represented in the format of an internal mental Number Line (Hubbard et al., 2005, Rotzer et al., 2009).
From a neurofunctional perspective, processing of numerical magnitudes has been identified in the parietal lobes and more specifically in the bilateral intraparietal sulcus (IPS; Ansari, 2008, Nieder and Dehaene, 2009, Piazza et al., 2004, Piazza et al., 2007; see Arsalidou and Taylor, 2011 for a meta-analysis). This area is sensitive to the distance effect in digit comparison tasks both for children and adults (Mussolin et al., 2010, Pinel et al., 2001) as well as being less sensitive in children with mathematical learning disability (Mussolin et al., 2010). The IPS is also found to be more active in calculation tasks compared to reading numerical symbols (Burbaud et al., 1999, Chochon et al., 1999, Pesenti et al., 2000), and more active for larger compared to smaller arithmetical problems (Ashkenazi et al., 2012, De Smedt et al., 2011). Moreover, the IPS shows greater activation during subtraction than multiplication (Chochon et al., 1999, Ischebeck et al., 2006, Prado et al., 2011, Prado et al., 2014) likely due to the fact that subtraction problems rely more on calculation procedures (Fayol and Thevenot, 2012) and quantity processing compared to multiplication problems (Dehaene et al., 2003, Prado et al., 2011, Prado et al., 2014).
Another parietal area often active in tasks requiring numerical manipulation is the posterior superior parietal lobule (PSPL) with mesial extension into the precuneus (PCu; Arsalidou and Taylor, 2011, Dehaene et al., 2003, Kaufmann et al., 2011). This area is active during number comparison (Pesenti et al., 2000, Pinel et al., 2001), approximation (Dehaene et al., 1999), subtraction of two digits (Knops et al., 2009, Lee, 2000), and counting (Piazza et al., 2002). Increased activation is found for more complex operations (Menon et al., 2000), and for subtraction problems compared to multiplication problems (Prado et al., 2011, Prado et al., 2014). However, this region also plays a role in several visuospatial tasks such as reaching, grasping, eye and/or attention orienting, mental rotation, and spatial working memory (Hubbard et al., 2005, Knops et al., 2009, Simon et al., 2002, Simon et al., 2004). Knops et al. (2009) investigated the relation between eye movement and arithmetic processing in adults using fMRI. They trained a multivariate classifier on saccade-related activity in the PSPL and were able to predict the type of mental operation (i.e., addition of subtraction), irrespective of notation (i.e., symbolic or non-symbolic). The authors suggest that mental arithmetic recruits parietal areas that are associated with visuospatial processing and that mental calculation may (at least partially) rely on the dynamic interplay between subsystems of the parietal cortex (i.e., IPS and PSPL).
On the one hand, imaging studies indicate that different subsystems in the parietal cortex (i.e., IPS for numerical magnitude and PSPL for visuospatial components of numerical and arithmetical processing) support mental calculation and, on the other hand, performance in the NL task is correlated and predictive of future arithmetic skill. However, to our knowledge, there is no direct evidence that estimating the position of a number on a line is related to functional areas used for calculation. Only two studies investigate the neural bases of the NL task. The first study investigated whether an intervention program using a NL-like game induces neurofunctional changes in areas involved in judging the relative magnitude of digits (Kucian et al., 2011). The game was intended to improve both estimation and arithmetical skills (i.e., numbers, sets of dots and arithmetical results had to be positioned on lines) in 9-year-old children with and without mathematical difficulty. Performance in the NL task was significantly improved for both groups. Using fMRI, children were required to judge whether triplets of digits were in ascending order. Training resulted in decreased activation in left IPS along with frontal areas, suggesting an increased efficiency in performing the task. Although these results support the effectiveness of the intervention, the authors showed improvement in magnitude judgment and not in arithmetical processing. The second study, using fMRI, investigates the brain regions involved during a classical NL task and a brightness estimation line task (i.e., continuous magnitude judgment) using shades of gray (Vogel et al., 2013). Results indicated some overlap in the right posterior part of IPS but, most importantly, the bilateral anterior part of IPS was specifically recruited for estimating symbolic numbers. This study is the first that directly investigates the NL task in an imaging paradigm and shows activations in areas typically involved in numerical magnitude processing. Unfortunately, no direct relation between these activations and arithmetical performance was investigated.
In the present study, we test whether performance on the NL task is related to activation found during simple arithmetical processing and if this relation relies on domain-specific processes. This would bring neuro-functional evidence to the behavioral relation found between performance to the NL task and arithmetic skill. We therefore collected behavioral performance on the NL task and functional (fMRI) data during a single-digit Subtraction task in 8- to 14-year-old children. In this age range, arithmetical learning and estimation abilities are still improving (e.g., Holloway and Ansari, 2009, Siegler and Booth, 2004) increasing the chance of observing a relation between the two tasks. We chose single-digit subtraction problems because they have shown to rely more on quantitative manipulation compared to other arithmetical operations (Dehaene, 1992, Dehaene et al., 2003, Fayol and Thevenot, 2012, Prado et al., 2011). In support of this, the symbolic distance effect, that is the ability to determine which of two digits is numerically larger, was found to be specifically related to subtraction problems in children (Lonnemann et al., 2011). To isolate domain-specific areas involved in numerical processing, participants were also asked to perform a numerosity judgment task in the scanner (i.e., numerical comparison of sets of dots). Parietal areas activated by this task were then used as regions of interest (ROI) to investigate the relation between performance on the NL task and activation to the Subtraction task. Indeed, the parietal cortex, and specifically the IPS, has shown modulation of activation specifically to changes in numerical information (Cantlon et al., 2006, Piazza et al., 2004).
Within the numerical processing areas, we expected to find a significant relation of performance on the NL task with bilateral IPS activation. The IPS responds specifically to magnitude information required in both the NL task and the Subtractions task (Piazza et al., 2004). Specifically, we expected children with better NL estimation abilities to show greater activation for larger compared to smaller problems (De Smedt et al., 2011, Stanescu-Cosson et al., 2000) indicating greater quantity manipulation for more effortful and less well learned problems. Because children in our study are still learning, only small problems are likely to be adequately mastered thus showing less activation compared to larger problems (Delazer et al., 2003, Ischebeck et al., 2006, Ischebeck et al., 2007). However, children with lower NL estimation abilities were expected to not show a problem size effect: children with lower arithmetical skill have shown less modulation with problem size or even an atypical reversed modulation (De Smedt et al., 2011). Additionally, because the NL task requires transposing into space a numerical value and moving visuo-spatial attention along the physical line, we expected to find a relation within the bilateral PSPL. The PSPL has been found to support different aspects of spatial processing in numerical tasks (Kaufmann et al., 2011, Knops et al., 2009, Lee, 2000, Pinel et al., 2001). In particular, common activation of bilateral PSPL was found for arithmetic problems and shifts of visuo-spatial attentions (Knops et al., 2009). Increased activity in subtraction problems was found for older compared to younger children (Prado et al., 2014) suggesting an increasing reliance on such processes with greater expertise. Moreover, a modulation of activation was found in the PSPL for more complex problems (Menon et al., 2000). Because spatial processes are engaged more in participants with greater expertise and because larger problems can be seen as more complex for children that are still learning, we expected participants with finer ability in estimating the position on the Number Line task to show greater reliance on PSPL.
Section snippets
Participants
Thirty-nine children (22 females, 17 males) between 8 and 13 years of age (mean age = 11:4, SD = 1:6, range = 8:5–13:7) were chosen based on standardized testing performance and fMRI scan quality. All participants had a full-scale IQ standard score greater than 85 on the Wechsler Abbreviated Scale of Intelligence (WASI; Wechsler, 1999) with a group average of 118 (SD = 11.3). To ensure that participants had no mathematical difficulty, children had to score 85 or above (mean = 105.2, SD = 11.4) on the Math
Behavioral
For the NL task, the average estimation error (PAE) was 6.1% (SD = 5.7%). Groups were divided for higher and lower estimation errors based on the median value: the lower NL estimation error group misjudged the position by 2.9% (SD = .9%, n = 19), and the higher NL estimation error group misjudged by 9.4% (SD = 6.7%, n = 20). The two groups were significantly different after controlling for unequal variances (t(19.6) = − 4.25, p < .001).
For the Subtraction task, overall accuracy was 91% (SD = 8 ms) and average RT
Discussion
It has been argued that variances in mathematical skill are to be found in individual differences in number specific processes (Butterworth et al., 2011, Dehaene et al., 2003). Among the various basic numerical tasks, performance to the NL task is considered to be indicative of the quality of a child's numerical representation and has shown to be related with and be predictive of future arithmetical skill (Siegler and Booth, 2004, Siegler and Opfer, 2003). However, no study has investigated
Acknowledgment
This project was funded by the National Institute of Child Health and Human Development (grant number HD059177) awarded to J.R.B.
References (97)
- et al.
Is 2 + 2 = 4? Meta-analyses of brain areas needed for numbers and calculations
NeuroImage
(2011) - et al.
Weak task-related modulation and stimulus representations during arithmetic problem solving in children with developmental dyscalculia
Dev. Cogn. Neurosci.
(2012) - et al.
Strategies in subtraction problem solving in children
J. Exp. Child Psychol.
(2008) - et al.
A functional magnetic resonance imaging study of mental subtraction in human subjects
Neurosci. Lett.
(1999) - et al.
The functional neuroanatomy of simple calculation and number repetition: a parametric PET activation study
NeuroImage
(2000) - et al.
Working memory and individual differences in mathematics achievement: a longitudinal study from first grade to second grade
J. Exp. Child Psychol.
(2009) - et al.
Effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency
NeuroImage
(2011) - et al.
How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior
Trends Neurosci. Educ.
(2013) Varieties of numerical abilities
Cognition
(1992)- et al.
Learning complex arithmetic—an fMRI study
Cogn. Brain Res.
(2003)
Causal role of spatial attention in arithmetic problem solving: Evidence from left unilateral neglect
Neuropsychologia
The use of procedural knowledge in simple addition and subtraction problems
Cognition
Developmental changes in performance monitoring: how electrophysiological data can enhance our understanding of error and feedback processing in childhood and adolescence
Behav. Brain Res.
Neural correlates of approximate quantification strategies in young and older adults: an fMRI study
Brain Res.
To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving
Neuropsychologia
Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement
J. Exp. Child Psychol.
How specifically do we learn? Imaging the learning of multiplication and subtraction
NeuroImage
Imaging early practice effects in arithmetic
NeuroImage
Dissociating the solution processes of small, large, and zero multiplications by means of fMRI
NeuroImage
Numerical ordering and symbolic arithmetic share frontal and parietal circuits in the right hemisphere
NeuroImage
The neural substrate of arithmetic operations and procedure complexity
Brain Res. Cogn. Brain Res.
Mental number line training in children with developmental dyscalculia
NeuroImage
The association between children's numerical magnitude processing and mental multi-digit subtraction
Acta Psychol. (Amst).
Symbolic and non-symbolic distance effects in children and their connection with arithmetic skills
J. Neurolinguistics
Dissociating prefrontal and parietal cortex activation during arithmetic processing
NeuroImage
The number domain — can we count on parietal cortex?
Neuron
Early number knowledge and cognitive ability affect early arithmetic ability
J. Exp. Child Psychol.
Are subitizing and counting implemented as separate or functionally overlapping processes?
NeuroImage
Tuning curves for approximate numerosity in the human intraparietal sulcus
Neuron
A magnitude code common to numerosities and number symbols in human intraparietal cortex
Neuron
Modulation of parietal activation by semantic distance in a number comparison task
NeuroImage
Impaired parietal magnitude processing in developmental dyscalculia
Curr. Biol.
Dysfunctional neural network of spatial working memory contributes to developmental dyscalculia
Neuropsychologia
Approximate number sense, symbolic number processing, or number–space mappings: what underlies mathematics achievement?
J. Exp. Child Psychol.
Topographical layout of hand, eye, calculation, and language-related areas in the human parietal lobe
Neuron
Automatized clustering and functional geometry of human parietofrontal networks for language, space, and number
NeuroImage
The anterior cingulate as a conflict monitor: fMRI and ERP studies
Physiol. Behav.
Overlapping and distinct brain regions involved in estimating the spatial position of numerical and non-numerical magnitudes: an fMRI study
Neuropsychologia
How verbal and spatial manipulation networks contribute to calculation: an fMRI study
Neuropsychologia
Effects of development and enculturation on number representation in the brain
Nat. Rev. Neurosci.
Neural correlates of symbolic number processing in children and adults
Neuroreport
Numerical estimation in preschoolers
Dev. Psychol.
Children with mathematical learning disability fail in recruiting verbal and numerical brain regions when solving simple multiplication problems
Cortex
Brain bases of learning and development of language and reading
Developmental and individual differences in pure numerical estimation
Dev. Psychol.
Numerical magnitude representations influence arithmetic learning
Child Dev.
Modality independence of word comprehension
Hum. Brain Mapp.
Dyscalculia: from brain to education
Science
Cited by (30)
The home mathematics environment and its relation to children's mathematical skills for Chinese families
2023, Learning and Individual DifferencesGray matter volume in left intraparietal sulcus predicts longitudinal gains in subtraction skill in elementary school
2021, NeuroImageCitation Excerpt :This dataset has been deposited in OpenNeuro (10.18112/openneuro.ds001486.v1.1.0) and a detailed description of the dataset is provided in Suárez-Pellicioni et al. (2019). Time-point 1 of this dataset is the basis of other publications including Berteletti and Booth (2015, 2015), Berteletti et al. (2014), Demir-Lira et al. (2019), Demir et al. (2014, 2015) and Prado et al. (2014). The longitudinal data of this dataset is the basis of other publications including Demir-Lira et al. (2016), Suárez-Pellicioni and Booth (2018), and Suárez-Pellicioni et al. (2018, 2019, 2020).
Structured versus free block play: the impact on arithmetic processing
2021, Trends in Neuroscience and EducationCitation Excerpt :However, studies have shown that finger use varies with arithmetic operation ([3,53]). For example, Berteletti and Booth [3] found that children recruited finger based motor areas more for subtraction than for multiplication suggesting that finger use supported subtraction but not multiplication. It should be noted that finger processing is related to visuospatial processing [15,38].
Temporo-frontal activation during phonological processing predicts gains in arithmetic facts in young children
2019, Developmental Cognitive NeuroscienceCitation Excerpt :It is important to note that we did not find brain activation during phonological processing to be concurrently associated with math fluency or multiplication skill at time 1 or time 2, as we only showed prediction of gains over time (see Fig. 5). This could be seen as contradictory with the Berteletti et al. (2014) finding that higher phonological awareness was concurrently associated with greater activation when solving small multiplication problems in verbal areas of the left temporal cortex. These differences likely result from the nature of the phonological and math measures.
Lack of improvement in multiplication is associated with reverting from verbal retrieval to numerical operations
2018, NeuroImageCitation Excerpt :Similarly, because of a specific a priori hypothesis of brain areas involved in quantity manipulation, the left and right IPL/SPL were used as anatomical masks. The choice of these anatomical masks was based on previous findings from our lab (e.g. Prado et al., 2011, 2013, 2014; Berteletti et al., 2014a; Berteletti et al., 2014b; Berteletti et al., 2014). 3dClustSim, available as part of the AFNI fMRI analysis package, was used to calculate cluster size threshold for significance for these two anatomically defined masks.