Number sense and mathematics: Which, when and how?

Nonsymbolic Estimation and Its Relationship With Mathematics

Nonsymbolic estimation involves nonverbal processing of quantities and numerosities without using numerals. For example, this ability enables us to select a queue with fewer people without counting. Research suggests that this type of numerosity processing depends on the absolute number of items in a set: Evaluation of individual sets including fewer items is more accurate compared with those containing more items (set-size effect; e.g., Gordon, 2004 ; Whalen, Gallistel, & Gelman, 1999 ). Furthermore, discrimination between two sets is more difficult when the discrepancy between the number of items in the sets is smaller (distance effect; e.g., Feigenson, Carey, & Hauser, 2002 ; Holloway & Ansari, 2009 ; Moyer & Landauer, 1967 ). These two effects are encompassed by Weber’s law, with the Weber Fraction indexing the minimum ratio between two sets reliably discernible by individuals ( Weber, 1834 ).

Numerosity processing can be carried out without formal knowledge of numbers or formal instruction (e.g., Pica, Lemer, Izard, & Dehaene, 2004 ) and, in humans, this skill improves with development. For example, 6-month-old babies can successfully discriminate only between large ratios, such as 8 versus 16 (ratios 1:2), with corresponding Weber Fraction of 1 ([(2 − 1)÷1]; e.g., Libertus & Brannon, 2010 ). Adults can discriminate larger numerosities and smaller ratios ( Halberda & Feigenson, 2008 ; Halberda, Ly, Wilmer, Naiman, & Germine, 2012 ).

People differ greatly in the speed and accuracy of estimation (e.g., Halberda, Mazzocco, & Feigenson, 2008 ). Individual differences in nonsymbolic estimation, assessed using different nonsymbolic tasks, have been found in preschoolers, school-age children, and adults (e.g., Barth et al., 2006 ; Gilmore, McCarthy, & Spelke, 2010 ; Halberda et al., 2012 ; Nys & Content, 2012 ). A few studies that looked at potential sex differences in nonsymbolic estimation found no average differences between males and females (e.g., 3–5-year-olds, Bonny & Lourenco, 2013 ; 5–6-year-olds, Gilmore et al., 2010 ; 4-year-olds, Libertus, Feigenson, & Halberda, 2011 ; 14–15-year-olds, Mazzocco, Feigenson, & Halberda, 2011a ). However, one study reported a small male advantage in 4-year-olds ( Soltész, Szücs, & Szücs, 2010 ).

Several longitudinal studies showed an association between individual differences in nonsymbolic estimation and mathematical performance in preschool children ( Gilmore et al., 2010 ; Mazzocco, Feigenson, & Halberda, 2011b ) and older children ( Halberda & Feigenson, 2008 ), with evidence suggesting a causal association ( Wang, Odic, Halberda, & Feigenson, 2016 ). However, other studies have failed to find a significant correlation (e.g., Holloway & Ansari, 2009 ; Rousselle & Noël, 2007 ; Sasanguie, Defever, Maertens, & Reynvoet, 2014 ).

Despite inconsistencies across individual studies, meta-analyses have shown that nonsymbolic estimation is prospectively and retrospectively, weakly, associated with mathematics across development ( r = .24 prospectively and .17 retrospectively, Chen & Li, 2014 ; r = .22, Fazio, Bailey, Thompson, & Siegler, 2014 ; r = .24, Schneider et al., 2017 ). Discrepancies across individual studies may have stemmed from: differences in age of participants ( Fazio et al., 2014 ; Schneider et al., 2017 ); measures of estimation used (for a discussion see Clayton, Gilmore, & Inglis, 2015 ); specific mathematics skills with which estimation is being correlated ( Mazzocco et al., 2011a ); mathematics achievement level of the participants ( Bonny & Lourenco, 2013 ; Mazzocco et al., 2011a ); and overall lack of statistical power to detect weak associations.

Symbolic Estimation and Its Relationship With Mathematics

Symbolic estimation relies on symbols, such as Arabic numerals ( Booth & Siegler, 2006 ; Cohen Kadosh, et al., 2008 ). For example, by relying on symbolic estimation people can tell that the solution to a numerical problem is incorrect without calculating an exact answer. The size and ratio effects observed for nonsymbolic estimation are also observed for symbolic estimation. Overall, people are faster in comparing two small numbers (1 and 2) than two large numbers (8 and 9) even when the distance between them is kept constant, suggesting that it is easier to process small numbers ( Moyer & Landauer, 1967 ). Moreover, adults and children are faster and more accurate in judging the difference between two numerical magnitudes when the numerical distance between the numerals is larger (1 vs. 9) than when it is smaller (6 vs. 8; e.g., Dehaene, Dupoux, & Mehler, 1990 ). The presence of size and ratio effects in symbolic estimation has been taken as indirect evidence that symbolic representation of numbers builds on the approximate representation of nonsymbolic numerosity ( Feigenson, Dehaene, & Spelke, 2004 ). The closeness between symbolic and nonsymbolic estimation seems also supported by reliance on partially overlapping neuronal activity in the intraparietal sulcus (IPS) and prefrontal cortex (for a discussion see Nieder & Dehaene, 2009 ). IPS areas are activated when attending to numerosity stimuli (e.g., Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004 ) or manipulating Arabic number symbols (e.g., Pinel, Dehaene, Riviere, & LeBihan, 2001 ). Different neurons in parietal regions, respond to a specific numerosity (tuning function); such tuning functions are organized sequentially, preserving the order of cardinality (numerosity of a set size) and following the Weber law ( Nieder & Merten, 2007 ). However, some neural pathways show differential activation during encoding of numerical magnitudes gathered from symbolic and nonsymbolic stimuli ( Holloway, Price, & Ansari, 2010 ). Furthermore, there is evidence of lateralization in IPS response to symbolic and nonsymbolic processing ( Holloway, Battista, Vogel, & Ansari, 2013 ).

It is thought that, as numerals are acquired, they map onto existing nonsymbolic representations and become mentally represented along a mental “number line” (e.g., Restle, 1970 ; Siegler & Opfer, 2003 ). This line is organized in ascending order, following a left-to-right direction in English-writing participants and right-to-left in Arabic-writing participants ( Dehaene, Bossini, & Giraux, 1993 ; cf. Ito & Hatta, 2004 ). It is hypothesized that numbers on the mental number line are initially logarithmically compressed (e.g., Dehaene & Mehler, 1992 ). With age, a gradual shift seems to occur from the less accurate logarithmic mental number representation to a more precise linear representation. The linear representation becomes dominant from the age of 6 to 8 years, as evidenced by improved performance on the number line task ( Siegler & Booth, 2004 ). However, performance on this task may be based on strategies such as reliance on midpoint (knowing that 50 is half of 100; Ashcraft & Moore, 2012 ) and reliance on proportion-judgment, as the position of a number on a number line is estimated relatively to the size of the whole line ( Barth & Paladino, 2011 ). Therefore, developmental changes may be due to the increasing use of a reference point rather than a log-to-linear shift. Another explanation for the increased accuracy on number line tasks takes into account familiarity with number symbols (e.g., Ebersbach, Luwel, Frick, Onghena, & Verschaffel, 2008 ; Moeller, Pixner, Kaufmann, & Nuerk, 2009 ). These explanations are not mutually exclusive ( Dackermann, Huber, Bahnmueller, Nuerk, & Moeller, 2015 ).

Several studies in different cultures have found a correlation between performance on number line tasks and mathematics skills (e.g., Booth & Siegler, 2006 ; Fazio et al., 2014 ; Fuchs et al., 2010a ; Geary, 2011 ; Siegler & Booth, 2004 ; Siegler & Mu, 2008 ). The mechanisms of the association are unclear. Research suggests that experience with numbers, such as playing numerical board games, can improve children’s estimation abilities on the number line ( Siegler & Booth, 2004 ). In turn, improvement of magnitude processing on the number line was found to be causally related to better arithmetic (addition problems) skills ( Booth & Siegler, 2008 ). However bidirectional effects are also likely. For example, it was found that access to numerical instruction can improve nonsymbolic estimation skills in Western adults ( Nys et al., 2013 ). In children, the association between nonsymbolic estimation and mathematics was found to be mediated by symbolic estimation skills, such as knowledge of number words and Arabic numerals and of their meaning (cf. Räsänen, Salminen, Wilson, Aunio, & Dehaene, 2009 ; van Marle, Chu, Li, & Geary, 2014 ). It is possible that number line activities contribute to the knowledge of symbolic quantities, which is one of the most powerful predictors of later achievement ( Duncan et al., 2007 ; Jordan, Kaplan, Ramineni, & Locuniak, 2009 ).

Similar to nonsymbolic estimation, there is evidence pointing to a small male advantage in number line estimation ( Hannula, 2003 ; LeFevre et al., 2010 ), although these results are not consistent ( Gunderson, Ramirez, Beilock, & Levine, 2012 ; Thompson & Opfer, 2008 ).

Nonsymbolic and Symbolic Estimation and Other Cognitive Abilities

A wealth of previous research has found associations between mathematics and other non-numerical abilities, such as working memory (e.g., Bull, Johnston, & Roy, 1999 ; Geary, 2011 ; McLean & Hitch, 1999 ; Siegel & Ryan, 1989 ; Swanson & Sachse-Lee, 2001 ); speed of processing ( Bull & Johnston, 1997 ; Bull et al., 1999 ; Case, Kurland, & Goldberg, 1982 ); and reading and general cognitive factors (e.g., Dirks, Spyer, van Lieshout, & de Sonneville, 2008 ; Fuchs et al., 2010b ; Kovas, Harlaar, Petrill, & Plomin, 2005 ; Kovas, Haworth, Petrill, & Plomin, 2007 ). Less is known about the role of these abilities in the link between mathematics and nonsymbolic and symbolic estimation.

One study found that the correlation between a nonsymbolic (dot) discrimination task at age 14 and mathematical ability at age 8 remained significant after controlling for 16 cognitive measures assessed at age 8, including visuospatial reasoning, working memory, reading, word knowledge, and object perception ( Halberda & Feigenson, 2008 ). Similarly, nonsymbolic estimation skills were significantly correlated with mathematics in over 10,000 11-to 85-year-old participants, after controlling for age, sex, as well as measures of science, writing and computer ability ( Halberda et al., 2012 ). In preschoolers nonsymbolic estimation skills were associated with mathematical abilities, but not with vocabulary or letter identification in early primary school ( Mazzocco et al., 2011b ). However, another study found that a nonsymbolic (dot) discrimination task correlated only with short-term memory but not with counting and number knowledge in 4–7-year-olds ( Soltész et al., 2010 ). Number line estimation has been linked to individual differences in IQ and in aspects of working memory in 7–8-year-old children ( Geary, Hoard, Nugent, & Byrd-Craven, 2008 ) and with visuospatial skills ( Bachot, Gevers, Fias, & Roeyers, 2005 ).

The Present Study

The body of knowledge on the links between estimation, other cognitive abilities, and mathematics is growing. However, most of the studies into symbolic and nonsymbolic estimation have been conducted in early to middle childhood. Furthermore, most studies have used only a few measures, and therefore meta-analyses draw conclusions based on widely differing measures and ages (see Schneider et al., 2017 ). Previous research provided inconsistent findings regarding the presence of sex differences in estimation abilities. It is therefore unclear whether sex differences in estimation, if found, may contribute to the observed sex differences in mathematical ability (e.g., Spelke, 2005 ).

The present study is a large-scale multivariate investigation into the relationship between two aspects of number sense and formal mathematics across development. The study has three major aims: (1) to examine the relationship between nonsymbolic and symbolic estimation abilities, as assessed by a dot estimation and a number line tasks at age 16; (2) to assess whether estimation abilities measured at age 16 are related with mathematical abilities measured at ages 7, 9, 10, 12, 14, and 16; (3) to assess whether the links between mathematical ability and estimation are present after accounting for a number of verbal and non-verbal abilities measured in the same children at 7, 9, 10, 12, 14, and 16 years of age. The large sample used in the study affords a statistically powerful evaluation of potential sex differences in estimation and in the extent to which sex differences in estimation are associated with sex differences in mathematical ability.

Validation

Prior to the main data collection, the tasks were piloted and tested for reliability and suitability for Web administration using samples of 16-years-old singleton and twin students. All tests proved to be suitable for Web administration (see SOM for details) and showed good internal consistency and test–retest reliability (see Table 2 ).

9 years

Data for cognitive abilities (Verbal Ability and Non-Verbal Ability) were collected using child-completed postal booklets. Mathematics school achievement was collected using teacher questionnaires.

10 years

Data for cognitive abilities (Verbal Ability, Non-Verbal Ability, Mathematics Web and Reading) were collected using an online test battery. Mathematics school achievement was collected using teacher questionnaires.

12 years

Data for cognitive abilities (Verbal Ability, Non-Verbal Ability, Mathematics Web, Spatial Ability, Language and Reading) were collected using a Web-based test battery. Mathematics school achievement was collected using teacher questionnaires.