Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis

TLDR: The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains.

Abstract: Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school-aged children and for adults.

Figures

Figure 2. Distribution of the effect sizes (Pearson correlations) by age and task format. Dot size is proportional to the sample size.

Figure 1. Funnel plot of the 284 effect sizes. The inclined lines indicate the 95% confidence interval.

Table 2. Correlations, 95% CI in square brackets, and number of effect sizes in round brackets by age group, task format and measure.

Table 1. Number of effect sizes (k), correlation (r), 95% confidence interval, heterogeneity index (I2) and p value for the test of heterogeneity.

Full text

NUMERICAL MAGNITUDE PROCESSING
Running head: NUMERICAL MAGNITUDE PROCESSING
Associations of Non-Symbolic and Symbolic Numerical Magnitude Processing with
Mathematical Competence: A Meta-analysis
Michael Schneider
1
, Kassandra Beeres
1
, Leyla Coban
1
, Simon Merz
1
, S. Susan Schmidt
1
,
Johannes Stricker
1
, Bert De Smedt
2
1
University of Trier, Germany
2
University of Leuven, Belgium
Second revision
Word count for main text: 7141
Author Note
Correspondence concerning this article should be addressed to Michael Schneider,
m.schneider@uni-trier.de.

NUMERICAL MAGNITUDE PROCESSING 2
Research Highlights
• This is the first meta-analysis on the association of non-symbolic and symbolic magnitude
comparison with mathematical competence.
• The meta-analysis synthesized 284 effect sizes from 17.201 participants by means of a
random-effects two-level regression model.
• Associations with mathematical competence were stronger for symbolic than for non-
symbolic measures.
• Measures of comparison and mathematical competence strongly moderated the
comparison-competence association.

NUMERICAL MAGNITUDE PROCESSING 3
Abstract
Many studies have investigated the association between numerical magnitude processing
skills, as assessed by the numerical magnitude comparison task, and broader mathematical
competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied
considerably in their strengths. It remains unclear whether and to what extent the strength of
these associations differs systematically between non-symbolic and symbolic magnitude
comparison tasks and whether age, magnitude comparison measures or mathematical
competence measures are additional moderators. We investigated these questions by means of
a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with
17.201 participants. Effect sizes were combined by means of a two-level random-effects
regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI
[.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison
task and decreased very slightly with age. The correlation was higher for solution rates and
Weber fractions than for alternative measures of comparison proficiency. It was higher for
mathematical competencies that rely more heavily on the processing of magnitudes (i.e.
mental arithmetic and early mathematical abilities) than for others. The results support the
view that magnitude processing is reliably associated with mathematical competence over the
lifespan in a wide range of tasks, measures and mathematical subdomains. The association is
stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic
magnitude processing might be a more eligible candidate to be targeted by diagnostic
screening instruments and interventions for school aged children and adults.
Keywords: magnitude comparison, mental magnitude representation, approximate number
system, mathematical competence, meta-analysis

NUMERICAL MAGNITUDE PROCESSING 4
Associations of Non-Symbolic and Symbolic Numerical Magnitude Processing with
Mathematical Competence: A Meta-analysis
A wealth of empirical studies investigated the association between the processing of
numerical magnitudes and broader mathematical competence. These studies have far-reaching
implications because numbers are of fundamental importance in our society. For example,
numerical skills are strong predictors of success in school (Duncan et al., 2007), of medical
decision making (Reyna, Nelson, Han, & Dieckmann, 2009), and of valuations of monetary
amounts (Schley & Peters, 2014). They are associated with socioeconomic status (Ritchie &
Bates, 2013) and mortgage default (Gerardi, Goette, & Meier, 2013).
While previous studies have converged on the conclusion that numerical magnitude
processing is an important foundation for higher-level mathematical competence, studies are
heterogeneous in respect to whether the processing of non-symbolic magnitude
representations (i.e., dots), symbolic magnitude representations (i.e., digits), or both are
relevant for the learning of more advanced mathematical competence. Moreover, it has been
suggested that the association between magnitude processing and broader mathematical
competence might also be moderated by participant age (e.g., Inglis, Attridge, Batchelor, &
Gilmore, 2011; Rousselle & Noël, 2008), by measures of magnitude comparison skills (e.g.,
Price, Palmer, Battista, & Ansari, 2012), or by measures of mathematical competence (e.g.,
De Smedt, Noël, Gilmore, & Ansari, 2013). This hampers the integration of empirical
findings across studies in narrative reviews of the literature (De Smedt et al., 2013; Feigenson,
Libertus, & Halberda, 2013).
Meta-analyses quantitatively integrate findings across studies and allow for explicit tests
of moderating influences of third variables. Therefore, meta-analyses can substantially
advance our understanding of the associations between numerical magnitude processing and
broader mathematical competence. Two recent meta-analyses have investigated the
associations between non-symbolic numerical magnitude processing and mathematical

NUMERICAL MAGNITUDE PROCESSING 5
competence. Chen and Li (2014) included 47 effect sizes and found an overall correlation of r
= .20, 95% CI [.14, .26] for cross-sectional studies. Fazio, Bailey, Thompson, and Siegler
(2014) included 34 effect sizes and found an overall correlation of r = .22, 95% CI [.20, .25].
These two meta-analyses provided strong evidence for a weak but reliable association
between non-symbolic magnitude processing and mathematical competence, but they were
limited in their scope. They included only one effect size from each sample. So it was not
possible to analyze differences between effect sizes found with the same sample, for example,
accuracy versus speed of the same participants. Even more crucially, these meta-analyses only
included findings obtained with non-symbolic magnitude processing tasks. No conclusions
about differences between non-symbolic and symbolic magnitude processing could be drawn.
The current meta-analysis closes this gap in the research literature. A better understanding of
how non-symbolic and symbolic magnitude processing relate to broader mathematical
competence might provide helpful background information for educational interventions
aiming at improving learners’ numerical processing skills as preparation for more advanced
mathematical learning (De Smedt et al., 2013; Feigenson et al., 2013). At a theoretical level, it
will aid us to evaluate the importance of evolutionary older non-symbolic magnitude
representations, which we share with many other species (Cantlon, 2012), compared to
uniquely human symbolic representations of numbers.
The current study therefore included effect sizes from non-symbolic and from symbolic
numerical magnitude processing tasks. We used a two-level regression model with effect sizes
(level 1) nested under independent samples (level 2), so that all effect sizes from all samples
could be included. This yielded a database of 284 effect sizes and allowed us to estimate the
overall effect sizes along with the moderating influences of a non-symbolic vs. symbolic task
format, participant age, comparison measures and mathematical competence measures.
In our meta-analysis we focused on the comparison task, which is the most frequently
used task to assess numerical magnitude processing (Ansari, 2008; Dehaene, Dupoux, &

Frequently asked questions

Q1. What are the contributions in "Numerical magnitude processing running head: numerical magnitude processing associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis" ?

The authors investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17. 201 participants. 

Q2. How many participants are needed to detect the correlation between non-symbolic magnitude comparison and?

According to power analyses conducted with G*Power 3 (Faul, Erdfelder, Lang, & Buchner, 2007), given the effect sizes found in their meta-analyses, a critical alpha error level of 5%, a statistical power of 80%, and one-sided testing, 102 participants are needed to detect the correlation between non-symbolicmagnitude comparison and math competence, but only 64 participants are needed to detect the correlation between symbolic magnitude comparison and mathematical competence. 

Q3. Why did the authors conduct meta-regressions with continuous variables?

In addition, the authors conducted meta-regressions with years of age coded as continuousvariable, because continuous variables are more sensitive to gradual changes. 

Q4. Why did the authors enter all moderator variables as level-1 predictors of effect sizes?

The authors entered all moderator variables as level-1 predictors of effect sizes into their two-levelmodel, because their values can differ between effect sizes within independent samples. 

Q5. What is the main reason why the study was underpowered?

Chen and Li (2014) have argued that, given the relatively small correlationbetween non-symbolic magnitude comparison and mathematical competence, many studies in this field of research were severely underpowered because they lacked the large sample sizes needed to detect small effects. 

Q6. What is the importance of a good understanding of the numbers in terms of their magnitudes?

A good understanding of the numbers in terms of their magnitudes can be helpful for choosing an efficient calculation strategy (e.g., Peters, Smedt, Torbeyns, Verschaffel, & Ghesquière, 2014). 

Q7. What is the dominant view of symbolic magnitude processing?

The dominant view assumes that symbolic representations are mapped onto non-symbolic ones, which are then further processed, in which case non-symbolic magnitude processing would be an important component of symbolic magnitude processing (see, e.g., Piazza, 2010, for a review). 

Q8. How many students solved the symbolic magnitude comparison task?

Träff found three effect sizes between 0.470 and 0.670 with 134 students from Grades 4 to 6 who solved the symbolic magnitude comparison task. 

Q9. What is the effect of the approximate number system on the learning of math?

How training on exact or approximate mentalrepresentations of number can enhance first-grade students' basic number processing and arithmetic skills. 

Q10. How much of the variance of effect sizes was explained by differences between measures of magnitude comparison skills?

In their analyses, differences between measures of magnitude comparison skills explained about 14% of the variance of effect sizes found with these measures. 

Q11. How many studies reported reliabilities of magnitude comparison measures?

The authors did correct for non-perfect reliabilities of magnitude comparison measures in their meta-analysis, but only for those 25% of studies that reported the reliabilities of the magnitude comparison measures.