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MATHEMATICS VOCABULARY AND MATHEMATICS 1
Understanding the Relation Between Mathematics Vocabulary and Mathematics Performance:
A Meta-analysis
Xin Lin1*, Peng Peng1*, & Jiangang Zeng2
1. The University of Texas at Austin, USA
2. Louisiana State University
Lin, X., Peng, P., & Zeng, J. G. (in press). A Meta-analysis on the relation between mathematics
language and mathematics performance meta-analytic structural equation modeling
approach. Elementary School Journal.
*Correspondence concerning this article should be addressed to Xin Lin, Department of Special
Education, The University of Texas at Austin, 78712, USA. E-mail: lxjy1105@hotmail.com Or
Dr. Peng Peng, Department of Special Education, The University of Texas at Austin, 78712,
USA. E-mail: kevpp2004@hotmail.com
MATHEMATICS VOCABULARY AND MATHEMATICS 2
Abstract
In this meta-analysis of 40 studies with 55 independent samples and 7,988 participants, we
examined the relation between mathematics vocabulary (MV) and mathematics performance,
and we investigated the mechanism underlying this relation. Our findings suggest that MV was
moderately related to mathematics performance, r = .49, 95% CI [.47, .51]. Higher-order
mathematics tasks that required multistep processes demonstrated a stronger correlation with
MV than did foundational mathematics tasks, and this pattern remained stable across
development. After partialling out comprehension skills, cognitive skills, or both, the correlation
between MV and mathematics performance remained moderate and significant, rpartial = .17 ~.41,
with a trend showing a stronger relation between MV and higher-order mathematics tasks than
between MV and foundational mathematics tasks. These findings suggest the role of MV is more
than direct retrieved mathematics conceptual knowledge. MV might serve as a medium that
could facilitate cognitive reasoning in mathematics learning. Implications of these findings for
theories and practice of MV are discussed.
Keywords: mathematics vocabulary, mathematics performance, cognition,
comprehension
MATHEMATICS VOCABULARY AND MATHEMATICS 3
Understanding the Relation between Mathematics Vocabulary and Mathematics Performance:
A Meta-analysis
Academic language consists of vocabulary, grammar, and linguistic functions that
students use to acquire knowledge and perform tasks within specific academic fields such as
mathematics (Cummins, 2000). Because discipline-specific ways of using language can help
students develop a sense of how a discipline organizes knowledge (Fang, 2012), students must
master the discipline’s academic language for successful academic development (Kleemans et
al., 2018; Townsend et al., 2012). Not surprisingly, many studies have demonstrated that
academic language is not only concurrently related to academic performance (Kleemans et al.,
2018) but also a predictor of later academic performance (e.g., Purpura, Logan, et al., 2017;
Purpura, Napoli, et al., 2017; Townsend et al., 2012).
Mathematics, with its mathematics-specific language, is a particularly challenging
discipline for many students (Berch & Mazzocco, 2007). A key component of mathematics
language is mathematics vocabulary (MV, Moschkovich, 2015; Simpson & Cole, 2015), which
refers to the understanding of mathematics-specific words (e.g., divide, fraction), phrases (e.g.,
improper fraction, greater than, interior angle of a triangle), abbreviations (e.g., min for
minutes), and symbols (e.g., «, %, ×, -, -∞, √) that are routinely used in mathematics textbooks,
instruction, and assessments (Monroe & Orme, 2002; Moschkovich, 2015).
MV first emerges as number words (e.g., zero, four, and ten), words representing
quantities (e.g., more, less, and little), and words describing spatial relations (e.g., below, under,
end). As students progress in learning mathematics, MV becomes accumulatively complicated.
MATHEMATICS VOCABULARY AND MATHEMATICS 4
According to Hughes et al. (2018) and Powell et al. (2017), students need to master hundreds of
verbal mathematical terms by middle school. Students’ understanding of complex MV in later
grades is often built on their mastery of foundational MV early on. For example, students need to
master the word divisor to further understand the phrase greatest common divisor of two or more
integers, which refers to the largest positive integer that can divide each of the integers (e.g., the
greatest common divisor of 8 and 12 is 4). Moreover, as Pierce and Fontaine (2009) have stated,
for many high-stakes mathematics assessments (which feature word problems almost
exclusively), the difficulty of MV is amplified. For example, a review of the Partnership for
Assessment of Readiness for College and Careers (PARCC, 2015) Grade 5 End of Year
Released Items for Math showed that only 8 out of 35 word problems were written in
straightforward, everyday language. Not surprisingly, even though MV is important for
mathematics, mastering MV is a challenge for many students (Rubenstein & Thompson, 2002).
Given the theoretical importance and the challenge of mastering MV, it is surprising that
the relation between MV and mathematics performance remains unclear. Some studies have
indicated a small or even nonsignificant correlation (e.g., Bradley, 1987; Pina et al., 2015); some
have reported a moderate correlation, rs = .30 ~.60 (Brenner, 1981; Fuchs et al., 2018; Peng &
Lin, 2019; Vukovic, 2006); and others have reported a very high correlation, rs > .80 (e.g., Bae,
2013; McEntire, 1981). Possible explanations of the mixed findings are various types of
mathematics tasks and samples of different ages across studies. Moreover, MV may serve as
proxy measures for comprehension and cognitive skills. Without controlling for comprehension
and cognitive skills, the mechanism of the relation between MV and mathematics performance is
unclear. Thus, the current meta-analysis aims to investigate the relation between MV and
mathematics and whether these potential moderators (i.e., types of mathematics tasks and
MATHEMATICS VOCABULARY AND MATHEMATICS 5
development) and confounding variables (comprehension skills, cognitive skills) influence this
relation.
Possible Moderators
Types of Mathematics Tasks
Foundational mathematics skills, such as number knowledge, number combinations,
operations, and algorithms, are essential to complete higher-order mathematics problems (Powell
et al., 2013; Sayeski & Paulsen, 2010). Higher-order mathematics skills, such as solving word-
problems, fractions, and algebra, require multiple procedural steps and the integration of multiple
concepts and, therefore, may require more working memory and nonverbal reasoning (Peng et
al., 2020). Because MV primarily represents mathematics conceptual knowledge that is stored in
long-term memory (Schleppegrell, 2007), we hypothesize that MV should be a critical factor in
changing cognitive demands among higher-order mathematics tasks that are inherently more
cognitively demanding (Bloom, 1956; Lai, 2011). If students have a good mastery of MV, they
should be more likely to spend more cognitive resources on non-conceptual-related processes
during higher-order mathematics performance; if students have a relatively poor mastery of MV,
they should spend more cognitive resources in figuring out/understanding the conceptual
knowledge of MV, which would interfere with their performance of procedural steps (Fuchs et
al., 2015; Peng & Lin, 2019; Lin, 2020; Peng et al., 2020).
For example, during word-problem solving, when students can directly retrieve the
meaning of MV from long-term memory, they can focus on the cognitively demanding parts of
the task, such as identifying problem types and establishing an equation for solution, and
allocating more cognitive resources on the procedural part of the task, such as solving the
equation. In contrast, MV may be less important for foundational mathematics tasks that mainly
MATHEMATICS VOCABULARY AND MATHEMATICS 6
require procedural accuracy and fluency of arithmetic operations (Fuchs et al., 2015; Lin, 2020;
Peng & Lin, 2019).
Development
An iterative view suggests that MV leads to subsequent increases in mathematics
performance and vice versa (Rittle-Johnson, 2017; Rittle-Johnson et al., 2001). On the one hand,
MV can help students construct, select, and appropriately execute procedures in problem solving,
which can facilitate their mathematics performance. On the other hand, the underlying process of
implementing procedures in mathematics performance may help students develop and deepen
their understanding of MV, especially if their practice is designed to make underlying concepts
more apparent (e.g., practice equations such as 3 + 4 = 4 + 3 to better understand the term
commutative property). Therefore, we would expect that the relation between MV and
mathematics performance may increase with age.
However, such an effect of age (if it does exist) may interact with different types of
mathematics tasks. The relation between MV and higher-order mathematics tasks may increase
with age because (a) the complexity of higher-order mathematics tasks and MV increases with
age (Common Core State Standards Initiative, 2010); (b) higher-order mathematics tasks require
more cognitive resources to solve (Lin, 2020; Peng et al., 2020), and MV, as mentioned earlier,
can facilitate the allocation of cognitive resources during mathematics performance; and (c) the
increasing amount of higher-order mathematics practice may serve as an “intervention” for
mathematics conceptual knowledge that is closely related to MV development (Peng & Kievit,
2020).
In contrast, the relation between MV and foundational mathematics tasks may remain
stable or decrease with development. With development, performing foundational mathematics
MATHEMATICS VOCABULARY AND MATHEMATICS 7
tasks generally becomes more procedural, more reliant on the retrieval of mathematics facts, and
less conceptual (Geary et al., 1996; Peng et al., 2020). Moreover, a strong focus on teaching
procedural knowledge in the classroom (Cai & Nie, 2007) may reduce the role of conceptual
knowledge (MV) in solving mathematics problems (e.g., students can correctly solve "n – 13 =
25, n = __" without understanding what algebraic equation means; Peng & Lin, 2019). Indeed,
Powell et al. (2017) have demonstrated that the relation between computation and MV decreases
with age. However, this is the only empirical study that has tested age effects on the relation
between MV and foundational mathematics tasks, and it did not include higher-order
mathematics tasks. Thus, the effect of age on the relation between MV and different types of
mathematics tasks warrants further investigation.
Comprehension and Cognition: The Third Variables
In general, MV represents students’ mathematics conceptual knowledge stored in long-
term memory. According to prior studies, however, MV may serve as proxy measures for
comprehension and cognitive skills (Peng & Lin, 2019; Peng et al., 2020; Purpura, Napoli, et al.,
2017). Therefore, it is still unclear to what extent MV is involved in the active process of
comprehension and thinking in building mathematics knowledge and performing mathematics
tasks, and to what extent MV serves as retrieved knowledge during mathematics performance. In
the present study, comprehension skills are those used to make sense of text, including reading
comprehension, listening comprehension, and general vocabulary (Hoover & Gough, 1990).
Although some MV has precise mathematics-specific meanings (Rubenstein & Thompson,
2002), comprehension skills are foundational for mastering MV (e.g., Hornburg et al., 2018;
Peng & Lin, 2019) and for learning mathematics with understanding (e.g., Barwell, 2005;
Barwell et al., 2005; O’Halloran, 2005). Not surprisingly, prior research has reported that
MATHEMATICS VOCABULARY AND MATHEMATICS 8
comprehension skills (i.e., reading comprehension, listening comprehension, general vocabulary)
are related to different types of mathematics performance as well as different types of MV (e.g.,
Fuchs et al., 2018; Hornburg et al., 2018; Peng et al., 2020; Peng & Lin, 2019).
Cognitive skills, such as working memory and nonverbal reasoning, may also explain the
relation between MV and mathematics. Working memory is the capacity to simultaneously
maintain information and process an additional task (Daneman & Carpenter, 1980). Nonverbal
reasoning (fluid intelligence) consists of the set of mental processes that are used in dealing with
relatively novel tasks and in acquiring knowledge (Cattell, 1963). For example, to understand the
term the height of a parallelogram, students need to figure out the relations among the height of
a parallelogram, parallelogram (a four-sided shape formed by two pairs of parallel lines), and
height. During the process of thinking, working memory maintains the meaning of the term
parallelogram while combining it with the meaning of height. Thus, success in comprehending
and mastering MV relies on the ability to actively maintain and integrate prior learned MV in
working memory and use reasoning to make sense of the relations among the MV terms.
Moreover, any type of language processing (including processing of MV) relies on a
certain amount of working memory and reasoning capacities (Fedorenko, 2014; Gathercole &
Baddeley, 1993; Just & Carpenter, 1992; Peng et al., 2020). Thus, simply memorizing abstract
definitions of and statements about MV makes it very challenging for students to master MV or
apply MV to solving mathematics problems (e.g., Forsyth & Powell, 2017; Powell et al., 2017).
Indeed, previous research provides evidence that working memory and nonverbal reasoning
predict MV and that MV can mediate the relation between cognitive skills (e.g., working
memory and nonverbal reasoning) and mathematics performance (Fuchs et al., 2015; Fuchs et
al., 2018; Peng & Lin, 2019), especially for older children (Lin, 2020).
MATHEMATICS VOCABULARY AND MATHEMATICS 9
Thus, the investigation of whether comprehension skills, cognitive skills, or both fully or
partially explain the relation between MV and mathematics can provide direct and indirect
evidence on the nature of MV for mathematics performance. If the (size of the) relation between
MV and mathematics performance remains unchanged (or slightly decreases) after partialling out
comprehension skills, cognitive skills, or both, MV may be considered to act mainly in the
retrieval of conceptual knowledge stored in long-term memory for mathematics performance. If,
however, the relation between MV and mathematics performance greatly decreases after
partialling out comprehension skills, cognitive skills, or both, MV not only functions in the
retrieval of conceptual knowledge but also facilitates comprehension and thinking processes
during mathematics tasks.
Purpose and Research Questions
In this study, we investigate three research questions. First, what is the magnitude of the
relation between MV and mathematics in general, and what is the relation between MV and
various mathematics skills (i.e., number knowledge, mathematics computation, word-problem
solving, fractions, and algebra)? Second, is the relation between MV and mathematics
performance affected by types of mathematics tasks (foundational mathematics tasks vs. higher-
order mathematics tasks)? If so, will development change the influence of MV on different types
of mathematics tasks? Third, is MV related to mathematics performance after partialling out the
variance explained by comprehension (reading comprehension, listening comprehension, general
vocabulary), cognition (working memory and nonverbal reasoning), or both?
Method
Literature Search
MATHEMATICS VOCABULARY AND MATHEMATICS 10
We conducted a systematic search to locate studies for meta-analysis. An overview of the
search and screening procedures is presented in Figure 1. First, we searched Communication &
Mass Media Complete, Education Source, ERIC, PsycINFO, and Medline for articles published
from 1904 to May 2020. We searched titles and keywords using the following search terms:
(language OR vocabulary OR word OR term) AND (math* OR arithmetic OR "number sense"
OR numerical OR calculation OR computation OR "word problem*" OR algebra OR geometry
OR calculus OR fraction). To obtain all related studies, the search was not limited to peer-
reviewed journal articles, but instead included dissertations, conference proposals, research
reports, and technical reports. Two additional studies were identified as eligible through manual
search, which included searching references of the included studies. The first author, a doctoral
student in education, screened all studies. For the first screening, titles and abstracts were read to
locate promising studies based on inclusion criteria. This preliminary screening excluded 16
studies that were not in English, 20,235 studies that were not relevant (e.g., non-empirical
studies, single-case studies, commentaries), and 1,411 duplicate studies.
Study Selection Criteria
For the second screening, full articles of the remaining 1,114 studies were closely
reviewed for eligibility based on the following inclusion criteria: (a) studies with at least one
quantitative task measuring MV and at least one quantitative task measuring mathematics
performance; (b) studies reporting at least one correlation (r) between any measure of MV and
any measure of mathematics, or the percentage of variance in mathematics accounted for (R2) by
MV only.
On the basis of these criteria, we excluded 978 studies because they did not have an MV
measure. Within the 978 excluded studies, 14 studies used a composite score from the test for
MATHEMATICS VOCABULARY AND MATHEMATICS 11
Quantitative Concepts on the Woodcock–Johnson III Tests of Achievement (Woodcock et al.,
2001; e.g., Diamantopoulou et al., 2012), which includes subtests for both concepts and number
series. We included four studies that used only the concepts subtest (e.g., Pina et al., 2015; Pina
et al., 2014; Schmerold et al., 2017; Vukovic, 2006). Sixty-nine studies were excluded because
they did not have an objective measure of mathematics performance (i.e., teachers’ ratings of
students’ mathematics performance). Twenty-seven studies assessed both MV and mathematics
performance, but they were excluded for not reporting a direct bivariate correlation (or if their
authors did not respond to our email request for that information). Taken together, a total of
1,074 studies were excluded from the 1,114 studies in the second screening, resulting in a final
total of 40 studies: 27 peer-reviewed articles and 13 unpublished dissertations, with 55 unique
samples for our meta-analysis.
Coding
We coded studies according to participants’ characteristics, tasks used to measure MV
and mathematics performance, and study quality. Measures were coded as MV if they reported
students’ understanding of verbal (terms and phrases) and symbolic (number words, symbols,
and abbreviations) vocabulary in mathematics (see Appendix B for example MV measures from
reviewed study).
Covariates and Moderators
We also coded the covariates that were controlled for in our moderation analyses. We
were not interested in these covariates, but we controlled for them in the analysis because of their
potential impacts on the relation between MV and mathematics performance as confounders
(Forsyth & Powell, 2017; Lin et al., 2019; Namkung et al., 2019; Peng et al., 2020; Peng et al.,
2019). These covariates included publication type (peer-reviewed vs. others), sample status
MATHEMATICS VOCABULARY AND MATHEMATICS 12
(typically developing vs. atypically developing), and family SES (Harwell & LeBeau, 2010;
Peng et al., 2019; Peng et al., 2020; below-middle SES or middle-or-above SES, based on direct
report of SES level or indirect relevant SES information including parental education level and
free-reduced lunch rate. Compared with the typically developing group, the atypically
developing group is quite heterogeneous, including different developmental or acquired
disorders such as learning disabilities and autism.
Coded moderators included types of mathematics tasks (foundational mathematics tasks
[basic mathematics skills, such as accuracy and fluency of arithmetic operations] vs. higher-order
mathematics tasks [more complex problem solving, such as word problems, fractions, and
algebra]) and the interaction term of types of mathematics tasks and age. We also coded general
vocabulary, reading comprehension, listening comprehension, working memory, and nonverbal
reasoning (we coded and included only nonverbal reasoning tasks, to control for language
effects), if provided by the included studies. (See Appendix A for definitions of involved skills
and examples of tasks.) In addition, we coded the number of participants for each correlation to
weight each effect size, so that correlations obtained from larger samples were given more
weight in the analysis than were those obtained from smaller samples.
Study Quality
To rate the quality of each study, we adapted Thompson et al.’s (2005) quality indicator
for correlational research. Scale items represented five quality indicators: (a) effect sizes either
reported or calculable for each outcome relevant to this review, including nonsignificant effect
sizes; (b) appropriate interpretation of structure coefficients; (c) use of multivariate methods in
the presence of multiple outcome variables; (d) use of the highest available scale of data (interval
data were not converted to nominal scales) unless such choices were justified and thoughtfully
MATHEMATICS VOCABULARY AND MATHEMATICS 13
considered; (e) presentation of evidence that statistical assumptions were sufficiently met (e.g.,
homogeneity of variance, normal distribution, measures of central tendency). Based on Nelson
and McMaster (2019), each study’s score could range from 0 (study addressed no study quality
indicators) to 5 (study addressed all study quality indicators); we then categorized studies’
quality as high (4 or 5 points), medium (2 or 3 points), or low (0 or 1 point).
All studies reported effect sizes relevant to this meta-analysis and used the highest
available scale of data (n = 40). Most studies used multivariate methods in the presence of
multiple outcome variables (n = 39), presented evidence that statistical assumptions were
sufficiently met (n = 36), and interpreted the weights of the general linear model appropriately (n
= 35). The average score on the scale of 0 to 5 for studies was 4.75 (SD = 0.43, range = 4 to 5).
Ten studies received a score of 4; and 30 studies received a score of 5 (high quality). Because all
studies were categorized as high quality (receiving 4 or 5 points), we did not further investigate
whether study quality moderated the relation between MV and mathematics performance.
Interrater Agreement
Before coding the included articles, the first author trained the second coder, a doctoral
student with a master's degree in education, to use the coding rubric with practice articles;
disagreements were resolved through discussion. The second coder was required to reach 90%
agreement on the practice articles, which themselves did not meet the inclusion criteria for this
study. Then, using the coding system, the first author and the second coder independently coded
all of the included studies. We calculated interrater agreement as follows: [agreements ÷
(agreements + disagreements) × 100]. The interrater agreement between two coders was .89 to
1.00 for all variables of interest, with an average of .94 (SD = 0.035). Disagreements were
resolved through discussion of the original articles.
MATHEMATICS VOCABULARY AND MATHEMATICS 14
Analytic Strategies
The effect size index used for all outcome measures was Pearson's r, the correlation
between MV and mathematics performance. Analyses were based on Borenstein et al.’s (2005)
recommendations. We converted the correlation coefficients to Fisher's z scale, and all analyses
were performed using transformed values. The results, such as the summary effect and its
confidence interval, were then converted back to correlation coefficients for presentation. Also,
because it was hypothesized that this body of research would report a distribution of correlation
coefficients with significant between-studies variance, as opposed to a group of studies with a
single estimated true correlation, a random-effects model was appropriate for the present study
(Lipsey &Wilson, 2001). To summarize correlation coefficients and to examine potential
moderators, we ran weighted, random-effects meta-regression models using Hedges et al.’s
(2010) corrections with “robumeta” package in R (Fisher & Tipton, 2015).
To examine the influence of comprehension (reading comprehension, listening
comprehension, general vocabulary) and cognition (working memory and reasoning) on the
relation between MV and mathematics performance, we calculated the correlations between MV
and mathematics performance, partialling out comprehension, cognition, and both. Specifically,
we used the first-order partial correlation formula in Equation 1 to partial out comprehension and
cognitive skills separately; we used the second-order partial correlation formula in Equation 2 to
control for comprehension and cognitive skills simultaneously.
Equation 1:
Equation 2:
MATHEMATICS VOCABULARY AND MATHEMATICS 15
The partial correlation was based on the correlation matrices retrieved from the original
studies (e.g., if a study provided two measures of mathematics performance, two measures of
MV, and two measures of working memory, we calculated six correlations between MV and
mathematics performance, partialling out working memory). We then synthesized these partial
correlations to indicate whether the relation between MV and mathematics performance was
influenced by comprehension, cognition, or both. Because the way in which we calculated partial
correlations produces many effect sizes nested within a sample, we accounted for statistical
dependencies using the random effects robust standard error estimation technique developed by
Hedges et al. (2010), as discussed earlier. Also, we calculated the proportion of variance in the
relation between MV and mathematics performance that is explained by comprehension and
cognition, using the formula 1- (rpartial/r) (Preacher & Kelly, 2011).
Results
We present the results in three subsections. First, we report general characteristics of the
studies included in the present review. Second, to address the first research question, we present
the magnitude of the relation between MV and overall mathematics, and the relation between
MV and various mathematics skills (number knowledge, mathematics computation, word
problems, fractions, algebra). Third, to address our second research question, we present the
testing results for the effects of types of mathematics skills, age, and the interaction between age
and types of mathematics skills. Finally, we report the partial correlation results to answer our
third research question.
General Characteristics of the Included Studies
MATHEMATICS VOCABULARY AND MATHEMATICS 16
The reviewed studies included 7,988 individual students, with sample sizes ranging from
15 to 1,258 (M = 145.24, SD = 167.99; median = 104). Five studies involved atypically
developing students—four studies with students who had learning difficulties, and one with
students who had autism. The total number of students with learning difficulties was 684, and the
number of students with autism was 20. Across samples, the students’ age ranged from 4.18 to
20.00 years. The age of students in most of the samples was between 4.78 and 11.33 years, and
the median age was 7.50 years. Countries of data collection included the United States (n = 31),
Canada (n = 3), China (n = 2), Spain (n = 2), the Netherlands (n = 1), Singapore (n = 1), and
Taiwan (n = 1); one study included samples from both the United States and China (Kung et al.,
2019).
The Relation Between Mathematics Vocabulary and Mathematics Performance
Table 1 shows the relation between MV and overall mathematics and the relation
between MV and various mathematics skills, for each subcategory of moderators and covariates.
Overall, the relation between MV and mathematics performance was r = .49, 95% CI [.47, .51].
The average correlation between MV and different mathematics skills was moderate and
significant, rs = .31~.58, ps < .05.
Regarding each subcategory of moderators and covariates, the average correlations
between MV and mathematics performance were significant for both the peer-reviewed articles,
r = .53, 95% CI [.51, .55], and the non-peer-reviewed articles, r = .38, 95% CI [.34, .41]. For
sample status, the average correlations for typically developing and atypically developing
participants were all significant. The average correlation between MV and mathematics for
typically developing participants was r = .49, 95% CI [.47, .51]; for atypically developing
participants, it was r = .54, 95% CI [.48, .59]. The average correlation between MV and
MATHEMATICS VOCABULARY AND MATHEMATICS 17
mathematics for participants from middle-or-above SES was r = .49, 95% CI [.47, .52]; for
participants from below-middle SES, it was r = .48, 95% CI [.46, .51]. The average correlation
between MV and foundational mathematics tasks was r = .48, 95% CI [.45, .51]; for higher-order
mathematics tasks, it was r = .50, 95% CI [.48, .52].
Moderation Effects
We ran the moderation analyses by putting all moderators of interest and covariates in a
meta-regression model simultaneously. As Table 2 shows, after controlling for covariates and
other moderators, the relation between higher-order mathematics tasks and MV was significantly
larger than the relation between foundational mathematics tasks and MV, ß = .25, p = .04. After
controlling for covariates and other moderators, age did not significantly affect the relations
between MV and different types of mathematics tasks (foundational mathematics tasks vs.
higher-order mathematics tasks), ß = -.02, p = .19. Among those covariates, SES was the only
significant factor explaining the relation between MV and mathematics, such that the relation
was significantly larger among participants from middle-or-above SES than among participants
from below-middle SES backgrounds, ß = .14, p = .02.
The Role of Comprehension and Cognitive Skills
Controlling for Comprehension
There were 20 studies with 169 effect sizes on the relation between MV and mathematics
performance, partialling out comprehension skills. Controlling for comprehension skills, the
correlation between MV and mathematics performance was r = .37, 95% CI [.35, .40]. That is,
comprehension skills explained 24% of the variance in the relation between MV and
mathematics performance at the effect size level.
Controlling for Cognition
MATHEMATICS VOCABULARY AND MATHEMATICS 18
There were 16 studies with 147 effect sizes on the correlations between MV and
mathematics performance, partialling out cognitive skills (nonverbal reasoning and working
memory). After controlling for cognitive skills, the correlation between MV and mathematics
performance was r = .41, 95% CI [.38, .44]. That is, cognitive skills explained 16% of the
variance in the relation between MV and mathematics performance at the effect size level.
Controlling for Comprehension and Cognition
There were 7 studies with 85 correlations between MV and mathematics performance,
partialling out comprehension (reading comprehension, listening comprehension, general
vocabulary) and cognition (nonverbal reasoning, working memory). After controlling for
comprehension skills and cognitive skills, the correlation between MV and mathematics
performance was r = .17, 95% CI [.15, .19]. That is, comprehension and cognitive skills
explained 65% of the variance in the relation between MV and mathematics performance at the
effect size level.
Moderation Analyses based on Partial Correlations
Next, we ran the moderation analyses for partial relations, controlling for comprehension,
cognitive skills, and both (see Table 3). Because there was no effect size for atypically
developing students after controlling for comprehension and cognitive skills, we did not control
sample status in the moderation analysis. After controlling for covariates, we did not find
significant moderators, which may be attributable to underpowered analyses with a relatively
small number of effect sizes. However, there was a clear trend showing that the relation between
higher-order mathematics tasks and MV was larger than the relation between foundational
mathematics tasks and MV after controlling for comprehension, cognitive skills, and both, with p
= .06, .08, and .06, respectively.
MATHEMATICS VOCABULARY AND MATHEMATICS 19
Discussion
The present meta-analysis suggests that MV was moderately related to mathematics.
Moderation and partial correlation analyses provided further insight into the nature of MV for
mathematics performance. In the following sections, we first discuss these findings in detail and
then discuss implications, limitations, and future directions.
The Relationship Between Mathematics and Mathematics Vocabulary
This study’s results indicate that overall, there was a positive, significant relation
(r = .49) between MV and mathematics performance. Word-problem solving showed the
strongest relation with MV (r = .58), whereas fractions, algebra, and number knowledge showed
relatively weaker relations with MV (r = .31, .44, .47, respectively).
Within foundational mathematics tasks, number knowledge demonstrated a relatively
stronger relation with MV than expected (r = .47). One explanation is that early mathematics
learning occurs very much through learning MV (Purpura, Logan, et al., 2017). Specifically, MV
first emerges as number terms, terms representing quantities, and terms representing spatial
relations, the understanding of which comprises the determinant factors for solving early
numeric problems. In other words, children need to master MV terms (i.e., more, few, before)
before being able to manipulate numbers in different ways (Gelman & Butterworth, 2005).
In contrast, the relations between MV and some higher-order mathematics skills such as
fractions and algebra were not as strong as expected, which may have been due to the
comprehensiveness of the MV measure. Specifically, one commonality among the four studies
involving fractions and algebra was that they included only content-specific MV (fraction terms
only, Bradley, 1987; algebra terms only, Kellar, 1939; Miles, 1999; Stegall, 2013), rather than a
comprehensive set of MV terms (MV terms selected from a textbook; Peng & Lin, 2019; Powell
MATHEMATICS VOCABULARY AND MATHEMATICS 20
et al., 2017). Given that students’ fraction and algebra performance is also based on their whole
number understanding and is likely to be embedded in a problem-solving format, simply
measuring fraction MV terms may not fully reflect the relationship between MV and higher-
order mathematics skills. Another possible explanation is that algebra and fractions were also
often measured in a procedural format (e.g., doing computation problems), which may tap
conceptual understanding to a lesser extent, as has been indicated by previous research (e.g., Cai
& Nie, 2007; Peng & Lin, 2019).
Types of Mathematics Tasks, Development, and Socioeconomic Status
The present study is the first meta-analysis to directly investigate whether MV is more
critical for certain types of mathematics skills across development and to do so after partialling
out comprehension and cognitive skills. In line with our hypothesis, we found that that the
relation between MV and higher-order mathematics is stronger than the relation between MV
and foundational mathematics. This finding is in line with Peng and Lin (2019), who compared
the relative importance of MV with mathematics computation and word-problem solving among
fourth-grade Chinese students. Their findings suggested that MV made a unique contribution to
word problems but not to computation.
Moreover, we examined whether age would interact with types of mathematics skills.
Contrary to our hypothesis, findings suggest that age did not significantly influence the relation
between MV and different types of mathematics. With respect to higher-order mathematics tasks,
a plausible explanation is that the importance of MV remains consistent because the cognitive
demand of higher-order mathematics tasks does not necessarily increase with age (Lin, 2020),
partly because both conceptual and procedural mathematics knowledge accumulate in long-term
memory with development. Students rely on direct retrieval as well as cognitive resources
MATHEMATICS VOCABULARY AND MATHEMATICS 21
simultaneously to solve increasingly complicated mathematics tasks (Geary et al., 2017; Lin,
2020; Peng et al., 2020). Regarding foundational mathematics tasks, the present findings are
somewhat in line with our hypothesis that the relation between MV and foundational
mathematics tasks remains stable with age.
Powell et al. (2017) suggested that the relation between foundational mathematics tasks
and MV decreases with age. However, Powell et al. (2017) used the same mathematics measures
(computation) for both third-graders and fifth-graders. As students accumulate mathematics
procedural knowledge with repeated practice, they may succeed in the same mathematics tasks
with less involvement of MV. That is, using the same mathematics measure for younger and
older students may cloud the relationship between MV and mathematics performance, because
that relationship may change with development. For example, Lin (2020) demonstrated that even
for the same mathematics skill (not assessed by the same measure), the involvement of MV and
cognitive skills vary with development. Thus, future empirical studies are needed to further
investigate whether the relation between foundational mathematics tasks and MV remains stable
or decreases with age, based on grade-specific or comprehensive MV measurement.
Although we did not focus on SES as a moderator, we did find that SES was the only
significant covariate. That is, the relation between MV and mathematics was weaker among
participants from below-middle SES backgrounds than among participants from middle-or-above
SES backgrounds. This finding implies that individuals from below-SES backgrounds may not
only be at risk for experiencing difficulties in mathematics and language processing (Fernald et
al., 2013; Rowe & Goldin-Meadow, 2009), but also may be less likely to build and enhance the
connection between MV and mathematics than their peers from middle-or-above SES.
Partialling Out Comprehension and Cognition
MATHEMATICS VOCABULARY AND MATHEMATICS 22
Findings from the partial correlations between MV and mathematics performance,
controlling for comprehension, cognitive skills, or both, further demonstrated that MV does not
simply serve as retrieved conceptual knowledge for mathematics problem-solving, but is also
actively involved in mathematics comprehension and thinking processes. In general, this finding
is generally in line with the Developmental Function Hypothesis of Language for Mathematics
proposed by Peng et al (2020), who have suggested that individuals may use language as a
medium to retrieve mathematics knowledge to perform mathematics tasks as well as facilitate
working memory and reasoning during mathematics learning. Our findings further contribute to
this framework (for the relation between general language and mathematics) by indicating that
the fluent use of MV for foundational mathematics tasks may save more cognitive resources for
the use of MV in higher-order mathematics learning/performance. Such use of mathematics
language boosts the mediating role of mathematics language in the mutual relations between
cognition and mathematics with development (Peng & Kievit, 2020; Peng et al,. 2020)
Additionally, this finding is in line with the direct and inferential mediation (DIME)
model of reading comprehension (Cromley & Azevedo, 2007; Cromley et al., 2010), which
suggests that vocabulary may bear direct and indirect roles in reading comprehension.
Analogically, the direct impact of MV on mathematics performance is just like the retrieval of
background knowledge through vocabulary for reading comprehension. Such a direct impact of
MV on mathematics is also in line with studies in which rapid naming (the ability to retrieve
knowledge from long-term memory) has partly explained the relation between MV and
mathematics performance (Purpura, Logan, et al., 2017), and rapid naming underlies the
development of both mathematics and reading (Georgiou et al., 2013). Thus, we think the
remaining unique variance shared between MV and mathematics performance after partialling
MATHEMATICS VOCABULARY AND MATHEMATICS 23
out comprehension, cognition, or both highlights MV as retrieved conceptual knowledge stored
in long-term memory during mathematics performance (e.g., “+” means “add”).
MV may also have an indirect effect (via comprehension and cognition) on mathematics
performance, analogous to vocabulary’s indirect effect on reading comprehension through
inference in the DIME model. First, students who use mental resources to figure out the meaning
of unfamiliar MV might be at a disadvantage in solving mathematics problems, compared with
those who already know the meaning of the MV and can allocate mental resources to other
procedures (i.e., inference making in word-problem solving). Second, the retrieval of conceptual
knowledge through MV may also serve as a medium for comprehension and cognitive processes
during mathematics performance (e.g., comprehension and reasoning is partly anchored in the
understanding of MV and the integration of MV and procedural knowledge). Together with
previous findings that MV can mediate the relation between comprehension/cognitive skills and
mathematics performance (e.g., Fuchs et al., 2018; Lin, 2020; Peng & Lin, 2019), our findings
suggest that MV, although a relatively independent construct of comprehension and cognition,
may actively interact with comprehension and cognition during mathematics performance.
Limitations
Our findings should be interpreted with several limitations in mind. First, we did not
further investigate whether different categories of MV would moderate the relation between MV
and mathematics performance, owing to the limited information provided by the reviewed
studies. Prior research has demonstrated that the contributions of MV to mathematics differ not
only by types of mathematics skills but also by types of MV. For example, Peng and Lin (2019)
have put forward a domain-specific versus domain-general hypothesis of MV, such that MV not
only has a domain-general nature (e.g., involving general language and cognition) but also has a
MATHEMATICS VOCABULARY AND MATHEMATICS 24
domain-specific nature such that MV in a specific domain (e.g., measurement and geometry, in
their study) has direct effects on one mathematics skill (e.g., word-problem solving), not another
(e.g., computation). Future studies should look further into such a domain-specific hypothesis
regarding MV, especially in later grades, to better understand the construct of MV.
Second, our results are correlational, and most of the research that we have reviewed here
assessed MV and mathematics performance concurrently. Therefore, readers should not interpret
our findings causally or assume directionality in relations. More longitudinal/experimental
studies are needed to investigate the relation/bidirectionality between MV and mathematics
further.
Third, we lacked the power to investigate possible heterogeneity among the reviewed
studies. For example, we included heterogeneous samples (typically and atypically developing
individuals). Although we controlled for sample status, we could not conduct further analyses for
a specific atypically developing group (e.g., autism spectrum disorder, learning disability), due to
the limited number of effect sizes. Also, foundational and higher-order mathematics tasks are
quite heterogeneous (i.e., foundational mathematics skills involves verbal counting, set
comparison, mathematics facts retrieval, standardized computation), and we could not further
disentangle those subcategories within these tasks, again because of the limited number of effect
sizes.
Last, although we searched for grey literature and implemented double coding of all
reviewed studies to control for bias, only the first author did the initial screening of the studies.
We did not double-screen 100% of articles as recommended by Talbott et al. (2018), which
might also have led to some bias or caused some missed studies.
Implications
MATHEMATICS VOCABULARY AND MATHEMATICS 25
Despite the aforementioned limitations, the present study’s findings have several
important implications for practice. First, MV instruction may be essential for mathematics
instruction. The finding that MV is not just a proxy of comprehension or cognition suggests the
importance of explicit MV instruction (in addition to general language/reading instruction) for
mathematics instruction. MV demonstrated stronger relations with higher-order mathematics
tasks, and such relations remain stable with development. Thus, MV instruction should be
constant and especially important in mathematics development in later grades when higher-order
mathematics tasks are emphasized (Lin, 2020).
Second, given that comprehension and cognitive skills are heavily involved in MV
learning and that students with learning difficulties in mathematics often demonstrate difficulties
in cognition and comprehension (Fuchs et al., 2008; Lin et al., 2019; Peng et al., 2018), the
present study provides support for using explicit routines in MV instruction to reduce
comprehension and cognitive load in performing mathematics tasks (Archer & Hughes, 2010;
Bay-Williams & Livers, 2009; Monroe & Orme, 2002). Explicit vocabulary instruction consists
of four steps: (a) introduce the word; (b) introduce the meaning of the word by providing a
student-friendly explanation; (c) illustrate with examples; (d) check students’ understanding.
Illustrating an MV term requires the use of a number of concrete, visual, verbal examples, with
the inclusion of all critical attributes in the examples. Instructors can also check students’
understanding by having students distinguish between examples and non-examples and explain
why. Students should demonstrate a complete understanding of the MV term while being asked
to explain why.
It is important to adopt approaches to reduce cognitive load during MV instruction, such
as by using visuals, mnemonic strategies (Riccomini et al., 2015), and the concreteness-fading
MATHEMATICS VOCABULARY AND MATHEMATICS 26
paradigm (Fyfe et al., 2014). Using mnemonic strategies and visuals could help students
remember the meaning of the MV (Rubenstein & Thompson, 2002). Also, given that using
mnemonic strategies could potentially help students remember MV meanings better, the direct
retrieval of MV meanings can possibly free up cognitive resources for other problem-solving
procedures.
Last, the weaker relation between MV and mathematics among children from below-
middle SES suggests that MV instruction may be especially crucial for these students, who may
be less likely to build and enhance the connection between MV and mathematics than their peers
from middle-or-above SES. For students with below-middle SES, embedding MV instruction in
different tiers of instruction within a response-to-intervention framework can maximally increase
their exposure to MV knowledge and boost their mastery and application of MV in mathematics
learning.
Directions for Future Research
Future research could differentiate different types of mathematics skills (e.g., whole-
number computation vs. fraction computation) or the comprehensiveness of MV measures
(content-specific/grade-specific vs. comprehensive) while examining the relation between MV
and mathematics. More longitudinal and experimental research is also needed because most
empirical studies summarized in the present study were cross-sectional and thus do not allow
strong causal inferences. Specifically, given the present discussion of MV, future experimental
research is needed to further validate the effects of MV on overall mathematical thinking and
learning processes, and the direct effect of MV on solving specific mathematics problems.
Conclusion
MATHEMATICS VOCABULARY AND MATHEMATICS 27
In sum, we have investigated the relation (and the mechanisms of this relation) between
MV and mathematics. Our findings provide new and updated information on MV as follows: (a)
MV is significantly and moderately related to mathematics, and this relation is stronger for
higher-order mathematics tasks than for foundational mathematics tasks. (b) Comprehension and
cognitive skills can each explain up to 24% of the variance in the MV–mathematics relation, and
together they can explain over 65% of the variance in this relation. (c) MV is more than essential
retrieved conceptual knowledge for solving mathematics problems; it also serves as an active
component in comprehending mathematics problems and thinking during the performance of
mathematics tasks. (d) MV is consistently important for mathematics performance across
preschool and post-secondary school, even after partialling out comprehension and cognitive
skills.
MATHEMATICS VOCABULARY AND MATHEMATICS 28
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