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Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies.

Journal of Educational Psychology
Improving Fraction Understanding in Sixth Graders With
Mathematics Difficulties: Effects of a Number Line
Approach Combined With Cognitive Learning Strategies
Christina A. Barbieri, Jessica Rodrigues, Nancy Dyson, and Nancy C. Jordan
Online First Publication, June 20, 2019.
Barbieri, C. A., Rodrigues, J., Dyson, N., & Jordan, N. C. (2019, June 20). Improving Fraction
Understanding in Sixth Graders With Mathematics Difficulties: Effects of a Number Line Approach
Combined With Cognitive Learning Strategies. Journal of Educational Psychology. Advance
online publication.
Improving Fraction Understanding in Sixth Graders With Mathematics
Difficulties: Effects of a Number Line Approach Combined With Cognitive
Learning Strategies
Christina A. Barbieri, Jessica Rodrigues, Nancy Dyson, and Nancy C. Jordan
University of Delaware
The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention
was centered on the number line and incorporated key principles from the science of learning. Sixth
graders (N51) who struggled with fraction concepts were randomly assigned at the student level to
the experimental intervention (n28) or to a business-as-usual control who received their school’s
intervention (n23). The experimental intervention occurred over 6 weeks (27 lessons). Fraction
number line estimation, magnitude comparisons, concepts, and arithmetic were assessed at pretest,
posttest, and delayed posttest. The experimental group demonstrated significantly more learning than the
control group from pretest to posttest, with meaningful effect sizes on measures of fraction concepts (g
1.09), number line estimation as measured by percent absolute error (g.85), and magnitude
comparisons (g.82). These improvements held at delayed posttest 7 weeks later. Exploratory analyses
showed a significant interaction between classroom attentive behavior and intervention group on fraction
concepts at posttest, suggesting a buffering effect of the experimental intervention on the normally
negative impact of low attentive behavior on learning. A number line– centered approach to teaching
fractions that also incorporates research-based learning strategies helps struggling learners to make
durable gains in their conceptual understanding of fractions.
Educational Impact and Implications Statement
A mathematics intervention that used a number line– centered approach and validated learning
principles to teach fraction concepts helped struggling sixth graders improve their fraction under-
standing. After participating in the intervention, students performed better on assessments of fraction
concepts, number line estimation, and magnitude comparisons than a group of students who received
their school’s regular intervention, and these improvements held seven weeks later. Findings suggest
that students who are struggling with fractions, even after receiving several years of formal fraction
instruction in school, can still make large gains in their understanding, preparing them for more
advanced mathematics and for success in STEM related fields.
Keywords: fraction, fraction magnitude, intervention, mathematics difficulties, number line
Supplemental materials:
A strong foundation in fractions helps students succeed in
mathematics (National Mathematics Advisory Panel, 2008). More
specifically, fraction magnitude understanding supports algebra
proficiency (Booth & Newton, 2012). Because Algebra is a req-
uisite skill for enrollment in advanced level mathematics courses
as well as many science courses, algebra proficiency leads to a
higher rate of college admittance, and eventual pursuit of careers
in STEM disciplines (Chen, 2009; Matthews & Farmer, 2008;
Schneider, Swanson, & Riegle-Crumb, 1998). Moreover, fraction
knowledge is needed for many non-STEM jobs (Handel, 2016) as
well as for everyday activities, such as managing money, cooking,
and doing home repairs.
Christina A. Barbieri, Jessica Rodrigues, Nancy Dyson, and Nancy C.
Jordan, College of Education and Human Development, University of
Jessica Rodrigues is now at College of Education, University of Mis-
The authors thank instructors Heather Suchanec Cooper, Kristiana Rios,
and Luke F. Rinne for assisting in administration of the intervention.
Funding for this research was provided by the Institute of Education
Sciences, U.S. Department of Education, Grants R324A160127 and
R305A100150. IRB Project Name: Developing a Fraction Sense Interven-
tion; IRBnet ID: 931086.
Correspondence concerning this article should be addressed to Christina
A. Barbieri, College of Education and Human Development, University of
Delaware, 113 Willard Hall Education Building, Newark, DE 19716.
This document is copyrighted by the American Psychological Association or one of its allied publishers.
This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.
Journal of Educational Psychology
© 2019 American Psychological Association 2019, Vol. 1, No. 999, 000
Unfortunately, many students enter sixth grade with a tenuous
grasp of fractions, even after several years of instruction on the
topic (Resnick et al., 2016). Students who enter seventh grade
without foundational knowledge of fractions face cascading math-
ematics difficulties (Mazzocco & Devlin, 2008). To address this
problem, we developed and evaluated an experimental intervention
for entering sixth graders who exhibit low knowledge of fraction
concepts. Our experimental intervention is informed by current
research on fraction learning; specifically, the importance of un-
derstanding fractions as magnitudes that can be represented on the
number line. Numerical magnitude knowledge uniquely predicts a
range of mathematical competencies (see Schneider, Thompson, &
Rittle-Johnson, 2018 for a review). Moreover, deficits in under-
standing of symbolic numerical magnitudes characterize students
with mathematics difficulties and disabilities (e.g., Butterworth,
2005; Butterworth & Reigosa-Crespo, 2007). Fraction magnitude
knowledge is especially predictive of mathematics proficiency,
beyond whole number skills and general cognitive competencies
(Resnick et al., 2016). The number line is an effective but often
underused tool for developing fraction magnitude knowledge in
struggling students (Dyson, Jordan, Rodrigues, Barbieri, & Rinne,
2018; Fuchs et al., 2014; Gersten, Schumacher, & Jordan, 2017;
Saxe, Diakow, & Gearhart, 2013). In addition to centering instruc-
tion on the number line, our approach aims to bolster students’
fraction skills through application of research-based learning prin-
ciples from cognitive science (Brown, Roediger, & McDaniel,
2014; Rittle-Johnson & Jordan, 2016).
Transitioning From Whole Numbers to Fractions
When learning fractions, students gradually expand their under-
standing of numbers; they take into account differences, as well as
similarities, between fractions and whole numbers. Whole num-
bers are represented linearly with each number being exactly one
more than the previous number and only one number represents
each magnitude. Fractions, on the other hand, can be represented
in countless ways (e.g., 1/5, 2/10, and so on). There are infinite
fractional parts between integers, and fractions can be less than,
equal to, or more than one (Resnick et al., 2016). When determin-
ing the magnitude of a fraction, the size of the numerator or
denominator cannot be considered in isolation as in separate whole
numbers. Larger numbers in the fraction do not always signify
larger magnitudes (e.g., 1/4 1/2). With fractions less than one,
multiplication does not lead to a product greater than a factor and
division does not lead to a quotient smaller than a dividend.
Students often incorrectly apply whole number logic to fractions
(DeWolf & Vosniadou, 2015; Siegler, Thompson, & Schneider,
2011; Vamvakoussi & Vosniadou, 2010), and the problem seems
to be especially pervasive in low-achieving students (Malone &
Fuchs, 2017). In a study of fraction arithmetic errors, it was found
that low-achieving students focused on the size of the fractional
parts rather than the relations among the parts (Malone & Fuchs,
2017). Importantly, these students’ errors reflected poor magnitude
understanding rather than difficulties with part–whole knowledge
more generally.
Reasoning About Fractions on the Number Line
Numerical magnitude reasoning is reflected by students’ ability
to estimate magnitudes on the number line. Understanding that all
real numbers are represented as magnitudes on a number line
provides a unifying framework for number learning (e.g., Siegler
et al., 2011). For example, a fraction of 1/19 is very close to 0
relative to 6/7 which is closer to 1, and 5/4 is greater than 1.
Students with stronger whole number estimation skills in third
grade are more likely to perform better on fraction concepts and
procedures measures in fourth and sixth grades (Bailey et al.,
2015; Fuchs et al., 2013).
Fraction magnitude knowledge predicts both broad and more
specific mathematics outcomes, over and above general cognitive
abilities and whole number skills. A longitudinal study found that
growth in fraction number line estimation (FNLE) acuity between
fourth and sixth grades predicts mathematics achievement at the
end of sixth grade, even when controlling for a constellation of
domain general and domain specific abilities (Resnick et al.,
2016). A troubling finding was that a significant number of stu-
dents showed little to no growth in fraction number line estimation
accuracy between fourth and sixth grade, even though they had
received three years of fraction instruction in school. Fraction
arithmetic is typically introduced in fourth grade with addition and
subtraction and remains the primary focus through sixth grade
(National Governors Association Center for Best Practices &
Council of Chief State School Officers, 2010). At this point,
formal fraction arithmetic instruction typically comes to a close
with division of fractions and students move on to prealgebra
instruction. FNLE acuity specifically predicts prealgebra skills
(Booth & Newton, 2012), a key element of sixth-grade achieve-
ment (Booth, Newton, & Twiss-Garrity, 2014). One proposed
explanation is that understanding rules that govern the relationship
between the numerator and denominator can later be translated to
reasoning about algebraic equations (e.g., “fractional representa-
tions of 1/3 will fit into the equation, x/y1/3, which is equivalent
to the equation y3x,” Empson & Levi, 2011, p. 134). Further,
fraction magnitude knowledge appears to fully mediate the relation
between early whole number magnitude knowledge and later frac-
tion arithmetic (Bailey, Siegler, & Geary, 2014). That is, children
who are highly accurate in their number line estimations of whole
numbers in first grade tend to have highly accurate number line
estimations of fractions in middle school which then predicts
higher fraction arithmetic skills.
Number lines are a visual, mathematically correct way to rep-
resent complex fraction concepts. The importance of the number
line has recently been noted in the Common Core State Standards
for Mathematics (CCSS-M) with students being asked as early as
Grade 3 to understand a fraction as a number on a number line
(National Governors Association Center for Best Practices &
Council of Chief State School Officers [NGACBP & CCSSO],
2010). Unfortunately, many interventions used in schools still
emphasize part–whole models for struggling learners, often to the
exclusion of number line models (Jordan, Resnick, Rodrigues,
Hansen, & Dyson, 2017). Part–whole models represent fractional
values as a shaded region of a whole or a subset of a group of
objects. For example, a popular part–whole model used when
teaching fractions is in the form of a pizza relating slices to
fractional values. These models are concrete representations that
build on intuitive understandings (Mix, Levine, & Huttenlocher,
1999). However, overemphasis of this approach may lead to a
limited way of thinking about fractions that does not encourage an
understanding of fractions as numbers with their own numerical
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magnitude (Gersten et al., 2017). In other words, fractions are
numerical values determined by the relationship between the nu-
merator and denominator and not just “parts of a whole.” Although
number lines can be thought of more broadly as part–whole
models when partitioning, the number line more naturally lends
itself to a continuous and unified representation of numerical
fraction magnitude than traditional part–whole models. The num-
ber line representation is more aligned with the understanding that
the numerator and denominator work together to determine the
value of the fraction and cannot be separated, as opposed to other
tools used to find a discrete part of a whole (e.g., pizza slices).
Hamdan and Gunderson (2017) found that even second and third
graders trained on a fraction number line estimation task demon-
strated greater transfer of knowledge to a novel fraction compar-
ison task than those who were trained using an area model.
Previous work with fourth graders deemed at risk for mathematics
difficulties showed that the number line is an effective tool for
building fraction concepts and skills (Fuchs et al., 2013, 2014),
although the extent to which the findings hold over time was not
Principles From the Science of Learning
General learning principles should guide any mathematical in-
tervention. Findings from the science of learning have yielded
many such principles that instructional designers can use in learn-
ing environments (Booth et al., 2017). Many factors related to
instructional technique, dosage, and timing can be combined in an
unlimited number of ways to yield varying effects (Koedinger,
Booth, & Klahr, 2013). In the present study, we chose to employ
several major principles that improve understanding and retention
for students with learning difficulties and disabilities. These in-
clude use of explicit instruction, representational gestures, dual
coding (i.e., words and visuals) for presenting information, spaced
and interleaved practice, and systematic feedback. The following
is a description of the learning principles that we capitalize on to
increase the effectiveness of our experimental intervention.
Explicit Instruction
Hanover Research (2014) conducted a curricular review of
seven mathematics intervention programs that have research sup-
port in an attempt to narrow in on key components of effective
mathematics interventions. The one common instructional compo-
nent of these interventions was explicit and systematic instruc-
tional methods. In a meta-analysis on mathematics instruction,
Gersten and colleagues (2009) also found that explicit instruction
was a highly effective instructional technique for students with
mathematics difficulties. Explicit instruction encompasses a wide
array of instructional approaches but, as explained by Gersten and
colleagues, involves teachers demonstrating a step-by-step strategy
to solve a specific type of problem and encouraging students to use
this particular strategy in their own work. To be effective, explicit
instruction must include clear teacher models of problem-solving,
opportunities for guided practice, and regular feedback (Doabler &
Fien, 2013). Bryant and colleagues (2008) found that an interven-
tion using explicit instruction focused on number, operations, and
quantitative reasoning improved mathematics achievement. The
instructor in Bryant and colleagues’ intervention explained and
modeled the steps needed to solve problems before asking students
to complete problems on their own. For example, when teaching
addition and subtraction involving doubles, the instructor said, “I
have 12 connected cubes. I break them into two equal parts. Count
with me how many in each (6)” as she made two rows of six cubes.
The instructor then said, “This is a double fact: 6 612.” After
modeling the strategy on the board, students were then given the
same materials and asked to model a similar double fact (e.g., 4
48) while the instructor continued to ask guiding questions and
was available for further support.
Representational Gestures and Physical Movements
When people speak, they naturally gesture or move their hands
to help convey an idea to listeners. Representational gestures
depict a spatial object, event, or abstract concept (Goldin-Meadow,
2011). For example, a teacher can indicate that two sides of an
equation are equal by using a sweeping hand motion under each
side of the equation while also explaining the equivalence of the
two sides out loud (Perry, Church, & Goldin-Meadow, 1988).
These kinds of gestures enhance learning for mathematical mate-
rial (e.g., Church, Ayman-Nolley, & Mahootian, 2004; Cook &
Goldin-Meadow, 2006; Ping & Goldin-Meadow, 2008). On a
Piagetian conservation task, a gestural demonstration of the width
of a glass with C-shaped hands placed side by side to approximate
the corresponding width improved children’s performance more
than verbal explanation alone (Ping & Goldin-Meadow, 2008).
Ping and Goldin-Meadow argue that these kinds of gestures during
instruction help students form abstract representations of the prob-
lem to be solved. Although gesturing may not be necessary in
mathematics problem-solving situations that involve concrete ob-
jects, it becomes particularly useful when students must go beyond
whole number arithmetic to working with abstract representations,
such as fractions. Carefully designed gestures and movements that
emphasize target fraction concepts (e.g., focusing on the magni-
tude of the fraction by using a range of hand and finger movements
to convey fractions of varying sizes) assist students in forming a
representation of the concept and focus attention on key visuals.
Dual Coding (Visual and Verbal) and Triple
Coding (Magnitude)
Dual coding purports that information is encoded into long-term
memory via two pathways: visual and verbal. These pathways are
physiologically interconnected but also function independently
(Paivio, 1986; Welcome, Paivio, McRae, & Joanisse, 2011). Evi-
dence suggests that retention of material is greatest when visual
information is presented in combination with verbal information
(Cuevas, 2016, for a review). For example, research comparing the
retention of abstract words (e.g., justice) with concrete words (e.g.,
hammer) reveals that verbal information that is associated with
images (i.e., concrete words) is more easily remembered than
abstract terms not easily linked to imagery (Bauch & Otten, 2012;
Welcome et al., 2011). This finding suggests that pairing verbal
concepts with visual cues should be particularly helpful in improv-
ing retention of that material. A related model more specific to
numerical cognition is Dehaene’s Triple Code model (Dehaene,
1992), which suggests that numbers are stored in three distinct but
related forms. Two of these forms are similar to dual processing:
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visual form (i.e., Arabic numeral) and auditory form (i.e., verbal
word/name). The additional form is that of an analog magnitude
representation akin to a mental number line. Our intervention
approach supports both of these models.
Spaced and Interleaved Practice
Practice is a critical component of any intervention aimed at
skill retention. However, the way in which practice is used affects
learning and retention. Practice sessions should be spaced out or
distributed over time, rather than practicing only after the corre-
sponding instructional unit. Distributing practice across multiple
lessons is more effective for producing long-term retention than
practicing in quick succession or in a single session (see Dunlosky,
Rawson, Marsh, Nathan, & Willingham, 2013, for a review).
Although distributed practice effects seem to be greatest for sim-
pler materials in free-recall tasks (e.g., multiplication facts; Don-
ovan & Radosevich, 1999; Rea & Modigliani, 1985), there also is
evidence demonstrating the benefits of distributed practice for
learning more complex mathematical concepts, such as determin-
ing simple permutations (Rohrer & Taylor, 2006, 2007).
Interleaved or layered practice involves mixing the types of
problems used during practice sessions rather than grouping them
according to problem type (Mayfield & Chase, 2002). For exam-
ple, intermediate-grade students who completed practice assign-
ments with different types of problems mixed together were more
accurate in their mathematics problem-solving than students who
were assigned blocked practice (i.e., problem sets presented by
problem type) over the same period of time (Rohrer, Dedrick, &
Stershic, 2015). Interleaved practice helps students evaluate the de-
mands of types of problems and choose the correct strategy according
to problem type (Taylor & Rohrer, 2010). Additionally, interleaving
practice problems across instructional units increases spacing of the
practice (Carpenter, Cepeda, Rohrer, Kang, & Pashler, 2012).
Feedback is particularly useful for children with low prior
knowledge (Fyfe, Rittle-Johnson, & DeCaro, 2012). However, the
timing and manner of the feedback must be considered (Shute,
2008, for a review). Low-achievers in particular benefit from
feedback that is immediate (Mason & Bruning, 2001), explicit, and
directive (Moreno, 2004). Low-performing students also benefit
from feedback that is structured and scaffolded; feedback that is
immediate and specific to the incorrect step taken in real time is
particularly effective (Graesser, McNamara, & VanLehn, 2005).
For example, Fyfe and Rittle-Johnson (2016) found that children
with low prior knowledge who incorrectly responded to a mathe-
matics equivalence problem benefitted more from receiving im-
mediate feedback including the correct answer compared with no
feedback or summative feedback presented at a delay once all
problems were solved.
Intervention Research on Fractions
The majority of published fraction intervention studies focus on
fraction arithmetic and solving word problems that require arith-
metic skill. For example, Shin and Bryant (2017) conducted a
small-scale case study on the effects of a computer-assisted pro-
gram (Fun Fractions) that included metacognitive problem-
solving strategies (i.e., teaching a heuristic they term Read-
Restate-Represent-Answer) on three students’ solving of word
problems based primarily on fraction arithmetic. Bottge and col-
leagues (Bottge et al., 2014) examined the effects of another
computer-assisted program (Fractions at Work) that centered in-
struction around video-based real-world problems (e.g., building a
skateboard ramp) to be solved by fictitious students within the
video along with hands-on applied projects. Though this program
used a range of virtual and concrete representations of fractions
(e.g., fraction strips, number lines), the primary outcomes of in-
terest were fraction arithmetic.
One experimentally studied fraction intervention focused on
students’ understanding of fraction magnitudes as a continuous
numerical quantity is Fraction Face-Off! by Fuchs and colleagues
(Fuchs, Schumacher, Malone, & Fuchs, 2011). Fuchs and col-
leagues (see Fuchs, Malone, Schumacher, Namkung, & Wang,
2017 for a review) have iteratively designed an effective fraction
intervention with large effect sizes at immediate posttest for
fourth-grade students at risk for mathematics difficulty. Their
fraction intervention is supported by domain-general cognitive
learning principles (e.g., schema-based instruction, supported self-
explaining). However, Fraction Face-Off! is designed for younger
students when fraction instruction typically begins in schools.
Moreover, the durability of the results was not evaluated after a
delay. Our intervention builds on promising findings by Fuchs and
colleagues through targeting sixth-graders who have already re-
ceived fraction instruction yet still struggle with fraction under-
standings and assessing effects over time.
The Present Study
The present randomized study evaluated the effectiveness of an
experimental intervention designed to build fraction knowledge in
students who reach sixth grade with low fraction skills. As noted,
the experimental intervention focused on the number line, although
other common representations of fractions were also introduced
(e.g., area models) and connected explicitly to the number line.
The experimental intervention also incorporated the key learning
principles described previously. It was carried out at the beginning
of the school year to help students benefit from their regular
fractions instruction, which was occurring concurrently. Students
were selected for inclusion based on low performance on a vali-
dated fraction screener (Rodrigues, Jordan, Hansen, Resnick, &
Ye, 2017). Relative to a business-as-usual control, we assessed
learning not only from pre- to immediate posttest but also at a
7-week delay. The delayed posttest is particularly important for
determining skill retention over time (e.g., Bailey et al., 2016).
Outcomes included assessments that measured specific fraction
skills of fraction number line estimation (FNLE), fraction magni-
tude comparisons, and fraction arithmetic as well as a broader
fraction concepts measure. Skills measured on the broad fraction
concepts measure as well as the FNLE task and fraction compar-
ison task were all directly targeted within intervention lessons. In
addition to expected improvements on the broad concepts and
magnitude measures, we were interested in the extent to which
students in the experimental intervention group would show
greater improvements than control group students on fraction
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arithmetic skill, which was not the explicit focus of the experi-
mental intervention.
We also assessed children’s general cognitive competencies.
Prior work suggests that fadeout effects (i.e., the finding that
mathematics intervention treatment effects often diminish over
time once the intervention has been complete) may be attributable
to preexisting differences (e.g., language; prior mathematics
knowledge) between students in treatment and control groups
(Bailey et al., 2016; Bailey, Fuchs, Gilbert, Geary, & Fuchs, 2018).
Although not a major focus of the current study, assessing students
on a range of cognitive measures enabled us to test and account for
differences in these competencies in our analyses. That is, we
collected these measures with the intention of using measures that
conditions differed upon as covariates. We assessed working mem-
ory, receptive vocabulary, inhibitory control, nonverbal matrix
reasoning, and nonsymbolic proportional reasoning, all known
predictors of mathematics achievement (e.g., Fuchs et al., 2014;
Hansen et al., 2015; Ye et al., 2016). We also assessed classroom
attention. Struggling students commonly display poor attention
(Fuchs et al., 2005), and students’ classroom attentive behavior has
been shown to influence fraction knowledge in particular (Hansen
et al., 2015; Hecht, Close, & Santisi, 2003; Resnick et al., 2016; Ye
et al., 2016). Demographic variables including gender, special
education status, and English language learner status were assessed
to ensure equivalence of conditions.
In sum, the current study evaluated the effectiveness of an
intervention centered primarily on the number line, one that also
incorporates cognitive learning strategies to improve learning and
retention. In addition to a broad measure of fraction concepts more
akin to school measures, we were concerned with the specific
conceptual skills of fraction magnitude estimation for sixth grade
students at risk for mathematics failure. We hypothesized that the
intervention would lead to meaningful immediate and longer-term
improvements on a range of fraction skills. We also explored
whether the experimental intervention would have differential
impacts on the learning of students with different cognitive or
behavioral competencies, focusing on classroom attentive behavior
in particular.
Students from two public middle schools in the Northeast region
of the United States were recruited to participate. Both schools
were racially and ethnically diverse (School 1: 36% Black non-
Hispanic, 32% white Hispanic, 12% white non-Hispanic, 12%
Other; School 2: 50% Black, non-Hispanic, 31% white Hispanic,
19% white non-Hispanic) and served students from low-income
families (School 1: 52% low income; School 2: 35% low income).
Qualifying for enrollment in the schools’ free or reduced lunch
program was used as a proxy for low-income status. Information
on SES was not available at the student level. In the previous
school year, proficiency level on the state test in mathematics was
45% for both schools. That is, the majority of students did not meet
state mathematics proficiency benchmarks.
Using G
Power (Faul, Erdfelder, Lang, & Buchner, 2007), we
conducted a priori power analyses to determine the minimum
sample size to detect a meaningful effect on fraction knowledge
using .05. Our preliminary work led us to expect large effects
(i.e., Hedges’ g0.8) of the experimental intervention on the
fraction concepts measures. A priori analyses suggested a sample
size of 54 would provide power of .81 to detect a large effect on
fraction concepts.
A fraction screener (described in the Method section) was
administered to all nonhonors sixth graders in both schools. Stu-
dents who scored at or below a validated cutpoint for mathematics
difficulties were invited to participate in the study (Rodrigues et
al., 2017). Ninety-nine of the 392 students (25%) screened met this
selection criteria and were invited to participate in the experiment.
Of this sample, parents of 56 students provided informed consent
(School 1: n27; School 2: n29). These students also assented
to participate.
Participants were randomly assigned either to the experimental
intervention condition or the business-as-usual control (BAU) con-
dition stratified by regular mathematics classroom. If there were
three participating students in a classroom, we assigned the extra
student to the experimental intervention group to ensure the target
intervention group size of three to four students, an optimal group
size for struggling learners (Fuchs et al., 2014). Participants within
the experimental intervention condition were then randomly as-
signed to one of four experimental intervention groups within each
of the schools (i.e., eight experimental intervention groups total),
disregarding mathematics classroom membership to reduce issues
of nesting (i.e., each group included students from several math-
ematics classes). Each small group was taught by one researcher-
instructor. Because of available resources, four researcher–
instructors taught one group each and two researcher–instructors
taught two groups (one in each school).
Of this original sample, three students in the experimental
intervention and one student in the control group moved away
before the experiment was completed; one experimental interven-
tion student was removed before the completion of the study
because of overly disruptive behavior. The final sample (N51;
20 males, 31 females) included students enrolled in seven different
classrooms across the two schools (School 1: n25; School 2:
n26). Five participants were English Language Learners (ELL;
School 1: n4; School 2: n1). There were 28 experimental
intervention students and 23 control group students.
Pretest and Posttest Fraction Measures
Fraction concepts. This broad fraction concepts measure was
made up of 24 released National Assessment of Educational Prog-
ress (NAEP) items that assessed various aspects of fraction con-
cepts. Nineteen of these items were used as a screener to select
students eligible for participation. The screener included six part–
whole area model items, one set model item, six equivalence items,
two fraction magnitude items, one estimation item, and three
comparison and ordering items. These 19 screening items were
validated in a prior study (Rodrigues et al., 2017), which used
receiver operating characteristic (ROC) analysis to assess the di-
agnostic accuracy of the screener for predicting students’ later
mathematics risk status. The fraction concepts screener yielded an
area under the curve (AUC) value of .881, signifying a very good
screener (Cummings & Smolkowski, 2015). The analysis identi-
fied that students who scored at a cut score of 10 or below on the
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19-item screener had an 87% chance of failing a standardized
mathematics test at the end of sixth grade.
Five additional, more challenging, NAEP items were added to
the fraction concepts measure to avoid potential ceiling effects and
were administered along with the 19 screener items at all time
points. The five items assessed part–whole understanding of an
area model, fraction equivalence, and fraction magnitudes. As
these items were not included in the prior screener validation
study, performance on these five items was not counted toward the
screener score. Students received one point for each correct item
for up to a total of 24 points for the entire measure (up to a total
of 19 points for the screener). Screener scores were used only for
inclusion criteria. Performance on the full 24-item measure was
used at all time points to assess the effectiveness of the interven-
tion. The full fraction concepts measure as well as the screener
items have displayed good internal consistency with sixth graders
in prior studies with larger samples from the same region (.86
and .78, respectively; Jordan et al., 2017; Ye et al., 2016). Sample
NAEP items are displayed in Figure 1.
Fraction comparisons. The paper and pencil fraction compari-
sons measure includes 24 items for which students are presented with
two fractions and asked to select the larger fraction. Students complete
as many problems as they can within three minutes. The measure
includes several types of comparisons including unit fractions (e.g.,
1/3 or 1/2), fractions with like denominators (e.g., 5/7 or 6/7), frac-
tions with like numerators (e.g., 2/4 or 2/5), reciprocal fractions (e.g.,
8/4 or 4/8), and fractions with different denominators and numerators
(e.g., 12/50 or 8/60). Students received one point for each correct item
for up to a total of 24 points. Internal reliability of the fractions
comparison measure in sixth grade with a larger sample from the
same region was high (.92; Jordan et al., 2017).
Fraction number line estimation. Two paper-and-pencil frac-
tion number line estimation (FNLE) tasks were used. The tasks
were adapted from a computer administered version (Hansen et al.,
2015) for logistical reasons. These tasks includeda0to1scale and
a 0 to 2 scale. The 0 to 1 scale included six individual number lines
and the 0 to 2 number line included eight individual number lines,
each 81 mm in length. The number 0 was placed below the left end
of the number line and the 1 or 2 was placed at the right end of the
number line. The target fraction was centered below each number
line. The number lines for each task were printed on the same sheet
of paper but were staggered so that participants could not easily
use their estimates on other number lines to inform their place-
ments. Students were instructed to “Mark a line to show where
each number belongs on the number line if the endpoints are 0 and
1 [or 2].” The target numbers on the 0 to 1 task were all proper
fractions (1/4, 1/5, 1/3, 1/2, 1/19, 5/6). The target numbers on the
0 to 2 task included proper fractions (3/8, 5/6, 1/2, 1/19), improper
fractions (7/4, 5/5), and mixed numbers (1 11/12, 1 1/2).
Prior work on whole number magnitude has been concerned
with the logarithmic to linear shift that occurs throughout devel-
opment with increasingly large scales. That is, whole number
magnitudes are represented in a compressed logarithmic distribu-
tion (i.e., overestimating the distance between smaller numbers
and underestimating the distance between larger values) and be-
come increasingly more linear throughout development with in-
creasingly larger scales (Berteletti, Lucangeli, Piazza, Dehaene, &
Zorzi, 2010; Booth & Siegler, 2006; Siegler & Booth, 2004;
Thompson & Opfer, 2010). This method requires fitting linear and
logarithmic functions for each participant’s estimates and then
determining which pattern is a better fit for that learner (Siegler &
Opfer, 2003). Whereas becoming increasingly linear in one’s
whole number estimates is the standard developmental trajectory
that children follow (leading to increases in accuracy with whole
numbers), the same is not true for fractions (e.g., Opfer & DeVries,
2008; Siegler, Thompson, & Opfer, 2009). Thus, most work on
Figure 1. Sample of released NAEP fraction conceptual items.
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fraction magnitude estimation uses percent absolute error (PAE) as
a performance measure on fraction number line tasks (e.g., Fazio,
Kennedy, & Siegler, 2016; Fuchs et al., 2017; Hamdan & Gunder-
son, 2017; Siegler & Pyke, 2013). The current study follows this
field standard.
To gauge students’ accuracy of their estimates, PAE was cal-
culated for each estimate by dividing the absolute difference
between the participant’s estimate and the accurate target location
by the scale of the estimates. The PAE for each of the 14 number
lines was averaged to determine the overall PAE. Lower scores
indicate more accurate performance. All number line estimates
were measured twice by the same two research assistants, and any
discrepancies were reconciled. Internal consistency for the fraction
number line task in sixth grade with a larger sample from the same
region was high (.87; Resnick et al., 2016).
Fraction arithmetic. There were 12 written fraction arithme-
tic items: four addition, five subtraction, and three multiplication.
Proper and improper fractions as well as mixed numbers were
used. Addition and subtraction items included addends and sub-
trahends both with like and unlike denominators (e.g., 3/4 2/3
___; 5/6 – 2/6 ___). In each of the three multiplication items,
one factor was a fraction and one factor was an integer (e.g., 3/4
12 ___). Students were asked to give their answers in simplest
form. Students received one point per correct item as well as one
point per correct response in simplest form. Thus, there were a
total of 24 points to be earned. The internal reliability for this
measure in sixth grade with a larger sample from the same region
was high (.82; Hansen et al., 2015).
General Competencies
We assessed working memory, receptive vocabulary, and inhib-
itory control with validated measures from the NIH (National
Institutes of Health) Toolbox (Gershon et al., 2013). All toolbox
tasks were administered individually on an iPad. All raw scores
were converted to scaled scores for ease of interpretation (M
100, SD 15). These three NIH toolbox measures have demon-
strated good to excellent test/retest reliability at 8 –15 years of age,
with alphas ranging from .81–.91. Convergent and discriminant
validity were also established (see Bauer & Zelazo, 2013).
Working memory. Participants were presented with pictures
of food and/or animals that were labeled with audio and text and
displayed for 2 s. Participants recalled items aloud from smallest to
largest, with the number of images increasing by one item per trial.
Students completed a unidimensional (either food or animals) and
a two-dimensional (both food and animals) list. Higher scores
represent more items correctly recalled from both lists.
Receptive vocabulary. Participants were asked to select the
correct image from a group of four that most closely matched the
meaning of the word presented. The Toolbox Picture Vocabulary
Test (TPVT) is a Computer Adaptive Test (CAT) that adjusts the
level of difficulty based on each student’s performance. Item
Response Theory (IRT) is used to score performance on the TPVT.
Higher theta scores represent better vocabulary.
Inhibitory control. Students completed a traditional flanker
task in which they were shown a series of arrows and asked to
choose the direction of the center arrow. Sometimes the direction
of the center arrow was congruent with the arrows flanking it and
sometimes the direction was incongruent. Higher scores repre-
sented greater speed and accuracy.
Nonverbal reasoning. Nonverbal reasoning was assessed us-
ing the Matrix Reasoning subtest of the Wechsler Abbreviated
Scale of Intelligence (WASI; Wechsler, 1999). Students were
shown a series of grids that contain pictures in three of the four
cells which begin a pattern and are asked to choose the next grid
that completes the pattern. Higher tscores (M50, SD 10)
represent more items correct. Test/retest reliability was adequate at
12–16 years of age (.74). Convergent, discriminant, and
construct validity were also established (see Wechsler, 1999).
Nonsymbolic proportional reasoning. Proportional reason-
ing was measured using an iPad adaptation of the nonsymbolic
scaling task used by Boyer and Levine (2012). Students are shown
a target “juice” mixture represented by a vertical bar with a portion
of red representing the red powder mix and a portion of blue
representing the water. Students are shown two additional bars that
are a different size than the target and tasked with choosing the
option that is a rescaled version of the original. Correct responses
required either scaling down from a larger target to a smaller
match or scaling up from a smaller target to a larger match. Higher
scores represent more correct trials. Internal reliability with a
larger sample from the same region was high (.93; Ye et al.,
Classroom attentive behavior. Classroom attention was mea-
sured using mathematics teacher reports on the Inattentive Behav-
ior subscale of the SWAN Rating scale (Swanson et al., 2006).
Attentive behavior is one’s level of attention exhibited in the
classroom as observed by a student’s regular classroom teacher.
This subscale included nine Likert-style items that followed Di-
agnostic and Statistical Manual of Mental Disorders (DSM) cri-
teria for attention-deficit/hyperactivity disorder. A sample item is
“Gives close attention to detail and avoids careless mistakes.”
Responses ranged from 1 (far below average)to7(far above
average). Lower scores represent poorer attention. Sum scores
range between 9 and 63. Students’ regular mathematics teachers
rated individual students’ behavior. Internal consistency is high
(.92; Lakes, Swanson, & Riggs, 2012).
Background Variables
Demographic variables including gender, special education sta-
tus and English Language Learner (ELL) status as well as whether
students were classified as having a math, reading, and/or behav-
ioral disability, were obtained through school records with permis-
sion from parents/caregivers. Income status, race, and ethnicity
were not available for individual students.
The study design included group administration of a fraction
pretest right before the intervention period, a posttest immediately
after the intervention period, and a delayed posttest seven weeks
later. The general cognitive measures were administered prior to
the intervention period with the exception of the SWAN attentive
behavior scale (Swanson et al., 2006).
As the intervention was administered at the start of the school
year, the SWAN Rating Scale was completed several weeks after
the intervention began to allow general classroom teachers to
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become acquainted with their new students so that they could
provide more accurate representations of their normal classroom
behavior. Posttest fraction measures were administered to the
experimental and control conditions at the same time in the same
The intervention took place during a 6-week period in which
all students received specialized help from a teacher within
their school. This designated 45-min intervention time is in
addition to students’ regular mathematics class. In their regular
mathematics classes, both schools used the same mathematics
curriculum: Connected Mathematics Project (CMP; Lappan,
Difanis Phillips, Fey, & Friel, 2014). The CMP curriculum is
aligned with CCSS-M (NGACBP & CCSSO, 2010). According
to its manual, it aims to help students develop mathematical
understanding by emphasizing connections between mathemat-
ical ideas and their real-world problem-solving applications.
When followed according to its intended design, CMP includes
minimal explicit instruction on specific strategy use but instead
encourages students to invent their own strategies for problem-
solving and discuss multiple strategies during whole group
discussions. For fractions in sixth grade, this curriculum covers
factors, models, and fraction operations.
During the additional class period dedicated to intervention,
students in the experimental intervention condition received 27
researcher-designed lessons (described further below). These les-
sons were administered to each of the small groups by one of the
trained instructors. Concurrently, students in the control condition
received their regular mathematics intervention provided by their
school. Both schools used a computer adaptive tutoring software
for their mathematics intervention period, on which students
worked individually. Students received individualized assistance
in mathematics from the computer adaptive software based on
their current level of performance. One school used Dreambox
Learning (2012). The other used i-Ready (2016). Both programs
are aligned with the CCSS-M and as such address fraction under-
standing including operations and some magnitude judgments
along with other sixth-grade mathematics content.
Instructor Training
Experimental intervention instructors were trained research as-
sistants who also participated in lesson design. Instructors varied in
prior teaching experience. Two instructors were doctoral students,
two were postdoctoral researchers, and two were previous certified
teachers. Each of the six instructors received more than 16 hr of the
same training in administration of the lessons from one of the
authors of the current paper. Training included practice in use of
gestures, proper strategies for providing feedback, instructor/stu-
dent dialogue, and behavior management. Experimental interven-
tion instructors also practiced teaching the lessons in pairs and
provided each other feedback for lesson improvement prior to
administering the lessons.
Experimental Intervention Design
Lesson structure. The lessons were situated in the context of a
color run—a race in which runners are showered with different
colored powders at stations along the way. The race course models a
number line, on which students can think about fractions and their
magnitudes in a real-world context for understanding fraction mag-
nitude (Rodrigues, Dyson, Hansen, & Jordan, 2016). Lessons focused
primarily on denominators that occur frequently in measurement
activities of daily life, including halves, thirds, fourths, sixths, eights,
and twelfths. However, students also practiced with other denomina-
tors during practice activities and games involving fraction magnitude
comparison. The lessons were carefully scripted to increase fidelity.
The scope and sequence of the experimental intervention lessons is
presented in Table 1. A general overview of the lesson structure is
presented in Table 2.
Prior to the explicit instructional time of each lesson, students
completed a warm-up worksheet in which they individually prac-
ticed material they learned from the previous lessons. The National
Mathematics Advisory Panel (NMAP, 2008) deems the ability to
recall basic mathematics facts a crucial prerequisite skill for math-
ematics success. Quick and accurate recall, or automaticity, of
basic mathematics facts is thought to free up the necessary atten-
Table 1
Scope and Sequence of the Fraction Intervention
Key topics
Lessons 1– 4 Lessons 5–9 Lessons 10–15 Lessons 20 –21 Lessons 22–24
Halves Halves fourths
Halves fourths
Thirds sixths
twelfths All studied denominators
Counting by unit fractions, by whole and mixed numbers
Partitioning using linear, area, and set models
Finding 1/bof a set (multiplication)
Finding a/bof a set when a1 (multiplication)
Adding/Subtracting fractions with common denominators
Locating mixed numbers on the number line
Equating to improper fractions and mixed numbers
Measuring with rulers marked with whole numbers and... •
Finding equivalent fractions with different denominators
Comparing fraction magnitudes using various strategies
Measuring with cups
Adding/Subtracting with unlike denominators
Activities involving comparison of fraction magnitudes provided opportunities for students to apply fraction comparison strategies (e.g., benchmark
strategy) to denominators not studied within the intervention, such as fifths.
Involved only denominators common to measuring cups: halves, thirds,
fourths, and eighths.
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tional resources that a learner needs to focus on more complex
aspects of a task. Students who have a learning disability and those
with low mathematics achievement typically struggle with both
accuracy and speed in mathematics fact recall (Geary, 2004; Jor-
dan, Hanich, & Kaplan, 2003). Therefore, after the warm-up,
students participated in practice of whole number multiplication
facts which targeted multiplicands used as the main denominator
of that day’s lesson. Practice during a given lesson included facts
learned in prior lessons and was interleaved for fact retention.
Next, as a group, students practiced counting aloud fractions of
like denominators along the number line using both proper and
improper fractions or whole and mixed numbers. These activities
were also aimed at automaticity in preparation for the explicit
instructional period focused on concepts which lasted about 15 to
20 min per lesson. These lessons were predominantly focused on
developing students’ understanding of fraction as a number or
magnitude on their mental number line. As such, this was accom-
plished mostly through the use of a visual number line (or “race
course”) but also through the use of fraction strips and other
representations. Although not the primary focus, some attention
was given to addition and subtraction in a conceptual manner as
represented by moving forward (addition) and backward (subtrac-
tion) along a number line. More details are presented in Table 1,
which displays the scope and sequence of the lessons provided
within the experimental intervention.
Each lesson concluded with short, fast-paced card games that
gave students opportunities to practice lesson goals related to
fraction concepts in particular. The games targeted fraction con-
cepts such as magnitude judgments and their corresponding strat-
egies (e.g., comparing two fractions to each other, to one half, etc.)
as well as fraction equivalencies (e.g., 3 is the same as how many
halves?; 3 halves is the same as how many fourths?). Lastly,
students completed an independent cool-down worksheet in which
they solved problems that assessed their knowledge of the con-
cepts presented in the instructional period as well as concepts from
previous lessons. Many of the practice problems were near transfer
problems that required students to apply previously learned skills
to find solutions to problems very similar to those practiced
throughout the lessons. However, far transfer (i.e., novel) prob-
lems were also included to encourage students to expand upon
their knowledge and modify methods learned to find a solution
(Barnett & Ceci, 2002).
Cognitive learning strategies. Strategies based on key learn-
ing principles were implemented comprehensively throughout the
lessons to target the range of skills covered within the lessons as
noted within the Scope and Sequence in Table 1. For example,
students completed spaced and distributed practice on numerous
skills including but not limited to fraction magnitude comparisons,
whole number multiplication facts, and partitioning and marking
number lines. The following sections provide examples of how
each principle was regularly implemented.
Explicit instruction. In each experimental intervention lesson,
students were explicitly taught concepts and step-by-step strategies
the majority of the time (i.e., 15–20 min per lesson) prior to
practicing them independently. This explicit instruction period
focused primarily on the number line representation of fractions
although complementary representations were also provided (e.g.,
fraction bars). For example, in Lesson 2 students were taught how
to partition a number line race course into half miles. The instruc-
tor displayed a blank number line that represents a four-mile race
course and then used a paper bar that represented one mile to mark
off each of the four miles. Once all students correctly partitioned
and marked their number lines into four miles, the instructor
demonstrated finding one half mile by finding the midpoint of the
first mile. The instructor placed a pencil under the line and esti-
mated at which point the segments on the left and right of the
pencil are equal. The instructor then made a mark at this point and
explained that this mark is the one-half mile mark, because it is one
of two equal portions of a whole. The instructor repeated this
process until the remaining halves were marked, ensuring that
students followed along on their own race course. This explicit
modeling was included to encourage students to adopt these effi-
cient partitioning strategies in their independent work. A similar
process was used for fractions with other denominators such as
fourths and eighths (see Figure 2).
Representational gestures. In each experimental intervention
lesson, gestures were used to represent corresponding concepts and
Table 2
Overview of Lesson Structure
Activity Description Time
Warm-up Individual worksheet practice of material from previous day 3 minutes
Multiplication practice Speeded practice of whole number multiplication facts using multiplicands that are aligned with
denominators in the corresponding lesson
3 minutes
Counting Practice of oral counting of fractions with like denominators (e.g., “one-fourth, two-fourths, three-
fourths . . .”) using the number line as reference
3 minutes
Targeted instruction Explicit instruction targeting the lesson’s learning goals and focused on the number line 20–25 minutes
Games Short, fast-paced card games targeting fraction magnitude judgements (e.g., comparing two fractions
to each other, to one-half) and fraction equivalencies (e.g., 3 is the same as how many halves?;
3 halves is the same as how many fourths?)
3 minutes
Cool down (independent practice
and formative assessment) Individual worksheet practice of material from that day’s lesson and prior content. 3 minutes
Figure 2. Example of number line used in lesson on fourths.
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ideas. A chunking gesture helped students visualize the magnitude
of a particular unit fraction along a number line. The instructor
placed her index finger and thumb at the start and endpoint of a
particular magnitude (e.g., 0 to 1/4) much like a bracket and held
her fingers in this manner while demonstrating moving along the
number line in consistent units as she counted aloud and explained
the meaning of the corresponding magnitude. This gesturing was
done to foster the understanding that the magnitude of any given
fraction on a number line is the distance between 0 and that
fraction—not simply an arbitrary label for the mark on the number
line. Other gestures highlighted the meaning of the numerator and
denominator. For example, when explaining the meaning of one
half, the instructor displayed the fraction on the board and used a
components gesture. This gesture involved pointing to the “1” with
one finger while saying “one of...”andpointing to the “2” with
two fingers while stating, “two equal portions in the whole.”
Dual coding. Verbal information was typically presented with
a visual aid. Along with the number line, the experimental inter-
vention used concrete materials such as fraction bars and other
magnetic manipulatives that were presented and used with corre-
sponding verbal explanations. For example, when discussing
equivalent fractions, the instructor presented students with mag-
netic fraction bars that displayed the same whole partitioned into
different numbers of parts and labeled accordingly (e.g., halves,
fourths, and eighths). When converting halves to fourths, the
instructor connected verbal explanations with visual representa-
tions by saying, “We know that each green magnet represents
one-half. Do you see an equivalent fraction for one-half? One-half
equals two-fourths. One-half and two-fourths are equivalent frac-
tions.” Students were encouraged to match up the one half magnet
with two one fourth magnets to connect these verbal explanations
with visual representations. An image of these magnets is dis-
played in Figure 3.
Spaced and interleaved practice. Distributed and interleaved
practice was used to enhance long-term retention of basic prereq-
uisite competencies (e.g., whole number multiplication facts) and
develop fluency in specific problem-solving strategies for more
conceptual tasks (e.g., fraction magnitude comparisons). For ex-
ample, students practiced whole number multiplication facts for
several minutes during each lesson over the course of the 27-day
experimental intervention. Students also practiced fraction magni-
tude comparisons during which they were shown two fractions and
asked to indicate which fraction is more. Fractions taught in
previous lessons (e.g., unit fractions) were interleaved with those
taught in later lessons (e.g., improper fractions). Students also
worked on earlier presented problem types both at the warm-up
and cool-down phases of the lessons. For example, during the
warm-up and cool-down activities within a lesson, students prac-
ticed partitioning a number line using a range of denominators and
completing fraction computations with fractions of varying de-
nominators, even though the bulk of each lesson was generally
structured around one specific denominator that varied on every
fifth lesson (e.g., Lessons 1– 4 focused on halves, Lessons 5– 8
focused on fourths).
Feedback. Instructors gave individual feedback on students’
independent work during the warm-up activity and throughout the
lessons. This feedback was both corrective and process-oriented,
allowing students to revise their strategies. One example of cor-
rective feedback was the process used for error correction while
practicing whole number multiplication facts. When a student gave
an incorrect product, the instructor briefly displayed the number
sentence with the correct product and then gave the student another
opportunity to answer the question. An example of process-
oriented feedback was the approach used for addressing incorrect
fraction comparison strategies. During practice on fraction com-
parison tasks, students were asked to note the best strategy to
use for each particular pair of fractions based on fraction type.
For example, when comparing 6/8 and 1/4, if a student named an
incorrect strategy (e.g., “more parts equals larger fraction”), the
instructor noted the best strategy (e.g., benchmarking – determine
which is closer to 1) and explained why this choice was
more efficient (e.g., “These fractions have different denominators
so we can’t simply compare number of parts. We know that 1/4 is
closer to 0 and 6/8 is closer to 1. Therefore, 6/8 is larger than 1/4”).
Common errors across student work (e.g., skipping the fractions
that are equivalent to whole numbers when labeling halves on a
number line) were discussed and corrected with the group during
Fidelity of Implementation
Lessons were scripted to ensure instructor fidelity of implemen-
tation. All lessons were audio recorded to obtain measures of
fidelity, which were transcribed and coded by trained researchers.
Each lesson had a checklist of an average of 80 items to check off per
lesson (approximately one check per 30 seconds of audio recording).
Eight of the 27 lessons were randomly selected for each of the six
instructors and coded according to conformity to the scripted
activities. Instructors administered an average of 99% of all
scripted experimental intervention activities to students. Experi-
mental intervention activities that were not administered mainly
resulted from lack of time as a result of natural classroom distur-
bances (e.g., announcements, fire drills, student tardiness, behav-
ioral issues, etc.).
Plan of Analysis
Evaluating the effects of intervention. The primary focus of
the current study was to evaluate the effectiveness of our experi-
mental fraction intervention on improving student performance on
a range of fraction skills from pretest to immediate posttest and to
determine whether these skills were retained at a delay. To assess
whether there were differential effects on fraction learning from
pre- to immediate and delayed posttest between experimental and
control students, four 2 (Group: Intervention and Control) 3
(Time: Pretest, Posttest, Delayed posttest) mixed analyses of co-
variance (ANCOVA) were planned on fraction concepts, FNLE,
fraction comparisons, and fraction arithmetic, respectively.
Figure 3. Fraction magnet manipulatives.
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A mixed ANOVA is a general linear model similar to a combina-
tion between a repeated measures ANOVA (RMANOVA) and a
one-way ANOVA (Murrar & Brauer, 2018). It is termed a mixed
ANCOVA once a covariate is added to the model. In the absence of
baseline balance, we planned to control for any cognitive or demo-
graphic variables that conditions differed upon at pretest. The planned
mixed ANCOVAs explain variance in the dependent variable be-
tween groups, within groups, and, most importantly, between groups
over time, while controlling for any time invariant covariate measured
only at pretest. The main focus of a mixed ANOVA is the Time
Condition interaction, which examines whether differences over time
in the dependent variable were significantly different between condi-
tions. Planned trend analyses on Time Condition interactions
involved fitting linear and quadratic terms to determine whether the
effect of condition on the outcome increased linearly over time and
whether the effect leveled off showing a retention at delayed posttest.
All statistical analyses were performed using SPSS v. 24.0 (IBM
Corp. 2016).
Individual differences exploratory analyses. We explored
whether the experimental intervention had differential impacts on
the learning of students with different cognitive or behavioral
competencies. A preliminary examination of the data suggested
that the effect of intervention on the fraction outcome measures at
posttest may be moderated by students’ attentive behavior (as
measured by the SWAN teacher rating scale). Thus, we examined
this possibility using ordinary least squares (OLS) regression mod-
els to test for an interaction between students’ attentive behavior
score and condition on each outcome measure that demonstrated
differential improvements based on the mixed ANCOVA results.
Because a significant amount of learning occurred specifically
between pre- and posttest, models were tested on immediate post-
test only. A dummy code variable was created for condition
(Experimental intervention 1; Control condition 0). SWAN
scores were centered to reduce the risk of multicollinearity that
may arise when including interaction terms in analyses.
An interaction term was created between the centered Attentive
Behavior variable and condition dummy code. A series of multiple
regressions predicting posttest scores were conducted. Four regres-
sions for each of the outcomes of interest were built starting with
a pretest-only model and adding one term to the prior model until
the final model included pretest, dummy code for intervention,
classroom attentive behavior, and an interaction term between
condition and classroom attentive behavior. The main effects mod-
els reestablish the relationship between posttest scores and condi-
tion, controlling for pretest scores on each corresponding measure.
The three moderation models tested whether the effect of condition
on each outcome measure varied by students’ attentive behavior as
represented by classroom teacher ratings on the SWAN. These
exploratory models are displayed in Table 10 and supplemental
tables S11 and S12 and results are discussed below.
Corrections for multiple tests. Because the current study was
proposed to include planned multiple comparisons within the
mixed ANCOVAs noted above, corrections are not necessary
(Armstrong, 2014; Perneger, 1998). Additionally, exploratory
analyses that do not test specific hypotheses but rather provide
suggestions for future work do not require corrections. However,
we have opted to take a more conservative approach in interpreting
our findings in both sets of analyses and have adjusted our alpha
levels for multiple tests. We employed Benjamini and Hochberg’s
(1995) correction procedure for multiple tests which decreases the
False Discovery Rate (FDR). The FDR is the expected proportion
of the rejected null hypotheses which are incorrectly rejected.
Unlike the classic Bonferroni correction (Bonferroni, 1936), which
adjusts the alpha level once to use for all comparisons, the BH
correction adjusts the alpha level down to an increasingly conser-
vative cutoff, using an ordered set of obtained mpvalues, only
after each statistically significant result and not after nonsignifi-
cant results After finding the largest pvalue that satisfies pk
m, all tests with smaller pvalues are declared significant. BH
corrections were applied to the four Time Condition interactions
within the mixed ANCOVAs with adjusted alpha levels of 0.05,
0.0375, 0.025, and 0.0125. In our exploratory analyses, we adopt
BH adjusted alpha levels of 0.05, 0.033, and 0.017 to interpret the
results of the three Attentive Behavior Condition interactions.
Nesting as nuisance. We did not have substantive questions
on the effects of classroom level variables in the current study. As
previously noted, participants were randomly assigned at the stu-
dent level within classroom to the intervention or control group.
Then, students in the intervention condition were randomly as-
signed to one of four intervention groups in each school. Thus, we
did not expect to find substantial classroom effects. Indeed, intra-
class correlations (ICC) were low on all outcomes (ICC .05) and
did not warrant multilevel modeling (MLM). Thus, clustering can
be considered a nuisance variable (Clarke, Crawford, Steele, &
Vignoles, 2010, p. 7). However, we tested fixed effects models
controlling for cluster (i.e., accounting for all cluster-level effects)
and to reduce the issue of the omitted variable bias (Huang, 2016;
Kennedy, 2003). These exploratory analyses revealed a consistent
pattern of results regarding the effects of the intervention on all
four fraction outcomes demonstrated in the mixed ANCOVAs as
well as the significant moderation of effect of condition by Atten-
tive Behavior in the exploratory analyses reported below.
Determining effect sizes. To determine the magnitude of the
intervention effects on each outcome, we utilize Hedges’ g, Co-
hen’s U3, and an improvement index, all presented in Table 9.
suggested by the WWC Procedures and Standards Handbook,
Version 3 (U.S. Department of Education, Institute of Education
Sciences, 2014), effect sizes of an education intervention that uses
the same measure for pre- and posttests should be calculated as the
difference between the pre- and posttest mean difference of the
experimental intervention condition and the pre- and posttest mean
difference of the control condition. Thus, effect sizes reported in
Table 9 represent the difference of the differences as opposed to
simply the difference between posttests. Hedges’ g provides a
better estimate of effect sizes in small samples than Cohen’s d
(Cummings, 2012). Hedges’ g can be interpreted using Cohen’s
(1988) standards of small (0.2), medium (0.5), or large (0.8; e.g.,
Lakens, 2013). However, Cohen suggested caution when using
rules of thumb and emphasized the importance of considering
effect size in context. According to the same WWC Procedures
and Standards Handbook, an effect size (Hedges’ g) of at least .25
Whereas Hedges’ gprovides a standardized index of a mean difference
of learning between the two groups, partial eta squared (p
2) provides a
measure of variance explained by each factor of the mixed ANCOVAs.
Therefore, p
2is also presented as an additional measure of effect size in the
tables for each of the four 3 2 mixed ANCOVAs conducted.
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is considered meaningful in education research, even if statistical
significance is not reached. An effect size of .25 indicates that the
experimental group performed one fourth of a standard deviation
higher than the control group, based on the pooled variance from
the sample. To provide another measure of practical importance,
we also converted the effect size of Hedges’ gto Cohen’s U3 to
yield an improvement index. The U3 represents the percentile rank
of a student in the control group who performed at the level of an
average experimental group student. By definition, the average
control group student would rank at the 50th percentile. Thus,
finding the difference between the computed value of the U3 for
the experimental group and 50% yields the improvement index.
Descriptive Statistics
Descriptive statistics for the entire sample as well as by condi-
tion are presented in Table 3 on demographic variables as well as
participants’ scores at pretest on fraction measures and cognitive
competencies. All continuous variables were normally distributed.
Prior to addressing our research questions, chi square analyses and
ttests were conducted to confirm equivalence between conditions
on demographic variables, teacher report of attentive behavior,
pretest fraction measures and cognitive measures. As demonstrated
in Table 3, there were no significant differences at pretest, with the
exception of receptive vocabulary. Students in the control condi-
tion had significantly higher receptive vocabulary scores (M
81.37, SD 6.99) than the experimental intervention (M77.55,
SD 5.60), t(49) 2.166, p.035. Thus, receptive vocabulary
was used as a covariate in the four planned mixed ANCOVAs to
control for these differences. No other covariates were used.
Effect of Intervention on Fraction Outcomes
Results from the four 2 (Condition: Intervention and Control)
3 (Time: Pretest, Posttest, Delayed posttest) mixed analyses of
covariance (ANCOVA) conducted on fraction concepts, FNLE,
fraction comparisons, and fraction arithmetic (controlling for re-
ceptive vocabulary) are presented in Table 4. Estimated marginal
means reported for each outcome are adjusted for receptive vo-
cabulary at M79.27. Greenhouse-Geisser corrections were used
to correct for violations of assumptions of sphericity. As previ-
ously noted, the Time Condition interactions are interpreted
after applying BH corrections (Benjamini & Hochberg, 1995).
Fraction Concepts
As displayed in Table 5, there was no main effect of time, but
there was a main effect of condition as well as a significant time
by condition interaction. A trend analysis explicating the time by
condition interaction revealed both a significant linear and qua-
dratic component. This trend is plotted in Figure 4, which displays
a general increase in scores from pretest to immediate posttest and
a leveling off between posttest and delay for experimental inter-
vention students. The control group shows a small but noticeable
improvement between immediate and delayed posttest. Post hoc
pairwise comparisons revealed that the experimental intervention
Table 3
Descriptives and Tests of Equivalence Between Conditions at Pretest
Business-as-usual control
Intervention group
(n28) Difference test
Sig. ttestVariable MSDMSD
Continuous variables
Fraction concepts 9.83 3.38 9.54 1.86 t.369, p.715, ns
Fraction arithmetic 3.65 2.62 4.07 3.03 t.522, p.604, ns
Fraction comparisons 10.35 3.80 12.00 5.73 t1.185, p.242, ns
FNLE (PAE) 24.25 9.71 20.20 7.32 t1.70, p.096, ns
Attentive behavior 35.65 13.39 34.04 11.02 t.473, p.638, ns
Receptive vocabulary 81.37 6.99 77.55 5.60 t2.166, p.035
Working memory 86.86 11.50 90.08 10.93 t1.009, p.318, ns
Inhibitory control 102.53 14.08 103.05 13.41 t.133, p.894, ns
Nonverbal reasoning 33.83 10.19 36.43 9.44 t.945, p.349, ns
Proportional reasoning 14.36 5.34 16.11 5.56 t1.120, p.268, ns
% % Sig.
Categorical variables
Male 34.78% 50%
1.192, p.275, ns
Special education (Spec Ed) 30.43% 32.14%
.017, p.896, ns
English language learner (ELL) 13.04% 7.14%
.497, p.481, ns
Math learning disability (MLD) 30.43% 25.00%
.187, p.665, ns
Reading learning disability (RLD) 30.43% 28.57%
.021, p.884, ns
Behavioral disability .00% 3.57%
.838, p.360, ns
Note. Fraction concepts, fraction arithmetic, and fraction comparison scores are sum scores with the highest possible score of 24. FNLE is percent absolute
error with lower scores indicating higher accuracy. Attentive behavior is a sum score ranging from 9 to 63. Receptive vocabulary, working memory, and
inhibitory control are scaled scores (M100, SD 15). Nonverbal reasoning is a tscore (M50, SD 10). Proportional reasoning is number correct
of 24 trials.
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group had higher fraction concepts scores than the control condi-
tion both at posttest (p.001) and delayed posttest (p.008).
Fraction Number Line Estimation
As displayed in Table 6, there was no main effect of time, but
a main effect of condition as well as a significant time by condition
interaction. A trend analysis explicating the time by condition
interaction revealed a significant quadratic component and a trend
toward a significant linear component. This trend is plotted in
Figure 5, which displays a general decrease in PAE (indicating
more accurate performance) from pretest to posttest for the exper-
imental intervention group and then a leveling off between posttest
and delay. Post hoc pairwise comparisons revealed that the exper-
imental intervention group had lower PAE than the control con-
dition, both at posttest (p.001) and delayed posttest (p.001).
Fraction Comparisons
As displayed in Table 7, there was no main effect of time, but
a main effect of condition and a significant time by condition
interaction. A trend analysis explicating the time by condition
interaction revealed a trend toward a significant quadratic compo-
nent but not a significant linear component. This trend is plotted in
Figure 6, which displays an increase in scores from pretest to
posttest for both conditions but a much greater increase for the
experimental intervention group. The experimental intervention
shows a minor decline in fraction comparison scores from post- to
delayed posttest but still significantly outperforms the control
condition. Post hoc pairwise comparisons revealed that the exper-
imental intervention group had higher fraction comparison scores
than the control condition at both immediate posttest (p.001)
and delayed posttest (p.005).
Fraction Arithmetic
As displayed in Table 8, there was no main effect of time or
condition. There was no significant interaction between time and
condition. Both conditions demonstrated comparably small im-
provements in fraction arithmetic scores between pretest and post-
test and comparable declines between immediate and delayed
Effect Sizes
The experimental intervention yielded large effects on fraction
concepts (g1.09), FNLE (g0.85), and fraction comparisons
(g0.82) at posttest. Effects of the experimental intervention on
these three measures at delayed posttest were medium to large
(gs0.66, 0.60, 0.61, respectively). Effect sizes for fraction
arithmetic were small at both posttests.
Table 4
Estimated Marginal Means (EMM) for Fraction Outcome Measures Controlling for Receptive Vocabulary
Pretest Posttest Delayed posttest
Control Intervention Control Intervention Control Intervention
Fraction concepts 9.45 0.52 9.85 0.47 11.09 0.65 15.14 0.59 12.20 0.70 14.84 0.63
FNLE 24.43 1.83 20.05 1.65 23.27 1.55 12.40 1.40 21.36 1.69 11.93 1.53
Fraction comparison 10.45 1.07 11.91 0.96 12.79 1.16 18.68 1.05 13.19 1.21 18.06 1.09
Fraction arithmetic 3.46 0.60 4.23 0.55 6.94 1.22 8.66 1.10 5.84 0.89 7.06 0.81
Note. Adjusted for vocabulary at M79.27. All scores are raw scores with the exception of fraction number line estimation (FNLE), which uses the
percent absolute error (PAE) where lower scores indicate better performance.
Table 5
Mixed 2 (Condition) 3 (Time) ANCOVA on Pre-, Post-, and
Delayed Posttest Fraction Concepts Scores by Condition
(Intervention vs. Control) Controlling for Vocabulary
Source df F p MSE p
Condition 1, 48 10.53 .002 18.32 .18
Time 1.74, 83.71 .49 .611 4.28 .01
Vocabulary 1, 48 11.85 .001 18.32 .20
Time Vocabulary 1.74, 83.71 .37 .662 4.28 .01
Time Condition 1.74, 83.71 10.41 .001 4.28 .18
Within-subject contrasts: Time
Condition interaction
Linear 1, 48 6.31 .015 4.59 .12
Quadratic 1, 48 16.93 .001 2.89 .26
Note.MSE mean square error.
Pre-test Post-test Delayed Post-test
Fraction Concepts (Number Correct)
Control Intervention
Figure 4. Estimated marginal means of fraction concepts controlling for
receptive vocabulary.
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Exploration of Attentive Behavior as Moderator of the
Effect of Intervention on Fraction Concepts
As displayed in Table 10 and in supplemental tables S11 and
S12, the exploratory moderation models revealed a significant
interaction between attentive behavior and intervention only when
predicting posttest fraction concepts (.30, p.048). To
explicate the significant interaction, simple effects were calcu-
lated. This was done with follow-up split regressions based on
condition using the same variables included in the main effect
model for fraction concepts. Posttest fraction concepts scores were
regressed onto attentive behavior scores controlling for pretest
fraction concepts. Results demonstrated that although attentive
behavior scores were trending toward significance for the control
condition (.330, p.085), they were unrelated to posttest
fraction concepts scores for the experimental intervention group
(.020, p.921). These results suggest that on fraction
concepts, students who normally demonstrated low attentive be-
havior in the classroom may have benefitted more from the ex-
perimental intervention than the regular school intervention pro-
vided to the business-as-usual control condition. This moderation
is displayed in Figure 7, which displays the predicted fraction
concepts posttest score of a student who was one standard devia-
tion above the mean and a student who was one standard deviation
below the mean on the SWAN scale.
We evaluated the effectiveness of an experimental intervention
designed to improve fraction learning in sixth-grade students with
mathematics difficulties. The experimental intervention builds on
a growing body of research stressing the importance of number
lines as representational tools for learning key mathematics con-
cepts (Fazio et al., 2016; Siegler et al., 2010). Understanding that
all real numbers can be represented as magnitudes on a number
line provides a unifying structure for most mathematical learning,
including fractions (Siegler et al., 2011). The experimental inter-
vention was further supported by validated strategies from studies
on the Science of Learning.
Pretest Posttest Delayed Posttest
Number Line Estimation (PAE)
Control Intervention
Figure 5. Estimated marginal means of fraction number line estimation
controlling for receptive vocabulary.
Pretest Posttest Delayed Posttest
Fraction Comparison (Number Correct)
Control Intervention
Figure 6. Estimated marginal means of fraction comparison controlling
for receptive vocabulary.
Table 6
Mixed 2 (Condition) 3 (Time) ANCOVA on Pre-, Post-, and
Delayed Posttest Fraction Number Line Estimation (PAE) by
Condition (Intervention vs. Control) Controlling for Vocabulary
Source df F p MSE p
Condition 1, 48 17.70 .001 132.06 .27
Time 1.72, 82.70 2.17 .128 32.39 .04
Vocabulary 1, 48 4.70 .035 132.06 .09
Time Vocabulary 1.72, 82.70 3.20 .053 32.39 .06
Time Condition 1.72, 82.70 4.80 .014 32.39 .09
Within-subject contrasts: Time
Condition interaction
Linear 1, 48 3.81 .057 38.65 .07
Quadratic 1, 48 7.05 .011 17.15 .13
Note.MSE mean square error.
Table 7
Mixed 2 (Condition) 3 (Time) ANCOVA on Pre-, Post-, and
Delayed Posttest Fraction Comparisons by Condition
(Intervention vs. Control) Controlling for Vocabulary
Source df F p MSE p
Condition 1, 48 10.69 .002 53.65 .18
Time 2, 96 1.21 .301 16.47 .03
Vocabulary 1, 48 .04 .837 53.65 .01
Time Vocabulary 2, 96 1.29 .282 16.47 .03
Time Condition 2, 96 3.77 .027 16.47 .07
Within-subject contrasts: Time
Condition interaction
Linear 1, 48 3.59 .064 18.73 .07
Quadratic 1, 48 4.01 .051 14.70 .08
Note.MSE mean square error.
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As predicted, the experimental intervention led to large and
meaningful improvements on student measures aligned specifi-
cally to the experimental intervention (i.e., fraction number line
estimation acuity and fraction magnitude comparisons) as well as
on a broader measure of fraction concepts. Pre- to posttest effect
sizes (Hedges’ g) were large, ranging from 0.82 to 1.09. Impor-
tantly, these improvements generally held at a 7-week delay.
Because of stratified random sampling and random assignment, we
are confident that receipt of the experimental intervention was the
only systematic difference between conditions. Conditions did not
differ on fraction pretest measures, demographics, and cognitive
measures (with the exception of vocabulary, which we controlled
for in all subsequent analyses). Further, high instructor fidelity
reflects consistent administration of the experimental intervention
across instructors. Additionally, we used a validated screener
(Rodrigues et al., 2017) to identify selected students at risk for
mathematics failure, ensuring that our sample was highly targeted
to increase the applicability of our current findings to the appro-
priate population.
Our findings add to the current literature on the effectiveness of
a number-line approach to teaching fractions in important ways.
First, the findings show that such an experimental intervention
successfully boosts fraction knowledge in older students (i.e., sixth
graders) whose mathematical difficulties are likely to be en-
trenched, relative to those of younger learners. Much of the pre-
vious fraction number line intervention work has focused on
at-risk fourth graders (Fuchs et al., 2014) because this is the grade
when formal fraction instruction first takes place. In the present
study, we undertook a more challenging task. That is, we chose to
focus on sixth-grade students who had already received several
years of formal instruction on fractions, but who still showed weak
performance on a validated fraction screener at sixth grade entry.
Mathematics achievement shows steep declines around the time
that students transition into middle school (Wang & Pomerantz,
2009), and fractions are a key element of mathematics achieve-
ment during this period. According to the 2015 National Assess-
ment of Educational Progress (NAEP), only 40% of fourth graders
in the U.S. are proficient in mathematics and this declines to 33%
in eighth grade. Strikingly, without intervention, studies have
shown that students with mathematics difficulties make minimal
gains in fraction learning between sixth and eighth grades (Maz-
zocco & Devlin, 2008; Siegler & Pyke, 2013). As such, sixth grade
is a critical time to catch students who may have fallen through the
cracks in fraction learning during the intermediate grades.
Second, experimental intervention students’ gains were sus-
tained after an almost two-month delay. We speculate that inte-
gration of strategies based on learning principles worked as in-
tended, leading to deeper learning that was retained at a 7-week
delay. Explicit instruction likely provided the appropriate teacher
modeling and student practice necessary to build students’ reper-
toire of problem-solving strategies (Doabler & Fien, 2013). Rep-
resentational gestures encouraged students to form abstract repre-
sentations of fractions, as suggested by Ping and Goldin-Meadow
(2008), leading to improved performance on the three conceptually
oriented fraction measures. Presenting information verbally and
visually likely fostered dual coding and improved retention of the
learning material (Cuevas, 2016) along with spaced and inter-
leaved practice (Rohrer et al., 2015). Finally, it is probable that
immediate and process-oriented feedback throughout the lessons
necessary for learners with low prior knowledge encouraged stu-
dents to refine their thinking and problem-solving skills (Graesser
et al., 2005; Fyfe & Rittle-Johnson, 2016). However, it is impor-
tant to note that we chose to employ these learning principles in
combination, rather than isolating the effects of each feature. Our
“engineering” approach aimed to maximize student learning
through the use of established, research-based learning strategies.
Future work would need to experimentally isolate these features to
determine causal effects of each principle on the development of
fraction understanding in particular.
Our sample demonstrated below-average performance on most
of the cognitive and behavioral measures, including working mem-
ory, nonverbal reasoning, receptive vocabulary, and classroom
attentive behavior. Even after employing random assignment strat-
Table 8
Mixed 2 (Condition) 3 (Time) ANCOVA on Pre-, Post-, and
Delayed Posttest Fraction Arithmetic by Condition (Intervention
vs. Control) Controlling for Vocabulary
Source df F p MSE p
Condition 1, 48 1.25 .269 42.11 .03
Time 2, 96 .02 .982 7.85 .01
Vocabulary 1, 48 2.22 .143 42.11 .04
Time Vocabulary 2, 96 .26 .769 7.85 .01
Time Condition 2, 96 .33 .719 7.85 .01
Within-subject contrasts: Time
Condition interaction
Linear 1, 48 .16 .689 7.54 .01
Quadratic 1, 48 .49 .489 8.16 .01
Note.MSE mean square error.
Table 9
Effect Sizes on Differences Between Estimated Marginal Means Between Conditions
Pre–Posttest Pre–Delayed posttest
Measure gU3
index gU3
Fraction concepts 1.09 .87 36.74% .66 .75 24.89%
Fraction NLE (PAE) .85 .19 30.81% .60 .27 23.01%
Fraction comparison .82 .80 29.76% .61 .73 23.17%
Fraction arithmetic .17 .57 6.71% .11 .54 4.42%
Note. Comparisons of estimated marginal means controlling for receptive vocabulary. Fraction number line
estimation (FNLE) is scored as percent absolute error (PAE) where lower scores indicate better performance.
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ified by classroom, our intervention and control group showed
significant differences on receptive vocabulary prior to the start of
the intervention. When controlling for receptive vocabulary we
found that vocabulary predicted performance on the general frac-
tion concepts measure and number line task but did not differen-
tially predict performance by condition. This is not surprising
when considering the fraction concepts measure which includes
several word problems. Vocabulary did not predict performance on
the fraction comparison or arithmetic measures. Although not a
focus of the current study, these findings may suggest a potentially
interesting area for future work. A recent study demonstrated that
syntactic ability was a significant predictor of first and second
graders’ mathematics performance whereas vocabulary was not
(Chow & Ekholm, 2019). Yet, earlier work suggests that a range
of oral language skills, including receptive vocabulary, are impor-
tant predictors of fraction performance in particular (Chow &
Jacobs, 2016). Future work should explore further the role that oral
language skills, and vocabulary in particular, may play in learning
fraction concepts.
One curious finding was that students demonstrated average
inhibitory control as measured by the flanker task, which was not
highly correlated with classroom attentive behavior (r.289).
This low correlation suggests that these measures (i.e., flanker task
and SWAN) assess related but distinct forms of attention. We
propose that the measure of classroom attentive behavior used in
the current study taps students’ self-regulation skills, whereas the
flanker task is an assessment of selective visual attention in par-
ticular. The exploratory analyses involving classroom attentive
behavior suggests the SWAN measure as a potentially important
measure of attention as described below.
A potentially interesting exploratory finding was the moderating
effect of attentive behavior on condition. That is, students who
exhibited inattentive behavior in their regular mathematics class-
room (based on teacher reports) tended to benefit more from the
experimental intervention than from the regular school interven-
tion that was provided to the business-as-usual control group (i.e.,
working with the computer adaptive mathematics software). High
attention in the classroom allows learners to stay on task and focus
on key parts of a problem (Finn, Pannozzo, & Voelkl, 1995). Prior
research has confirmed the direct and indirect influences of atten-
tive behavior on broad fraction skills (e.g., Hecht et al., 2003).
Classroom attention has been linked to specific fraction skills
including fraction conceptual knowledge (Hansen et al., 2015),
fraction magnitude estimation (Ye et al., 2016), and fraction arith-
metic (Ye et al., 2016). Arguably, the explicit supports provided in
our experimental intervention helped students attend more often
than the school control intervention administered individually on
computers. It is also possible that the small, synergistic groups that
our experimental intervention provided were important in this
respect. In any case, the results should be interpreted cautiously,
considering that this moderating effect was present only on the
broad fraction concepts measure—not on the number line estima-
tion or comparison tasks. These exploratory findings should spark
future systematic study with a larger sample on the relationship
between classroom attentive behavior and the effects of fractions
Although our primary hypotheses received support, we did not
find improvements in fraction arithmetic for the experimental
intervention students. Rather, growth in fraction arithmetic fol-
lowed a pattern similar to that of the control condition, with slight
increases between pre- and posttest and then a slight decline at
A correlation matrix between pretest fraction measures and cognitive
competencies is available in a supplemental online appendix.
Low (-1 SD) High (+ 1 SD)
Post-Fraction Concepts (# correct)
Attentive Behavior (SWAN)
Control Intervention
Figure 7. Moderating effect of attentive behavior on predicted fraction
concepts score by condition.
Table 10
Prediction Models of Posttest Fraction Concepts Scores
Fraction concepts
Pretest model Main effects model Covariate model Moderation model
Measure B(SE)B(SE)B(SE)B(SE)
Pretest fraction concepts .47
.65 (.18) .49
.67 (.15) .42
.58 (.16) .35
.48 (.17)
Experimental intervention .46
3.38 (.80) .47
3.44 (.79) .47
3.43 (.76)
Attentive behavior .18 .05 (.04) .41
.13 (.05)
Attentive behavior Intervention .30
.14 (.07)
.22 .43 .46 .50
from prior model .21
.03 .05
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delayed posttest. It is possible that the minor changes in students’
fraction arithmetic were associated with students’ regular class-
room instruction rather than differential experiences during the
intervention period. The finding is not surprising given that our
experimental intervention focused on developing fraction con-
cepts, although some research suggests that instruction in fraction
magnitudes leads to improved fraction arithmetic as well as con-
cepts (Bailey, Hoard, Nugent, & Geary, 2012). However, a post
hoc power analysis suggested we were underpowered for detecting
small effects. Another issue is that our arithmetic assessment may
not have been sufficiently sensitive to growth. Many of the items
did not match the content of the limited arithmetic instruction
provided in our experimental intervention, leading to few oppor-
tunities for students to demonstrate the arithmetic they may have
learned. For example, although modeling multiplication of a frac-
tion and a whole number was covered in two lessons of focused
instruction and interleaved review over seven lessons, only three of
the arithmetic items assessed this skill. Examination of perfor-
mance at the item level showed that experimental intervention
students answered three particular arithmetic items more accu-
rately at posttest than did controls: two addition with common
denominators and one multiplication (1/2 24). For each of these
problems, 20% to 30% more of the students in the experimental
intervention group answered correctly than those in the control
condition. Future iterations of our experimental intervention
should use measures that align fraction arithmetic problems more
closely to the content that is taught as well as address the concep-
tual underpinnings of procedures taught in the classroom to help
students avoid misapplication (Rittle-Johnson, 2017).
Our lack of specific information about the instruction offered to
the business-as-usual control during their corresponding school
intervention period as well as in regular mathematics classes limits
interpretation of the results. Although we can confirm that students
in the control condition were simultaneously receiving an individ-
ualized mathematics intervention offered by their schools, the
software programs were adaptive. As such, the specific content
each control student was exposed to likely varied, but this was not
made available to us. We also were not given specific information
on how fractions are normally taught in their regular mathematics
classes, although both participating schools followed CCSS-M
benchmarks (NGACBP & CCSSO, 2010) and used the same
curriculum (Connected Mathematics Project). The sixth grade
CCSS-M benchmarks emphasize understanding of rational num-
bers on a number line, but it is unclear to what extent our sample’s
regular mathematics teachers incorporated the number line into
their regular instruction. The finding that children in both groups
gained at the same rate in arithmetic but not on fraction magnitude
concepts (e.g., fraction number line estimation) suggests that arith-
metic may have been the focus in regular mathematics classes. Still,
if differences did exist in the treatment of fractions in their classrooms,
these differences would be randomly distributed across conditions and
should not systematically influence the impact of our experimental
intervention on students’ fraction knowledge.
One interesting finding is that, although students in the exper-
imental intervention condition outperformed the control group at
delayed posttest on the broad fraction concepts measure (with
medium to large effects), the control group showed a small but
noticeable improvement between immediate and delayed posttest
while the intervention condition levels off. Formal fraction instruc-
tion within students’ regular mathematics class would have con-
cluded at the point of immediate posttest. Even if fraction instruc-
tion continued in some of the regular mathematics classes, this
would not have systematically differed by condition as students
from both the intervention and control groups were distributed
evenly within each regular mathematics class. One plausible ex-
planation is that students in the control group as well as their
parents, mathematics teachers, and intervention period teachers
were each made aware of their need for mathematics intervention
particularly in the area of fractions. Students were also aware that
they would be taking the fraction assessment again. It is possible
that the control group students’ mathematics teachers assigned
these students additional practice with fractions either during class
or as homework between immediate and delayed posttest to make
up for the lack of the comprehensive fraction intervention that the
experimental students received. Unfortunately, we do not have
access to detailed records of regular classroom instruction that
occurred over the course of data collection for the current study.
However, school administrators and teachers did not mention this
issue to our research team. Future work should include collection
of more detailed information on classroom instruction received
over the course of the study to help interpret results more contex-
The pattern of results for the broad fraction concepts measure
suggests that the control group may eventually catch up to the
intervention group. Future work should include one or more ad-
ditional time points to assess longer-term retention of improve-
ments by condition. Because of the lack of attention fractions
receives in formal instruction within the regular sixth-grade math-
ematics class, students may need a longer-term intervention to show
continued improvements, either in the form of a longer intervention
period, additional refresher sessions scheduled later in the school year,
or the same number of sessions but spread out across the school year.
These possibilities should be explored further.
To understand the practical significance of the current findings,
we examined how students in our experimental intervention group
compared with their higher-performing peers at the completion of
the intervention period. A previous longitudinal study with a
sample representative of the population that the current study
participants were sampled from found three distinct growth trajec-
tories in fraction number line estimation skills (Resnick et al.,
2016), using a similar FNLE measure. As noted earlier, one group
of students in the 2016 study started accurate in the fourth grade
and ended accurate in the sixth grade (Class 1). These students
were more likely to have high sixth grade mathematics achieve-
ment scores. Two groups of students started off with inaccurate
number line estimates but one group showed steep improvements
(Class 2) while the other showed minimal growth (Class 3). The
group that started off inaccurate but showed substantial gains
between fourth and sixth grades was more likely to have average
sixth-grade mathematics achievement scores. The group that
started off inaccurate and showed minimal growth was more likely
to have low sixth grade mathematics achievement scores. In the
present study sample, students made inaccurate number line esti-
mates prior to the start of the intervention, with scores similar to
the two inaccurate classes in the aforementioned 2016 longitudinal
study. However, at the completion of the present intervention and
at delayed posttest, the experimental intervention group showed
steep growth with scores similar to Class 2, which was the group
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that was more likely to have average mathematics achievement
scores at the end of sixth grade. The control condition showed no
growth and had scores similar to Class 1 in the aforementioned
2016 study, which was the group that was likely to have low
mathematics achievement scores. We suspect that our experimen-
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between low and average-performing students, although further
work is needed to confirm this suggestion.
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Received June 1, 2018
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Accepted May 13, 2019
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... Specifically, participants in Fuchs et al. (2016) and Malone et al. (2019) were fourth-grade students with general math difficulties (below the 35 th percentile in a standardized math assessment). Participants in Barbieri et al. (2020) and Dyson et al. (2020) were sixth graders who had low fraction skills in a screening assessment comprised of fraction items from Braithwaite & Siegler (2021) the NAEP. Similarly, participants in Jayanthi et al. (2021) were fifth graders with low fraction skills (15 th to 37 th percentile) according Test for Understanding of Fractions. ...
... However, tasks varied on the type of arithmetic operations included. Fuchs et al. (2016) and Malone et al. (2019) used only addition and subtraction problems; Dyson et al. (2020) and Barbieri et al. (2020) also included multiplication problems; and Jayanthi et al. (2021) included the four operations. Moreover, not all interventions included fraction arithmetic instructions, specifically Malone and Fuchs focused on fraction concepts, making fraction arithmetic a far transfer task. ...
... Schematic renderings of training stimuli and tasks. A) InBarbieri et al. (2020), numberlines were introduced in the context of a racing game. B) InAbreu-Mendoza et al. (2021), students represented proportions using Cuisenaire then used symbols (letters) to represent them. ...
Full-text available
By the end of elementary school, students should be able to solve fraction arithmetic and comparison problems. However, less than 30 % of students in the United States have accomplished these educational milestones by the eighth grade. Not only is fraction knowledge as vital to learn more complex math, like algebra, it also has major implications on individuals' health and employment outcomes. Thus, it has become crucial to develop educational methods to enhance students' fraction understanding. In contrast to these struggles with symbolic fractions, young children can work surprisingly well with nonsymbolic representations of proportions, and nonsymbolic ratio processing skills are also positively related to fraction ability in older children and adults. These findings have led to educational interventions which leveraged nonsymbolic skills, primarily via numberlines, to enhance symbolic fraction understanding. This paper aims to offer a systematic review of these emerging intervention and training studies. We identified 19 papers, representing 22 studies, which we grouped in three categories: classroom-based interventions, multiple-session trainings, and single-session training studies. These studies provide an optimistic picture of the malleability of students' fractions skills: Across categories, placing nonsymbolic and symbolic representations of fractions on numberlines successfully enhanced fraction skills of low and typically achieving students, resulting in small-to-large effect sizes. These results suggest that fostering nonsymbolic skills may be important in addressing the persistent bottleneck that fractions pose in mathematical knowledge.
... To develop a comprehensive understanding of the measure construct of fractions, the student should be able to locate the fraction on the number line as well as identify it on a specific point on the number line (Smith, 2002). Intervention studies also show that practicing number line tasks may lead to improved knowledge of fractions (Barbieri et al., 2019;Fuchs et al., 2016;Saxe et al., 2013). Conclusively, the above-mentioned five subconstructs focus on the different properties of fractions that are required to attain mastery in this domain. ...
... Importantly, the number line can be used to teach students the unique property of density of rational numbers (Behr et al., 1993). Indeed, intervention studies show that practicing number line tasks lead to improved fraction knowledge (Barbieri et al., 2019;Fuchs et al., 2016;Saxe et al., 2013). Therefore, the game includes four different task types that make use of number lines. ...
Fraction knowledge is critical for the holistic development of mathematical knowledge. While fraction instruction typically begins in elementary school, children often encounter relational numerical concepts much earlier in their environment (e.g. sharing candies, varying the sweetness of a drink, baking muffins). A recently proposed theory, the Ratio Processing System (RPS) theory, posits that the understanding of symbolic fractions and non-symbolic relational magnitudes are fundamentally intertwined. However, research on the RPS theory and fraction learning interventions in the classroom are limited. In this dissertation, we examine both symbolic and non-symbolic relative magnitude processing from the perspectives of both cognitive neuroscience and educational game-based approach. First, performance accuracy on a match-to-sample task reveals individuals with varied mathematics skills to be perceptually sensitive to non-symbolic ratios but not to symbolic fractions. Second, univariate and multivariate analyses of neural activity patterns using a fMRI-adaptation paradigm suggest for an absence of overlapping brain activations for symbolic and non-symbolic magnitudes. Third, analyses of fifth grader’s fraction knowledge after playing a fraction educational game developed in the context of this thesis (Math Matthews Fractions) revealed that the game did not improve overall fraction skills above traditional classroom instruction. However, it was successful at improving decimal knowledge. The results of this thesis lead us to argue for fraction instruction focused on both perceptual methods as well as building connections between the multiple constructs of fractions. Future research holds great potential for examining fraction games that support teachers in building a holistic fraction understanding, rooted in the percept-concept links.
... Such training could be extended to younger children who could potentially use their understanding of transitive inferences to develop their knowledge of some mathematical concepts that rely on relational processing and deduction. As for fraction knowledge, carefully designed instruction (Barbieri et al., 2020) and educational games (Fazio et al., 2016) have been developed, and these interventions have been shown to bring about significant improvement in the understanding of fractions. Finally, the acquisition of the Relation to Operands principle may be facilitated through exposing children to a mixture of correct and incorrect arithmetic problems as well as having them to compare the relative magnitudes of the numbers in arithmetic problems (Prather & Alibali, 2011). ...
... for developing an accurate sense of fraction magnitude (e.g., Fuchs et al., 2013;Rodrigues, Dyson, Hansen, & Jordan, 2017). For example, an intervention centering on learning fractions using a number line improved sixth graders' understanding of fraction magnitude and fraction concepts (Barbieri, Rodrigues, Dyson, & Jordan, 2019). In another interventionbased study, number line training transferred to second and third grader's performance on fraction magnitude comparison, whereas area models (i.e., part-whole models) did not (Hamdan & Gunderson, 2017). ...
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The integrated theory of numerical development provides a unified approach to understanding numerical development, including acquisition of knowledge about whole numbers, fractions, decimals, percentages, negatives, and relations among all of these types of numbers (Siegler, Thompson, & Schneider, 2011). Although, considerable progress has been made toward many aspects of this integration (Siegler, Im, Schiller, Tian, & Braithwaite, 2020), the role of percentages has received much less attention than that of the other types of numbers. This chapter is an effort to redress this imbalance by reporting data on understanding of percentages and their relations to other types of numbers. We first describe the integrated theory; then summarize what is known about development of understanding of whole numbers, fractions, and decimals; then describe recent progress in understanding the role of percentages; and finally consider instructional implications of the theory and research.
... Throughout practice, teachers should actively help students select and properly use the strategy appropriate for a particular problem type. After all strategies within a unit are introduced, problem types for every other problem can be interleaved to encourage students to select the appropriate strategy and fine-tune their organizational framework for problem-solving within that domain (Barbieri, Rodrigues, Dyson, & Jordan, 2019). ...
Mathematics difficulties (MD) are widespread. Higher mathematics achievement is associated with college success and greater job opportunities. In this chapter, we address the intersection between cognitive research on the science of learning and the education of students with MD. First, a conceptual framework for mathematics learning is provided, including consideration of individual differences in domain general and domain specific learning processes. Next, learning principles that promote successful learning in mathematics are described, along with examples of how the principles can be translated into educational practice for students with MD. We argue that in addition to providing developmentally appropriate instruction in content knowledge, such as whole number and fraction knowledge, application of more general learning principles validated by research will lead to deeper and more durable learning for struggling students. These learning principles include studying and comparing correct as well as incorrect worked out problem-solutions; using integrated visual and verbal models to reduce splitting attention; interleaving or varying practice with problems of different types; providing frequent cumulative practice or review that is spaced out over time; connecting and integrating concrete and symbolic representations; presenting arithmetic problems in different formats; using physical movements and gestures to promote learning; and incorporating activities with number lines to increase learning of whole numbers and fractions.
Understanding rational numbers is critical for secondary mathematics achievement. However, students with mathematics difficulties (MD) struggle with rational number topics, including fractions, decimals, and percentages. The purpose of this systematic review was to describe the instructional foci of rational number interventions, determine the overall effect size, and explore potential moderators. Forty-three studies were included and 150 effect sizes were meta-analyzed using robust variance estimation. The majority of studies focused on teaching fraction magnitude and arithmetic. An overall effect size of g = 1.02 [0.80, 1.25] was found for rational number interventions favoring treatment conditions over business as usual control. Proximal measures contributed to higher effect sizes than distal measures. Limitations included a high number of fraction interventions contributing to the overall effect size and a large amount of heterogeneity among study effect sizes.
An understanding of concepts related to geometric measurement is considered to be critical to the development of individuals’ mathematics knowledge. Specifically, the National Mathematics Advisory Panel’s 2008 report listed the skills of area and perimeter as foundational for algebra readiness. Yet, this content knowledge continues to be an underdeveloped skill area for many school–age children and especially those with learning disabilities. This article provides educators with the following four strategies that are grounded in research and can be implemented during instruction on area and perimeter: (a) instruction using manipulatives, (b) focusing on salient variables within problems, (c) utilizing visual–chunking representations, and (d) incorporating contextualized scenarios and experiences.
El estudio tiene por objetivo identificar los tipos de pensamiento computacional presentes en la resolución de problemas con fracciones, así como explorar el rol del aprendizaje adaptativo en esta actividad, en niños de cuarto grado de educación básica. El diseño de investigación es de estudio de caso y consistió en la resolución de ejercicios con fracciones por un grupo de estudiantes, a través de una aplicación móvil basada en el aprendizaje adaptativo desarrollada para realizar las pruebas. Los resultados muestran que las modalidades de pensamiento computacional presentes en el test son las de ensayo y error, iteración y recursividad, con una predominancia de esta última. Así mismo, la adaptación de los ejercicios por el sistema, en función de las capacidades individuales de los niños, señala los beneficios que tiene para el aprendizaje de las fracciones el apoyarse en este tipo de tecnología.
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The articles in this special issue provide state-of-the-art reviews of the brain and cognitive systems that are engaged during some aspects of mathematical learning, as well as the self-beliefs, anxiety, and social factors that influence engagement with mathematics, along with discussion of any associated sex differences. These issues are integrated into an evolutionary perspective that includes discussion of how evolved brain and cognitive systems might be co-opted for learning in the evolutionarily novel domain of mathematics. Attitudes and beliefs about mathematics are considered in the context of the evolution of self-awareness that in turn explains why many students do not value mathematics, despites its importance in the modern world, as highly as many other personal traits, such as their physical appearance. The overall argument is that reflecting on academic learning and attitudes from an evolutionary perspective provides insights into student learning and self-beliefs about learning that might otherwise elude explanation.
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We present 1st-grade, 2nd-grade, and 3rd-grade impacts for a 1st-grade intervention targeting the conceptual and procedural bases that support arithmetic. At-risk students (average age at pretest = 6.5) were randomly assigned to 3 conditions: a control group (n = 224) and 2 variants of the intervention (same conceptual instruction but different forms of practice: speeded [n = 211] vs. non-speeded [n = 204]). Impacts on all 1st-grade content outcomes were significant and positive, but no follow-up impacts were significant. Many intervention children achieved average mathematics achievement at the end of 3rd grade, and prior math and reading assessment performance predicted which students will require sustained intervention. Finally, projecting impacts two years later based on non-experimental estimates of effects of 1st-grade math skills overestimates long-term intervention effects.
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Students in the United States consistently underperform on state tests of mathe- matical pro ciency (e.g., Kim, Schneider, Engec, & Siskind, 2006; Pennsylvania Department of Education, 2011) and in international comparisons on the Trends in International Mathematics and Science Study (TIMSS; e.g., Mullis, Martin, Foy, & Arora, 2012) and Programme for International Student Assessment (PISA; e.g., Fleischman, Hopstock, Pelczar, & Shelley, 2010). Among other issues, aspects of early mathematics instruction can interfere with later learn- ing (McNeil et al., 2006; see also McNeil et al., Chapter 8; Van Hoof et al., Chapter 5) and students are not adequately prepared to tackle dif cult gate- keepers, such as fractions (Booth & Newton, 2012) and algebra (Department of Education, 1997), that in turn prevent them from advancing in the elds of Science, Technology, Engineering, and Mathematics (STEM). To address these issues, educators and researchers have repeatedly called for mathematics instruction in the United States to be based on evidence (e.g., CCSSI, 2010; NCLB, 2002; NMAP, 2008).Over the past few decades, the eld of cognitive science has identi ed fac- tors that have the potential to substantively improve students’ learning. These principles have been detailed in several reviews (e.g., Dunlosky, Rawson, Marsh, Nathan, & Willingham, 2013; Koedinger, Booth, & Klahr, 2013; Pashler et al., 2007), and many involve a comparison of different approaches (such as spaced vs. massed practice), showing that one type of instructional technique is superior to another type of instructional technique (Koedinger et al., 2013). Many of these cognitive science principles are potentially useful for improving mathematics instruction, perhaps especially those concerning abstract and con- crete representations, analogical comparison, feedback, error re ection, scaf- folding, distributed practice, interleaved practice, and worked examples.
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The magnitude comparison task and the number line estimation task are widely used in the literatures on numerical cognition, mathematical development and mathematics education. It has been suggested that these tasks assess a central component of mathematical competence and, thus, are useful tools for diagnosing mathematical competence and development. In the current chapter, we provide a comparative literature review of how strongly and under which conditions magnitude comparison and number line estimation are associated with broader mathematical competence. We found that both tasks reliably correlate with counting, arithmetic, and mathematical school achievement. Participants use different sets of solutions strategies in the two tasks, and the correlations with competence are higher for number line estimation than for magnitude comparison suggesting that the two tasks assess related but partly different aspects of mathematical competence. Number line estimation might be a more useful tool for diagnosing and predicting broader mathematical competence than magnitude comparison.
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Developing strong knowledge about mathematics is important for success academically, economically, and in life, but many children fail to become proficient in math. Research on the developmental relations between conceptual and procedural knowledge of math provides insights into the development of knowledge about math. First, competency in math requires children to develop conceptual knowledge, procedural knowledge, and procedural flexibility. Second, conceptual and procedural knowledge often develop in a bidirectional, iterative fashion, with improvements in one type of knowledge-supporting improvements in the other, as well as procedural flexibility. Third, learning techniques such as comparing, explaining, and exploring promote more than one type of knowledge about math, indicating that each is an important learning process. Researchers need to develop and validate measurement tools, devise more comprehensive theories of math development, and build more bridges between research and educational practice.
The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses — the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferronitype procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.
The purpose of this study was to examine the associations between child language ability and mathematics performance. First- and second-grade children (N = 365) were assessed on language ability and mathematics performance. Structural equation models revealed that receptive syntax and a broad screening tool significantly predicted math performance, while vocabulary did not. Path analyses corroborate these findings, with receptive syntax emerging as the only significant predictor of all indicators of mathematics performance. We conclude that syntax is a strong predictor of mathematics performance while vocabulary is not. Further, although many studies use receptive vocabulary to index language, it may not be the most predictive of or the best proxy for language ability in young children in the context of mathematics learning.
Children's ability to place fractions on a number line strongly correlates with math achievement. But does the number line play a causal role in fraction learning or does it simply index more advanced fraction knowledge? The number line may be a particularly effective representation for fraction learning because its properties align with the desired mental representation and take advantage of preexisting spatial-numeric biases. Using a pretest-training-posttest design, we examined second and third graders' fraction learning in 3 conditions: number line training, area model training, and a non-numerical control. Children who received number line training improved at representing fractions with number lines, and children who received area model training improved at representing fractions with area models. Critically, only the number line training led to transfer to an untrained fraction magnitude comparison task. We conclude that the number line plays a causal role in children's fraction magnitude understanding, and is more beneficial than the widely used area model. (PsycINFO Database Record