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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. http://dx.doi.org/10.1037/edu0000384

CITATION

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. http://dx.doi.org/10.1037/edu0000384

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: http://dx.doi.org/10.1037/edu0000384.supp

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

Delaware.

Jessica Rodrigues is now at College of Education, University of Mis-

souri.

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.

E-mail: barbieri@udel.edu

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

0022-0663/19/$12.00 http://dx.doi.org/10.1037/edu0000384

1

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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2BARBIERI, RODRIGUES, DYSON, AND JORDAN

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

investigated.

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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3

IMPROVING FRACTION UNDERSTANDING

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

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

4BARBIERI, RODRIGUES, DYSON, AND JORDAN

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.

Method

Participants

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

ⴱ

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

5

IMPROVING FRACTION UNDERSTANDING

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.

6BARBIERI, RODRIGUES, DYSON, AND JORDAN

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.,

2016).

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.

Procedure

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

7

IMPROVING FRACTION UNDERSTANDING

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

classroom.

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

eighths

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

a

••

Measuring with cups

b

•

Adding/Subtracting with unlike denominators •

a

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.

b

Involved only denominators common to measuring cups: halves, thirds,

fourths, and eighths.

8BARBIERI, RODRIGUES, DYSON, AND JORDAN

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.

9

IMPROVING FRACTION UNDERSTANDING

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

lessons.

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.

10 BARBIERI, RODRIGUES, DYSON, AND JORDAN

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 pkⱕ

k

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.

1

As

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

1

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.

11

IMPROVING FRACTION UNDERSTANDING

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.

Results

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

(n23)

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 t1.185, 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

ⴱ

Working memory 86.86 11.50 90.08 10.93 t1.009, 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 t1.120, p.268, ns

% % Sig.

2

Categorical variables

Male 34.78% 50%

2

1.192, p.275, ns

Special education (Spec Ed) 30.43% 32.14%

2

.017, p.896, ns

English language learner (ELL) 13.04% 7.14%

2

.497, p.481, ns

Math learning disability (MLD) 30.43% 25.00%

2

.187, p.665, ns

Reading learning disability (RLD) 30.43% 28.57%

2

.021, p.884, ns

Behavioral disability .00% 3.57%

2

.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.

ⴱ

p.05.

12 BARBIERI, RODRIGUES, DYSON, AND JORDAN

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

posttest.

Effect Sizes

The experimental intervention yielded large effects on fraction

concepts (g1.09), FNLE (g0.85), 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

Measure EMM SE EMM SE EMM SE EMM SE EMM SE EMM SE

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

2

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.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

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.

13

IMPROVING FRACTION UNDERSTANDING

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.

Discussion

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.

0.00

4.00

8.00

12.00

16.00

20.00

24.00

28.00

32.00

36.00

Pretest Posttest Delayed Posttest

Number Line Estimation (PAE)

Control Intervention

Figure 5. Estimated marginal means of fraction number line estimation

controlling for receptive vocabulary.

0.00

4.00

8.00

12.00

16.00

20.00

24.00

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

2

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

2

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.

14 BARBIERI, RODRIGUES, DYSON, AND JORDAN

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

2

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

Improvement

index gU3

Improvement

index

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.

15

IMPROVING FRACTION UNDERSTANDING

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).

2

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

interventions.

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

2

A correlation matrix between pretest fraction measures and cognitive

competencies is available in a supplemental online appendix.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

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

ⴱⴱ

.65 (.18) .49

ⴱⴱ

.67 (.15) .42

ⴱⴱ

.58 (.16) .35

ⴱⴱ

.48 (.17)

Experimental intervention .46

ⴱⴱ

3.38 (.80) .47

ⴱⴱ

3.44 (.79) .47

ⴱⴱ

3.43 (.76)

Attentive behavior .18 .05 (.04) .41

ⴱ

.13 (.05)

Attentive behavior Intervention .30

ⴱ

.14 (.07)

R

2

.22 .43 .46 .50

R

2

from prior model .21

ⴱⴱ

.03 .05

ⴱ

F13.54

ⴱⴱ

18.16

ⴱⴱ

13.22

ⴱⴱ

11.61

ⴱⴱ

ⴱ

p.05.

ⴱⴱ

p.01.

16 BARBIERI, RODRIGUES, DYSON, AND JORDAN

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-

tually.

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

17

IMPROVING FRACTION UNDERSTANDING

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-

tal intervention helps minimize the mathematics achievement gap

between low and average-performing students, although further

work is needed to confirm this suggestion.

Overall, our intervention findings hold promise for strengthen-

ing fraction skills in struggling students, which then should bolster

later mathematics achievement more generally (Bailey et al., 2012)

and algebra readiness in particular (Booth & Newton, 2012). We

saw large improvements in fraction concepts, fraction number line

estimation, and fraction comparison skills after just six weeks of

targeted instruction, and these improvements held seven weeks

later. Translating research on learning principles into real-world

classrooms for struggling students while keeping the number line

representation of fractions at the forefront seemed to have a

powerful effect on the development of foundational mathematics

knowledge. Perhaps the most important lesson to be learned from

this study is that at-risk students can show marked improvements

in their fraction knowledge several years after receiving initial

instruction, if provided with a purposeful intervention designed to

capitalize on learning principles from cognitive science and math-

ematics development.

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Received June 1, 2018

Revision received March 25, 2019

Accepted May 13, 2019 䡲

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IMPROVING FRACTION UNDERSTANDING