learning disability quarterly 2014 orosco 45 53

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Learning Disability Quarterly

2014, Vol 37(1) 4553 Hammill Institute on Disabilities 2013

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DOI: 10.1177/0731948713504206ldq.sagepub.com

Article

Although publications aimed at improving instruction for

English language learners (ELLs) has grown within the past

decade (e.g., Crdenas-Hagan, Carlson, & Pollard-Durodola,

2007; Mathes, Pollard-Durodola, Crdenas-Hagan, Linan-Thompson, & Vaughn, 2007; Pollard-Durodola & Simmons,

2009), to date there is surprisingly little research on assistive

instruction with teachers of ELLs on English language

development in content areas such as math (Janzen, 2008).

Further instructional support that improves math compre-

hension for ELLs is still needed. However, lower oral lan-

guage proficiency in English (e.g., English vocabulary skills

and content knowledge) is often overlooked as a part of this

added instruction (Goldenberg, 2011). Research indicates

that although basic math skills such as computation are

taught well enough for ELLs to perform as well as their

native English-speaking peers, teachers are not showing

ELLs how to grasp specific oral language development (i.e.,math vocabulary and concepts) that impacts word-problem-

solving comprehension, at proficiency levels equivalent to

native English speakers (Orosco, Swanson, OConnor, &

Lussier, 2013). Well-developed oral proficiency in English

may be a critical step to improving word-problem-solving

skills for ELLs (Orosco et al., 2013). Specifically, English

vocabulary knowledge, listening comprehension, and the

ability to manage contextual aspects of language (i.e., pro-

viding direct and explicit instruction with math terminology

and concepts) may need to be linked to improving compre-

hension (Rupley & Nichols, 2005), especially when devel-

oping word-problem-solving skills.

Also, elementary school teachers are particularly chal-lenged and need to summon extra resources because of the

range of instructional needs of ELLs for whom math con-

tent in a second language is more arduous due to limited (a)

experiences in vocabulary development, (b) prior math con-

tent knowledge, and (c) strategies to improve word-problem-

solving skills. In addition, students at risk for math

disabilities (MD) may need more intensive, individualized,

or small-group instruction that is highly structured and

explicit to mediate word-problem-solving content success-

fully. Furthermore, given the multistep process of the word-

problem-solving process, strategy instruction continues to

be an important intervention approach to improve solution

accuracy.The literature has proposed a set of instructional math

practices that have been validated by research on English

native speakers with or at risk for MD. First, math instruction

206 LDQXXX10.1177/0731948713504206Learning Disability QuarterlyOrosco

1University of California, Riverside, USA

Corresponding Author:

Michael J. Orosco, Area of Special Education, Graduate School of

Education, University of California, Riverside, CA 92521, USA.

Email: [email protected]

Word Problem Strategy for Latino EnglishLanguage Learners at Risk for MathDisabilities

Michael J. Orosco, PhD1

Abstract

English Language Learners(ELLs) at risk for math disabilities(MD) are challenged in solving word problems for numerous

reasons such as (a) learning English as a second language, (b) limited experience using math vocabulary, and (c) lack of

strategies to improve word-problem-solving skills. As a result of these difficulties, ELLs may not only need math supportbut also oral language and reading development assistance. The purpose of this study was to assess the effectiveness

of a math comprehension strategy procedure based on a dynamic assessment (DA) framework. The strategy provided

scaffolding support based on the students reading and language comprehension levels. A multiple baseline was used to

assess 6 third-grade Latino ELLs at risk for MD. As compared with baseline, the strategy increased problem-solving abilityfor all participants. All students level of performance was maintained during follow-up sessions. Results suggest that a focus

on comprehension strategies may help facilitate math skills development for ELLs at risk for MD.

Keywords

content-area instruction, instructional strategies, single-subject methods, research design or utilization, at risk, thinking/

cognition, education, Teacher

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46 Learning Disability Quarterly 37(1)

should include (a) methods of explicit and direct instruction

that teaches conceptual understanding of math concepts and

principles of a word problem (e.g., Fuchs, Fuchs, Finelli,

Courey, & Hamlett, 2004; Griffin & Jitendra, 2008; Jitendra,

DiPipi, & Perron-Jones, 2002; Jitendra, Griffin, Deatline-

Buchman, & Sczesniak, 2007; Swanson, Hoskyn, & Lee,

1999), (b) visual representation techniques designed to bridge

a connection from verbal information to symbolic under-

standing by creating a mental model (e.g., Jitendra et al.,

2007; Jitendra & Xin, 1997; Van Garderen & Montague,

2003), (c) using instructional feedback with peer-assisted

learning strategies during instruction (e.g., Fuchs, Compton,

et al., 2008; Fuchs, Fuchs, Yazdin, & Powell, 2002; Fuchs,

Seethaler, et al., 2008), and (d) small group instruction,

instructional modeling, corrective feedback, and student ver-

balizations (Baker, Gersten, & Lee, 2002; Gersten et al.,

2009; Swanson et al., 1999). Although this effective instruc-

tion with English only students may serve as a foundation for

teaching ELLs, however, this general effective instruction isprobably not sufficient to promote accelerated math learning

among ELLs at risk for MD. Nevertheless, it is most likely a

necessary basis.

In addition, another discrepancy that arises from the

math literature is the high dependency on standardized

measures (e.g., administered pre- and post-test) which test a

students word-problem-solving performance by presenting

scripted tasks that require the student to access previous

learned knowledge with little teacher input (e.g., Grigorenko,

2009; Haywood & Lidz, 2007; Swanson & Lussier, 2001).

Because of this, static assessments have not been able to

incorporate teacherstudent interaction as part of the testing

process, where such feedback can help alleviate studentproblem-solving difficulty. As a result of this, there is an

opportunity in the math literature in developing an assess-

ment model that can be used to identify word-problem-

solving challenges, make diagnostic decisions, and propose

instructional strategies that address learning difficulties.

The purpose of this study was to assess the effectiveness of

a math strategy procedure based on a dynamic assessment

(DA) framework.

DA diverges with traditional models of assessment and

instruction, in which only the students current competen-

cies are measured, and the tester does not intervene so as

not to influence the results (Orosco et al., 2013). In con-

trast, DA determines whether substantive changes that

occur in student performance were due to instructional scaf-

folding across an array of tasks. During DA, a teacher facil-

itates a students ability to build on prior knowledge through

studentteacher interaction, and uses this mediation process

as a way to help the student internalize new information

(Vygotsky, 1962, 1978). The teacher uses a standardized

protocol to measure learning potential. The learning poten-

tial is a higher level of academic ability that a student can

achieve after new knowledge is presented instructionally.

Specifically, learning potential is measured in terms of the

difference between, and/or change from a students level of

performance when unassisted versus their achievable level

of performance with assistance (i.e., Zone of Proximal

Development[ZPD]; Sternberg & Grigorenko, 2002). Thus,DA allows for the measurement of students developing

skills in determining which of these abilities are modifiable,

and what interventions or mediational supports may be

developed to improve their cognitive processes (Englert &

Mariage, 2013).

Although DA has been suggested as an alternative to tra-

ditional math assessment (e.g., Fuchs, Compton, et al.,

2008; Seethaler, Fuchs, Fuchs, & Compton, 2012), there are

few published studies with Latino ELLs at risk for MD. The

purpose of this study was to investigate a word-problem-

solving strategy called Dynamic Strategic Math (DSM).

DSM was operationally defined throughout this study as the

tester modifying word-problem solving via a four-levelvocabulary modification procedure (Table 1) to the stu-

dents level of word-problem-solving cognition, and then

providing intervention with probes that assessed students

word-problem-solving ability. This study addressed two

research questions with Latino ELLs.

Research Question 1:To what level does DA improve

students word-problem-solving skills as measured by

Table 1. Four-Level Math Vocabulary Modification Procedure.

Level Description Example

Basic (Level 1) Math terms used in everydayconversation

before, after, combine, extra together, more, greater than,total, fewer, sort, fewer than, take away

Intermediate (Level 2) Math terms not directly associatedwith a specific math content area

addition, digits, division, multiplication, factor, factors,subtraction, sum

Advance intermediate (Level 3) Math terms directly associated with aspecific content area

quotient, divisor, divisible by, dividend, least commondenominator, least common multiple

Technical vocabulary (Level 4) Math terms associated with a specificmath terminology

perimeter, area, cylinder, inch, meter, centimeter, mile,rectangle, square, triangle, cube, right triangle

Source.Adapted from Ernst-Slavit and Slavit (2007); Orosco, Swanson, OConnor, and Lussier (2013).



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word-problem-solving achievement (compared with

baseline level)?

Research Question 2:To what degree does DA main-

tain students word-problem-solving skills in generaliza-

tion sessions?

Method

Setting and Participants

Six third-grade Latino ELLs at risk for MD from a southern

California (English/Spanish) elementary classroom partici-

pated in this study. This study defines Latino ELLs as stu-

dents who speak Spanish as their native language, are

identified as coming from Latin American descendants

(e.g., Mexican, Mexican American), and are in the process

of acquiring English as a second language, and who have

not achieved full English proficiency. The schools popula-

tion consisted of 453 students (55% Hispanic [39% Latino

ELLs], 22% Black/African Americans, 14% White (non-Hispanic), 5% Asian, and 4% Other [categories created

based on the U.S. Census]). According to district informa-

tion, 75% of the schools population was in the free or

reduced-price lunch program.

While there is controversy over the definition of learn-

ing disabilities, this study adhered to the growing consen-

sus among researchers that it is best to use an absolute

definition of learning disabilities (cutoff score on achieve-

ment) rather than a discrepancy between achievement and

IQ (e.g., Fletcher et al., 1989). In determining the criteria at

risk for MD, we first considered (a) teacher recommenda-

tion for intervention based on students being exposed to at

least 3 years of math instruction; (b) students who had con-tinued to experience word-problem-solving difficulties in

English; (c) students who had performed in the lower 25th

percentile on district math tests previously; (d) Spanish

spoken as their native language, as determined by the

schools home language survey; (e) the California English

Language Development Test (CELDT; Marr, Rodden, &

Woods, 2009) was used to define ELL status; (f) Woodcock

Johnson NU Tests of Achievement 3rd Edition, Achievement

Test: Applied Problems (WJ NU III-ACH Test 10;

Woodcock, McGrew, & Mather, 2007; students who per-

formed in the lower 25th percentile were included in the

sample); and (g) parent consent. The WJ NU III has a

reported internal reliability coefficient of .85 for ACH Test

10 ages 8 to 10 (Woodcock et al., 2007). The test was

administered at pre- and post-test, and the test data were

compared with multiple baseline data, in determining

whether the math intervention positively mediated stu-

dents word-problem-solving skills. Finally, the CELDT

measures English proficiency (listening, speaking, reading,

and writing), and reliability scores for this test are between

.73 and .94 across grade levels (Marr et al., 2009).

Instrument

DSM is built on a conceptual foundation of reciprocal

teaching (Palincsar & Brown, 1984) and many features

associated with effective instruction (e.g., collaborative

group work, interactive dialogue, and explicit teaching

strategies; for example, Baker et al., 2002; National

Mathematics Advisory Panel, 2008; National ResearchCouncil, 2001). DSM includes instructional practices asso-

ciated with improved reading comprehension: (a) building

vocabulary so that students can contextualize and bring

meaning to math language; (b) teaching students to moni-

tor their comprehension and procedures for adjusting when

word-problem-solving difficulties arise; (c) using cooper-

ating learning practices so that students not only practice

their English language skills, but also their problem-solving

skills; (d) providing support for questioning strategies that

assist students in answering critical questions about the

word problem, feedback to students regarding their answers

to questions, and opportunities for students to ask andanswer questions about the word problem; and (e) teaching

students to write down and reflect on important ideas in the

problem-solving process.

Finally, DSM has shown preliminary evidence as a

research-validated practice. The initial research was con-

ducted with 6 second-grade Latino ELLs (Orosco et al.,

2013). In this single-subject study, students were taught by

the homeroom teacher to use the DSM with school math cur-

riculum. Students made significant improvements in their

word-problem-solving and demonstrated high levels of aca-

demic engagement. They assisted each other with word

meanings, main ideas, and understanding word problem

text. Teacher feedback indicated that students word-problem-solving comprehension gains were associated with the qual-

ity and quantity of DSM training and implementation.

Experimental Design

A changing criterion multiple baseline across subjects

design (Kennedy, 2005) was used for evaluating the effects

of a problem-solving intervention that aimed to change

(accelerate)word-problem-solving efficiency in a sys-

tematic stepwise method. In this study, each intervention

session is associated with a stepwise criterion that targets

a word-problem-solving level of difficulty (four levels or

steps). The student advances to a higher level of difficulty

after solving a set of four word problems correctly at their

ZPD level. To prevent selection bias (Kratochwill &

Levin, 2010), participants were selected and categorized

based on teacher recommendation (all students had low

math and reading scores and a need for intervention), and

a list was randomly generated (based on Woodcock

Johnson [WCJ] scores and reading rank) so as not to place

the student with the most serious word-problem-solving

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difficulties (e.g., WCJ math scores) first (e.g., Natasha

went first). After students math performance was stable in

the baseline phase, the independent variable was staggered

across subjects and the number of sessions necessary to

establish response stability (minimum of three sessions

above the baseline mean). All participants were individu-

ally administered four word problems per session similar

to those used during the baseline phase. This study was

conducted as a pullout program for 17 sessions (average

2025 min per session) over a 5-week period and was a

supplementary intervention to the general education math

curriculum students received (50 min/day).

Word problems. In this study, word problems applied were

similar to those used in daily instruction. These word prob-

lems were linguistically modified based on a scaffolding

ladder that premised the language of math into four one-

dimensional language levels, each providing a scaffold that

supported the next higher level of word-problem-solving

development. Level 1 word problems were embedded inbasic math terminology used in everyday discourse (high-

frequency words), Level 2 word problems incorporated

math terms not directly associated with a specific math con-

tent (general math words), Level 3 word problems included

math words directly associated with a specific math content

area (specialized math vocabulary), and Level 4 incorpo-

rated math vocabulary associated with a specific math

content area topic (technical vocabulary). As an example of

this scaffolding (Table 2), a Level 2 word problem may

have asked the following: Luis likes to spend time at theme

parks. He spent three hours in Disneyland and four hours at

Universal Studies. What was the sum of time spent at theme

parks in all? In this example, the word problem was made

less linguistically complex by taking the Level 2 math term

(sum), and teaching a Level 1 meaning (total) without alter-

ing the math concept being taught.

Probing. In this study, a probing procedure was developed

by the researcher (see Table 3), in which the dependent vari-

able was word-problem level achieved with the strategy

intervention. The tool was designed to scaffold differing

levels of students word-problem-solving skills through the

application of five prompts in determining a students abil-

ity with and without assistance. Scoring of the five levels

involved the assignment of points at each prompt (0 =

incorrect response, 1 = correct response). As part of the

probing procedure, each student was asked to solve four lin-guistically modified word problems based on their problem-

solving level. After 3-min duration, if the student was

having difficulty solving the problem, the student was given

the prompts with 1 min to answer each prompt. The admin-

istration of prompts averaged 4- to 5-min duration, and the

number of prompts administered to solve the problem

established the type of intervention students received.

Table 2. Dynamic Strategic Math Cue Sheet (Abbreviated Example).

Word problem example: Luis likes to spend time at theme parks. He spent three hours in Disneyland and four hours at UniversalStudies. What was the sum of time spent in theme parks in all?

Examiner, A word problem asks a question (point to the question): What was the sum of time spent in theme parks in all? Next, Iwill underline the important words in the question.

Examiner, I know that the word sum means an amount obtained as a result of adding numbers. What does the concept sum of timemean? Sum of time means to add (+) time together. Yes, this makes sense. Sum of time can also mean to add (+) time together.

Examiner, What was the sum of time spent in theme parks in all? The word problem states: Luis likes to spend time at themeparks. He spent three hours in Disneyland and four hours at Universal Studies. I am going to circle these numbers, as these are thenumbers I need to solve this problem. Okay, lets solve the problem. He needs to sum or add time, writing 3 hours + 4 hours = 7hours. My answer is 7 hours. Luis spent 7 hours at theme parks.

Examiner, Okay, I need to check my answer. In the ones place 3 plus 4 equals 7. Luis spent 7 hours at theme parks. Now it is your turn.

Table 3. Dynamic Strategic Math Probe (DSMAP).

Examiner, A word problem asks a question. Can you find the question in the following word problem?

Examiner, In each question there are always important words. Can you underline words in this question that you think are importantto solving this problem?

Examiner, In each math problem there are always numbers that you need to solve the problem. Can you circle the numbers that you

need to solve this problem?Examiner, Numbers are used to set-up and solve a math problem. Can you use these numbers to set-up the problem so that you cansolve the word problem?

Examiner, After solving the math problem, you need to check your answer. Can you check your answer?



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Procedure

Baseline phase. At the baseline level, each participant was

individually administered four grade-level word problems

that contained four progressive levels of word problem dif-

ficulty. Students were told to do their best, and given as

much time as possible to solve the problems. None of the

participants required more than 10 min in attempting tosolve the problems. This established the baseline level as

the word-problem level each student could accurately solve

without assistance. This also created the entry level/starting

point for the word problems for the intervention. All six

participants started at word-problem-solving Level 1.

Intervention Phase 1: Preteaching concepts and vocabu-

lary. ELLs were first pre-taught specific math concepts,

vocabulary, and terminology for the word-problem-solving

lesson by direct and explicit modeling (see the appendix for

description).

Intervention Phase 2: Teaching the strategies. At this level,

DSM integrated five common problem-solving strategies

(What Do I Know, What Can I Find, What Is The Set-Up,

Solve It, and Check For Understanding) that showed stu-

dents how to solve the word problem that were modeled

through direct and explicit instruction (see appendix for

description).

Intervention Phase 3: Co-operative learning and/or student pair-

ing. Once students were knowledgeable in strategy usage,

they were provided a collaborative approach (in pairs one

student and one teacher), which allowed the students to

practice these strategy methods (see appendix for descrip-tion). If word-problem-solving difficulties persisted in this

stage, the teacher re-taught specific strategies by reciprocal

teaching again until the students understood them.

Social Validity

At the conclusion of the study the social validity of the

intervention was assessed using a three-question interview

protocol. During this interview, participants were asked

questions regarding their satisfaction with DSM (e.g., Do

you think the strategy helped you understand word prob-

lems better? Please explain.)

Interobserver Agreement and Treatment

Integrity

An ELL/bilingual trained classroom teacher and researcher

alternated sessions administering the intervention. To check

on the degree to which intervention techniques were being

applied in teacher interactions with students, a treatment

integrity checklist based on the sequence of probe state-

ments (e.g., pacing, quality of instruction, and scaffolding)

for each intervention was applied. The checklist was com-

pleted by a classroom observer (the researcher alternated

with the ELL teacher as implementer/observer) for a total of

33% of all phases, including 33% for baseline, 33% for

intervention, and 33% for maintenance. The observer would

code for fidelity via a checklist and score yes or no for

each probed observed. A total agreement calculation method

for each session (i.e., dividing the number of agreements

between the probe responses by the number of disagree-

ments and then multiplying by 100) indicated the consistent

presence of intervention behaviors for all sessions by the

two observers at baseline (85%), intervention (100%), and

maintenance behaviors (90%).

Results

Figure 1 displays word-problem level achieved for eachparticipant as a function of baseline, intervention, and main-

tenance sessions. Visual analysis supports strong evidence

of a causal relationship (Kratochwill et al., 2010) between

the DSM intervention on word-problem-solving accuracy.

In addition, there was a clear consistency of level, trend,

and variability in baseline, intervention, and maintenance

phases. Also shown are WJ NU III-ACH Test 10 pre and

post-test scores (Table 4).

Baseline Performance

During baseline, all six participants started at a baseline

Level 1. Although the participants performance on word

problem solving demonstrated a pattern of stability at base-

line, the low performance on more language complex and

difficult word problems for all the participants indicated a

need for intervention.

Intervention

As compared with baseline, the intervention condition pro-

duced an increased trend (effect) in accuracy and level of

word-problem difficulty solved for all students. After each

session, each participant was administered a set of four

word problems based on the intervention level received. Allstudents showed immediate effects from DSM intervention

because they received learning opportunities that were open

and constrained to their background knowledge, oral

language development, vocabulary, and problem-solving

needs. First, students were directly and explicitly taught

math concepts and vocabulary that connected to everyday

words. This helped to build their linguistic math register.

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Next, students were taught comprehension strategies that

involved building on students current knowledge for solv-

ing word problems that depended on understanding the

quantities involved in the problems and the procedural flu-

ency required in solving these problems. Finally, students

were given carefully directed practice, with feedback, to

develop the intervention method through the use of probes.

These probes built on students thinking and helped them

practice their math and language skills, and also helped

them attend to relationships between the problems and solu-

tions to word problems.

Maintenance

Finally, to determine maintenance of treatment skills, all

students were administered four math word problems simi-

lar to those used during the baseline phase for three ses-

sions. During this phase, all students sustained a consistent

word-problem-solving accuracy level similar to the inter-

vention phase. Arthur maintained the highest level of

performance a Level 4, while, April, Natasha, Victoria,

Tomas, and Rudy maintained a Level 3. Post-test scores on

the WJ NU III-ACH Test 10 also indicated student gains as

a result of intervention.

Social Validity

Interview data indicated that all the participants were in

agreement (100%) that intervention procedures were rea-

sonable and effective. Several students commented around

the theme I like the teaching; we could talk about math.

The teacher commented, I really liked the simplicity of the

strategy, and how easily it integrated math content and

vocabulary with ELL reading pedagogy. The students rec-

ommended more collaboration and practice time. Also,

the teacher would have liked more professional develop-

ment and planning time to think about how to improve on

her word-problem-solving instruction.

Discussion

The purpose of this study was to investigate the effects of

a math intervention, DSM, on Latino ELLs word-prob-

lem-solving achievement. Visual analysis of graphed data

indicated a functional relationship between DSM and

increased word-problem-solving ability. As students

knowledge of DSM improved, their skills to solve word

problems with increasingly complex vocabulary and con-

tent improved during the intervention in comparison with

that during the baseline phase. In addition, students could

maintain knowledge of the DSM process, during three

maintenance sessions after intervention and relevant to

baseline phase. Overall, results provided support for the

impact of DSM on the improvement of word-problem-

solving skills.

The evidence from this study lends support that DSM

improved ELLs word-problem-solving ability during the

treatment phase in comparison to baseline phase, and this

performance was maintained during generalization sessions.

The DA literature infers that scaffolding interventions can

positively mediate word-problem-solving performance over

time because it can give mediation support based on stu-

dents actual developmental level as determined by indepen-

dent problem-solving performance, and the highest level of

potential development achieved through teacher assistanceor collaboration with more capable peers (Vygotsky, 1978).

The results (e.g., increased word-problem-solving ability) of

this study demonstrated a positive mediation trend during

intervention. Although students in this project had the basic

math skills to perform algorithmic computations adequately,

while encountering word problems, the participants needed

assistance in understanding math vocabulary and concepts

to improve their word-problem-solving performance. The

findings from this study were consistent with those of the

Figure 1. Word-problem-solving level achieved per session.



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Orosco 51

emerging literature in this and area that dynamic math

assessment may be an effective model for improving ELLs

word-problem-solving performance.

Finally, the evidence from this study provides addi-

tional support for the integration of math content with

reading comprehension strategies as an effective peda-

gogy for teaching ELLs word-problem-solving skills.

Evidence from this study indicates that reading compre-

hension strategies (e.g., reciprocal teaching) may be an

effective method because it provides a dialogic form of

scaffolding instruction that integrates students reading

prompts (e.g., consider their background knowledge on

the topic they are reading, to summarize key ideas, and to

self-question while they read) with math content. The

results from this study indicate that DSM may be an

effective word problem solving tool because it can

improve ELLs word problem solving accuracy, promotes

their academic language development, and provides equi-table access for them to general math education. In addi-

tion, this method gives teachers a deeper understanding of

the underlying processes, essential elements, develop-

ment trajectory, skill levels, and potential of their stu-

dents math achievement through improved instruction.

Limitations

While the findings from this study demonstrate that ELLs

word-problem-solving skills can be positively mediated to

higher development levels (e.g., criterion changing math

terminology difficulty) by providing scaffolding support,

there were limitations to this study. This was a small-scale

study (N= 6) in which individual data were collected for

duration of 17 sessions, and therefore, the extent to which

the intervention may have positively mediated word-

problem-solving skills in other ELLs for this duration of

time is unknown. And because of this, generalizing inter-

vention effectiveness to other ELL populations is limited.

To date, there have been few math studies that combine DA

with reading instruction with ELLs at the elementary in

comparing intervention effectiveness.

Implications

The evidence from this study has implications for ELLs at

risk for MD. These findings indicate that although partici-

pants had number sense, school data collected prior to the

study indicated that the participants skill level was limitedto number calculations with basic math problems and heav-

ily influenced by the context in which the numbers

appeared. This below basic grade-level performance indi-

cated that DSM provided ELLs with (a) a differentiated

math instruction, one that was matched to the students

English language proficiency; (b) instructional approaches,

which built on students pre-existing background knowl-

edge and made a connection with prior learning experi-

ences; (c) teacher and student joint collaborative activities;

(d) an opportunity to use sentence frames and models to

help students talk about math content; and (e) gave addi-

tional math practice and/or math time for discussion of key

concepts. Word-problem-solving data indicated that stu-

dents could acquire higher-level word-problem-solving

proficiency with scaffolding support in vocabulary and

reading comprehension development.

Appendix

Intervention Procedure

Baseline phase. The baseline level was established as the

word problem level (4 possible levels) the student could

accurately solve without assistance. This established the

word problem level/difficulty starting point of the

intervention.

Intervention phase 1: Preteaching concepts and

vocabulary.Each student was provided 3 5 index

cards to practice concepts and vocabulary. The teacher

modeled the activity by holding up a card, providing

a definition, elaborating on the definition through

contextualization, writing this vocabulary on chart

paper/blackboard, and applying usage of the word in

a math problem while solving it.

Table 4. Demographic, School-Related Data, and WCJ Test 10 Pre- and Post-Test Scores.

Student Gender AgeDistrict readingassessment level

District mathassessment level

WCJ pre-testpercentile (%)

WCJ pre-teststandard score

WCJ post-testpercentile (%)

WCJ post-teststandard score

Natasha F 8.7 2.4 Below basic 16 85 18 86

Victoria F 8.6 2.7 Below basic 18 86 19 87

April F 8.5 2.6 Below basic 14 84 16 85

Tomas M 8.8 2.8 Below basic 18 86 19 87Rudy M 9.0 2.5 Below basic 16 85 18 86

Arthur M 8.7 2.7 Below basic 18 86 21 88

M 16.67 18.50

SD 1.63 1.64

Note.WCJ = WoodcockJohnson NU Tests of Achievement 3rd Edition, Achievement Test: Applied Problems; F = Female; M = Male.



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1. Definition:The teacher stated: This is the word sum.

It means to combine, add, or total.[On the chart paper

the teacher wrote these sentences and next to them,

the math symbol +.]

2. Contextualization: The teacher contextualized the

vocabulary by stating: Luis likes to spend time at

theme parks. He spent three hours in Disneyland and

four hours at Universal Studies. What was the sum of

time spent in theme parks in all? What does sum

mean? [Referring to the chart paper.] Sum means to

combine (+). Sum means to add (+). Sum means to

total (+).

3. Writing & Using: The teacher repeats reading the

word problem, and then solves the problem on board.

3 + 4 = 7, the sum of time spent at theme parks was

seven. Now it is your turn. [Students repeat this

process.]

As students became familiar with this information, the

teacher then began to integrate and embed comprehensionstrategy instruction.

Intervention phase 2: Teaching the strategies.At this

level, DSM integrated five common problem-solving

strategies (Know, Find, Set-Up, Solve, and Check

Understanding):

1. Know -What do I know about the question occurred

after reading the word problem and consisted of

brainstorming (activating background knowledge)

what was already known about the word problem,

and predicting how it may be solved.

2. Find -Find the important vocabulary and numbers

occurred during reading and referred to the process

of finding key information for meaning and

understanding.

At this point, students were taught to use strategies to

help them figure out unknown words or concepts.

3. Set Up -Students also began to set it up during read-

ing by stopping after each sentence to find the main

idea and to check to see if this information was rele-

vant to solving the problem.

4. Solve -Finally, solve it and check it took place after

reading the word problem.5. Check Understanding -Next, students checked their

understanding by generating and answering questions

about what they had already read and reviewed what

they had learned by summarizing the key ideas pre-

sented in the word problem, solving it, and checking

it to see if solved correctly.

Intervention phase 3: Cooperative learning and/or

student pairing.Once students were knowledgeable

in strategy usage, they were provided a collaborative

approach (in pairs), which allowed the students to

practice this method. In this stage, the student assumed

the leadership role and imitated the teachers role.

Within this process:

1. The student generated and asked questions to check

for understanding.

2. The student solved the problem and checked to see if it

was answered correctly. If answered incorrectly, the prob-

lem-solving process was repeated between the teacher

and student again, to see where mistakes were made.

Within this process:

1. The teacher monitored the students effectiveness by

providing probes as needed (e.g., reading words, clar-

ifying math concepts, or reminding students of a strat-

egy skipped). (See Table 2 for probes and Table 3 for

strategy cues.)2. If word-problem-solving difficulties persisted, the

teacher then re-taught specific strategies by recipro-

cal teaching again until the student understood them.

Declaration of Conflicting Interest

The author(s) declared no potential conflicts of interest with respect

to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support

for the research, authorship, and/or publication of this article: This

paper is based on a study funded by the U.S. Department ofEducation, Cognition and Student Learning in Special Education

(USDE R324A090002), Institute of Education Sciences, awarded

to the author.

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