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Instructional practices in mathematics

for students with significant cognitive

disabilities have historically focused on

teaching mathematics within activities

of daily living, such as shopping (e.g.,

Aeschleman & Schladenauffen, 1984; Browder,

Snell, & Wildonger, 1988; Haring, Kennedy,Adams, & Pitts-Conway, 1987). Educators have

sometimes also used assistive devices that mini-

mize the need for mathematical competence (e.g.,

Fitzgerald & Koury, 1996; Frederick-Dugan,

Test, & Varn, 1991; Lancioni, Singh, OReilly, &

Oliva, 2003). In contrast, current federal policies

such as the No Child Left Behind Act of 2001 re-

quire assessing mathematical achievement on

state standards for all students. Although alter-

nate assessments can be used for students with

significant cognitive disabilities, these assessments

must be linked to the states academic content

standards (U.S. Department of Education, 2003).

The challenge that arises in educational planningis to address academic content standards that en-

compass more domains of mathematics than the

skills typical of a purely functional approach.

The National Council of Teachers of Mathe-

matics (NCTM, 2000) provides a comprehensive

set of standards based on mathematics goals that

address the achievement of all students. In

407Exceptional Children

Vol. 74, No. 4, pp. 407-432.2008 Council for Exceptional Children.

A Meta-Analysis on TeachingMathematics to Students WithSignificant Cognitive Disabilities

DIANE M. BROWDER

FRED SPOONER

LYNN AHLGRIM-DELZELL

AMBER A. HARRIS

SHAWNEE WAKEMAN

University of North Carolina at Charlotte

ABSTRACT: This article reports on a comprehensive literature review and meta-analysis of 68 exper-

iments on teaching mathematics to individuals with significant cognitive disabilities. Most of the

studies in the review addressed numbers and computation or measurement. Within the computa-

tion studies identified, most focused on counting, calculation, or number matching. For the mea-

surement studies, nearly all focused on money skills. Of the 54 single subject design studies, 19 were

classified as having all quality indicators for research design (13 representing the National Councilof Teachers of Mathematics Measurement standard and 6 representing the Numbers and Opera-

tions standard). These studies offer strong evidence for using systematic instruction to teach mathe-

matics skills and for using in vivo settings.

Exceptional Children


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408 Summer 2008

response to low mathematics performances of stu-dents throughout the United States, mathematicseducators and other professionals identified five

main components of mathematics instruction, in-cluding (a) number and operations, the ability tounderstand and represent numbers, relationshipsamong numbers, and number systems (e.g., thenumber 24 is 2 tens and 4 ones); (b) measure-ment, the ability to understand measurable at-tributes of objects (e.g., time and money); (c) dataanalysis and probability, the ability to collect, or-ganize, and display relevant data to answer ques-tions with appropriate statistical methods (e.g.,graphing); (d) geometry, the ability to analyzecharacteristics and properties of two- and three-dimensional shapes, apply transformation and use

symmetry to analyze mathematical situations(e.g., length, width, area, and volume); and (e) al-gebra, the ability to understand patterns, rela-tions, and functions of numbers, and usemathematical models to represent and understandquantitative relationships (e.g.,x+ 4 = 7, solve forx). Most states standards include similar domainsfor achievement in mathematics. In contrast,Maccini and Gagnon (2002) found that teachersof students with learning and behavioral disabili-ties were not familiar with these standards andmostly taught basic skills versus algebra and ge-ometry. The respondents reported the lack of ade-

quate materials as the primary barrier to teachingbroader standards. This lack of knowledge and re-sources may be compounded for teachers of stu-dents with significant cognitive disabilities whoalso must now address the multiple domains oftheir state standards to prepare students for alter-nate assessments.

If educators turn to the research literature forguidance, they can find some information onteaching mathematics. A comprehensive review ofstudies on mathematics interventions for elemen-tary-aged students with mild disabilities (Kroes-

bergen & Van Luit, 2003) identified 58 studiesthat primarily focused on basic number and oper-ation skills. A meta-analysis of these studies sug-gested that direct instruction was the mosteffective intervention for increasing basic mathe-matics skills, whereas self-instruction was more ef-fective in enhancing problem-solving skills. Thisreview of literature also revealed that directteacher-led instruction was more effective in

teaching numeracy and/or computation skillsthan peer- or computer-assisted learning tech-niques. Browder and Grasso (1999) reviewed 43

studies to consider interventions for five basiccomponents of money management (i.e., compu-tation, banking, budgeting, purchasing, and sav-ing) for individuals with significant cognitivedisabilities. Most studies in this review targetedmoney computation and purchasing. In many ofthe studies, money skills were taught in the con-text of real-life activities using systematic prompt-ing and feedback for specific target skills, such asselecting the correct dollar amount (Test, Howell,Burkhart, & Beroth, 1993).

Other investigators have also synthesizedmathematics related instruction for students with

cognitive disabilities (e.g., Mastropieri, Bakken,& Scruggs, 1991; Xin, Grasso, Dipipi-Hoy, & Ji-tendra, 2005). The Mastropieri et al. synthesis of25 studies identified research conducted in threemajor areas: basic skills and concepts; rule learn-ing and problem-solving; and applications, in-cluding use of time, money and measurementskills. Xin et al., in examining 28 studies on func-tional mathematics instruction (e.g., purchasing)for students with mild to severe cognitive disabili-ties, found a moderate positive effect for acquisi-tion of purchasing skills but a large effect formaintenance of this skills and a moderate effect

for their generalization. Reviews on other compo-nents of mathematics, such as algebra, typicallyhave focused on students with learning disabilitiesor other high incidence disabilities (Jitendra &

Xin, 1997; Maccini & Gagnon, 2000; Maccini,McNaughton, & Ruhl, 1999; Miller, Butler, &Lee, 1998; Xin & Jitendra, 1999).

Although these reviews offer some directionfor teaching mathematics to students with learn-ing disabilities or other high incidence disabilities,there are three major limitations in the reviews todate. First, none of the reviews on mathematics

focused on students with significant cognitive dis-abilities; instead the reviews have considered thefull range of developmental disabilities or havebeen targeted to high incidence disabilities. Be-cause of this more general focus, it is not clearhow well students with significant cognitive dis-abilities are represented in the findings. A secondlimitation is that the reviews have not focused onthe multiple domains of mathematics (e.g., geom-


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etry, money management) represented in moststate standards, so it is not clear what evidenceexists for teaching skills such as geometry or data

analysis, as well as computation or money man-agement. A third limitation is that the reviewshave not considered the criteria that should be ap-plied to evaluate research prior to recommendingan intervention for practice. Odom and col-leagues (2005) recently have provided guidelinesfor the quality and quantity of research neededfor a practice to be considered evidence-based.

None of the reviews on mathematicsfocused on students with significant

cognitive disabilities; instead the reviews

have considered the full range ofdevelopmental disabilities or have beentargeted to high incidence disabilities.

The purpose of this study was to gain under-standing of the components of mathematics thathave been taught to students with significant cog-nitive disabilities and to evaluate this research toidentify evidence-based practices. We organizedthe literature by NCTM (2000) components ofmathematics and evaluated the quality of researchusing a subset of indicators recommended for sin-gle subject (Horner et al., 2005) and groupresearch (Gersten et al., 2005) within special edu-cation. A meta-analysis of the effect of instruc-tional components of studies with all andmost of this subset of quality indicators identi-fied promising instructional practices for individ-uals with significant cognitive disabilities.

The research questions posed for this studywere:

1. What NCTM components of mathematicsare represented in this evidence?

2. What types of skills are represented withthese NCTM components being taught toindividuals with significant cognitive disabili-ties?

3. What evidence exists that individuals withsignificant cognitive disabilities can learnmathematics?

4. What evidence-based instructional practiceshave been successful in the acquisition of

mathematics skills of individuals with signifi-

cant cognitive disabilities?

M E T H O D

L I TER A TU R E SEA R C H PR O C ED U R ES

We used a list of 148 terms or combinations of

terms (e.g., moderate mental retardation, severe dis-abilities, autism, mathematics, money, counting,graphing, geometry, and measurement) to identifythe research base of academics and students with

significant cognitive disabilities. We also searched

literature on self-monitoring to identify studies

that taught data analysis and graphing even

though the primary focus of the research was notspecifically on mathematics instruction. Both

electronic and print resources were used to deter-

mine which articles to include within the review.

Electronic databases included InfoTrac, Masterfile

Premier, ERIC, PsychInfo, Academic Search Elite,

and ProQuest for Dissertations. We also manually

searched several journals (Education and Trainingin Mental Retardation and Developmental Disabili-ties, Remedial and Special Education, Research andPractice for Persons with Severe Disabilities, andJournal of Applied Behavior Analysis) table of con-tents to include references that may have been

overlooked during the electronic search.

IN C LU SI O N CR I TER I A

We developed four inclusion criteria for selecting

articles within the review: (a) The article had to

be published in a peer-reviewed journal in English

between the years of 1975 and 2005 or be a dis-

sertation; (b) subjects within each article had to

include at least one participant diagnosed as hav-

ing a significant cognitive disability (i.e., autism,

developmental disability, or moderate, severe orprofound mental retardation); (c) the study had

to have an intervention that focused on teaching

academic mathematics skills and report first-hand

data (reviews of literature were not included); and

(d) the study must have used an experimental or

quasi-experimental design for either group or sin-

gle subject studies.



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CO D I N G

We first coded individual studies meeting the pre-vious inclusion criteria using an original review

form developed to record the characteristics thatwould be analyzed. These characteristics included(a) mathematics content, (b) instructionalmethod(s), (c) type of prompting and promptfading, (d) reinforcement, (e) setting, (f) type ofinstructor, (g) research design, (h) reliability andvalidity, (i) procedural fidelity, and (j) number ofparticipants. Interrater reliability checked for cod-ing agreement for one third of the review forms:

A second rater independently coded 22 studiescalculating interrater reliability using a point-by-point method; the number of agreements was di-

vided by total number of coded items, thenmultiplied by 100 to convert it to a percentage.Using a subset of quality indicators devel-

oped for group (Gersten et al., 2005) and singlesubject (Horner et al., 2005) designs, we deter-mined which studies met the criteria for being ev-idence-based. Single subject criteria comprised:(a) dependent variable operationally defined andincluded data on interrater reliability, (b) methodsincluding the independent variable were ade-quately described, (c) adequate procedural fidelityabove 80%, and (d) baseline and experimentalcontrol (between or within participant replica-

tions). We classified studies as having all fourcriteria, most (with three of four criteria), orweak (two or fewer criteria). The final step inthe Horner et al. criteria was to determine

whether each intervention was represented in atleast 5 studies conducted in at least three differentlocations with at least three different researchers

with a minimum of 20 participants. In cases ofmore than one author for a given study, werecorded only the lead author and state where thelead author was located. For the group studies, weapplied the Gersten et al. essential criteria, whichincluded (a) multiple outcome measures includ-ing ones aligned with the intervention and thosemeasuring generalized performance with evidenceof reliability and validity; (b) intervention clearlydefined with adequate procedural fidelity and in-terrater agreement; (c) sufficient description ofthe participants with procedures promoting cre-ation of comparable groups; and (d) data analysislinked to the research questions that describe

treatment effects. Again, we classified studies ashaving all four criteria, most (with three offour criteria), or weak (two or fewer criteria).

We calculated interrater reliability for coding thestrength of a study using the same procedure asdescribed in coding the content of the studies.

EF F E CT I V EN E S S O F IN STR U C TI O N A LIN TER V EN TI O N S

Our meta-analysis was intended to determine theeffectiveness of the interventions used within theselected studies. Using a nonparametric approachto meta-analysis, we calculated the percentage ofnonoverlapping data (PND) between the baselineand treatment phases to determine the effects of

interventions used in the single-subject studies when a readable graph of the intervention andbaseline was provided (Scruggs, Mastropieri, &Castro, 1987). A number of alternative methodsfor calculating effectiveness of single-subject de-signs have been proposed, such as mean baselinereduction, percentage of zero data, PND, andstandard mean difference. Comparisons of thesealternate methods to analyze single-subject datafound consistent interpretations across the differ-ent methods (Campbell, 2004; Olive & Smith,2005). Campbell found the regression-based deffect size to be difficult to interpret and an over-

estimate of the effect of treatment, and recom-mended that it not be used with single-subject.

We selected the PND estimate because it is acommonly used and easily interpretable method.For each study we calculated and recorded a PNDfor each instructional variable. (For example, if astudy investigated the effects of constant timedelay, error correction, and reinforcement, we cal-culated and recorded one PND for each of thethree variables.) We used only the data points inthe baseline and intervention phases because notall studies included generalization or maintenancedata. An intervention data point was consideredto be nonoverlapping if it was above the highestbaseline point. We excluded studies that displayedinappropriate baseline trends such as when thebaseline trend was not stabilized prior to the in-tervention and was already heading in the direc-tion of the expected effect of the treatmentintervention. To compute the PND, the numberof nonoverlapping data points in an intervention

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was divided by the total number of interventionpoints for each data series. Interrater reliability

was also used to check for agreement on PND

calculations. We computed a median PND foreach group of studies investigating the same in-structional variable. The median PND is pre-ferred over the mean because PNDs are notnormally distributed (Scruggs, Mastropieri, Cook,& Escobar, 1986). These median PND values

were interpreted based on Scruggs and Mas-tropieri (1998) criteria: a PND greater than 90%is classified as a large effect, a PND value between70.1% and 90% is classified as a moderate effect,a PND value between 50.1% and 70% is classi-fied as a low effect, and a PND value of 50% orless is classified as not effective. Interrater reliabil-

ity was calculated on 30% (n = 23) of the PNDs.For the group studies, we planned to use the

standardized mean difference Cohens d methodfor calculating effect size to determine the effec-tiveness of the interventions when the study pro-vided mean, standard deviation, or inferentialstatistic (i.e., ttest or Ftest) and sample size. Al-though the reporting of effect size is becoming in-creasingly common, there is some controversyregarding its interpretation. Cohen (1988) ini-tially provided operational definitions of effectsizes such that d= .2 was considered as small, d=.5 as medium, and d= .8 as large. The current no-

tion is that effect size should be interpreted withconsideration of effect sizes reported in relatedstudies. Lipsey (1990) provides benchmarks forevaluation of effect size when few previous studiesin a given area exist. Given the lack of research inevaluating effect size for mathematics for students

with significant cognitive disabilities, we usedLipseys empirical interpretative guidelines as fol-lows: (a) .15 or less as a small effect, (b) .16 to .45as a moderate effect, and (c) .46 to .90 as a largeeffect.

Each study was entered into a statistical

database program (SPSS). To describe the popula-tion of studies in the database, we calculated fre-quencies for strength of the studies and for typesof (a) academic category (which of five keyNCTM components of mathematics taught), (b)specific skill, (c) error correction, (d) promptingstrategy, (e) reinforcement, (f) instructional for-mat, (g) setting, (h) instructor, and (i) partici-pants. Using only the studies with all and

most quality indicators for research design, weconducted an analysis of the PNDs and effect sizeof the various interventions. We omitted weak

studies (few quality indicators for research design)from the PND analysis based on the recommen-dations by Horner et al. (2005) that evidence-based practice be derived from research withacceptable standards of quality.

Some studies provided multiple comparisons.Several studies included multiple dependent vari-ables such as steps in a purchasing task analysis,computation problems to solve, counting objects,and telling time (e.g., Matson & Long, 1986;Shapiro & Ackerman, 1983; Vacc & Cannon,1991). Some studies also contained multiple in-dependent variables with separate analyses such as

one-more-than dollar strategy versus mixed prac-tice (Denny & Test, 1995); other studies used aninstructional package with several instructionalmethods that were analyzed as a whole (e.g., Testet al., 1993). When this occurred, we enteredeach dependent and independent variable sepa-rately into the database.

R E S U L T S

IN TER R A TER RELI A BI LI TY

Agreement for inclusion of articles was throughconsensus by two researchers and no disagree-ments occurred. Overall interrater reliability forcoding the article components was 91.6%, with arange of 85% to 100% per article. Overall inter-rater reliability for coding of the strength of thearticle was 98.8%, with a range of 80% to 100%per article. Interrater reliability for the PND cal-culations was 82.3%. All disagreements in PND

were reviewed by the two raters so that there was100% agreement on the final score used for theanalysis.

DE SC RI PT IO N O F T HE IN C LU D ED STU D I ES

We located a total of 101 articles and reviewedthem to establish the degree to which each metthe mathematics inclusion criteria. We excluded36 articles from the meta-analysis for the follow-ing reasons. Two dissertations could not be ob-tained from ProQuest or the institution where thestudy was conducted, and an Internet search for



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these authors proved unproductive. Nine articleswere not intervention studies. Fourteen articlesdid not include any data on the mathematics

component of the study. Eleven articles were ex-cluded because none of the participants was clas-sified as having a moderate or severe disability.The remaining 65 articles located in 22 differentpublications met the inclusion criteria and wereincluded in the analysis. Two articles reported twoseparate experiments (Lalli, Mace, Browder, &Brown, 1989; Test et al., 1993) and 1 study in-corporated both group and single-subject analyses(Shapiro & Ackerman, 1983), for a total pool of68 studies. Of the 68 studies, 54 applied a singlesubject design and 14 used a group design. Be-cause some of these studies used multiple compar-

isons, there were a total of 89 instructionalcomparisons.

A total of 493 individuals with disabilitiesparticipated in these studies. Forty-five studies in-cluded 336 individuals with moderate cognitivedisability; 17 studies included 64 individuals withsevere cognitive disability; 12 studies included 24individuals with autism; 4 studies included 13 in-dividuals with unspecified developmental disabil-ity; and 1 study described 1 individual withmultiple disabilities. Nine studies (with an addi-tional 73 participants) provided only group infor-

mation regarding level and type of disability, soindividual frequencies could not be obtained.Fifty-nine studies reported participants gender,

with 205 males and 129 females. Thirty-five stud-ies included participants of high school/transitionage 15 to 21 years old. Twenty-nine studies in-cluded middle school students, 26 studies in-cluded elementary students, 16 studies includedadults over 22 years of age, and 4 studies includedpreschool children. Two studies did not specifythe age of the participants. It was possible for par-ticipants to have more than one disability and forstudies to include more than one age group.

The majority of these studies took place inspecial education classrooms (56.7%) or in thecommunity (26.9%). A few were conducted ingeneral education classrooms (35.8%) or at home(13.4%). Other settings included employmentsettings (4.5%) and residential facilities (4.5%).Sixteen studies were conducted in multiple set-tings.

QU ESTI O N S 1 AND 2: NCTMCO M PO NE NT S A N D MATHEMATICSSK I LLS RE PR ES EN TE D I N T HE

IN C LU D ED STU D I ES We conducted a frequency analysis of the codedNCTM components and corresponding mathskills to address our first two research questions.

Although all components were represented, mostof the studies focused on teaching the NCTMmathematical standards of Numbers and Opera-tions and Measurement (see Figure 1).

Thirty-seven studies (40.3%) included Num-bers and Operations skills such as counting, cal-culations, and matching numbers (e.g., Lalli etal., 1989; Matson & Long, 1986; Morin &

Miller, 1998). Thirty-six studies (53.7%) in-cluded Measurement skills involving money andtime (e.g., Borakove & Cuvo, 1977; Colyer &Collins, 1996; McDonnell, Horner, & Williams,1984). Two studies (3.0%) focused on skills re-lated to the Algebra standardsolving wordproblems, determining equivalence, and quantify-ing sets (Miser, 1985; Neef, Nelles, Iwata, &Page, 2003). Two studies (3.0%) targeted the Ge-ometry standard, focusing on recognizing andmatching shapes (Hitchcock & Noonan, 2000;Mackay, Soraci, Carlin, Dennis, & Strawbridge,2002). Two studies (3.0%) focused on the Data

Analysis and Probability standard, involvinggraphing within a self-monitoring skill(Copeland, Hughes, Agran, Wehmeyer, & Fowler,2002; Lovett & Haring, 1989). An additional 5studies also had a self-monitoring component,but the mathematics skills were either counting ormatching and were coded as Numbers and Oper-ations. Some studies taught more than one skillincorporating more than one NCTM Standard.

Because most of the studies were conductedwith individuals with moderate intellectual dis-ability, we gave additional consideration to thecomponents of mathematics found in research

with individuals with severe and profound intel-lectual disability. Of the 17 studies utilizing par-ticipants with severe and profound intellectualdisability, 6 focused on Numbers and Operationsand 12 focused on the Measurement standard.One study incorporated two standards. Numbersand Operations skills included calculating (n = 2),counting (n = 2) and matching numbers (n = 2).

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Measurement skills included money (n = 11) andtelling time (n = 1). Money skills consisted ofskills such as purchasing (e.g., Alcantara, 1994;Gaule, Nietupski, & Certo, 1985; McDonnell,1987), teaching the next dollar strategy (Schloss,

Kobza, & Apler, 1997), and coin identification(Llorente & Gaffan, 1989).

QU ESTI O N 3: EV I D EN C E T H AT IN D I VI D UA L S W I T H SI G N I F I C A N TCO G N I TI V E D I S A BI L IT I E S C A N LE A R N MATHEMATICS

In order to answer Research Question 3, we com-puted descriptive statistics for the PNDs of thesingle-subject studies and Cohens dfor the groupstudies, separately. The median PND of the 54single-subject studies was 92.15, with a mean of82.51 and a standard deviation of 25.79. PNDvalues ranged from 0% to 100%. The medianCohens deffect size of the 14 group studies was.79, with a mean of 2.18 and a standard deviationof 2.90. The d values ranged from .04 to 7.31.Overall, without consideration of quality of thestudy, type of mathematical skill, or instructionalstrategy, there is a large effect for teaching mathe-

matics to students with significant cognitive dis-abilities.

QU ESTI O N 4: EV I D EN C E-BASEDPR AC T IC E S F OR SI N G L E -SU BJ EC T STU D I ES

In order to address research question 4, we classi-fied the 54 single-subject design studies as havingall or most of the quality indicators for single-subject research in special education provided byHorner et al. (2005). From this analysis, 19 metthe initial criteria to be classified as having allquality indicators for research design. Thirty stud-ies met three of the four initial criteria and wereclassified as having most. Five studies were clas-sified weak (two or fewer indicators) and ex-cluded from further analysis. Twenty-eight of the30 most indicators studies did not include pro-cedural fidelity. Two studies included proceduralfidelity, but did not provide baseline data. Fivestudies were classified as a weak quality studymissing two criteria. Three of these studies didnot include procedural fidelity and baseline data.Two of the 6 studies did not include proceduralfidelity and reliability data. The final step forthose studies that met the initial quality indicator


F I G U R E 1

Mathematics Components Addressed in Studies Completed With Students With SignificantCognitive Disabilities

40

35

30

25

20

15

10

5

0

other, 7

count, 9

match, 9

calculate,12

time, 3

money,33

other, 2 other, 2

Numbers andOperations

Measurement Geometry Data AnalysisAlgebraother, 2


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criteria was to determine if the interventions wererepresented in at least 5 studies conducted by atleast three researchers in at least three different lo-

cations with at least 20 participants. Seventeen re-searchers in 10 locations with a total of 63participants were represented in the 19 studiesclassified as having all indicators. The com-monly used evidence-based practice in these 19studies was systematic instruction with explicitprompting and feedback for a defined response(or set of responses) taught across days.

Table 1 summarizes the evidence-based prac-tice found in the single subject studies and is or-ganized by component of mathematics. The 19all indicator studies fall into two of the NCTM(2000) standards of Numbers and Operations (n

= 6) and Measurement (n = 13). Money (e.g.,matching coins, counting and identification ofcoins and bills), purchasing, and computationalskills were the most common mathematics skills

within these two components (see Table 1).We conducted further analysis to identify if

the evidence-based practice of systematic instruc-tion was found for individuals classified as havingsevere or profound intellectual disability. Only 3of the 19 studies included such individuals (Bo-rakove & Cuvo, 1977; McDonnell, 1987; Schlosset al., 1997). Two of the studies taught purchas-ing skills and 1 taught money skills. Given that

Horner et al. (2005) recommend at least 5 studiesfor a practice to be evidence-based, these 3 highquality studies provide emerging evidence that in-dividuals with severe or profound disabilities canlearn money skills through systematic instruction.

Next we performed a meta-analysis for thecomponents of the systematic instructional prac-tices represented in the single-subject studies withall indicators. One of the 19 studies was ex-cluded from analysis because the single-subjectdata was provided in a summary histogram for-mat precluding calculation of PND. Two partici-

pants on another study were excluded due tounstable, increasing baseline trend at the intro-duction of the intervention. Because some re-searchers made multiple comparisons in a singlestudy, these 18 studies employed 28 instructionalcomparisons.

Table 2 presents results of the PND analysesfor the components of systematic instruction instudies with all indicators. PNDs for the 28

comparisons ranged from 59.0% to 100%, with amedian PND value of 92.15%. There were suffi-cient replications (i.e., five or more studies per

Horner et al., 2005) for all but three componentsof these treatment packages: in vivo instruction (n= 4), use of picture cues (n = 3), and saying noto correct errors (n = 4). These three methods didmeet the criteria for number of researchers, loca-tions, and participants. All but two of the instruc-tional techniques (massed trial and telling astudent no when an error is made) could be la-beled as a large effect based on the Mastropieriand Scruggs (1989) criteria. These two practices

would be labeled as moderate effects. In vivotraining had the highest median PND at 100%.Reinforcing a students correct response, stimulus

prompting, and physical guidance had medianPND values above 97%.

In order to further examine the strengths and weaknesses among this group of components ofsystematic instruction treatment packages, weconducted a series of Mann-Whitney U analysesfor each of the components using the median asthe measure of central tendency. The in vivomethod of instructional format was statisticallysignificant (U= 12.5,p = .032); studies that usedthis technique had higher median PND valuesthan studies that did not use it. The massed trialformat method of instructional format was statis-

tically significant (U= 6.0, p = .032), indicatingthat studies that used this technique had lowermedian PND values than studies that did not useit. The physical guidance prompting technique

was also statistically significant (U = 24.0, p =.023) indicating that studies that used this tech-nique had higher median PND values than stud-ies that did not use it. All the other techniqueslisted in Table 2 were not statistically significant.

Due to the limited number of studies withall indicators, those that were missing one of theHorner et al. (2005) criteria (most) were com-

bined with the all indicators evidence-basedstudies in a second analysis. These results are pre-sented in Table 3. Given the Horner et al. require-ment of five replications with three differentresearchers in three different locations and at least20 participants, only those comparisons with afrequency of at least five were included in Table 3.

We conducted a nonparametric Krusal-Wallis testusing the median as the measure of central mea-

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TA

B

LE

1

NCTMComponentsReflectedinSingle-SubjectResearchStudiesWithStudentsWithSignificantCognitiveDisabilities(AllorMostQuality

IndicatorsforResearchDesign)

Skills

No.ofStudies

Represented

No.ofStudiesfor

No.of

NCTM

All

Most

inthe

Each

Skilla

GeographicLocations

b

No.ofAuthorsb

No.ofParticipantsb

Component

n=19

n=34

Component

All

Most

All

Most

All

Most

All

Most

Measurement

13

18

Money

7

10

3

9

6

10

25

40

Purchasing

5

6

4

6

4

6

29

19

Time

1

1

1

1

1

1

2

4

Numberand

6

11

Calculations

4

5

4

5

4

5

8

11

Operations

NumberID

3

0

3

0

3

0

16

0

Counting

1

2

1

2

1

2

5

25

DataAnalysis

0

2

Graphing

0

2

NA

2

NA

2

NA

6

andProbability

Geometry

0

2

Shapes

0

2

NA

2

NA

2

NA

7

Algebra

0

1

Wordproblems

0

1

NA

1

NA

1

NA

1

Note.Studiesclassifiedasall(a)operationallydefinedthedependentva

riableandincludeddataoninterraterreliability,(b)adequatelydescribedmethodsincludingthein-

dependentvariable,(c)hada

dequateproceduralfidelityabove80%,and

(d)hadbaselineandexperimentalcontrol(betweenorwithinparticipantreplications).Studieswith

mosthadofthesecriteria.

aSomestudiesincorporatedm

orethanoneskill.bThesestudiesrepresent

atotalof17researchers,in10locationswit

h63participants.


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416 Summer 2008

TA

B

LE

2

MedianandMeanPNDValuesforInstructionalComponentsRepresentedinMathematicsEvidence-BasedSin

gle-SubjectResearch

Confidence

No.of

No.of

No.of

No.of

No.of

InstructionalComponent

Median

Mean

Interva

l

Studies

Comparisonsa

Researchers

Locations

Participants

Sig.

InstructionalFormat

Invivo

100.0

98.1

91.8

100

4

4

4

3

21

.032

Totaltaskchaining

95.4

94.1

89.9

98.1

6

10

5

4

23

ns

Massedtrial

81.3

79.7

71.8

87.5

9

9

8

6

32

.001

Prompting

Stimulus

97.6

95.9

90.3

100

5

6

5

5

39

ns

Physicalguidance

97.2

97.0

94.5

99.5

6

7

5

4

25

.023

Picturecue

95.9

91.2

70.9

100

3

3

4

3

21

ns

Physicalmodel

95.5

91.0

79.5

100

8

8

7

6

31

ns

Verbalmodel

95.4

89.4

80.8

97.8

9

12

8

7

35

ns

Post-response

91.2

85.4

73.4

97.3

5

7

4

3

17

ns

PromptFading

Leastintrusive

93.8

90.2

80.0

100

7

7

6

5

33

ns

Constanttimedelay

92.2

89.2

81.8

96.6

8

9

7

6

32

ns

ErrorCorrection

Nextlevelprompt

93.8

91.6

82.9

100

7

7

6

5

30

ns

Teacherdemonstration

91.7

89.0

77.5

100

9

8

8

7

28

ns

Studentrepeatcorrect

91.5

87.5

76.3

98.6

8

8

7

6

30

ns

Sayno

89.0

88.3

81.2

95.3

4

4

4

3

22

ns

ReinforceCorrectResponse

97.8

90.3

85.6

95.1

17

22

15

9

65

ns

Note.ns=notstatisticallysig

nificant.

aSomeresearchersemployed

multiplecomparisonsofinstructionalvariab

lesinasinglestudy.Testsofsignificancewereusedforcomparisonsofstudiesthatdidincludethis

individualcomponentinthe

interventiontreatmentpackagewiththosethatdidnot.


11/26


TA

B

LE

3

MedianandMeanPNDV

aluesforMathematicsInstructionalComponentsinSingle-SubjectResearchWithM

ostandAllQualityIndicatorsforResearc

hDesign

Confidence

No.of

No.of

N

o.of

No.of

No.of


Median

Mean

Interval

Studies

Comparisonsa

Researchers

Locations

Participants

Sig.

Invivo

100

94.7

86.2

9

100

8

11

7

6

36

.013

Totaltaskchaining

96.9

85.7

73.2

7

98.0

13

19

9

6

47

ns

Massedtrial

90.0

79.8

70.5

0

89.0

4

24

36

22

15

73

ns

Prompting

Physicalguidance

96.9

89.8

81.4

4

98.2

7

10

17

9

7

38

ns

Physicalmodel

95.6

85.7

76.0

1

95.3

2

22

30

17

12

73

ns

Stimulus

95.5

85.1

73.9

2

96.2

2

18

27

15

11

65

ns

Verbalmodel

94.4

84.1

75.8

5

92.3

3

27

41

23

16

90

ns

Postresponse

90.0

80.9

69.5

4

92.3

8

16

22

13

8

45

ns

Picturecue

82.9

82.9

69.1

0

95.4

8

8

13

8

6

31

ns

PromptFading

Leastintrusive

97.2

87.0

76.2

7

97.7

5

13

21

10

8

56

ns

Constanttimedelay

92.7

84.0

68.1

6

99.8

4

10

12

9

6

40

ns

ErrorCorrection

Nextlevelprompt

93.9

84.8

74.5

1

95.1

7

16

27

13

10

56

ns


93.8

90.6

84.8

9

96.2

0

20

26

16

9

53

ns


92.9

87.9

81.0

94.8

7

18

22

16

11

60

ns

Sayno

91.9

91.0

84.8

2

97.1

8

8

12

8

7

31

ns


92.9

84.6

78.55

90.6

5

44

61

38

22

140

ns

Note.ns=notstatisticallysignificant.Studiesclassifiedasall(a)operat

ionallydefinedthedependentvariableandincludeddataoninterraterreliability,(b)ade

quately

describedmethodsincludingtheindependentvariable,(c)hadadequate

proceduralfidelityabove80%,and(d)had

baselineandexperimentalcontrol(between

orwithin

participantreplications).Stu

dieswithmosthadthreefourthsofthesecriteria.Testsofsignificancewereusedtocomparestudiesthathadtheindividualcompo

nentinthe

interventiontreatmentpackagewiththosethatdidnot.


multiplecomparisonsofinstructionalvariablesinasinglestudy.


12/26

sure of tendency for each of the components ofsystematic instruction to determine if any signifi-cant differences existed between the PND values

of the studies with most indicators and PNDvalues of the all indicators studies. None of theKrusal-Wallis tests was significant, indicating that

we could assume that the PND values for bothsets of studies were similar. Therefore, the medianPND values; confidence intervals; and number ofstudies, comparisons, locations, and researchersfor each of the components of systematic instruc-tion were reanalyzed using this combineddatabase. This database yielded 49 studies with 62instructional comparisons. Two additional studies

with most indicators were eliminated from thePND analysis because they did not provide ade-

quate baseline data. The PND values for the re-maining studies ranged from 79.15% to 100%,

with a median of 92.60%. All of the PNDs ex-cept for one can be classified as a large effect usingthe Mastropieri and Scruggs (1989) criteria. Thepicture cue technique was rated as a moderate ef-fect. It had been rated as a large effect in the pre-vious analysis of all indicators studies. Althoughnot considered as a statistically significant differ-ence, the two instructional methods identified asa moderate effect in the all indicators studydatabase (saying no and massed trial) were clas-sified as a strong effect in the combined database.

We conducted a second series of Mann-Whitney U analyses for each of the componentsof the systematic instruction treatment packagesusing the combined database and median as themeasure of central tendency. Once again, the invivo method of instructional format was statisti-cally significant (U = 148.0, p = .013); studiesthat used this technique had higher median PNDvalues than studies that did not use it. None ofthe other Mann-Whitney U analyses was statisti-cally significant (see Table 3).

In our final set of analyses, we evaluated

whether the strong evidence of effectiveness forsystematic instruction would be found for each ofthe mathematics skills with 5 or more studies in-cluding money (coin and bill identification andvalues), purchasing, and calculation (See Tables46). The median PNDs for money (84.80%)and calculation (89.80%) are classified as a mod-erate effect whereas the median PND for purchas-ing (97.60%) is classified as a large effect. Thus,

the evidence for systematic instruction as an evi-dence-based practice is stronger for purchasingthan money use (not combined with purchasing)

or calculation. Mann-Whitney U analyses foreach of the components of the systematic instruc-tion treatment packages were nonsignificant. Forexample, in vivo instruction was not found toproduce significant effects within the purchasingstudies alone, but only the purchasing studiesused in vivo.

As Tables 4 through 6 reveal, not all subcom-ponents of the systematic instructional treatmentpackages had the recommended 5 studies and 20participants as well as not revealing significant ef-fects in comparisons of components. In contrast,a review of the types of components and effect

sizes for components provide some hints for whatmay have contributed to instructional effective-ness. For example, the money studies (Table 4)suggested a reliance on error correction (see mod-erate effect sizes for error correction strategies)and massed trial instruction (also moderate ef-fect), with weaker effects for the types of promptsthan the purchasing studies. The purchasing stud-ies (Table 5) not only used prompting but also ex-plicit prompt fading strategies like the system ofleast prompts or time delay. This provides a possi-ble clue that it is not just the use of definedprompts that makes systematic instruction effec-

tive, but the use of explicit prompt fading strate-gies. In contrast, the system of least promptsapplied in the calculation studies is associated

with weaker effects than teacher demonstration.Given that there were only 3 studies that used thesystem of least prompts, it is important to viewthis outcome with caution. What may be sug-gested is that some mathematical concepts mayneed more than simply guiding a student to makea specific response (e.g., guiding their hand totouch the answer). Some may need explicit con-ceptual demonstrations (e.g., demonstrations of

place value).Evidence-Based Practices for Evidence-BasedGroup Studies. In order to address Research Ques-tion 4 for the group studies, we applied the Ger-sten et al. (2005) criteria to the 14 group designstudies, again using the all or most classifica-tion of the quality indicators for group research inspecial education. Of the 14 group studies identi-fied in the comprehensive literature review, none

418 Summer 2008


13/26


TA

B

L

E4

MedianandMeanPNDValuesforInstructionalComponentsRepresentedinMostandAllSingle-SubjectResearch(MoneyStudies)

Confidence

No.of

No.of

No.of

No.of

No.of


Median

Mean

Interval

Studies

Comparisonsa

Researchers

Locations

Participants

Sig.

InstructionalFormat

Massedtrial

84.8

0

74.2

3

57.2

3

91.2

2

11

12

11

8

37

ns

Totaltaskchaining

74.3

0

62.2

0

0

100

4

4

4

4

16

ns

Prompting

Stimulus

81.0

0

69.2

9

43.0

7

95.5

1

8

8

8

8

24

ns

Physicalmodel

75.8

0

66.5

7

36.8

3

96.3

1

7

7

7

7

21

ns

Verbalmodel

75.8

0

72.4

0

49.4

8

95.3

2

9

9

9

8

28

ns

Picturecue

75.8

0

79.1

0

57.3

8

100

5

5

5

3

20

ns

ErrorCorrection

Sayno

91.7

0

90.2

7

81.3

9

99.1

4

3

3

3

3

11

ns


89.7

0

82.8

1

71.8

1

93.8

1

8

9

7

6

30

ns


86.2

0

79.9

8

67.2

0

92.7

6

8

9

8

6

25

ns

Nextlevelprompt

74.4

0

63.8

2

28.3

1

99.3

3

6

6

6

5

19

ns


84.8

0

76.3

1

63.3

0

89.3

3

15

16

15

11

50

ns

Total

84.8

0

76.3

1

63.3

0

89.3

3

15

16

15

11

50


nificant.



lesinasinglestudy.Testsofsignificancewereusedtocomparestudiesthathadtheindividual

componentintheinterventiontreatmentpackagewiththosethatdidno

t.


14/26

420 Summer 2008

TA

B

LE

5

MedianandMeanPNDValuesforInstructionalComponentsRepresentedinSingle-SubjectResearch(PurchasingStudies)

Confidence

No.of

No.of

No.of

No.of

No.of


Median

Mean

Interva

l

Studies

Comparisonsa

Researchers

Locations

Participants

Sig.

InstructionalFormat

Totaltaskchaining

97.2

0

94.6

4

89.2

1

100

6

7

6

5

21

ns

Invivo

95.3

5

87.1

8

55.7

6

100

4

4

4

4

24

ns

Prompting

Stimulus

99.0

0

98.8

0

96.5

4

100

4

4

4

4

20

ns

Verbalmodel

98.0

0

91.9

7

77.9

8

100

7

7

5

5

27

ns

Physicalguidance

97.2

0

89.0

6

67.3

4

100

5

5

4

4

23

ns

Physicalmodel

97.6

0

90.8

8

73.8

1

100

6

7

5

5

21

ns

PromptFading

Systemofleastprompts

97.2

0

79.2

9

47.8

8

100

7

7

5

5

34

ns

Constanttimedelay

92.7

5

79.2

8

43.1

5

100

5

6

4

3

25

ns

ErrorCorrection

Nextlevelprompt

97.6

0

90.8

8

73.8

1

100

6

6

5

5

23

ns


96.0

5

87.5

3

55.8

6

100

4

4

4

4

16

ns


97.6

0

85.7

5

68.7

9

100

11

12

10

10

45

ns

Total

97.6

0

85.7

5

68.7

9

100

11

12

10

10

45

ns


nificant.



lesinasinglestudy.Testsofsignificancewereusedtocomparestudiesthathadtheindividualcom-

ponentintheinterventiontreatmentpackagewiththosethatdidnot.


15/26


TA

B

LE

6

MedianandMeanPNDValuesforInstructionalComponentsRepresentedinSingle-SubjectResearch(CalculationStudies)

Confidence

No.of

No.of

No.of

No.of

No.of


Median

Mean

Interva

l

Studies

Comparisonsa

Researchers

Locations

Participants

Sig.

InstructionalFormat

Massedtrial

76.1

0

82.9

3

45.9

8

100

3

3

3

3

6

ns

PromptFading

Systemofleastprompts

76.1

0

82.9

3

45.9

8

100

3

3

3

3

6

ns

ErrorCorrection


92.1

0

93.2

0

77.5

0

100

3

3

3

3

4

ns


92.1

0

83.8

1

61.6

9

100

7

7

7

6

12

ns

Total

89.8

0

82.8

5

64.2

0

100

8

8

8

6

12

Note.Testsofsignificancewereusedtocomparestudiesthathadtheindividualcomponentintheinterventiontreatm

entpackagewiththosethatdidnot.

ns=notstatisticallysignifica

nt.


16/26

was found to meet all four of Gersten et al.s crite-ria for quality indicators in group design research.Four studies were identified as having most cri-

teria (three out of four). All of the studies incor-porated a measure that directly linked to theinstruction, but only 2 incorporated multiplemeasures that included a measure of generalizedperformance. Four group studies included proce-dural fidelity and 7 included interrater reliability.Two group studies used a computer program in

which procedural fidelity and interrater reliabilitymay be assumed in the program, but such was notstated. None of the group studies discussed relia-bility or validity of the outcome measures. All ofthe studies provided data analysis connected tothe research questions, 10 provided inferential

statistics, but none provided an effect size. An ef-fect size could be calculated from the availablestatistics for 6 of the studies. The 4 studies that

were classified with most indicators providednine instructional comparisons (1 study had sixcomparisons). Given the limited number of stud-ies with the number of possible instructionalmethods, each method resulted in at most twocomparisons. Therefore, due to the insufficientdatabase, we did not conduct additional analysesconducted on the group studies.

D I S C U S S I O N

L I MI TA TI O N S

Before discussing the findings to consider whatguidance might be drawn for mathematics in-struction, it is important to consider three limita-tions of this meta-analysis. One limitation for anycomprehensive literature review is the file drawerproblem: studies with nonsignificant results arenot published. We tried to address this problemby including unpublished dissertations, but real-ize this is only a partial solution. Although somelower PND values were reported in several of thearticles represented in this study (i.e., 58.3%,73.0%, and 75.8%), by and large the PNDs fell

within the 90% to 100% range. Statistical teststhat provide an estimate of the number of studiesneeded to accommodate for this problem includeusingZscores or Cohens deffect size. A compa-rable technique for single-subject studies utilizing

PNDs could not be found. Given the number ofstudies with large effects using the Mastropieriand Scruggs (1989) criteria represented in this

study, the possibility of the file drawer problemimpacting these results must be considered. Also,although considerable effort went into locatingarticles on teaching mathematics to individuals

with significant cognitive disabilities, it is alsopossible we overlooked one or more studies.

It is not just the use of defined prompts thatmakes systematic instruction effective, butthe use of explicit prompt fading strategies.

Second, the application of quality indicatorsfor research as proposed by Horner et al. (2005)and Gersten et al. (2005) is a recent phe-nomenon. Overall, we found a small body of re-search using group experimental procedures withso much variation that no analyses could be com-pleted to identify evidence-based interventions

within this set. The lack of group research may beunderstandable given that this is a low incidencepopulation, which makes the logistics of identify-ing groups for intervention difficult. What also isnotable is that the quality of most of this research

was not acceptable using the Gersten et al. crite-

ria. Similarly, Browder, Wakeman, Spooner,Ahlgrim-Delzell, and Algozzine (2006) were un-able to include group studies in their meta-analy-sis of reading for this population due to their lownumber and quality. The need exists for furtherdiscussion among researchers about the applica-tion of high quality group design to low incidencepopulations.

In contrast, the application of Horner et al.s(2005) quality indicators for the single-subject re-search was feasible and provided a small but ade-quate body of studies for further analysis (n = 19).

Although we chose a fine-grain analysis for eachcomponent of the treatment packages used inthese studies, it is important to remember thatnone of these variables (e.g., massed trials, con-stant time delay) was used in isolation. Instead, inevery study, there were either massed or dis-tributed trials of instruction with prompting andfeedback. The clearest evidence-based practice wassystematic instruction, a procedure derived from

422 Summer 2008


17/26

applied behavior analysis including operationaliz-ing specific responses to be acquired through theuse of systematic prompting and feedback

(Collins, 2007). All 19 studies with all quality in-dicators and the 34 total with all or mostquality indicators used systematic instruction.Thus, the overall finding is for moderate to strongeffects for the use of systematic instruction toteach mathematics to students with significantcognitive disabilities. Because systematic instruc-tion is typically a treatment package, we thoughtit would be interesting to try to identify whichcomponents of these packages produce the largesteffect size. The limitation of this approach is thata much smaller number of studies were thenavailable for consideration of each component.

We also do not know from our data the extent towhich the combination of treatments influencedeach other. For example, a procedure like in vivoinstruction, although producing strong effects,

was not used in isolation, but with systematicprompting and feedback. All of the meta-analyses

were descriptive in nature (summarizing and de-scribing the characteristics of the numbers), notinferential; therefore, causality cannot be inferred.It would be erroneous to conclude, for example,that in vivo instruction caused the larger effectsize. Thus, the strongest overall finding of thismeta-analysis is support for a systematic instruc-

tion treatment package. The analysis of individualcomponents associated with stronger effectsshould be viewed with caution.

A third caution is the possible overestimationof effects that may be unique to studies with stu-dents with significant cognitive disabilities due toboth the frequent occurrence of zero baselinesand the measurement of a small scope of re-sponses. Often in studies with students with sig-nificant cognitive disabilities, the baseline phase iszero or near zero performance. If students learn atleast one new response in the first couple of ses-

sions and do not regress to zero, the resultingPND is strong. Sometimes the responses beinggraphed, although useful to participants, are ex-tremely limited in the scope of mathematicallearning. For example, the participants may onlylearn to count out single dollars up to 10 or dis-criminate between a $1 and a $5 bill. Future ap-plications of effect size may need to include someconsideration of the magnitude of the outcomes

for this population. For example, solving an alge-braic equation and identifying numbers couldtheoretically have the same number of responses

being measured (e.g., 10 responses), but the mag-nitude of mathematical learning is not necessarilythe same.

CO MP ON EN T S O F MA T H E MA T I CS A N DTY PE S OF SK I LLS

The first and second research questions for this re-view were concerned with the components ofmathematics and specific types of skills that havebeen acquired by this population. The limitedscope of current research on teaching mathematicsto this population leaves deficits in understanding

how to access the general curriculum in this area.As shown in Figure 1, researchers have emphasizedmeasurement skills (including money) and com-putation. By including the self-monitoring litera-ture, we were also able to locate two studies in

which participants received instruction in how tocompile and evaluate data that reported data onthis instruction even though these were not iden-tified as mathematics studies per se. We foundonly three geometry studies and six algebraic ap-plications, but these did not have sufficient qual-ity to draw conclusions about evidence forpractice. In contrast, it would be difficult to find

a state that does not have content standards in ge-ometry and algebra. The question that arises is

whether all components of mathematics are rele-vant for this population. Access to the generalcurriculum requires setting priorities for students

who need intensive instruction to master andgeneralize skills; it may be argued, however, thatthis prioritization should include targets in eachof the five major component areas of mathemat-ics. Perhaps students should have some priorities

within geometry or algebra, albeit not necessarilythe entire scope and sequence of this domain.

Whether or not this content has relevance for thispopulation needs both stakeholder input and em-pirical investigation. In the absence of research toeither support or refute teaching these skills, somestates (e.g., Kansas and South Dakota) have in-cluded geometry or algebra content in their alter-nate assessments, which implies that theirdiscussions led their stakeholders to choose broadgeneral curriculum access (see Kansas State



18/26

Department of Education, 2007; South DakotaDepartment of Education, 2007).

WH AT T HE RESEA R C H RE V E A L S ABO U TMATHEMATICAL LE AR NI NG F OR T HI S PO P U LA TI O N

Given the limitations noted, some conclusionscan be drawn from the research about mathemati-cal learning for this population. The third ques-tion we asked of this literature was what evidenceexists that students with significant cognitive dis-abilities can learn mathematics. We found thatstudents with both moderate and severe cognitivedisabilities can learn specific target skills like com-putation (e.g., Matson & Long, 1986); graphing

(e.g., Ackerman & Shapiro, 1984); matchingshapes (e.g., Mackay et al., 2002); and countingmoney (e.g., Gardill & Browder, 1995). In stud-ies that focused on functional applications, partic-ipants also demonstrated their ability to applyskills like money use to real-life contexts (e.g.,Colyer & Collins, 1996; Test et al., 1993). Thereis more evidence that students can learn computa-tion and measurement (including money) thandata analysis, geometry, and algebra, but this isdue more to the lack of research in these otherareas. More evidence also exists for students withmoderate versus severe disabilities to acquire

mathematical skills, but again, this can be at-tributed to the smaller number of studies includ-ing students with more severe disabilities. Giventhe inadequate number of studies to draw firmerconclusions about teaching some strands of math-ematics and including students with more severedisabilities, the few studies that do exist offersome promise for pursuing additional research.

We also do not know whether students canbuild these skills towards some generalized mathe-matic semantical competence (numeracy); no re-search to date has evaluated a comprehensiveprogram across years for this population. To gaincompetence in numeracy, students need to masternot only the content of mathematics but also itsprocesses; we found no studies that pinpointedthese processes. NCTM (2000) identifies fiveprocesses of mathematics including problem solv-ing, reasoning and proof, communication, con-nections, and representation. Prior research hasshown that individuals with moderate and severe

disabilities can learn problem-solving skills(Rusch & Hughes, 1989), but these skills havenot been applied in studies that include mathe-

matical content except in a couple of studies withweak to most quality.

EV I D EN C E-BASED PR A C TI C E

The fourth question we posed was what evidence-based practices can be recommended for teachingmathematics to students with significant cogni-tive disabilities. Table 3 provides a list of the in-structional components and specific practices foreach component sorted by median effect size. Asmentioned earlier, the strongest evidence is theoverall outcome for the use of systematic instruc-

tion. Specifically, we found 34 studies with allor most of Horner et al.s (2005) research qual-ity indicators that used systematic instruction toteach mathematics to students with significantcognitive disabilities. For example, Colyer andCollins (1996) had students with moderate dis-abilities practice counting an amount of moneyneeded using flash card training in the classroom.The teacher showed prices ranging from $.01 to$5.00 on each flash card and the student countedout $1 bills rounding up to the nearest dollar. Foreach card, the teacher followed a sequence of

prompts beginning with stating the price (like acashier would), then telling the student howmany dollars were needed (e.g., for $4.95, theteacher would say Give me four dollars and onemore). If the student still did not respond cor-rectly, the teacher modeled counting out theamount and had the student follow the model.Students then applied their new money countingskills to make community purchases. Two stu-dents with moderate disabilities mastered count-ing dollars; one did not. In contrast, in a studythat included students with a severe intellectualdisability, McDonnell and Laughlin (1989) gave

participants preselected dollar amounts and fo-cused on teaching purchasing skills. Gardill andBrowder (1995) chose an interim skill of teachingstudents with moderate intellectual disabilitiesand severe behavior disorders who had notlearned to count out the next dollar amount todiscriminate between types of money to makepurchases (e.g., $5 bill for lunch; $1 bill for soda).

424 Summer 2008


19/26

Each of these three studies had some in vivoinstruction; that is, participants applied theirlearning to real-world settings (e.g., store, restau-

rant) utilizing the mathematics skills. In our anal-yses of specific components of the treatmentpackages used in this body of research, we foundstrong support for in vivo instruction. Studiesthat used this instructional component had highquality designs, large effect sizes, and effects that

were significantly higher than those that did notuse the strategy. Note that all three examples(Colyer & Collins, 1996; Gardill & Browder,1985; McDonnell & Laughlin, 1989) also usedsome time of systematic prompting with an ex-plicit fading procedure. Constant time delay andthe system of least prompts were found to have

strong effect sizes, although these were not signifi-cantly different from studies that did not usethese two strategies. In contrast, enough studiesand participants were found in the purchasingstudies to identify explicit fading strategies as anevidence-based practice for purchasing skills(Table 5).

Systematic instruction should includethe use of a specific prompt fading

procedure such as least intrusive prompts

or time delay with feedback to teacha set of defined responses across time.

It is difficult to understand why the studieswith physical guidance had larger effects thanstudies that did not use this method (Table 2).Because physical guidance is typically used incombination with a prompt fading procedure liketime delay or least intrusive prompting, this find-ing should not be viewed as endorsement of usingthis method alone or without consideration of thestudents abilities (e.g., some students do not needphysical guidance). Studies that used massed trialinstruction had significantly weaker effects thanstudies that did not use this strategy. This out-come should also be viewed with caution; massedtrials are often used to help students master a setof difficult discriminations like matching dollaramounts to purchase prices (e.g., Colyer &Collins, 1996). In contrast, teachers might notuse massed trials when teaching students to use

one specific mathematics response in the contextof an activity (e.g., selecting a dollar to use avending machine). As described in the Limita-

tions section, effect size needs to be consideredconcurrent with the magnitude of mathematicallearning.

One finding that did not relate to one of theresearch questions is the lack of statistical differ-ence between the PNDs and effect sizes of studiesthat included all of the quality indicators forspecial education research and those that includedmost of the quality indicators. This finding hasbeen noted by other researchers as well (Graham& Perin, 2007; Pressley, Graham, & Harris,2006). Most of the studies in this analysis that

were classified as having most of the quality in-

dicators (i.e., missing only one of the criteria) didnot include information regarding procedural fi-delity of the implementation of the intervention.One possible explanation is that procedural fi-delity was conducted within the study, but not re-ported in the published article; procedural fidelityhas only been emphasized in journal publicationfor the past 10 years (Graham & Perin). Addi-tional research is needed to determine the rela-tionship between quality of research and studyoutcomes.

IM P L I C AT I O N S F O R

PR A C TI C E

The most important implication for practice fromthis research is that systematic instruction is anevidence-based practice for teaching mathematicsto students with significant cognitive disabilities.This systematic instruction should include the useof a specific prompt fading procedure such asleast intrusive prompts or time delay with feed-back to teach a set of defined responses acrosstime. Evidence-based practice will also includeopportunities to learn and use these skills in vivo:in the activities in which these mathematical skillsare typically applied (e.g., shopping, job applica-tions, etc.). This evidence-based practice is sum-marized in Table 7.

The challenge that arises is that the intensityof instruction that this population may needoften differs from the nature of instruction ingeneral education contexts (Billingsley & Kelley,1994). Students may need additional support tomaster specific target responses with systematic



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426 Summer 2008

T A B L E 7

Summary of Evidence-Based Practice for Teaching Mathematics to Students With SignificantCognitive Disabilities

Extension to OtherPractice Approach Examples Mathematics Components

In vivo Teach a real lifeapplication forthe mathematicsprinciple to belearned; teach in,as well as for, reallife settings

Using money to make a purchase Making relative comparisons

among prices while shopping

Applying graphing skills to familiaractivities such as behavior goals

Applying concepts ofgeometry to plan aroute to destination

Using algebraic equationsin computing unknownamount (how many morestamps needed to mailinvitations)

Using graphs tosummarize informationlearned in general

education (e.g., weatherpatterns; election results)

Systematicinstruction

Define a specificresponse or set ofresponses and teachto mastery usingdefined, consistentprompting andfeedback andexplicit promptfading

Use a system of least prompts; forexample, waiting for the student torespond, then providing a specificverbal direction, then a model, andthen physical guidance as needed(physical guidance may be neededfor some students to learn to makethe correct response)

OR

Use time delay: select one specificprompt like a model; introduce it

concurrent with the stimulusmaterial for early teaching trials(no delay) and fade it acrossteaching trials by delaying the time(a few seconds) between presentingthe material and providing theprompt

Praise correct responses and tell thestudent specifically what was correct(Good, you counted five dollars.)

Correct errors using simple verbalfeedback and additional prompting(No, this is how you count fivedollars. Now you try.)

Opportunitiesto respond

Give the studentnumerousopportunities tolearn and practicethe new responses,but use caution inapplying a massedtrial format

Student may need daily practice forseveral weeks to learn to identify ordraw a line segment in geometry;doing so with a variety of activitiesand materials may be more effectivethan simply pointing to a linesegment 10 times


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instruction. Careful consideration also needs tobe given to the skills targeted within the generalcurriculum. Although research to date has focused

primarily on computation and measurement, stu-dents may be able to learn skills in other contentstrands. What will be important is to define skills

within content areas like algebra and data analysisto operationalize the outcomes for learning.

Not all students will be able to acquire thesame level of mathematical understanding, but

with support may achieve the end goal. In plan-ning access to general mathematics curriculum, it

will be important to have multiple points of ac-cess. Some students may be able to master theconcept or perform the operation (e.g., count outmoney). Others may do so with adaptive devices

or other forms of support (e.g., discriminate be-tween currency); and others may need to focus ona life application while still learning some aspectof the content (e.g., that money is exchanged forgoods). Similarly, some students might learn tosolve simple algebraic equations (e.g., 2 +x= 5 bycounting from 2 to 5 on a number line); othersmay be able to apply the symbols using a real-lifeapplication. Others may simply learn the basicidea thatxis a missing quantity. Currently in re-search, participants are typically all taught thesame target response. Future research is needed to

show how achievement might be differentiatedfor students with differing levels of ability withinthe mathematical content.

FU TU R E RESEA R C H NE E D S

Future research is needed on mathematics learningfor this population that (a) extends currently effec-tive methods to other subgroups within the popu-lation, (b) provides more examples of effectivemethods overall, and (c) expands the scope ofmathematical content addressed. Although a sub-group of the studies with high quality and strongeffects included students with severe intellectualdisabilities, these were of insufficient number to beable to draw conclusions. Few participants had themultiple complex disabilities also found withinthis population (e.g., significant cognitive disabil-ity concurrent with deaf-blindness or cerebralpalsy). Future research is needed if the systematicinstruction procedures found within this literaturealso produce strong effects with these subgroups.

More research is needed on effective strate-gies overall. Even the strongest research to date in-cludes a few students who do not master the

target skills (e.g., Ackerman & Shapiro, 1984;Colyer & Collins, 1996; Kapadia & Fantuzzo,1988; Morrison & Rosales-Ruiz, 1997). Althoughthe number of studies was large enough to drawsome inferences, it is still small enough to leavemany unanswered questions, such as what alter-native methods may have worked for some ofthese students. As mentioned earlier, additionalresearch is needed to explore teaching otherstrands of mathematical content such as algebraand geometry in ways that are both effective andmeaningful for the population.

S U M M A R Y

The research to date provides evidence that stu-dents with significant cognitive disabilities canlearn mathematics content. Systematic instructionincluding the use of a prompt fading method

with feedback is an evidence-based procedure forteaching this content. This instruction may bemore effective if it includes opportunities to applyskills in real-life contexts.

This evidence primarily stems from research

on teaching computation and measurement, onlya small portion of the scope and sequence of gen-eral mathematics education. Currently, teachersare addressing areas of mathematics to promotecurricular inclusion of students and address skillsneeded to show achievement in alternate assess-ments for which there is minimal guidance inresearch. Research is needed in all of the compo-nents of mathematics that can evaluate the extentto which students acquire numeracy and whetherthey generalize these skills for enhanced life out-comes. Interventions are needed that illustratecomprehensive approaches to mathematics in-

struction and progress across grades. Studentsmay need to learn to use traditional mathematicsmaterials and symbols, not only real-life materials,to be able to more fully access the content. Withthese innovations, it will be critical to ensure thatthis mathematics learning continues to be consis-tent with ongoing values that students are learn-ing skills relevant to their preferences and needs.



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a meta-analysis on teaching math w mi

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