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Running head: MATH INTERVENTIONS
EDUC 841:
A Review of Literature on Middle School Math Interventions
Anne Brawand
George Mason University
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Abstract
Eleven investigations pertaining to math interventions used in the middle school setting
for students with and without disabilities were part of an integrative review utilizing metaanalytic research integration techniques. All of these studies appeared in eight different journals
published from 2001-2010. Effect sizes were calculated on all study outcomes. Regarding
disability type in relationship to setting, the effects of math interventions for students with
learning disabilities (LD) were most successful when they took place in a room in the school by
the researcher. The highest effect size was evident in the 7th grade for math interventions having
greater than 30 sessions, and lasting more than 76 minutes for each session. Additionally,
interventions with less than 20 students in the multiple baseline comparison had a strong effect.
Enhanced Anchored Instruction (EAI) and math fluency were the most effective interventions.
However, the Explicit Inquiry Routine (EIR) intervention had the highest effect size for students
with LD. Additionally, math fluency was the most effective intervention for students with mild
intellectual disabilities and self-regulation strategy in math had the highest effect size for
students without disabilities.
Keywords: math, intervention, students with disabilities, middle school
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A Review of Literature on Middle School Math Interventions
Mathematics is essential for daily living needs that require such skills as budgeting, time
management, and cooking, as well as for educational and occupational opportunities which
reflect an increasingly technological society that necessitates problem solving and reasoning
skills (Maccini, Mulcahy, & Wilson, 2007). The No Child Left Behind Act of 2001 (NCLB)
frameworks a national initiative to improve the link between elementary and secondary
education and high-stakes testing. In response to this legislation, the National Council of
Teachers of Mathematics (NCTM, 2000) has called for standards that range beyond skills of
basic procedural competency. According to Miller and Hudson (2007), it’s important to provide
balanced instruction across mathematics standards and to address conceptual, procedural, and
declarative knowledge within a comprehensive math curriculum. General strategies for
addressing achievement problems in secondary mathematics include organizing explicit teaching
of important concepts, providing numerous examples of new concepts that address the overall
range of the concepts, direct teaching of relevant cognitive routines, and systematically teaching
of prioritized objectives (Montague & Jitendra, 2007).
Many students with learning disabilities may exhibit difficulties in the area of memory
and general strategy use, literacy and communication, specific processes and strategies
associated with math problems, and low motivation and affect (Bryant & Bryant, 2008). A
number of students may also have difficulty with the English language and communication
aspects of mathematics (Lang & Pagliaro, 2007). Rao and Mallow (2009) expressed concern
that a lack of knowledge of basic math facts is a common impediment to learning higher-level
math for all students including those with disabilities and that students with this deficiency in
knowledge may learn neither math computation nor higher order mathematics. Additionally,
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students who are taught math skills until they achieve fluency tend to maintain their skills (Axtell
et al., 2009).
In 2007, Maccini et al. extended a previous review by Maccini and Hughes (1997) on
math interventions for secondary students with LD in order to determine the nature and focus of
current math interventions that were effective for assisting secondary students with LD. Maccini
et al. found that many practices produced significant gains for secondary students with a learning
disability in math including: mnemonic strategy instruction, graduated instructional approach,
cognitive strategy instruction involving planning, schema-based instruction, and contextualized
videodisc instruction. They also concluded that the nature and focus of many math interventions
at the secondary level tended to favor remedial interventions, neglecting to address middle
school and high school curriculum standards. However, compared to Maccini and Hughes
(1997); the percentage of studies targeting secondary math content increased from 35% to 57%.
Finally, Maccini et al. (2007) also suggested that future math interventions include more detailed
descriptions of participants, include larger sample sizes, and examine effects of math
interventions in the general education classroom.
A preliminary literature search was conducted to obtain a count of math intervention
studies that appear to be elementary vs. secondary, which yielded 73% elementary level and only
27% at the secondary level. The purpose of this paper is to review research involving effective
math interventions for students with and without disabilities in the middle school setting.
Specific questions to explore include: 1) Are math interventions more effective when participants
are identified as LD, mild intellectual disability, or non-disabled, in a specific setting with a
specific intervener? 2) Do math interventions work best for middle school students for a certain
comparison type (experimental vs. control, A vs. B, multiple baseline) and sample size? 3) How
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do design and group delivery size of the intervention influence the outcome? and 4) What impact
do study validity and quality have on the strength of the intervention?
Method
Literature Search Procedures
The database PsychInfo was searched from the years 2001 to the present using the
following descriptors as keywords: math, intervention, students with disabilities, middle school.
Due to a limited number of articles on this topic, the search was expanded to include middle
school students with and without disabilities in the last 10 years. Names of prominent scholars
in this topic were also entered in the same data bases to identify any additional sources. These
authors included Bottge and Jitendra. In addition, an ancestry search was conducted using the
articles identified from the databases.
Criteria for Inclusion and Exclusion
These search procedures identified 36 articles. Articles were examined for relevancy.
Specifically, articles had to be peer-reviewed intervention research studies in math including
middle school students with or without disabilities. Sufficient data had to be provided to
compute effect sizes for group experimental research or percent of non-overlapping data (PND)
for single subject design studies. Studies that measured progress of students, analyzed
perceptions of teachers/students, evaluated a math curriculum school or district wide, or involved
multiple subject areas were excluded.
Final Sample
This resulted in a pool of 11 articles published from 2001-2010 in the following journals:
Remedial and Special Education, Exceptional Children, Journal of Special Education, Learning
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Disabilities Research and Practice, Learning Disabilities Quarterly, Education and Training in
Developmental Disabilities, and Psychology in the Schools; that met all inclusion criteria.
Coding Instrument
Sixty-six variables were coded. Basic variable areas included general article identifying
information such as author, year of publication, and name of journal. Information on the
participating sample was also coded. This section included sample size, demographic data on the
sample such as gender, race, age, and grade level. Examples of variables included on the coding
sheet are: type of comparison for effect size, total sample size, grade level, type of disability,
number of sessions, number of minutes per session, type of dependent measures, type of design,
and study quality. For instance, studies coded as “1,” had random assignment of students for
type of design. Coding conventions or rules were developed as coding progressed to assist with
consistent decision making and coding. For example, weighted averages were always computed
when coding sample sizes by grade, age, number of sessions, and minutes per session. So
participants in grades 7 and 8 would have a grade level identified as 7.5. Also for sessions
lasting 20-30 minutes the amount of time recorded for that variable was 25 minutes.
Results
All data from the coding sheets were entered into the Statistical Package for the Social
Sciences (SPSS) computer program for analyses. The overall characteristics of the data set are
presented first. Then, effect sizes and additional study characteristics are presented followed by
overall quality of the studies.
Overall Characteristics of the Data Set
These studies resulted in a sample of 462 middle school students from sixth, seventh, and
eighth grade; with a mean grade level of 7.1. The mean age for the participants sample was
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12.97, with a mean IQ of 79.04. In terms of students with disabilities as displayed in Figure 1,
the majority of studies included students with LD (N = 6), and the remaining studies consisted of
students with mild intellectual disabilities (N = 2), and non-disabled students (N = 3). As
displayed in Figure 2, almost half of the studies were published in 2009 (N = 5). In addition, 4
of the 11 studies selected were published in Education and Training In Developmental
Disabilities, and The Journal of Special Education had the next highest amount of studies
included (N = 3). The comparison groups included in the math intervention studies selected
were almost evenly spread across the 3 types of as displayed in Figure 3: experimental vs.
control (N = 3), A vs. B (N = 4), and multiple baseline (N = 4). The range for number of
participants included in the selected of studies was from 2 to 128 students. Thirty–six percent of
the studies included a sample size with less than 20 students, and 36% of the studies had between
forty and sixty students for sample size. Nine out of the 11 studies did not specify
socioeconomic status, and the population density identified was rural, as well as location of the
Midwest yielded the highest amount of studies (N = 4 for both population density, and
geographic location). However, 2 of the 11 studies didn’t specify geographic region and 3
studies did not report population density.
The intervention categories identified in the studies consist of math fluency; the Explicit
Inquiry Routine (EIR); math technology; Enhanced Anchored Instruction (EAI); self-regulated
training in math; problem solving; and concrete, representational, abstract (CRA). The EIR
intervention integrates teaching practices from both general and special education to engage
students in an inquiry process across concrete, representational, and abstract modes to develop an
understanding of the concept (Schuermann, Deshler, & Schumaker, 2009). The math technology
intervention used was a FLY Pentop Computer by Leapfrog (Bouck et. al, 2009), and EAI refers
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to the presentation of problems in a multimedia format and then application of what students
learn in a hands-on format (Bottge et al, 2007).
Word problems were the most prominent dependent measure (N = 4), and algebra
concepts were the least (N = 1). The other types of dependent variables were math achievement
(N = 3), and fractions (N = 3). The special education classroom was utilized the most for the
middle school math interventions studied (N = 5), and the general education teacher delivered
the intervention in the majority of studies (N = 6). The group delivery size ranged from 1 to 4
students, with a mean total of 2.73 students. The mean number of experimental sessions is 30.2,
with a mean amount of 51 minutes for each session. Finally, the overall effect size for the
selection of studies included is significant (ES = .86), as well as an overall percent of nonoverlapping data (PND) result of 84.3%.
Effect Sizes by Disability, Setting and Teacher
As displayed in Table 1, math interventions in the experimental condition were most
effective when they took place in a room in the school by the researcher for students with LD
(ES = 2.32, PND = 93%). Students with mild intellectual disabilities also yielded strong results
when the intervention was implemented by the researcher in a separate room in the school (PND
= 100%). Students with LD who received the treatment in a regular class, with a regular teacher,
were close behind these outcomes, ES = 1.27. Finally, students without disabilities also had high
effect sizes after treatment in a regular class with the regular teacher conducting the intervention,
ES = .61.
Effect Sizes by Type of Comparison and Sample Size
Math interventions implemented utilizing a multiple baseline comparison had the
strongest effect sizes (ES = 2.32, PND = 84%). The comparison of A vs. B also yielded high
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effects, ES = .69, followed by experimental vs. control, ES = .55. A breakdown analysis to study
sample size in relation to these results had the highest effect size for the smallest sample size
range of less than 20 students in the multiple baseline comparison (ES = 2.32). The sample size
values of 2 to 128 students were recoded to reflect these ranges of effect sizes in Table 2.
Additionally, a sample size of 21 to 40 students for the experimental vs. control comparison type
had the next highest effect (ES = 1.2), followed by a sample size of more than 41 students in the
A vs. B comparison type (ES = .68).
Effect Sizes by Disability and Intervention
Overall, EAI was the most effective intervention (ES = 2.32, PND = 93%), followed by
math fluency (ES = 1.2, PND = 100%). Table 3 displays a breakdown analysis to detail the most
effective math interventions by disability type. The EIR intervention had the highest effect size
for students with LD (ES = 2.32, PND = 93%). Math fluency was the most effective
intervention for students with mild intellectual disabilities (PND = 100%), and self-regulation
strategy in math had the highest effect size for students without disabilities, ES = .51.
Effects by Grade, Intensity and Duration
Grade 7 had the highest effect sizes for math interventions overall, ES = 1.04, and PND =
97%. Further analysis was conducted including intensity and duration to attempt to explain this
data. The only 6th grade study did not include number of minutes so only 7th grade effects by
duration and intensity are reported in Table 4. The highest effect size is evident in the 7th grade
for math interventions having greater than 30 sessions, and lasting more than 76 minutes for each
session, ES = 2.16.
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Effect sizes by Design and Group Delivery Size
There was not a significant effect between the types of design implemented in the studies,
as they were both identified as having high significance. The random assignment design resulted
in an effect size of .76, and non-random/relevant matching design yielded and effect size equal to
.76. Effect sizes for group delivery size were pretty evenly spread, with small groups of 2 to 8
students having the most significant effect, ES = 1.16. Interventions implemented individually
resulted in a strong PND of 91%, followed by whole class group delivery size, ES = .76.
Effect Sizes by Validity and Quality of Study
Overall study quality was coded in two ways by validity and quality. If the study
included randomization it was classified as having “high” validity if 2 or more classes to each
condition it was of “medium” validity, and “low” validity was identified as just 1 class per
condition. A quality coding rubric was also used to rate the overall quality of the study in
relation to 9 key components consisting of: participant/setting descriptions, ability to be
replicated, random assignment, detailed intervention procedure, appropriate design for research
questions, technique used for data analysis, fidelity of implementation, effect sizes given, and
reliability measured (see Appendix). Studies that included all 9 of these aforementioned
components were coded as “high” quality, at least 6 components were “medium” quality, and at
least 3 components were identified as “low” quality. Studies with high validity/random
assignment and high quality had the highest effect, ES = 1.2. Studies with medium validity and
medium quality had the next strongest effect, ES = 1.09. Finally low validity and low quality
studies yielded an effect size of .51.
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Summary and Conclusions
A meta-analysis of 11 experimental studies revealed that effect sizes vary for the math
interventions for middle school students with and without disabilities according to specific
variables and areas being measured. As previously mentioned, Maccini et al. (2007) suggested
that math interventions include more detailed descriptions of participants, larger sample sizes,
and examine the effects of math interventions in the general education classroom. It seems that
the majority of studies reviewed adhered to these suggestions as they were able to be coded so
effect sizes could be compared. Regarding disability type in relationship to setting, the effects of
math interventions for students with LD were most successful when they took place in a room in
the school by the researcher. This finding is important because a high percentage of middle
school students with LD are included in co-taught environments, so they may be more successful
working on certain skills in a smaller environment. An additional finding relative to sample size
was that less than 20 students in the multiple baseline comparison had the highest effect size.
So, although in a smaller environment, one-on-one instruction for a math intervention is
preferred.
EAI and math fluency were the most effective interventions. However, the EIR
intervention had the highest effect size for students with LD. This shift to interventions that
measure algebra concepts, fractions and problem solving is of significance because a recent
review by Maccini et al. (2007) concluded that many math interventions at the secondary level
were focused on remedial support, and neglected to address middle school and high school
curriculum standards. Additionally, math fluency was the most effective intervention for
students with mild intellectual disabilities and self-regulation strategy in math had the highest
effect size for students without disabilities. This finding can be attributed to instructional setting
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in that math fluency interventions tend to be delivered individually or in a smaller environment
such as a self-contained classroom, and self-regulation is a strategy that can be practiced with a
whole class. It’s necessary to know the math skills that individual students need to develop and
how these individual students may be more successful with certain math interventions according
to the setting utilized. Another factor relative to setting is the duration and time of the
intervention. Highest effects were also evident in 7th grade for math interventions having greater
than 30 sessions, and lasting more than 76 minutes for each session.
Finally, studies with high validity/random assignment and high quality had the highest
effect, and studies with medium validity and medium quality had the next strongest effect. This
verifies that studies that follow quality indicators have a higher success rate. In conclusion,
interventions for students with similar disabilities in small groups have proven to be effective;
however, to generalize that finding to implementations in classroom teaching isn’t always
realistic in today’s larger inclusive settings. Teachers need to do their best to differentiate by
student learning styles or disability characteristics and employ small groups often, especially for
the presentation of new concepts.
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References
Axtell, P. K., McCallum, R. S., Bell, S. M., & Poncy, B. (2009). Developing math automaticity
using a classwide fluency building procedure for middle school students: A preliminary
study. Psychology in the Schools, 46, 526-538. doi: 10.1002/pits.20395
Bryant, B. R., & Bryant, D. P. (2008). Introduction to the special series: Mathematics and
learning disabilities. Learning Disability Quarterly, 31, 3-10.
http://www.highbeam.com/doc/1G1-176203759.html
Bottge, B. A., Heinrichs, M., Chan, S., & Serlin, R. C. (2001). Anchoring adoloscents
understanding of math concepts in rich problem-solving environments. Remedial and
Special Education, 22, 299-314. doi: 10.1177/074193250102200505
Bottge, B. A., Heinrichs, M., Mehta, Z. D., & Hung, Y. (2002). Weighing the benefits of
anchored math instruction for students with disabilties in general education classes. The
Journal of Special Education, 35, 186-200.
Bottge, B. A., Rueda, E., Serlin, R. C., Hung, Y., & Kwon, J. M. (2007). Shrinking
achievement differences with anchored math problems: challenges and possibilties. The
Journal of Special Education, 41, 31-49. doi: 10.1177/00224669070410010301
Bottge, B. A., Stephens, A. C., Rueda, E., Laroque, P. T., & Grant, T. S. (2010). Anchoring
problem-solving and computation instruction in context-rich learning environments.
Exceptional Children, 76, 417-437.
Bouck, E. C., Bassette, L., Taber-Doughty, T., Flanagan, S. F., & Szwed, K. (2009). Pentop
computers as tools for teaching multiplication to students with mild intellectual
disabilties. Education and Training in Developmental Disabilties, 44, 367-380.
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Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for
students with mathematics disabilties: comparing two teaching sequences. Learning
disabilties Research & Practice, 18, 99-111. doi: 10.1111/1540-5826.00066
Jitendra, A., DiPipi, C. M., Perron-Jones, N. (2002). An exploratory of schema-based wordproblem-solving instruction of middle school students with learning disabilities: An
emphasis on conceptual and procedural understanding. The Journal of Special
Education, 36, 23-38. doi: 10.1177/00224669020360010301
Lang, H. & Pagliaro, C. (2007). Factors predicting recall of mathematics terms by deaf
students: Implications for teaching. Journal of Deaf Studies and Deaf Education, 12,
449-460. doi: 10.1093/deafed/enm021
Maccini, P., Mulcahy, C. A. & Wilson, M. G. (2007). A follow-up of mathematics
interventions for secondary students with learning disabilities. Learning Disabilities
Research and Practice, 22, 58-74. doi: 10.1111/j.1540-5826.2007.00231.x
Miller, S. P., & Hudson, P. J. (2007). Using evidence-based practices to build mathematics
competence related to conceptual, procedural, and declarative knowledge. Learning
Disabilities Research and Practice, 22, 47-57.
doi: 10.1111/j.1540-5826.2007.00230.x
Montague, M. & Jitendra, A. (2007). Teaching mathematics to middle school students with
learning difficulties. New York: Guiliford.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA. Author.
No Child Left Behind Act of 2001, 20 U.S.C. 70 § 6301 et seq. (2002).
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Perels, F., Dignath, C., & Schmitz, B. (2009). Is it possible to improve mathematical
achievement by means of self-regulation startegies? Evaluation of an intervention in
regular math classes. European Journal of Psychology of Education, 24, 17-31. doi:
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Rao, S., & Mallow, L. (2009). Using simultaneous prompting procedure to promote recall of
multiplication facts by middle school students with cognitive impairment. Education and
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Table 1
Effects by Disability, Setting, and Intervener
Disability Area
Setting
Intervener
ES
LD
regular ed class
Regular Ed Teacher
1.27
special ed class
Special Ed Teacher
.37
Regular Ed Teacher
Mild Intellectual
Non-disabled
PND
82%
room in the school
Researcher
2.32
93%
special ed class
Special Ed Teacher
62%
room in the school
Researcher
100%
regular ed class
Regular Ed Teacher
.61
special ed class
Regular Ed Teacher
.26
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Table 2
Effect Sizes by Type of Comparison and Sample Size
Type of Comparison
Sample Size
ES
Exp. vs. Control
21-40
1.2
>40
.39
A vs. B
>40
.69
Multiple Baseline
<20
2.32
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Table 3
Effect Sizes by Disability and Intervention
Disability
Intervention
ES
Learning Disabilities (LD)
math fluency
1.2
EIR
2.32
EAI
.94
problem solving
CRA
Mild Intellectual
Non-disabled
PND
93%
82.3%
.15
math fluency
100%
technology/math
62%
EAI
.49
self-regulation/math
.51
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Table 4
Effects by Grade, Intensity and Duration
Grade
# sessions
# minutes
ES
7
<15
<50
.13
15-30
51-75
1.07
>30
<50
>30
<76
<15
<50
<15
51-75
8
PND
93%
100%
2.16
82.3%
.06
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Table 5
Effect Sizes by Validity and Quality of Study
Validity
Quality
ES
high (random assignment)
high
1.2
medium
.65
medium
1.09
low
.15
medium (2 or more classes)
low (1 class per condition)
PND
88%
high
62%
medium
100%
low
.51
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Figure 2
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Figure 3
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Appendix
Quality Index Coding and Rubric
HIGH QUALITY Score of “1” (9 out of 9 components)
______ Operational definitions of participants/setting
______ Dependent variable allows for replication (observable)
______ Attempt at random participant assignment
______ Intervention clearly described (detailed procedure)
______ Design and technique match research questions
______ Data analysis technique appropriate
______ Fidelity of Implementation described
______ Effect size/PND calculated
______ Interobserver agreement/ reliability measured
MEDIUM QUALITY Score of “2” (6 of above components)
LOW QUALITY Score of “3” (3 of above components)
_____________