Abstract
The purpose of this study was to examine geometry interventions for students with learning disabilities. We synthesized nine intervention studies by focusing on the geometry concepts and skills taught to students with learning disabilities, intervention types used, instructional components embedded, and the methodological rigor of the studies. Intervention studies were mainly single-case designs. The geometry topics included angle recognition, and perimeter, area, and volume problems. The findings of this synthesis contribute to the current literature by showing that geometry interventions for students with learning disabilities incorporated proven effective instructional components (e.g., multiple representations, skill modeling). Regarding the methodological rigor, the results showed several issues, including the lack of description of interventionist training, intervention treatment fidelity, and adequate technical information for student outcome measures. Limitations of the studies and suggestions for future intervention research are discussed.
Geometry is a core content area in school mathematics education, and students need to have an essential understanding of measurement and geometry at an early age to support complex mathematics understanding in later life (Goldenberg & Clements, 2014). Geometry provides a natural foundation for the development of reasoning and justification skills (National Council of Teachers of Mathematics, 2000); it also provides opportunities to enhance cognitive performance, communication processes, and language comprehension (Cawley et al., 2009). Moreover, geometry is one of 10 knowledge domains associated with STEM (science, technology, engineering, and mathematics)-based occupations (Carnevale et al., 2011), and improving the quality of geometry education may strengthen the preparation of STEM-capable individuals.
According to the data from the Trends in International Mathematics and Science Study in 2015, poor geometry skills represent a growing national issue. The study measured and compared student achievement in mathematics and sciences among educational systems worldwide. U.S. fourth-graders who participated in the study ranked 14th among 49 education systems worldwide in mathematics on a composite scale that combined multiple mathematics content areas, including geometry; more specifically, U.S. students ranked 23rd out of the 49 education systems in the geometry domain. For eighth graders, the results were similar: U.S. students ranked 10th in mathematics among 39 education systems and 15th in geometry. These results implied that geometry was not a relative strength for U.S. K−12 students.
For students with learning disabilities (LD), geometry is important—geometry is closely related to other content areas in mathematics, and geometry questions play a significant role in high-stakes tests. According to the Office of Special Education Programs (2017), students with LD make up about 34.4% of the total students who are eligible for special education services in the United States. They experience significant problems in one or more academic areas, including mathematics, due to cognitive processing difficulties (Gartland & Strosnider, 2018). Because many students with LD study in general education classes and take the same tests with other students, it is necessary to develop effective geometry instructional practices to meet the learning needs of students with LD.
The Common Core State Standards for Mathematics (CCSSM; National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) were created to ensure that students throughout the United States receive the mathematics education needed at each grade level and graduate prepared for later schooling or a career. Currently, 41 states have adopted the standards. The Common Core Standards Writing Team (2013), together with other mathematicians and educational researchers, has published Progressions for the CCSSM, which explains the connections between the standards and provides detailed interpretations in geometry. The CCSSM and additional resources related to the standards provide mathematics teachers and researchers with guidance regarding how to educate K−12 students, including students with LD. Specifically, in the area of geometry, the CCSSM require students to be able to (a) understand shapes in kindergarten; (b) reason about the shapes and their attributes from Grades 1 through 3; (c) discriminate and analyze shapes with the help of lines, angles, coordinate planes, or physical models to calculate area and volume from Grades 4 to 8; and (d) formalize geometric knowledge and view geometry through a careful and systematic perspective starting in high school.
The van Hiele Theoretical Model of Geometric Thought Development
Effective mathematics instruction requires that teachers understand what students know, what they need to know, and how to challenge and support them in the ways they need (National Council of Teachers of Mathematics, 2000). The van Hiele model of the development of geometric thought can be used to assess a student’s ability and guide a teacher’s instructional suggestions (Crowley, 1987). The model consists of five levels: visualization, analysis, information deduction, formal deduction, and rigor. Through sufficient experiences in geometry and teachers’ instructions, students can have a better understanding of geometry problems. However, in reality, many teachers face challenges when teaching geometry (Goldenberg & Clements, 2014), and some teachers are not prepared to teach geometry (Lin et al., 2011). Lin and colleagues compared U.S. preservice teachers’ geometry knowledge and van Hiele geometry thinking levels with that of Taiwan preservice teachers using the Entering Geometry Test, a geometry content knowledge assessment tool. Results indicated that the geometry knowledge and van Hiele thinking levels were significantly lower for U.S. preservice teachers than for those from Taiwan.
Instructional Components (ICs) for Teaching Students With LD
As indicated in a report of the National Mathematics Advisory Panel (2008), better understanding of the effective teaching skills and practices for all mathematics teachers is still urgently needed, especially for elementary and middle school levels. In the interest of identifying the separable elements of treatment techniques, researchers have conducted studies to find the ICs that could induce better educational outcomes. Swanson and Carson (1996) used a list of ICs to identify several effective “teaching approaches” (e.g., direct instruction) that studies using group designs showed to be positively associated with better academic performance (reading, math) of students with LD. Later, Swanson and Sachse-Lee (2000) found that specific ICs (e.g., small-group instruction) predicted better academic outcomes in a meta-analysis of single-case design studies for students with LD. Based on this ICs list, researchers (Dennis et al., 2016; Kroesbergen & Luit, 2003; Swanson & Hoskyn, 2001) successfully identified the effective ICs embedded in the instructional models in different studies for students from diverse groups (e.g., students with mathematics LD).
Previous Syntheses in Geometry for Students With Disabilities
Only one synthesis focused on geometry interventions with students who had disabilities. Through their synthesis, Bergstrom and Zhang (2016) provided a review of geometry interventions for K−12 students with and without disabilities. The researchers summarized the intervention methods, study features, and geometry outcomes of the included studies. Bergstrom and Zhang found that nine of the 32 studies included in this review focused on teaching geometry to students with disabilities. The disability type varied, including learning difficulties, intellectual disabilities, and attention-deficit/hyperactivity disorder. Bergstrom and Zhang also examined the instructional method with students with special needs and categorized the intervention methods into three types: curriculum programs, educational technology, and instructional strategies. According to Bergstrom and Zhang, “curriculum programs” referred to a specific course to teach the academic content, and the new program was usually compared with a traditional curriculum as a control group to examine the effects of the new one. The curriculum could include learning objectives, lesson plans, assignments, projects, videos, and other components. “Educational technology” described a specific technology program as a part of the geometry instruction. “Instructional strategies” referred to the strategies that a teacher used to present knowledge, engage the students, and teach the concepts (e.g., the use of direct instruction and cognitive strategies). Bergstrom and Zhang (2016) found that 16 of 32 studies employed instructional strategies, and the remaining studies employed either technology or a new geometry curriculum.
Even though findings indicated that students’ geometry outcomes improved, Bergstrom and Zhang did not report students’ performance based on the disability type. The data on students with LD within those studies were not disaggregated from other participants, and no findings indicated whether the interventions specifically improved the outcomes of students with LD. Moreover, Bergstrom and Zhang did not analyze the ICs of geometry interventions or report on the methodological rigor of the studies.
Current Study and Research Questions
To date, no synthesis exists that specifically centers on students with LD and geometry instruction. Published geometry studies in special education had a wide variability in target participants’ disability categories. The disabilities discussed in published studies included LD (Cihak & Bowlin, 2009; Strickland & Maccini, 2012), autism (Dixon et al., 2016), attention-deficit/hyperactivity disorder (Kang & Zentall, 2011), and mild intellectual disabilities (Creech-Galloway et al., 2013).
The goal of the present study was to extend the current literature by focusing on students with LD, analyzing the ICs of geometry interventions, and examining the effectiveness and quality of the studies (Council for Exceptional Children [CEC], 2014) from 1975 to 2019. Public law included the definition of LD for the first time in 1975 (Education for All Handicapped Children Act, 1975). Five research questions guided this synthesis:
Method
Inclusion Criteria and Selection Procedures
In this synthesis, we selected and analyzed only intervention studies, following the recommendations in position statements by the National Council of Teachers of Mathematics (2011) to use intensive interventions for students who struggle in mathematics. The rigorous methods and procedures in the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (www.prisma-statement.org; see Figure S1, online supplemental data) were used for conducting a systematic review of the literature in this study. The inclusion criteria were studies that (a) employed an experimental, quasi-experimental, or a single-case design; (b) included at least one student in kindergarten to Grade 12 with LD; (c) were written in English; (d) were peer-reviewed journal articles or dissertations between 1975 and February 2019; (e) included at least one intervention designed to teach geometry concepts and skills as the independent variable; (f) included a dependent measure for geometry concepts and skills; and (g) included disaggregated data for students with LD if students without LD were part of the sample.
First, we conducted an online search using electronic databases, including ProQuest Dissertations and Theses Global, Education Resources Information Center, PsycINFO, Education Source, and Education Complete. The search, using the following keywords and keyword combinations, yielded 5,379 studies between 1975 and 2019: learning dis*, learning diff*, learning problem, math* disabilit*, math* disorder*, dyscalculia, special education, intervention, instruction, strategy, geometry, geometric, area problem*, shapes, coordinates, lines, perimeter, volume, surface area, surface, congruence, angle, plane, polygon, quadrilateral, spatial, symmetry, tangram, theorem, transform, triangle, 2-dimension, and 3-dimension. Second, we performed a hand search of major peer-reviewed journals between 2012 and 2019 to double-check the electronic database search. The journals that published studies on students with LD included Exceptional Children, Journal of Learning Disabilities, Journal for Research in Mathematics Education, The Journal of Special Education, Learning Disabilities Research and Practice, Learning Disability Quarterly, Remedial and Special Education, and International Journal of Education. Third, we conducted a search of Google Scholar using similar keywords to those above to identify additional studies.
Using these three searching steps, we found 4,430 studies after removing duplicates. A screening of the titles, keywords, and abstracts yielded 107 studies for further review. After trying to get access to the full-text studies and conducting a full-text screening, we refined the list of studies to 44. Nine studies met inclusion criteria.
Studies were excluded if they (a) used a case design (e.g., Shaw & Durden, 1998) or a qualitative design (e.g., Casey et al., 2016), (b) included participants who were gifted and talented (e.g., Kok & Davasligl, 2014) or at risk (e.g., Clarke et al., 2011), (c) included participants who were teachers (e.g., Griffin et al., 2013), (d) were not an intervention but a test accommodation (e.g., Zhang, 2017; Zhang et al., 2012, 2014), (e) had no independent measure for geometry performance (e.g., Bottge et al., 2015), (f) did not disaggregate the geometry outcomes of students with LD from other participants (e.g., Bouck et al., 2015), or (g) were not journal articles or dissertations (Schneider, 2018).
To ensure the consistency of the study selection process, two doctoral-level researchers in special education independently coded nine (20%) studies, resulting in a 100% interrater reliability rate (IRR). The formula for calculating IRR was to divide the number of agreements by the total number of agreements plus disagreements and multiply by 100. The two researchers then independently coded the rest of the studies (k = 35) that potentially met the inclusion criteria. The final IRR of full-text screening was 95.45%, with two discrepancies resolved after discussion between researchers. We used Cohen’s Kappa (Cohen, 1960) as an additional statistic to report the interrater reliability by correcting the agreement rates that might occur by chance. The interrater reliability is interpreted as “poor” if the Cohen’s Kappa value is below 0.40, “fair” if between 0.40 and 0.59, “good” if between 0.60 and 0.75, and “excellent” if greater than 0.75 (Cohen, 1988). Cohen’s Kappa of this study was .87, indicating a strong interrater reliability.
Next, we conducted an ancestry search of the included studies. We also reviewed the studies included in Bergstrom and Zhang’s (2016) previous synthesis. No additional studies were identified through these procedures.
Coding Procedures
For this synthesis, we developed two coding sheets to review the literature systematically. The items in the coding sheets were adapted from Shin and Bryant (2015) and finalized based on the research questions. The first round of coding focused on the essential features of each study, and the second round focused on the items related to methodological rigor based on the CEC (2014) standards.
Study features
We used the following categories to describe the essential features of the studies based on the research questions: (a) student demographic information (number of students with LD, student age, description of the identification criteria for LD), (b) design type, (c) intervention components (i.e., CCSSM geometry skills, intervention type, ICs, outcome measures), and (d) results (effect size [ES]).
Intervention type and ICs
Based on Bergstrom and Zhang (2016), we coded the geometry intervention types in our synthesis. Because there is no previous literature on effective ICs for geometry interventions, we reviewed previous studies to construct a list of ICs for this synthesis. We initially included 18 ICs for teaching mathematics (Swanson & Hoskyn, 1998, 2001). The detailed descriptions of each item are available in Swanson and Hoskyn (2001). The 18 ICs were advance organizers, attributions, control difficulty or processing demands of tasks, elaboration, explicit practice, large-group learning, novelty in implementing or presenting new teaching materials, one-on-one instruction, peer modeling, questioning, reinforcement, sequencing, skill modeling, small-group instruction, strategy cues, supplement to teacher involvement, task reduction, and technology. Gersten et al. (2009) noted that teachers’ use of visual representations and heuristic strategies also yielded high ESs in teaching mathematics to students with LD, and Bergstrom and Zhang indicated that these two components were effective in geometry interventions for students with and without disabilities. Therefore, we added visual representations and heuristic strategies to the list, resulting in a total of 20 ICs. The detailed descriptions of these two additional ICs are available in Gersten et al.
Quality indicators (QIs)
The CEC (2014) standards are an integrated set of QIs for both single-case designs and group designs (Cook et al., 2015). The CEC standards are built on the pioneering and foundational works of Gersten et al. (2005) and R. H. Horner et al. (2005). There are eight core categories, including Context and Setting, Participants, Intervention Agent, Description of Practice, Implementation Fidelity, Interval Validity, Outcome Measures/Dependent Variables, and Data Analysis, with a total of 28 QIs. Of those QIs, 24 applied to group designs, 22 applied to single-case designs, and 18 applied to both single-case and group designs.
To determine whether a study met each QI, we created rating criteria based on Common et al. (2017). In addition, we calculated the quality score of each study based on Watts et al.’s (2019) method. Each QI was scored as 1 if the study met the criterion and 0 if it did not. The formula for calculating the overall quality score for each study was to divide the obtained score by the total possible score and multiply by 100. The range of the quality score of a study was 0 to 100.
Coding reliability
Two trained special education doctoral-level researchers who had taken courses in research designs double-coded all studies. The initial coding reliability of two studies (22%) was high, with an IRR of 96% and a Cohen’s Kappa value of 0.87. After discussing and solving the discrepancies, the two coders independently coded all the studies and reached a final IRR of 93% and a final Kappa value of 0.70 (>0.60), indicating a good interrater reliability (Cohen, 1988).
For the ICs, we also contacted the authors to ensure the accuracy of the coding results; however, we were unable to contact the authors of two studies. One study (C. M. Horner, 1984) was published in 1984 with no updated contact information available; the authors of the other study listed an email address that was no longer in use (Cass et al., 2003). After contacting the authors of the remaining seven studies, we received coding results and prepared the data for analysis. The IRR between our coding results and the authors’ was 100%, and the Kappa value was 1.
Data Analysis
As a new nonparametric statistical measure specific to single-case designs, Tau-U gained popularity by representing the percentage of nonoverlap between phases, an alternative to both regression-based and simpler nonoverlap models (Parker et al., 2011). Therefore, we chose Tau-U to analyze the single-case designs for this study. A value between 0 and 0.2 represents a small ES, 0.2 and 0.6 represents a moderate ES, 0.6 and 0.8 represents a large ES, and 0.8 and greater indicates a very large ES (Parker et al., 2011). For higher reliability and validity, we used the WebPlotDigitizer digitizing program to complete the data extraction from the graphs (Moeyaert et al., 2016).
Hedges’s g was widely used to compute the ES of group designs (Hedges, 1981) and to adjust for potential errors for studies with a small sample size (What Works Clearinghouse, 2017). To adjust for the pretest performance between groups, we calculated the ES by dividing an adjusted mean difference of the pretest and posttest between the treatment and comparison groups with the pooled pretest and posttest standard deviations. To correct the bias upward for small samples (under 50), we multiplied Hedges’s g with a factor ofω = (1 – 3/[4N – 9]), with N being the total number of participants in the treatment and comparison groups (Morris, 2008). The interpretation criteria of Hedges’s g is the same as those of Cohen’s convention: The ES is categorized as small (0.2), medium (0.5), and large (0.8) (Cohen, 1988).
Results
Of the nine studies included in this synthesis, four studies (Cass et al., 2003; Satsangi & Bouck, 2015; Strickland & Maccini, 2012; Xin & Hord, 2013) were previously identified in Bergstrom and Zhang’s (2016) synthesis. Five additional studies (Cihak & Bowlin, 2009; C. M. Horner, 1984; Kozulin & Kazaz, 2016; Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019) were included in this synthesis. Except for one dissertation (C. M. Horner, 1984), the studies were peer-reviewed articles. Seven studies used single-case designs (Cass et al., 2003; Cihak & Bowlin, 2009; Satsangi & Bouck, 2015; Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019; Strickland & Maccini, 2012; Xin & Hord, 2013) and two used quasi-experimental group designs (C. M. Horner, 1984; Kozulin & Kazaz, 2016). Six of the single-case studies employed a multiple-baseline design, and one (i.e., Satsangi, Hammer, & Hogan, 2019) used an alternative treatment design. The total number of participants with LD was 71, and the researchers reported various LD identification criteria. The study features of the geometry interventions are presented in Table 1. The results of the QI examination are presented in Table S2 (see online supplemental data).
Summary of Study Features.
Note. CCSSM = Common Core State Standards for Mathematics; LD = learning disabilities; ICs = instructional components; IC-3 = control difficulty or processing demands of tasks; IC-4 = elaboration; IC-5 = explicit practice; IC-7 = novelty; IC-8 = one-on-one instruction; IC-12 = sequencing; IC-13 = skill modeling; IC-15 = strategy cues; IC-16 = supplement to teacher involvement; IC-18 = technology; IC-19 = use of heuristics; IC-20 = multiple representations; IC-10 = questioning; IC-17 = task reduction; IC-14 = small-group instruction; IC-1 = advance organizers; IC-2 = attributions; IC-6 = large-group learning; IC-9 = peer modeling; IC-11 = reinforcement; RTI = response to intervention; p = perimeter; A = area; PA = perimeter and area; CRA = concrete-representational-abstract instructional sequence; MD = Measurement & Data; COMP = conceptual model-based problem solving.
CCSSM Geometry Topics
All geometry topics were aligned with the CCSSM. C. M. Horner (1984) studied seventh and eighth graders with LD who learned angle-recognition skills (CCSSM 4. MD. C6), and the remaining researchers focused on perimeter, area, and volume. Cihak and Bowlin (2009) taught perimeter (CCSSM 3. MD. D8) of polygons to high school students with LD. Strickland and Maccini (2012) targeted area word problem skills by applying the area equation of rectangles (CCSSM, 4. MD. A3) with eighth and ninth graders with LD. Five studies (Cass et al., 2003; Kozulin & Kazaz, 2016; Satsangi & Bouck, 2015; Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019) included junior and high school participants with LD who learned to solve both perimeter and area (CCSSM 3. MD. C5, 6, and 7) problems featuring shapes with only right angles. Xin and Hord (2013) studied the teaching of perimeter, area, and volume problems using shapes with right angles (CCSSM 5. MD. C5) to fourth and fifth graders with LD. Some researchers examined other content areas. In three studies (Cihak & Bowlin, 2009; Satsangi, Hammer, & Bouck, 2019; Xin & Hord, 2013), researchers asked students to calculate the value of an unknown side given the area or perimeter value and one side of the shape (CCSSM 3. MD. D8). In two studies (Cass et al., 2003; Satsangi, Hammer, & Bouck, 2019), teachers provided students with practice opportunities for real-world mathematics problems (CCSSM 4. MD. A3; 7. G. B6).
Of the nine studies, eight included participants from junior and high schools (n = 69; 97%), and one study (Xin & Hord, 2013) included fourth and fifth graders. However, the geometry skills taught were mostly elementary school level (e.g., perimeter and area problems in the CCSSM in Grade 3, angle recognition in Grade 4, and volume of a rectangular prism in Grade 5), with the exception of one study (Satsangi, Hammer, & Bouck, 2019), which covered geometry word problems in Grade 7. Two studies (Satsangi, Hammer, & Bouck, 2019; Xin & Hord, 2013) taught grade-aligned geometry skills.
Intervention Type
Researchers of four studies implemented geometry interventions using instructional strategies. Cass et al. (2003) used geoboards to instruct the concepts of perimeter and area. Kozulin and Kazaz (2016) provided students with multiple activities of manipulating various concrete models to understand measurement and to solve perimeter and area problems. Strickland and Maccini (2012) used linear equations to solve area word problems. Xin and Hord (2013) taught geometry problem-solving skills with the COMP cognitive model and the concrete-representational-abstract instructional sequence. In four studies, researchers used educational technology. Cihak and Bowlin (2009) and Satsangi, Hammer, and Bouck (2019) used video modeling with the steps to solve a problem via a laptop. C. M. Horner (1984) used the LOGO computer program to teach angle-recognition skills. Satsangi and Bouck (2015) used a virtual manipulative program to help students with perimeter and area problems. Only one study included both instructional strategies and educational technology. Satsangi, Hammer, and Hogan (2019) compared the effects of explicit instruction and video modeling when teaching students perimeter and area problems. None of the included studies used a geometry curriculum.
ICs
On average, we coded 12.4 ICs for each study. Among the studies, the number of ICs ranged from 11 to 15. Strickland and Maccini (2012) included the highest number of ICs (n = 15). Three studies (Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019; Xin & Hord, 2013) included 13 ICs. Researchers in three studies (C. M. Horner, 1984; Kozulin & Kazaz, 2016; Satsangi & Bouck, 2015) incorporated 12 ICs. Cass et al. (2003) and Cihak and Bowlin (2009) included 11 ICs. Furthermore, 18 out of 20 ICs were found at least once among the included studies. Many ICs appeared in multiple geometry interventions, whereas two ICs were never used (large-group instruction and peer modeling). The following seven ICs were used in all included studies: control difficulty, explicit practice, novelty, skill modeling, strategy cues, heuristic instruction, and multiple representations. Elaboration, one-on-one instruction, and sequencing appeared in eight studies. Seven studies included task reduction.
Effects of the Studies
Single-case studies
To assess the effects of the seven single-case designs, Tau-U was calculated for all except one study (Strickland & Maccini, 2012), which did not provide enough raw data points during the intervention phase for the three participants with LD. However, all three participants in Strickland and Maccini’s study increased their overall geometry performance, improving 77.5, 77.5, and 69.8 percentage points.
Researchers of six studies examined the effectiveness of the interventions in improving the perimeter problem-solving skills of the participants (Cass et al., 2003; Cihak & Bowlin, 2009; Satsangi & Bouck, 2015; Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019; Xin & Hord, 2013). Researchers of three studies (Cass et al., 2003; Cihak & Bowlin, 2009; Satsangi & Bouck, 2015) reported the perimeter outcomes separately. Among these three studies, all participants with LD mastered target skills and achieved accuracy of 100% at the end of the intervention, with the Tau-U ranging from 0.75 to 1.61, indicating large and very large ESs (Parker et al., 2011). Particularly, in Cihak and Bowlin’s study, all participants increased perimeter problem-solving skills of various polygons, with the Tau-U values ranging from 1 to 1.61. In contrast, three studies (Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019; Xin & Hord, 2013) reported the outcomes of perimeter and area problems together. The results of two studies (Satsangi, Hammer, & Bouck, 2019; Satsangi, Hammer, & Hogan, 2019) were positive (Tau-U > 0.8), and one study’s results (Xin & Hord, 2013) were mixed, with a moderate (Tau-U = 0.44) and very large (Tau-U = 0.87) ES for two participants with LD. Even though the volume of the rectangular prism was one of the target skills in Xin and Hord’s study, the ES of solving volume problems was not calculated due to the insufficient data points of this measure in the intervention phase. For area problem-solving skills, two studies (Cass et al., 2003; Satsangi & Bouck, 2015) showed very large ESs (Tau-U > 0.8).
Among the included geometry interventions, all geometry measures were researcher developed with limited validity or reliability information. Also, all single-case studies except for one (Satsangi, Hammer, & Hogan, 2019) included a maintenance phase. Only two studies included the generalization phase (Satsangi & Bouck, 2015; Strickland & Maccini, 2012).
Group designs
In Kozulin and Kazaz (2016), the treatment group of students with LD significantly improved their perimeter and area problem-solving skills compared with the control group of students without LD (t = 2.26, p < .01). After adjusting for the pretest means for both groups, the ES for the perimeter problem-solving skills was medium to large (g = 0.64; Cohen, 1988) and the ES for the area problem-solving skills was very large (g = 1.08). The outcomes of Kozulin and Kazaz’s study indicated that students with LD even outperformed their typical peers at the end of the intervention. In C. M. Horner (1984), the students with LD in the treatment group outperformed the students with LD in the control group, and the ES of angle-recognition skills was medium to large (g = 0.6).
Quality of the Studies
To compare the quality of the selected studies, we examined the QIs (see Table S2) and calculated the quality scores of both single-case and group designs. The average score of all studies was 88. The average quality score for single-case designs (k = 7) was 91. The quality score for two group designs was 77.
Single-case studies
Taken together, one study (Satsangi, Hammer, & Bouck, 2019) met all criteria with a quality score of 100. Researchers of the rest of the single-case studies did not provide sufficient information on the following categories according to the CEC (2014) standards: 3.1 Intervention Agent Description, 3.2 Intervention Agent Training Description, 5.1 Implementation Fidelity, 5.3 Fidelity Assessed Throughout Study, 6.6 Baseline Data Points, and 7.4 Measured Repeatedly. Researchers of most single-case designs provided no demographic information of the interventionists (QI 3.1), even though these studies mentioned who implemented the intervention (e.g., a teacher, a researcher). In addition, authors in four studies (Satsangi & Bouck, 2015; Satsangi, Hammer, & Hogan, 2019; Strickland & Maccini, 2012; Xin & Hord, 2013) described neither any training information before or during the intervention nor the qualification of the intervention agents (QI 3.2). Moreover, Xin and Hord did not report the fidelity of the implementation or whether the fidelity check occurred throughout the study across participants (QI 5.1; 5.3), even though they provided the implementation duration of the intervention. Strickland and Maccini did not provide enough data points in the baseline phase or the intervention phase (QI 6.6; 7.4).
Group designs
Researchers of both group design studies (C. M. Horner, 1984; Kozulin & Kazaz, 2016) did not provide detailed descriptions on the following QIs: 3.1 Intervention Agent Description, 3.2 Intervention Agent Training Description, 5.1 Implementation Fidelity, 5.3 Fidelity Assessed Throughout Study, and 7.5 Adequate Interrater Reliability. In addition, Kozulin and Kazaz provided no validity information of the measure used in the study (QI 7.6).
Discussion
The purpose of this study was to synthesize the research-based practices of geometry interventions administered to students with LD. We extended the literature by expanding the search year compared with the previous synthesis (Bergstrom & Zhang, 2016) and including a dissertation (C. M. Horner, 1984). We also examined the ICs, effectiveness of the interventions, and the QIs based on the CEC (2014) standards. We provide findings about geometry interventions for students with LD, what works, and areas that still need exploration.
CCSSM Geometry Topics
The findings of this synthesis are consistent with previous research. Previously, Bergstrom and Zhang (2016) indicated that the geometry skills taught to students with disabilities were only “basic skills.” Results of our synthesis demonstrated limited geometry topics covered for students with LD, including angle-recognition, and perimeter, area, and volume problems. Even though 97% of participants with LD were at the junior high level, the geometry skills taught were mostly lower than their actual grade levels. Many geometry concepts and skills listed in the CCSSM are still understudied (e.g., the Pythagorean Theorem).
Intervention Type and ICs
Unlike previous findings on intervention types, our results indicated that geometry interventions included only instructional strategies and the use of technology (Bergstrom & Zhang, 2016). We found that almost all studies in this category incorporated concrete or semi-concrete models, such as the concrete-representational-abstract instructional sequence or manipulative instructions using geoboards. Researchers successfully delivered geometry instruction through video modeling or computer programs, such as virtual manipulatives and LOGO. During the baseline phases or pretests, participants demonstrated limited understanding of geometry concepts that should be required during elementary grades (e.g., concepts of perimeter and area), which indicated a low level of geometric thought based on the van Hiele model (i.e., visualization level, analysis level). The use of instructional strategies and technology provided multiple opportunities working with shape models to build on students’ knowledge of various shapes. By observing and manipulating models, students could form mental representations of geometric shapes, which support spatial reasoning and problem-solving skills (Common Core Standards Writing Team, 2013).
For the third research question, results indicated that seven ICs in the included geometry interventions yielded positive results, including skills modeling and explicit instruction. These components are consistent with the geometry learning model that van Hiele proposed in the 1980s, which emphasizes the importance of sufficient geometry experiences (Crowley, 1987). Moreover, the examination of the ICs in this synthesis, in part, responds to Gersten et al. (2009), who encouraged the analysis of the ICs addressed in other mathematics topics.
Effects of the Studies
The analysis of the intervention effects extended current literature (Bergstrom & Zhang, 2016) by providing the effects of the interventions for students with LD. We found that instructional strategies that used multiple representations and technology-based interventions were effective in teaching geometry concepts and skills. Students receiving one-on-one instruction in single-case designs improved their geometry performance, with mostly large ESs (Tau-U > 0.8). Also, students with LD receiving instruction in groups showed positive outcomes, with medium to large ESs. These results should encourage educators and researchers to have high expectations that students with LD can learn geometry well.
Strategies that are effective for students with and at risk for disabilities include understanding the students’ current level of geometry thought, identifying related resources based on students’ learning needs, and adopting the right type of assessment and instruction when developing interventions (Witzel & Little, 2016). Despite the positive effects of the geometry interventions in this synthesis, almost all studies used only one type of researcher-developed measure, with no validity or reliability information. As a bridge to provide useful feedback about the lessons, the use of multiple measures (e.g., proximal and distal measures) with adequate validity or reliability information is crucial to evaluate the effects of the interventions or the generalizability of the skills to broader types of problems. Besides, students better understand the spatial aspect of the world and build confidence in geometry by applying skills to real-life situations (Lappan, 1999). We found that two studies (Cass et al., 2003; Satsangi, Hammer, & Bouck, 2019) incorporated real-life examples to teach geometry, and these studies had large ESs. Future researchers should consider these issues while developing their designs.
Quality of the Studies
This synthesis contributed to the literature by evaluating the methodological rigor of geometry interventions in special education. Findings indicated that the study quality was generally high and that the quality of single-case designs was relatively higher than that of group designs. We found that many special education researchers were aware of the necessity of reporting core QIs in their studies, even though several issues remained—for example, the lack of descriptions in implementation fidelity or interventionist training. Methodologically, sound studies enable reviewers and other researchers to determine whether there is sufficient evidence to establish a functional relation (Kennedy, 2005) or whether an instructional strategy can be considered an evidence-based practice (What Works Clearinghouse, 2017). It is important to be aware of these issues because studies with poor reporting can affect many consumers, including practitioners, policy makers, grant funders, and journal editors (Talbott et al., 2018). When stakeholders choose an intervention, the study quality matters. We need more high-quality studies that can help close research-to-practice gaps (Cook et al., 2015) and make the process of implementation and replication easier.
Limitations and Future Research
Several limitations of the present synthesis can inform future research. The first limitation is that this synthesis was based on a small number of geometry interventions for students with LD. The findings suggest that many areas in K−12 geometry education are understudied and require future research. Additional research should replicate or create new interventions, including using geometry curriculum programs, to extend the current literature. For example, an investigation of geometry–vocabulary knowledge for K−12 students may enhance student understanding in geometry and increase mathematics achievement (Bryant et al., 2003; Powell & Nelson, 2017).
Second, even though the results of this synthesis indicated that seven ICs were favorably used in geometry interventions for students with LD, the interpretation of the results must be made with caution. Because students with LD tend to exhibit weakness in various mathematics skills based on their learning needs (Bryant et al., 2000), future studies should examine the effectiveness of various ICs, including the ones used less frequently (e.g., large-group intervention, peer modeling). Additional studies that are designed to systematically manipulate specific ICs to identify their importance for the interventions are needed.
Third, when evaluating the nine studies in this synthesis, we found the interpretation and implementation of the QIs challenging. We tried to make the best judgments based on the CEC (2014) standards and criteria aligned with previous research (Common et al., 2017). For example, for QI 2.1, no operationalized definition of “sufficient” is available in the CEC standards when deciding whether “a study provides sufficient information on the population of participants.” Researchers recommended that the criterion was met if a study reported “at least one demographic element (age, gender, race/ethnicity, socioeconomic status)” (Common et al., 2017, p. 335). However, this bar set for quality may appear low to many researchers. We encourage more researchers to add to this discussion and provide better application suggestions for these standards. In addition, even though researchers might have implemented particular QIs but did not report them due to variables in the publication process (e.g., publications’ page limits), we evaluate the interventions based on their written work. The goal of examining the methodological rigor was to provide information for future researchers on geometry interventions so that more attention can be extended to addressing most if not all of the QIs.
Implications for Practice
This synthesis has several implications for practice, despite the limitations. First, the findings show that students with LD can improve geometry concepts and skills through geometry interventions. In particular, students with LD may benefit from well-designed intervention studies embedded with effective ICs. Therefore, teachers can consider using ICs (e.g., explicit instruction, practice opportunities in a sequential manner) when introducing geometry concepts and skills to students with LD. The use of multiple representations, technology-based instruction, and real-life examples for practicing (e.g., pools, desks) is also highly recommended because students may better understand the purpose of learning and establish an interest in geometry, which may have long-term benefits. Second, it should be noted that elementary geometry education is crucial, because geometry knowledge at early grades can help students with LD lay a good foundation for more advanced geometry topics in junior and high school level. Elementary teachers are suggested to use methodologically sound geometry intervention programs and implement them with high fidelity.
Supplemental Material
Geometry_Interventions_for_Students_With_Learning_Disabilities_Table_and_Figure_Supplemental_Files – Supplemental material for Geometry Interventions for Students With Learning Disabilities: A Research Synthesis
Supplemental material, Geometry_Interventions_for_Students_With_Learning_Disabilities_Table_and_Figure_Supplemental_Files for Geometry Interventions for Students With Learning Disabilities: A Research Synthesis by Meijia Liu, Diane Pedrotty Bryant, Elly Kiru and Maryam Nozari in Learning Disability Quarterly
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
References
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