Teaching Geometry to Students With Learning Disabilities: Introduction to the Special Series

Abstract

This article introduces and contextualizes the four articles that constitute the thematic special series on geometry instruction for students with learning disabilities or difficulties. The four articles, each emphasizing one important aspect of geometry learning and instruction for students with learning difficulties or disabilities, are aimed to answer critical questions raised by special education/math education researchers and practitioners on how to teach geometry to students with learning disabilities or difficulties.

Keywords

geometry learning disabilities geometry interventions mathematics learning difficulties

The intent of this introductory article is to provide a framework for understanding the contributions of the four articles published in this Geometry Learning & Instruction for Students with Learning Disabilities special issue of Learning Disability Quarterly. First, I will discuss the limited research base on geometry instruction and students with learning disabilities and how it is situated within the wider research base on mathematics learning disabilities. Second, I will trace the development of educators’ growing interest in geometry education, review the research base specific to geometry learning and instruction, and outline some of the accomplishments and limitations of the existing research approaches. Last, I will briefly introduce the four studies included in this special issue and the relations among these four studies, with suggestions for how the field can move forward with respect to both research and practice related to geometry and students with learning disabilities or difficulties.

Geometry Instruction for Students With Learning Disabilities: An Understudied Area

Geometry instruction within K-12 mathematics education involves helping students to learn spatial relations and properties of shapes, which are crucial for students to succeed in science, technology, engineering, and mathematics (STEM) areas (Common Core State Standards in Mathematics, 2010). Geometry interprets and reflects on the physical environment and is fundamental for learning advanced topics in mathematics, science, geography, architecture, art, design, technology, and engineering in college or postgraduate studies. Learning geometry helps students develop multiple skills, including visual imagery conjecturing, deductive reasoning, logical argument, and proof. For example, geometric representations improve students’ learning in other areas of mathematics, such as the linear representations of the number system, the relationships between the graphs of functions, and graphical representations of data in statistics (Jones, 2002).

Many students manifest particular difficulties with learning geometry (Gal & Linchevski, 2010). Programme for International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS) results report that U.S. students score lower on the geometry domain compared with the numerical domains such as algebra and data (Organisation for Economic Co-Operation and Development, 2014; Provasnik et al., 2012), indicating inadequate geometry instruction in the U.S. educational system.

Historically, much less attention has been paid to students’ learning difficulties and development in geometry than to their numerical development. This could reflect an underestimate of the importance of geometry instruction in U.S. education history (N. Sinclair, 2008): The study of geometry was not required for application and admission to college until 1844, and geometry content in the U.S. mathematics curricula continues to be understudied. Only 0.24% of state standards and 5.73% of Common Core Standards in Mathematics (Common Core State Standards Initiative [CCSSM], 2010) address geometry concepts, and 9.27% of state standards and 1.64% of CCSSM address advanced geometry (Porter et al., 2011). This means that the percentage of instructional time for geometry is much lower than the percentages for numerical subjects in K-12 education. In contrast, geometry content comprises a larger percentage of the entire mathematics curriculum in some Asian countries with higher achievement outcomes. For example, based on the China National Standards in Mathematics (CCSSM, 2010), geometry is one of three primary domains (i.e., arithmetic and algebra, geometry, stats and probabilities) mandated in the mathematics curriculum for Grades 1 to 9; and in the high school math curriculum, geometry topics (e.g., 3D geometry, analytic geometry, and trigonometry) comprise over 40% of all required topics from Grades 10 to 12.

In addition, geometry curricula in the United States have been less rigorous than in many countries whose students achieve high scores in TIMSS and PISA (e.g., Singapore, China, Shanghai, Korea, Japan, and Hong Kong). The mathematics curricula in the United States has been criticized as “a mile wide and an inch deep.” Although students in the United States learn a broad range of geometry concepts (e.g., 2-D geometry, geometry transformations, congruence and similarity, and 3D geometry) simultaneously beginning in Grade 1 (Schmidt et al., 2005), the depth level has been questioned. For example, many advanced geometry skills, such as the ability to “prove geometric theorems (CCSS.Math.Content.HSG.CO.C.9, 10, 11),” to “prove theorems involving similarity (CCSS.Math.Content.HSG.SRT.B.4, 5),” and to “understand and apply theorems about circles (CCSS.Math.Content.HSG.C.A.1, 2, 3, 4),” are not required until high school in the United States (CCSSM, 2010), whereas learning these rigorous theorems is required by middle school standards in many countries with higher math-achieving students, such as China (China Department of Education, 2011) and Hong Kong (The Curriculum Development Council of the Hong Kong Education Bureau, 2017). Indeed, for more than a decade, international comparison research (Ma, 1999; Schmidt et al., 2005) has suggested that mathematics education in the United States, including geometry education, should become substantially more focused, rigorous, and coherent to improve the mathematics achievement of American students. Schmidt et al. (2005) compared mathematics curricula of the United States with the standards of countries with high achievement outcomes and suggested that the national mathematics standards in the countries with high achievement demonstrate a clear three-stage pattern: Stage 2 (Grades 5 and 6) serves as transition in which attention to topics such as coordinate geometry leads to the formal instruction of rigorous algebra and geometry that is characteristic of the third stage (Grades 7 and 8). In contrast, the three-stage structure does not appear in the NCTM standards and mathematics state standards in the United States.

The relatively rare research on geometry instruction for students with learning disabilities could also be related to current methods of identifying students with learning disabilities. Both the RtI and the traditional IQ-achievement discrepancy models rely heavily on early-elementary-level academic assessment to identify students with mathematics learning disabilities. Within the traditional discrepancy model, as of the third grade when high-stake testing begins and many students with learning disabilities are identified, geometry comprises a relatively smaller component of mathematics curricula in the United States. Within the RtI model, the current assessment instruments used for Tier 1 universal screening also primarily focus on numerical development in the early elementary grades. Geometry becomes an increasingly important and difficult subject as students move to higher grade levels, especially during the middle and high school years; however, in practice, fewer learning disabilities cases are identified at the high school level. Consequently, most special education literature focuses on mathematics subjects at elementary and middle school levels, whereas there is sparse research on advanced mathematics subjects in the upper grades, including advanced geometry in the high school mathematics curricula.

Existing Research on Geometry Education

In spite of the limited existing research regarding geometry instruction for students with learning disabilities in the field of special education, cognitive psychologists, educational psychologists, and mathematics education researchers have accumulated considerable findings that address students’ geometry learning processes. The first set of research aims to explain students’ individual differences in learning geometry. One line of research identified significant cognitive and noncognitive contributors to explain students’ individual differences in geometry abilities. Identified cognitive factors include spatial abilities (Clements et al., 1997; Geary, 1996; Jeung et al., 1997; Kyttälä & Lehto, 2008; Purcell & Gero, 1998; Spelke et al., 2010; Verstijnen et al., 1998), fluid intelligence and reasoning skills (Battista, 1990; Dawkins, 2015; Giofrè et al., 2014), working memory (Giofrè et al., 2013, 2014), geometry content knowledge (Bokosmaty et al., 2015; Lawson & Chinnappan, 1994), and meta-cognition (Aydın & Ubuz, 2010). In addition, motivation, persistence, and student emotions (e.g., boredom and enjoyment) are frequently cited to interpret students’ geometry learning difficulties and individual differences (Bailey et al., 2014; Super & Bachrach, 1957). Some pedagogical variables also have been documented, such as the geometry curriculum (Battista & Clements, 1995; Clements & Battista, 1986; Halat et al., 2008; Oner, 2008), teachers’ instructional approaches (Kuzniak & Rauscher, 2011), and task complexity level (Hsu & Silver, 2014).

Gender is another important factor in interpreting individual differences in geometry achievement. Previous literature reported a male superiority factor in geometry performance at the secondary and college levels (Battista, 1990; Hyde et al., 1990; Nordvik & Amponsah, 1998). This factor can be interpreted from multiple perspectives. First, spatial ability is the cognitive domain that shows the most robust gender differences favoring males, especially in the area of spatial rotations (Feng et al., 2007; Kimura, 1999; Terlecki & Newcombe, 2005). Second, research has found that females prefer to analyze spatial relations by manipulating elements and features, whereas males prefer to use global shape and holistic strategies when analyzing spatial information (Hegarty, 2018). Third, female students generally have lower self-efficacy in mathematics domains, including geometry (Erdoğan et al., 2011; Erkek & Işiksal-Bostan, 2015; Louis & Mistele, 2012). Female students with lower self-efficacy are less motivated and perseverant in solving geometry problems in test settings where motivation significantly influences geometry achievement (Bailey et al., 2014).

The second set of prior research focuses on geometry teaching and learning. One line of research, most of which is qualitative in nature and conducted by mathematics education researchers, investigates students’ reasoning processes and addresses how to foster students’ understanding and facilitate their geometric thinking (Dawkins, 2015; Herbst & Brach, 2006; Lavy & Bershadsky, 2003; Palha et al., 2013; Rahim & Olson, 1998). It is noteworthy that technology has placed more emphasis on the role of geometry instruction in facilitating students’ geometric reasoning (Labored, 2006). Since the 1960s, the Logo-driven programs have been widely researched (Boychev, 1999; diSessa & Lay, 1986; Harvey, 2005; Kalas, 2001; Kynigos, 2002; Silverman, 1999; Wilensky, 1999), which provide a computational environment in which users give commands through the “Logo” language. During the past two decades, considerable research has examined students’ geometric reasoning in a dynamic environment in which students can use computer programs to construct a diagram, and when an element of such a diagram is dragged with the mouse, the diagram is modified while the whole geometric relations are preserved (Laborde et al., 2006). Numerous studies (Arzaello, 2000; Healy, 2000; Hollebrands, 2002; Hölzl, 2001; Leung & Lopez-Real, 2002; Olivero, 2002; M. P. Sinclair, 2005) have shown that the dragging function of the dynamic geometry environment helps students to form conjectures, prove by contradiction, and justify the conjectures.

The above findings, principles, and practices from the mathematics education, educational psychology, cognitive psychology, and educational technology literature can inform the field on improving geometry outcomes for students with learning disabilities. More discourse and research are needed to better understand how to address the geometry learning needs of students with mathematics difficulties or disabilities. In this special issue, we describe where the field is currently with respect to geometry instruction and students with learning disabilities and offer suggestions for how the field can move forward to improve geometry outcomes for students with learning disabilities.

Article Foci

In this section, I provide a brief summary of each of the four articles that constitute this special series on teaching geometry to students with learning disabilities/difficulties. This special issue represents a collection of the most up-to-date research on geometry learning and teaching for students with learning disabilities from a group of multidisciplinary experts. With a general aim to help special educators to increase their knowledge and interest in this understudied area, this special issue specifically addresses the following questions:

1. Who Are the Students With Geometry Learning Difficulties?

In the first article, Chen, Li, and Zhang explore the characteristics of students who encounter difficulties in solving geometry problems through analysis of the TIMSS data on fourth-grade U.S. students’ mathematics performance. Fourth graders’ data were chosen because the fourth-grade assessment is closer to “intuitive geometry” that is independent from instruction, experiences, and culture (Giofrè et al., 2013). Three mathematics domains were assessed: data display, geometric shapes and measures, and number. It is interesting to note that this three-domain structure matches well with the structure of the national mathematics curriculum standards in some Asian countries with high achievement outcomes. Chen and colleagues performed the latent profile analysis to cluster all U.S. participants into different groups based on their performance across assessment of the three mathematics domains. The first interesting finding was that geometry is the weakest area for U.S. students among all three mathematics domains, which confirms the previous discussion of the underrepresentation of geometry content in U.S. mathematics standards. Another contribution of this study lies in the exploration of the relation between geometry difficulties and students’ performance in other mathematics domains. The cluster analysis did not find a cluster that had low scores in geometry only; rather, the analysis showed that the lowest geometry-performing group had the lowest scores across all three mathematics domains and also showed the largest discrepancy between geometry scores and performance in other mathematics domains. Such a domain discrepancy did not exist within high-achieving and average-achieving groups. In other words, the lowest geometry-performing students typically also had difficulties in other mathematics domains, and their weakest area was geometry. This study also examined the demographic background variables associated with this particular group of students and reported that gender, age, home language, race, and preference for mathematics and science significantly influenced the probability of being classified in the group. In sum, this study provides an initial profile of the group who experiences the most geometry difficulties and will stimulate more research to better understand this group of students.

2. What We Know And Where We Are? Existing Geometry Interventions for Students With Learning Disabilities.

The second article by Liu, Bryant, Kiru, and Nozari presents a very thorough research synthesis of the existing research base on effective interventions for teaching geometry students with learning disabilities. This study updated the findings of a previous synthesis on geometry instruction for all students (Bergstrom & Zhang, 2016) by specifically focusing on students with learning disabilities and adding an examination of the research rigor of the included studies. In this study, authors systematically synthesized seven intervention studies. They coded three categories of variables: study features, intervention type and instructional components, and quality indicators. Effect sizes were calculated. A significant contribution of this synthesis is a thorough analysis of the intervention components in each of the reviewed studies, which is very informative to educators who teach geometry to students with learning disabilities. Regarding methodological rigor, the results showed several issues related to the lack of training description of the interventionists, intervention treatment fidelity, and adequate technical information for student outcome measures. Results provide suggestions for how the field can move forward with respect to both research and practice related to geometry and students with learning disabilities.

An interesting yet important note is that among the seven included studies, only one investigated angle recognition skills; the remaining six focused on the topics of perimeter, area, and volume. It is noteworthy that mastery of these basic skills is required during Grades 3 to 5 according to CCSSM. However, an analysis of the participants’ grade/age characteristics showed that of the seven studies, six included participants from middle and high school levels and one included fourth- and fifth-grade participants. In other words, most of the intervention studies featured remediation programs to help students with learning disabilities to make up skills that were supposed to have been mastered during elementary grades rather than programs to supplement ongoing inclusive classroom teaching on advanced topics required by middle school or high school curricula. This finding implies a direction that needs more attention in the future research.

3. How Effective Is Using Cutting-Edge Technologies in Helping Students With Learning Disabilities to Learn Geometry? — Evaluating an Intervention.

The third article by Choo and colleagues presents an exploration of an intervention program with a quasi-experimental design. A highlight of this study is the integration of 3D printing techniques into the intervention. Based on pedagogical concepts of enhanced anchored instruction, authors developed 3D geometry lessons using computer-aided design (CAD) software to help students visualize and construct 3D models and tested the feasibility of delivering the lessons for teaching geometry standards. This study addressed an understudied problem, namely the potential of applying 3D printing technology to teach students with learning disabilities to develop their specific spatial thinking skills. This study also addresses the lack of research on the geometry and spatial thinking skills that students with mathematics learning disabilities are likely to struggle to learn. Based on pedagogical concepts of enhanced anchored instruction, the authors developed a series of technology-based geometry lessons that employed explicit instruction to teach foundational concepts of basic geometry and contextualized instruction using 3D printing projects to teach CAD skills. Results from this exploratory study have implications for classroom practices and future researchers for developing intervention programs using 3D and CAD techniques to engage students with learning disabilities in geometry problem solving.

4. How to Help Students With Learning Difficulties to Remove Cognitive Barriers in Geometry Problems Solving? Exploring a Testing Accommodation Support.

The last article by Zhang, Indyk, and Greenstein explores the effectiveness of a testing accommodation support, namely schematic chunking, on the performance of students with mathematics learning difficulties and students at risk for mathematics failure. Accommodation supports aim to help students with special needs gain access to the problems by removing barriers attributed to their disabilities. Different from interventions or testing modifications, accommodations do not teach students specific strategies, content knowledge, or skills, nor do they alter task difficulties or lower expectations. Zhang and colleagues (2012, 2014) have conducted a series of studies on chunking accommodation support for elementary students with learning disabilities or difficulties in basic geometry problems requiring visual imagery. Using the cognitive load theory as the theoretical framework, this study extended the exploration of chunking representation from basic intuitive geometry problems at the elementary level to schematic visual chunking in higher order geometry problem solving at the high school level. In particular, this study also investigated the moderating effects of task difficulty/complexity level and the role of participants’ expertise or knowledge of the geometry content. The accommodation method developed and validated in this study offers a potential support for teachers to use in formative or summative assessment.

Summary

Taken together, the four articles in this special series provide a window into the ongoing work carried out by special education researchers who are interested in geometry instruction for students with learning disabilities or difficulties. The intent is that these articles will inspire more special education scholars, as well as teachers and policy makers, to attend to this important yet understudied area. We also hope these articles will spark further, substantial multidisciplinary collaboration among experts who can share expertise through different theoretical lenses.

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.

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