Final Commentary to the Cross-Disciplinary Thematic Special Series: Special Education and Mathematics Education

Abstract

Four studies into characteristics and instructional needs of students with learning disabilities are summarized in this article. These studies are also reviewed in the wider context of mathematics intervention research in special education. These studies generally rely on qualitative methodology, and they are best understood in light of a constructive approach to learning. Two themes bind the four studies reviewed here. The first is what distinguishes students with math difficulties from those that may be considered as having a math disability. The second is the nature of curriculum and how the work described in this special issue departs from structured approaches to interventions that have a long and rich history in special education.

Keywords

math difficulties mathematics constructivism curriculum design

The purpose of this commentary is to provide a contemporary context for thinking about the four studies of Lewis, Hunt, Xin and Tzur et al., and Liu and Xin that appeared in this special issue. One of the themes that run through these studies is the use of alternative theoretical orientations. For example, Hunt, as well as Xin and Tzur et al. drew upon a long history of Piagetian-oriented thinking (Piaget, 1985) and on recent work in mathematics education (Simon & Tzur, 2004; Steffe & Cobb, 1998; Steffe & Olive, 2010; Tzur et al., 2013). Their approaches deviate from much of the special education literature, in which behaviorally oriented pedagogy, such as explicit instruction (see Gersten, Chard, et al., 2009), is a common frame of reference. Instead, their approaches pay close attention to the conceptual characteristics that are inferred to underlie, and thus explain, students’ mathematical behaviors.

To explain the conceptual foundations of students’ behaviors, another theme common to these papers is the qualitative dimension of research on knowing and learning exhibited by students with mathematics learning disabilities. Lewis and Hunt use in-depth clinical interviews as their primary research tools, yielding insights about student competence that typically do not appear in the broader, quantitative orientation to research found in contemporary, mathematics-related special education literature. Liu and Xin use single subject methodology, which includes careful analysis of student written artifacts. These different methodologies remind us that a continuum of research perspectives—from highly qualitative research, through individual students, to large-scale randomized control studies (Randomized Control Trial [RCT])—offer valuable insights into characteristics, needs, and effectiveness of instructional methods for students with mathematics learning disabilities.

In addition to what has been mentioned above, two broad themes seem fruitful for contextualizing the body of work found in the four studies. The first theme accentuates a spate of recent research that distinguishes between at risk learners, those who are “very low performers,” and those with a mathematics disability (MD). The latter category often overlaps with a mathematics learning disability (MLD), and as a consequence, MD is used in the remainder of the article to characterize this learning condition. Space limitations limit the ability to summarize the findings from the MD research to date; thus, this commentary draws attention to some underlying assumptions found in the current MD literature. Particular attention is placed on research efforts to determine who among primary grade students has MD and who is “at risk” for mathematics failure.

The second theme for contextualizing the four articles in this special issue is more nuanced. Findings from the four studies accentuate, and perhaps challenge, the way many special educators think about curriculum, particularly a “research-validated” curriculum (or intervention) that is embedded within large-scale, RCT studies. In large-scale research, the term curriculum is often restricted to highly structured print or electronic materials. The role of pedagogical content knowledge of interventionists who use these materials, particularly about students’ available conceptions, is rarely discussed in the RCT literature for at-risk students. It will be argued that typical assumptions about curriculum may be problematic because of what is needed on the part of interventionists to address student conceptions, teach for conceptual understanding, and foster the kinds of thinking outlined in the Mathematical Practices dimension of the Common Core Standards.

Theme 1: Distinguishing At-Risk From MD Students

Regarding the first theme, Gersten and Chard’s (1999) essay on number sense signified an important shift in the discussion of MD and the way in which MLD had been characterized in earlier decades. For years, special education researchers focused less on an underlying etiology of MD and more on skills-based analyses and/or interventions (Woodward, 2012). Gersten and Chard’s description of number sense as an analog to phonemic awareness was wide-ranging, albeit incomplete, because of the state of the research at the time. Yet they effectively drew attention to a wave of cognitive, developmental research that underscored the importance of early number skills, such as numeral identification, counting strategies, and number magnitude. Their essay also ventured into other areas of early math proficiency, such as fluency with addition and subtraction facts and the “mental number line.”

Subsequent early number sense research was often conducted by reading researchers who were interested in distinguishing between students who had a reading and MD from those who only possessed the latter (e.g., Clarke & Shinn, 2004; Fuchs et al., 2007). This body of work shifted the discussion of MD, largely focused on number sense, to methods for identifying students who were at risk for failure in mathematics at an early age. Core ideas behind curriculum-based measurement were transposed into “brief and easy to administer” measures that assessed numeral identification, counting strategies, and number magnitude, along with other skills such as figuring out missing numbers in a sequence (e.g., 17 ___ 19). Other more time-consuming assessments included nonverbal addition and subtraction number combinations along with simple addition/subtraction word problems (see Jordan, Glutting, Ramineni, & Watkins, 2010).

Commercial versions of these math screening instruments have evolved over the last 10 years, and they continue to rely on counting, magnitude, and missing number sequences for identifying at risk students. For example, easyCBM (2012) and DIBELS Math (2012) not only addressed primary grade number sense, but their measures extend into progress monitoring assessments that place a significant emphasis upon math computations and basic concepts. We note that mathematics education research on how children develop a robust concept of number and additive reasoning (Fuson, 1992; Steffe & von Glasersfeld, 1985; Tzur & Lambert, 2011) goes beyond the computational focus of these assessments.

Practical and conceptual issues seem to persist in these efforts to identify MD and, more broadly, students at risk for early failure in mathematics. Gersten et al. (2012) astutely observed that many of these screening measures tended to yield far too many false positives. That is, students were falsely classified as needing added assistance in math. In addition to the traditional concern around mislabeling students, excessive numbers of false positives had the potential of leading to a misallocation of scarce intervention resources. Gersten et al. also noted that brief, easy to administer measures have relatively low predictive validity, at least as compared with more traditional diagnostic measures. This was a recurrent compromise associated with rather brief instruments that could be easily administered by paraprofessionals. Consequently, the literature on primary grade and even intermediate grade students now abounds with a variety of terms beyond MD. They include categories such as at risk, low performing, and “very low performing” (see Fuchs et al., in press).

It is in this context that Lewis and Hunt’s studies (in the previous issue) are salient. Lewis noted that much of the research on MD over the last two decades has focused on whole number calculations and basic number processing. Little of the research was directed at higher-level mathematical concepts (e.g., fractions). Moreover, by concentrating on early classification and intervention, attention has been diverted away from students who exhibit more pronounced, long-term difficulties in mathematics. Lewis’s two postsecondary participants exhibited a range of fractional understandings that could only be identified through extended, clinical interviews. Current screening and progress monitoring tools described above have not been designed to capture the kinds of persistent, cognitively rooted error patterns studied by Lewis and by Hunt (e.g., how a student might attend to the complement of the non-unit fraction 3/8 when the visual model for the fraction represents 5/8).

Lewis’s methodology is also reminiscent of dynamic assessment (Tzuriel, 2001), where a subject’s erroneous understanding is revealed through probes and responses to instructional tasks. Again, this kind of diagnostic framework is rooted in an assessment tradition in special education that runs counter to the screening and progress monitoring measures mentioned earlier in terms of the amount of time, the nature of the tasks, and the qualifications of the assessor. Put simply, the methods that Lewis used are time-consuming and expensive, because they rely a great deal on expert judgment.

It should also be noted that the persistent misunderstandings of a simple, fraction area model that Lewis’s two subjects exhibited are far deeper and more complicated than the functional effects of an “impoverished part-whole” representation of fractions on student understanding as they are often characterized in the literature (e.g., Fuchs et al., in press; Jordan, Resnick, & Carrique, 2016). In other words, it is difficult to imagine that a systematic misunderstanding of the complement of a fraction could be traced entirely to the typical shaded pie graph that would have been used briefly when these students were first taught fractions years earlier in their respective educations. The “complement of a fraction” conception is extraordinary, and Lewis’s study provokes further pondering on how pervasive this kind of thinking is with students with MD.

Hunt’s study covered a similar terrain insofar as clinical interviews, along with a dynamic adaptation of tasks, allow her to investigate the depth of student understanding in fractions. Hunt’s work focuses on what students do know as well as what they are yet to learn. This focus differs from typical assumptions about MD, which tend to focus on student deficiencies (e.g., skills knowledge, attention, working memory) and how to correct them. Hunt’s study is but one within a line of work based on close collaboration with mathematics educators whose research on students’ construction of fractional concepts informs intervention that builds on precise assessment of what students do know (see Hunt, 2015; Hunt & Empson, 2015; Hunt, Tzur, & Westenskow, 2016; Tzur, 2007; Tzur & Hunt, 2015).

By focusing on what students do know, Hunt’s findings add to a literature where the boundaries between MD, “at risk,” low performing, and very low performing are fluid. Although there are descriptive differences between her MD and Tier 2 students, the differences are not statistically significant. In recent years, poverty and race became important background variables to response to intervention (RtI), which is an emerging theme in special education research on students with learning disabilities (see Harry & Klingner, 2014; Woodward, 2016b). Likewise, Hunt’s study point to the need to add cognitive characteristics of what students do know as a foundational aspect of interventions that target students based on analysis of their ways of reasoning.

Theme 2: Views of Curriculum (or Interventions)

A second thematic way to think about the set of articles in this special series is through the interventions for MD, at risk, and low-performing students, specifically as they are described in the RCT literature. This context for thinking about interventions is important because of the emphasis, particularly since No Child Left Behind (NCLB) act, on research-based curriculum and “gold-standard” research methods. Curriculum in RCT studies is often characterized solely as print or electronic media, and these materials are often “validated” as a function of the research.

For example, recent research (e.g., Clarke, Baker, & Smolkowski, 2015; Doabler et al., 2015; Fuchs et al., in press) often used scripted materials as part of their RCT intervention studies. Accordingly, intervention specialists received professional development on these materials prior to the study so fidelity of implementing the curriculum could be achieved and measured. Doabler et al., in particular, made a strong argument for a highly structured and externally driven approach to contemporary curriculum development for at-risk and MD students, one that complies with the spirit of the original NCLB mandate. They argued for validated frameworks for designing curricula, field testing, and the use of pedagogical methods such as explicit instruction. Much of this thinking comes from an earlier era of direct instruction research (see Carnine, Jitendra, & Silbert, 1997), and it is worth noting that their print materials rely on detailed scripts to assist interventionists in using the curricula with fidelity.

What is often understated in much of the current RCT mathematics research for students with mathematics difficulties is that the interventionists vary considerably in their teaching qualifications, and may not be versed with best practices envisioned by mathematics educators (see Clarke et al., 2015; Fuchs et al., 2005; Fuchs et al., in press). This variation in teaching experience and knowledge level is partly a function of the scale of research and the desire to ensure that the curriculum is used in a consistent manner from one research site to the next. Another reason to use intervention specialists who may not be highly trained in mathematical pedagogy and content knowledge could be that these individuals represent the modal interventionist in our schools today (Woodward, 2016a). Put simply, there are not enough well trained mathematics educators in schools today to deliver Tier 2 or Tier 3 intervention services. In this regard, much of the current RCT research reflects the ecological conditions of mathematics interventions in the United States.

What makes the four articles in this special series of importance is the offering of alternative lenses to examine curriculum design and implementation—both in research and in practice. All four studies seem predicated on highly adaptive assessment and instruction that does not follow a predefined script. Specifically, these four studies raise questions about how far interventionists with limited pedagogical content knowledge can use scripted materials to remedy deep seated mathematical misconceptions, such as those articulated in Lewis’s (in the previous issue) study, or can help MD students develop self-explanation skills described by Liu and Xin (IN PRESS). Xin and Tzur et al.’s (IN PRESS) research into intelligent tutoring systems is an instructive way to consider the concept of math curriculum, as research on students’ conceptual progression, by both mathematics educators and special educators, drives interventions tailored to individuals’ conceptions.

The appeal of computer based instruction for at risk, low performing, and students with learning disabilities has been part of the special education literature for decades. Software programs can adapt instruction based on student performance, review concepts systematically, and they have the potential to capture a considerable amount of data (e.g., types of items missed, number of completed problems in a session). Xin and Tzur et al. (IN PRESS) describe an intelligent tutoring program designed to teach multiplicative thinking that draws on a constructivist framework. This framework explicitly articulates knowing and learning as cognitive change (Simon & Tzur, 2004; Tzur & Simon, 2004), and provides a corresponding pedagogy that capitalizes on what each student does know (Tzur, 2011, 2013; Tzur et al., 2013; Tzur & Lambert, 2011). This framework is augmented by the Conceptual Model-Based Problem Solving (COMPS; Xin, 2012) approach, which fosters students’ meaningful representation of word problem structures in mathematical model equations. We consider this framework, and studies done in full, genuine collaboration between special and mathematics educators, to be a refreshing and promising way forward.

Where, specifically, the intelligent tutoring program that Xin and Tzur et al. (IN PRESS) describe departs from the tradition of computer based instruction in special education can be found at the end of their article. They are careful in suggesting that the PGBM-COMPS program is not a stand-alone instructional system. Rather, it is to be used in concert with a teacher whose understanding of children’s ways of reasoning is instrumental for supporting student learning. In other words, the way in which teachers’ work synergizes with the computer program to foster new, proper conceptions builds off of student conceptions, and problem solving seems paramount to curriculum and instructional design.

The work of Liu and Xin (IN PRESS) further reinforces the role of a teacher’s (or interventionist’s) pedagogical content knowledge. Rooted in a synthesis of high-quality research on self-explanation (see Gersten, Beckmann, et al., 2009), their work addresses another dimension of the Common Core Standards less visible in current RCT special education, mathematics research: the Mathematical Practices. The pedagogical skills needed to help students with MD “repair” their initial statements (hence, reasoning linked with those statements) in an iterative manner is considerable, because it arises from inferring into deeply seated student conceptions. Accordingly, interventionists in this study needed to determine not only or mainly whether a student’s explanation was correct, but rather how reasoning could be improved and if it attended to precision. It is difficult to imagine that the kind of scripts associated with curricula described in many recent RCT studies could account for the teacher-student interactions that Liu and Xin describe. The key reason for this is the unpredictable way that context and teacher-student interactions can drive pedagogical decision making and what happens next in a classroom. Moreover, it is feasible that two highly skilled interventionists might respond differently to the same instructional conditions. The learning trajectories approach (see Sztajn, Confrey, Wilson, & Edgington, 2012; Woodward, 2016a) captures this kind of dynamic framework for instructional decision making.

We contend that the four studies presented in this special series, and the constructivist and/or sociocultural frameworks that guided them, provide a plausible way forward in special education. The studies by Hunt, by Xin and Tzur et al., and by Liu and Xin demonstrated the fruits of ongoing collaborations among special education and mathematics education scholars of learning and teaching. Their work is, thus, a worthwhile contribution when considering what intervention in mathematics might mean in Tier 2 and Tier 3 settings. Indeed, there are times when at risk, low performing, or MD students can work on procedural tasks with guidance that does not demand high levels of pedagogical content knowledge. However, as Shapiro, Zigmond, Wallace, and Marston (2011) reminded us, the fundamental nature of RtI is to decrease group size and increase what they call instructional intensity. We argue that, in the era of the Common Core Standards, as well as today’s broad vision of mathematics education, intense instruction must go beyond the acquisition of basic skills. As the authors in this special series implicitly argue, this means an in-depth understanding of what students do and do not know, which creates the foundation for interventions that carefully attend to student thinking and communication. In this sense, the four articles in this special series can greatly inform and help advance the contemporary literature on mathematics in special education.

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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