Four advisory panels have been appointed since 1999 alone to advise the nation on how best to teach mathematics: The National Commission on Mathematics and Science Teaching for the 21st Century, National Research Council, the RAND Mathematics Study Panel, and the National Mathematics Advisory Panel. The reports emerging from each are detailed, often technical, but well worth reading, especially for those involved in math education, because they capture what each expert panel concludes schools must teach and students learn in math. What these reports make clear is that mathematics teaching and learning are complex undertakings. The National Research Council, for example, refers to “mathematical proficiency” as five intertwined strands, described in the box on page 3 (Kilpatrick, Swafford, & Findell, 2001). Learning each of these strands is an ongoing process that builds on itself. As new concepts and skills are learned, new terms and symbols must also be learned and older skills remembered and applied.

The final report of the National Mathematics Advisory Panel (2008) speaks clearly to the need for math curricula that fosters student success in algebra (and beyond) and experienced math teachers who use researched-based instructional strategies. The report also stresses the “mutually reinforcing benefits of conceptual understanding, procedural fluency, and automatic recall of facts” (National Mathematics Advisory Panel, 2008, p. xiv). Math teachers know this already—and recognize the very real consequences of students not achieving a level of mastery with foundational math concepts. Disability can further compromise student learning (Spear-Swerling, 2005), especially if the disability affects recall of information and the generalization of skills from one learning situation to another.

Which brings us to the next two parts in this Evidence for Education: How disabilities can affect math learning and how to effectively address these special learning needs.

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Many different disabilities can affect children’s math learning and performance, but none more than disabilities that affect cognition—mental retardation, traumatic brain injury, attention-deficit/hyperactivity disorder, and learning disabilities, to name a few. Several specific areas of disability are clearly connected to math learning difficulties. Visual processing, visual memory, and visual-spatial relationships all impact math proficiency in that they are threads in the fabric of conceptual understanding and procedural fluency (Kilpatrick et al., 2001). Specific math learning difficulties also can affect a student’s ability to formulate, represent, and solve math problems (known as strategic competence).

The term learning disabilities (LD) certainly appears throughout the literature on math difficulties. This is not especially surprising: LD is the most frequently referenced disability affecting math learning and performance, with a well-documented impact on the learning of 5% to 10% of children in grades K-12 (Fuchs & Fuchs, 2002; Garnett, 1998; Geary, 2001, 2004; Mazzocco & Thompson, 2005).

That’s more than 2.8 million children (U.S. Department of Education, 2007). While some of these children are primarily affected in their ability to read or write, many others struggle predominantly in the math arena, a manifestation of LD known as dyscalculia. This can be seen in the Federal definition of LD, which captures well the variable impact of the disability:

Specific learning disability means a disorder in one or more of the basic psychological processes involved in understanding or in using language, spoken or written, that may manifest itself in the imperfect ability to listen, think, speak, read, write, spell, or to do mathematical calculations....

34 C.F.R.§ 300.8(c)(10)(i)

The “imperfect ability to…do mathematical calculations” accurately describes how LD affects many students. However, not all children with LD have math troubles, and not all children with math troubles have a learning disability. The commonality of interest here, then, is trouble with math, not what disability a child may have. That’s one very good reason to look beyond labels and focus on what teachers can do, instructionally speaking, to support students who are struggling in math. Which we’re going to do right now.

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We know a great deal about effective math instruction for students with disabilities, especially students who have LD. There have been five meta-analyses on the subject, reviewing a total of 183 research studies (Adams & Carnine, 2003; Baker, Gersten, & Lee, 2002; Browder, Spooner, Ahlgrim-Delzell, Harris, & Wakeman, 2008; Kroesbergen & Van Luit, 2003; Xin & Jitendra, 1999). The studies combined in these meta-analyses involved students with a variety of disabilities—most notably, LD, but other disabilities as well, including mild mental retardation, AD/HD, behavioral disorders, and students with significant cognitive disabilities. The meta-analyses found strong evidence of instructional approaches that appear to help students with disabilities improve their math achievement. We now also have the National Mathematics Advisory Panel Report (2008) that further investigates successful mathematical teaching strategies and provides additional support for the research results.

According to these studies, four methods of instruction show the most promise. These are:

• Systematic and explicit instruction, a detailed instructional approach in which teachers guide students through a defined instructional sequence. Within systematic and explicit instruction students learn to regularly apply strategies that effective learners use as a fundamental part of mastering concepts.
• Self-instruction, through which students learn to manage their own learning with specific prompting or solution-oriented questions.
• Peer tutoring, an approach that involves pairing students together to learn or practice an academic task.
• Visual representation, which uses manipulatives, pictures, number lines, and graphs of functions and relationships to teach mathematical concepts.

Of course, to make use of this information, an educator would need to know much more about each approach. So let’s take a closer look.

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Explicit instruction, often called direct instruction, refers to an instructional practice that carefully constructs interactions between students and their teacher. Teachers clearly state a teaching objective and follow a defined instructional sequence. They assess how much students already know on the subject and tailor subsequent instruction, based upon that initial evaluation of student skills. Students move through the curriculum, both individually and in groups, repeatedly practicing skills at a pace determined by the teacher’s understanding of student needs and progress (Swanson, 2001). Explicit instruction has been found to be especially successful when a child has problems with a specific or isolated skill (Kroesbergen & Van Luit, 2003).

The Center for Applied Special Technology (CAST) offers a helpful snapshot of an explicit instructional episode (Hall, 2002), shown in Figure 1 below. Consistent communication between teacher and student creates the foundation for the instructional process. Instructional episodes involve pacing a lesson appropriately, allowing adequate processing and feedback time, encouraging frequent student reponses, and listening and monitoring throughout a lesson.

Systematic instruction focuses on teaching students how to learn by giving them the tools and techniques that efficient learners use to understand and learn new material or skills. Systematic instruction, sometimes called “strategy instruction,” refers to the strategies students learn that help them integrate new information with what is already known in a way that makes sense and be able to recall the information or skill later, even in a different situation or place. Typically, teachers model strategy use for students, including thinking aloud through the problem-solving process, so students can see when and how to use a particular strategy and what they can gain by doing so. Systematic instruction is particularly helpful in strengthening essential skills such as organization and attention, and often includes:

• Memory devices, to help students remember the strategy (e.g., a first-letter mnemonic created by forming a word from the beginning letters of other words);
• Strategy steps stated in everyday language and beginning with action verbs (e.g., read the problem carefully);
• Strategy steps stated in the order in which they are to be used (e.g., students are cued to read the word problem carefully before trying to solving the problem);
• Strategy steps that prompt students to use cognitive abilities (e.g., the critical steps needed in solving a problem) (Lenz, Ellis, & Scanlon, 1996, as cited in Maccini & Gagnon, n.d.).

All students can benefit from a systematic approach to instruction, not just those with disabilities. That’s why many of the textbooks being published today include overt systematic approaches to instruction in their explanations and learning activities. It’s also why NICHCY’s first Evidence for Education was devoted to the power of strategy instruction. The research into systematic and explicit instruction is clear—the approaches taken together positively impact student learning (Swanson, in press). The National Mathematics Advisory Panel Report (2008) found that explicit instruction was primarily effective for computation (i.e., basic math operations), but not as effective for higher order problem solving. That being understood, meta-analyses and research reviews by Swanson (1999, 2001) and Swanson and Hoskyn (1998) assert that breaking down instruction into steps, working in small groups, questioning students directly, and promoting ongoing practice and feedback seem to have greater impact when combined with systematic “strategies.”

What does a combined systematic and explicit instructional approach look like in practice? Tammy Cihylik, a learning support teacher at Harry S. Truman Elementary School in Allentown, Pennsylvania, describes a first-grade lesson that uses money to explore mathematical concepts:

“[Students] use manipulatives,” she explains, “looking at the penny, identifying the penny.” Cihylik prompts the students with explicit questions: “What does the penny look like? How much is it worth?” Then she provides the answers herself, with statements like, “The penny is brown, and is worth one cent.” Cihylik encourages students to repeat the descriptive phrases after her, and then leads them in applying that basic understanding in a systematic fashion. After counting out five pennies and demonstrating their worth of five cents, she instructs the students to count out six pennies and report their worth. She repeats this activity each day, and incorporates other coins and questions as students master the idea of value.

Within this example, the relationship between explicit and systematic instruction becomes clear. The teacher is leading the instructional process through continually checking in, demonstration, and scaffolding/extending ideas as students build understanding. She uses specific strategies involving prompts that remind students the value of the coins, simply stated action verbs, and metacognitive cues that ask students to monitor their money. Montague (2007) suggests, “The instructional method underlying cognitive strategy instruction is explicit instruction.”

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Self-instruction refers to a variety of self-regulation strategies that students can use to manage themselves as learners and direct their own behavior, including their attention (Graham, Harris, & Reid, 1992). Learning is essentially broken down into elements that contribute to success:

• setting goals
• keeping on task
• checking your work as you go
• remembering to use a specific strategy
• monitoring your own progress
• being alert to confusion or distraction and taking corrective action
• checking your answer to make sure it makes sense and that the math calculations were correctly done.

When students discuss the nature of learning in this way, they develop both a detailed picture of themselves as learners (known as metacognitive awareness) and the self-regulation skills that good learners use to manage and take charge of the learning process. Some examples of self-instruction statements are shown on the next page.

To teach students to “talk to themselves” while learning new information, solving a math problem, or completing a task, teachers first model self-instruction aloud. They take a task and think aloud while working through it, crafting a monologue that overtly includes the mental behaviors associated with effective learning: goal-setting, self-monitoring, self-questioning, and self-checking. Montague (2004) suggests that both correct and incorrect problem-solving behaviors be modeled.

Modeling of correct behaviors helps students understand how good problem solvers use the processes and strategies appropriately. Modeling of incorrect behaviors allows students to learn how to use self-regulation strategies to monitor their performance and locate and correct errors. Self-regulation strategies are learned and practiced in the actual context of problem solving. When students learn the modeling routine, they then can exchange places with the teacher and become models for their peers. (p. 5)

The self-statements that students use to talk themselves through the problem-solving process are actually prompting students to use a range of strategies and to recognize that certain strategies need to be deployed at certain times (e.g., self-evaluation when you’re done, to check your work). Because learning is a very personal experience, it’s important that teachers and students work together to generate self-statements that are not only appropriate to the math tasks at hand but also to individual students. Instruction also needs to include frequent opportunities to practice their use, with feedback (Graham et al., 1992) until students have internalized the process.

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Peer tutoring is a term that’s been used to describe a wide array of tutoring arrangements, but most of the research on its success refers to students working in pairs to help one another learn material or practice an academic task. Peer tutoring works best when students of different ability levels work together (Kunsch, Jitendra, & Sood, 2007). During a peer tutoring assignment, it is common for the teacher to have students switch roles partway through, so the tutor becomes the tutee. Since explaining a concept to another person helps extend one’s own learning, this practice gives both students the opportunity to better understand the material being studied.

Research has also shown that a variety of peer-tutoring programs are effective in teaching mathematics, including Classwide Peer Tutoring (CWPT), Peer-Assisted Learning Strategies (PALS), and Reciprocal Peer Tutoring (RPT) (Barley et al., 2002). Successful peer-tutoring approaches may involve the use of different materials, reward systems, and reinforcement procedures, but at their core they share the following characteristics (Barley et al., 2002):

• The teacher trains the students to act both as tutors and tutees, so they are prepared to tutor, and receive tutoring from, their peers. Before engaging in a peer-tutoring program, students need to understand how the peer- tutoring process works and what is expected of them in each role.
• Peer-tutoring programs benefit from using highly structured activities. Structured activities may include teacher-prepared materials and lessons (as in Classwide Peer Tutoring) or structured teaching routines that students follow when it is their turn to be the teacher (as in Reciprocal Peer Tutoring).
• Materials used for the lesson (e.g., flashcards, worksheets, manipulatives, and assessment materials) should be provided to the students. Students engaging in peer tutoring require the same materials to teach each other as a teacher would use for the lesson.
• Continual monitoring and feedback from the teacher help students engaged in peer tutoring stay focused on the lesson and improve their tutoring and learning skills.

Finally, there is mounting research evidence to suggest that, while low-achieving students may receive moderate benefits from peer tutoring, effects for students specifically identified with LD may be less noticeable unless care is taken to pair these students with a more proficient peer who can model and guide learning objectives (Kunsch, Jitendra, & Sood, 2007).

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Mathematics instruction is a complex process that attempts to make abstract concepts tangible, difficult ideas understandable, and multifaceted problems solvable. Visual representations bring research-based options, tools, and alternatives to bear in meeting the instructional challenge of mathematics education (Gersten et al., 2008).

Visual representations, broadly defined, can include manipulatives, pictures, number lines, and graphs of functions and relationships. “Representation approaches to solving mathematical problems include pictorial (e.g., diagramming); concrete (e.g., manipulatives); verbal (linguistic training); and mapping instruction (schema-based)” (Xin & Jitendra, 1999, p. 211). Research has explored the ways in which visual representations can be used in solving story problems (Walker & Poteet, 1989); learning basic math skills such as addition, subtraction, multiplication, and division (Manalo, Bunnell, & Stillman, 2000); and mastering fractions (Butler, Miller, Crehan, Babbitt, & Pierce, 2003) and algebra (Witzel, Mercer, & Miller, 2003).

Concrete-Representational-Abstract (CRA) techniques are probably the most common example of mathematics instruction incorporating visual representations. The CRA technique actually refers to a simple concept that has proven to be a very effective method of teaching math to students with disabilities (Butler et al., 2003; Morin & Miller, 1998). CRA is a three-part instructional strategy in which the teacher first uses concrete materials (such as colored chips, base-ten blocks, geometric figures, pattern blocks, or unifix cubes) to model the mathematical concept to be learned, then demonstrates the concept in representational terms (such as drawing pictures), and finally in abstract or symbolic terms (such as numbers, notation, or mathematical symbols).

During a fraction lesson using CRA techniques, for example, the teacher might first show the students plastic pie pieces, and explain that, when the circle is split into 4 pieces, each of those pieces is ¼ of the whole, and when a circle is split into 8 pieces, each piece is ⅛ of the whole. After seeing the teacher demonstrate fraction concepts using concrete manipulatives, students would then be given plastic circles split into equal pieces and asked what portion of the whole one section of that circle would be. By holding the objects in their hands and working with them concretely, students are actually building a mental image of the reality being explored physically.

After introducing the concept of fractions with concrete manipulatives, the teacher would model the concept in representational terms, either by drawing pictures or by giving students a worksheet of unfilled-in circles split into different fractions and asking students to shade in segments to show the fraction of the circle the teacher names.

In the final stage of the CRA technique, the teacher demonstrates how fractions are written using abstract terms such as numbers and symbols (e.g., ¼ or ½). The teacher would explain what the numerator and denominator are and allow students to practice writing different fractions
on their own.

As the Access Center (2004) points out, CRA works well with individual students, in small groups, and with an entire class. It’s also appropriate at both the elementary and secondary levels. The National Council of Teachers of Mathematics (NCTM) recommends that, when using CRA, teachers make sure that students understand what has been taught at each step before moving instruction to the next stage (Berkas & Pattison, 2007). In some cases, students may need to continue using manipulatives in the representational and abstract stages, as a way of demonstrating understanding.

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