Effective Mathematics Instruction

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Jonathan James Tue, Jan 11, 2022 05:06 PM UTC

This Evidence for Education is about helping students stay on track in math, building concepts upon the concept in a steady progression of skills. This is as much a national priority as it is a practical necessity for the students themselves because daily life involves math—from the check-out counter at the school store to the express line in the grocery, from our most routine jobs to the high-paying, high-profile ones in engineering, technology, and science (Lee, Grigg, & Dion, 2007; U.S. Government Accountability Office, 2005). So—two questions naturally arise. What do students need to know how to do, mathematically? And what instructional approaches are effective in teaching those skills?

This Evidence for Education addresses these questions, and one more: What do we do when disability affects a student’s ability to learn math skills? That’s the reality for literally millions of students in our schools; certain disabilities do add to the challenge of learning an already challenging subject. Therefore, what the research has to say about effective math instruction for students with disabilities is a vital tool in the hands of school personnel responsible for designing and delivering math programming. This publication offers just such research-based tools and guidance to teachers, administrators, and families.

1. nichcy.org http://nichcy.org/research/ee/math
Effective Mathematics Instruction
by Kathlyn Steedly, Ph.D., Kyrie Dragoo, M.Ed., Sousan Arefeh, Ph.D., &
Stephen D. Luke, Ed.D.
Evidence for Education • Volume III • Issue I • 2008
Links updated, December 2011
Table of Contents
What Students Need to Know and We Need To Teach
How Disabilities Can Affect Math Achievement
Effective Math Instruction for Students with Learning Difficulties
Choosing and Using Effective Math Intervention Strategies
“Can you do addition?” the White Queen asked. “What’s one and one
and one and one and one and one and one and one and one and one?”
“I don’t know,” said Alice. “I lost count.”
—Lewis Carroll
Through the Looking Glass
We start this Evidence for Education with an odd, little quote that illustrates several things quickly about
math. It’s easy to get lost, especially if the question comes at you too fast, and once you get lost, well…
Well, we don’t want students to get lost in math. This Evidence for Education is about helping students stay
on track in math, building concept upon concept in a steady progression of skills. This is as much a national
priority as it is a practical necessity for the students themselves, because daily life involves math—from the
check-out counter at the school store to the express line in the grocery, from our most routine jobs to the
high-paying, high-profile ones in engineering, technology, and science (Lee, Grigg, & Dion, 2007; U.S.
Government Accountability Office, 2005).
So—two questions naturally arise. What do students need to know how to do, mathematically? And what
instructional approaches are effective in teaching those skills?
This Evidence for Education addresses these questions, and one more: What do we do when disability
affects a student’s ability to learn math skills? That’s the reality for literally millions of students in our schools;
certain disabilities do add to the challenge of learning an already challenging subject. Therefore, what the
research has to say about effective math instruction for students with disabilities is a vital tool in the hands of
school personnel responsible for designing and delivering math programming. This publication offers just
such research-based tools and guidance to teachers, administrators, and families.
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What Students Need to Know and We Need to Teach | The Work of
Expert Panels
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

2. 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.
Mathematics Advisory Panels and Their Reports
National Commission on Mathematics and Science Teaching for the 21st Century—Before It’s
Too Late
http://www2.ed.gov/inits/Math/glenn/toc.html
National Research Council—Adding It Up: Helping Children Learn Mathematics
http://www.nap.edu/catalog.php?record_id=9822
RAND Mathematics Study Panel—Mathematical Proficiency for All Students
http://www.rand.org/pubs/monograph_reports/MR1643/index.html
Foundations for Success: The Final Report of the National Mathematics Advisory Panel
http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
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How Disabilities Can Affect Math Achievement
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.§
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.

3. 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.
The National Research Council’s Concept of “Mathematical Proficiency”
“The integrated and balanced development of all five strands of mathematical proficiency [shown below] …
should guide the teaching and learning of school mathematics.”
Conceptual understanding— comprehension of mathematical concepts, operations, and relations
Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
Strategic competence—ability to formulate, represent, and solve mathematical problems
Adaptive reasoning—a capacity for logical thought, reflection, explanation, and justification
Productive disposition— habitual inclination to see mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one’s own efficacy. (Kilpatrick, et al., 2001, p. 11)
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Effective Mathematics Instruction for Students with Learning Difficulties
in Math — Four Approaches That Improve Results
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 and Systematic Instruction
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,
The Center for Applied Special Technology (CAST) offers a helpful snapshot of an explicit instructional

4. 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.”
For M ore Information on
Explicit and Systematic Instruction
National Institute for Direct Instruction (NIFDI)
The Access Center’s Direct or Explicit Instruction and Mathematics
Special Connections’ Direct Instruction: Math
NICHCY’s The Power of Strategy Instruction
The Access Center’s Strategy/Implicit Instruction and Mathematics

5. University of Nebraska-Lincoln’s Cognitive Strategy Instruction: Math
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

6. 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.
For M ore Information and Guidance on Self-Instruction
Penn State’s Self-Regulation Abilities, Beyond Intelligence, Play Major Role in Early Achievement
The National Research Center on the Gifted and Talented (NRC/GT)’s module, Self-Regulation
The Access Center’s Math Problem Solving for Primary Elementary Students with Disabilities
The Access Center’s Math Problem Solving for Upper Elementary Students with Disabilities
Marjorie Montague’s 2007 article, “Self-Regulation and Mathematics Instruction” in Learning Disabilities
Research & Practice, 22(1), 75–83. (This article appears in a special issue of the journal devoted to math
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Peer Tutoring
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

7. learning objectives (Kunsch, Jitendra, & Sood, 2007).
For M ore Information on Peer Tutoring
Vanderbilt’s Peer-Assisted Learning Strategies (PALS)
The Institute of Education Sciences’ Intervention: Peer-Assisted Learning Strategies (PALS)
The Institute of Education Sciences’ Intervention: ClassWide Peer Tutoring (CWPT)
The Access Center’s Using Peer Tutoring for Math
The Access Center’s Using Peer Tutoring to Facilitate Access
(Reviews the use of CWPT, RPT, and PALS in teaching mathematics and other subjects)
Fulk and King’s Classwide Peer Tutoring at Work
Special Connection’s An Introduction to ClassWide Peer Tutoring
The Center for Effective Collaboration and Practice’s Classwide Peer Tutoring: Information for Families
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Visual Representations
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

8. 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.
For M ore Information and Guidance on Visual Representations
The Access Center’s Concrete-Representational-Abstract Instructional Approach
TeachingLD’s Teaching Students Math Problem-Solving Through Graphic Representations
Special Connection’s Concrete-Representational-Abstract (C-R-A) Instruction
MathVIDS’ Concrete-Representational-Abstract Sequence of Instruction
Using Research-based Methods to Teach Fraction Concepts: What REALLY Works
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Putting the Research to Work: Choosing and Using Effective Math
Intervention Strategies
We’ve briefly examined four approaches to teaching mathematics to students with disabilities that research
has shown to be effective (Adams & Carnine, 2003; Baker, Gersten, & Lee, 2002; Kroesbergen & Van Luit,
2003; Xin & Jitendra, 1999). Each is worthy of study in its own right, so we hope that the sources of
additional information we’ve provided will help teachers, administrators, and families bring these research-
based practices into the math classroom.
When it comes time to determine how you can best teach math to your students, select an instructional
intervention that supports the educational goals of those students based on age, needs, and abilities.
Research findings can and do help identify effective and promising practices, but it’s essential to consider
how well-matched any research actually is to your local situation and whether or not a specific practice will be
useful or appropriate for a particular classroom or child. Interventions are likely to be most effective when
they are applied to similar content, in similar settings, and with the age groups intended for them. That’s why
it’s important to look closely at the components of any research study to determine whether the overall
findings provide appropriate guidance for your specific students, subjects, and grades—apples to apples, so
to speak.
Of great value to those seeking to better understand the evidence base for math (and other) educational

9. interventions are these three sources of information:
What Works Clearinghouse
http://ies.ed.gov/ncee/wwc/
Best Evidence Encyclopedia (BEE)
http://www.bestevidence.org
Center on Instruction
http://www.centeroninstruction.org/index.cfm
Each of these sources looks closely at existing research on educational interventions and reports on their
effectiveness. Each has evaluated a wide range of commercial math series and materials in use around the
country and has categorized them by how much evidence of effectiveness they show. What Works uses six
categories (strong positive, potentially positive, mixed, potentially negative, strong negative, and no
discernible evidence), while the Best Evidence Encyclopedia uses five (strong, moderate, limited, insufficient
to make a judgment, and no qualifying studies), either of which offers valuable information to decision
makers. To help teachers and administrators investigate those interventions most relevant to their local
situation and need, interventions are also broken out by level (elementary and middle and/or high school).
The instructional approaches on which we have focused take place in classrooms where they often coexist,
support, strengthen, and work together to effectively teach students. These approaches can be seen as
threads in the fabric of the classroom. In the complex world of the classroom, students benefit from one-on-
one guidance from a teacher who can model problem-solving techniques and control the difficulty of tasks
through feedback and cues.
And speaking of feedback…consistent and ongoing feedback has been shown to be quite effective in
improving student performance. That’s why it’s an aspect that should be incorporated into all classrooms,
regardless of the intervention. In particular, the value of immediate feedback stands out. Regular feedback
helps students guide and improve their own practice, even as giving feedback helps teachers guide and
tailor their own instruction.
When you consider the interventions described here, it’s exciting to realize that they are ready tools in our
hands and in our classrooms, and can serve us well as a means of improving students’ math proficiency and
outcomes. We hope they do just that.
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Concrete-Representational-Abstract Techniques – a threepart instructional strategy where the teacher
uses concrete materials (manipulatives) to model a concept to be learned, then uses representational terms
(pictures), and finally uses abstract, symbolic terms (numbers, math symbols).
Explicit and Systematic Instruction – teacher-led demonstrations of various strategies that students, in
turn, use to guide their way through the problem-solving process.
Dyscalculia – a form of learning disability that causes an individual to have difficulty in understanding
concepts of quantity, time, place, value, and sequencing, and in successfully manipulating numbers or their
representations in mathematical operations.
Intervention – specific services, activities, or products developed and implemented to change or improve
student knowledge, attitudes, behavior, or awareness.
M athematical Proficiency – a term developed by the National Research Council that describes five
interrelated strands of knowledge, skills, abilities, and beliefs that allow for mathematics manipulation and
achievement across all mathematical domains (e.g., conceptual understanding, procedural fluency, strategic
competence, adaptive reasoning, and productive disposition) (Kilpatrick et al., 2001).
Operations – mathematical procedures such as addition, subtraction, multiplication, and division.
Peer Tutoring – an instructional arrangement where students work in pairs to help one another learn
material or practice an academic task.
Representation Techniques – visually or schematically presenting the ideas or information contained in

10. word problems in language or visual media.
Self-Instruction – an instructional approach that teaches students to use a variety of self-statements or
verbal prompts to guide themselves through the problem-solving process.
Strategy Instruction/Strategy Training – an instructional approach that teaches students how to use the
same tools and techniques that efficient learners use to understand and learn new material or skills.
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Access Center. (2004). Concrete-representational-abstract instructional approach. Retrieved March 21,
2008, from the Access Center Web site:
Adams, G., & Carnine, D. (2003). Direct instruction. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.),
Handbook of learning disabilities (pp. 403–416). New York: Guilford Press.
Baker, S., Gersten, R., & Lee, D. (2002). A synthesis of empirical research on teaching mathematics to low-
achieving students. The Elementary School Journal, 103(1), 51–73.
Barley, Z., Lauer, P. A., Arens, S. A., Apthorp, H. S., Englert, K. S., Snow, D., & Akiba, M. (2002). Helping at-
risk students meet standards: A synthesis of evidence-based classroom practices. Retrieved March 20,
2008, from the Midcontinent Research for Education and Learning Web site:
http://www.mcrel.org/PDF/Synthesis/5022RR_RSHelpingAtRisk.pdf
Berkas, N., & Pattison, C. (2007, November). Manipulatives: More than a special education intervention.
NCTM News Bulletin. Retrieved March 20, 2008, from the National Council of Teachers of Mathematics Web
site: http://www.nctm.org/news/release_list.aspx?id=12698
Browder, D. M., Spooner, F., Ahlgrim-Delzell, L., Harris, A., & Wakeman, S. Y., (2008). A meta-analysis on
teaching mathematics to students with significant cognitive disabilities. Exceptional Children, 74(4), 407-432.
Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with
mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice,
18(2), 99-111.
Fuchs, L. S., & Fuchs, D. (2002). Mathematical problem solving profiles of students with mathematics
disabilities with and without co-morbid reading disabilities. Journal of Learning Disabilities, 35(6), 563–573.
Garnett, K. (1998). Math learning disabilities. Retrieved November 10, 2006, from the LD Online Web site:
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