Math Methodology

Part 1: Instruction Essay (Page 3 of 3)
Addressing Needs of Students with Math Difficulties

NCLB mandates that states and districts adopt programs and policies supported by scientifically based research, which will influence instructional strategies that educators use.  In a standards-based classroom four instructional strategies are key: Inquiry and problem solving Collaborative learning Assessment embedded in instruction Higher order questioning Math Methodology is a three part series on instruction, assessment, and curriculum.  Sections contains relevant essays and resources: Part 1: Math Methodology: Instruction The Instruction Essay (Page 1 of 3) contains the following subsections: Introduction to Teaching Challenges including teacher goal setting and perspectives on improving instruction Bloom's Taxonomy and Levels of Understanding The Instruction Essay (Page 2 of 3): Teaching Mathematics Right the First Time: Learning for Understanding, a focus on strategies for putting research into practice The Instruction Essay (Page 3 of 3) on this page addresses the needs of students with math difficulties.  It contains the following subsections: Sources of Math Difficulties Prevention and Intervention Principles Assessing Mathematics Learning Needs and Associated Teaching Strategies Math Methodology Instruction Resources also includes resources for special needs students (e.g., deaf, visually impaired, learning disabilities, English language learners). Part 2: The Role of Assessment and resources Part 3: Curriculum: Content and Mapping and resources

Teaching and Math Methodology

Instruction

Addressing Needs of Students with Math Difficulties

Introduction to Sources of Math Difficulties

A learning disability is a life-long condition that manifests itself "by significant difficulties in acquisition and use of listening, speaking, reading, writing, reasoning, or mathematical abilities, or of social skills" (Kenyon, 2000, sec: Definition).  According to Amy Brodesky, Caroline Parker, Elizabeth Murray, and Lauren Katzman, students' success in  mathematics will depend on their strengths and needs related to cognitive processing, language, visual-spatial processing, organization, memory, attention, psycho-social, and fine-motor skills (2002). 

When students are having difficulty with mathematics, teachers need to be able to identify the source of the problem. Some problems result from physical, cognitive, sensory, and learning disabilities in general, which have been diagnosed by professional staff and relayed to the teacher so that appropriate accommodations and/or assistive technologies can be used.  Other problems might not have been diagnosed and the teacher observes those after working with students for a period of time.  The list is not exhaustive, but teachers might be alerted to potential math or learning disabilities from the following examples noted by Rochelle Kenyon (2000) and at Misunderstood Minds:

• Inability to recall basic number facts, or easily forgetting rules, procedures, formulas, or where they are or what they are doing when solving problems.

• Computational weaknesses.  These might also arise because students are not writing numerals clearly, are misreading operation signs in a problem.  They might be writing numbers backwards.  They might have difficulty keeping score in a game.

• Inability to connect abstract or conceptual representations with concrete representations or reality.

• Inability to make connections of math to real life experiences.  For example, they might know a number but not see its relation to an actual quantity.  They might have difficulty telling time.

• Difficulty with the language, such as with math terms that do not fit their everyday language, or following directions, or reading the math textbook.

• Difficulty comprehending the visual-spatial and perceptual aspects of math (e.g., perceptions of changes when objects are moved from one place to another, or working with 2-D representations of 3-D geometric objects).

• Problems with organization, such as when working with multi-step problems, identifying relevant information in word problems, losing sight of the final goal in problem solving, appreciating the reasonableness of a solution, inability to copy problems correctly, or overload when too many problems are presented at one time on a page to solve.

Silver, Strong, and Perini (2007) indicate lack of attention to learning styles (mastery, understanding, interpersonal, and self-expressive) also may lead to math difficulties, and these might be overcome by varying and using multiple instructional strategies.  Mastery learners like drills, lectures, demonstrations, and practice.  They "may experience difficulty when learning becomes too abstract or involves open-ended questions."  Understanding learners appreciate logic, debate, and inquiry and value research projects and independent study and reading.  They "may experience difficulty when there is a focus on the social environment of the classroom (e.g., cooperative learning)."  Interpersonal learners would value the social environment with cooperative learning, group experiences, discussion, and role playing and may experience difficulty with "independent seat work or when learning lacks real world application."  Finally, self-expressive learners like creativity, "open-ended and nonroutine problems" and examining what ifs.  For them, difficulties may arise with "drill and practice and rote problem solving" (sec: Part One: Introduction, Figure C).  The key here is to strike a balance in a selecting instructional strategies, as students can work in all four styles.

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Prevention and Intervention Principles

Lynn Fuchs and Douglas Fuchs (2001) say that prevention of math difficulties in this country is generally ineffective for all students, including the learning disabled (p. 85).  Part of the problem might lie with textbooks used, which form the basis for the majority of instruction that takes place in classrooms.  Texts might not adhere to important instructional principles that affect learning.  For example, those principles include: “providing clear objectives, teaching 1 new concept or skill at a time, reviewing background knowledge, providing explicit explanations, structuring the use of instructional time efficiently, providing adequate practice, structuring appropriate review, and organizing effective feedback” (p. 85).

However, there is research on intervention providing evidence of methods to prevent and treat math difficulties. Fuchs and Fuchs (2001) discuss Principles for the Prevention and Intervention of Mathematics Difficulties at three levels.  In essence, primary prevention focuses on universal design; secondary prevention (i.e., prereferral intervention), focuses on adaptations within the regular classroom; and tertiary prevention (i.e., intervention) focuses on highly individualized intensive and explicit contextualization of skill-based instruction. 

According to the National Mathematics Advisory Panel (2008):

Explicit systematic instruction typically entails teachers explaining and demonstrating specific strategies and allowing students many opportunities to ask and answer questions and to think aloud about the decisions they make while solving problems. It also entails careful sequencing of problems by the teacher or through instructional materials to highlight critical features. (p. 48)

After its review of 26 high-quality studies related to teaching low achieving students and students with learning disabilities, mostly using randomized control designs, the National Mathematics Advisory Panel concluded that explicit methods of instruction are effective with both groups of students.  In particular, "Explicit systematic instruction was found to improve the performance of students with learning disabilities in computation, solving word problems, and solving problems that require the application of mathematics to novel situations" (p. 48). Although the Panel recommends some explicit systematic instruction, "This kind of instruction should not comprise all the mathematics instruction these students receive" (p. 49).

Primary Prevention

Universal Design for Learning from the Center for Applied Special Technology calls for students to have multiple means of expression, representation, and engagement in their learning.  Instructional media should provide those elements and have scaffolds built in (Deubel, 2003).  Within a universal design framework, Fuchs and Fuchs (2001, pp. 86-87) present four principles of primary prevention that can be used with all students, including learning disabled:

1. Quick pace with varied instructional activities and high levels of engagement.  Students benefit from active involvement (e.g., discussing, writing, computing, problem solving) within “a greater range of grouping arrangements” for carrying out activities. 
2. Challenging standards for achievement.  Motivating statements convey high expectations that everyone will learn, and convey more than just trying to convince students that activities will be fun and interesting. 
3. Self-verbalization methods.  Self-verbalization strategies for approaching and solving problems benefit students with learning disabilities, and low-, average-, and high-performing students. For example, problem solving performance has been shown to improve by memorizing and verbalizing seven cognitive steps: “read the problem, paraphrase, visualize with a picture or diagram, hypothesize a plan to solve the problem, estimate the answer, compute, and check” (p. 87). 
4. Physical and visual representations of number concepts or problem-solving situations.  The physical and visual representations help build conceptual understanding, facilitate application of procedural knowledge, and long term retention of procedural competence.

Secondary Prevention

There are three principles for secondary prevention of math difficulties within the classroom: adaptations cannot be disruptive to the target learner, must be unobtrusive for others in the class, and must be feasible for the teacher to implement within the normal classroom routine.  At this stage, the teacher might benefit from additional structure and instructional strategies from special educators, school psychologists, collaboration with fellow teachers, or student-support groups (Fuchs & Fuchs, 2001).

Secondary prevention strategies that might work at this level include goal setting, self-monitoring of task completion and work quality, computer-assisted instruction, concrete representations of numbers and number concepts, and reinforcement.  However, unresponsive students might yet need the tertiary level of intervention (Fuchs & Fuchs, 2001).

Tertiary Prevention

Primary and secondary preventions have not been successful for the group of students needing tertiary prevention.  This level, also known as intervention, is typically performed by special educators who employ of a broader range of instructional strategies that might not be feasible within a regular classroom setting.  Intervention is characterized by three principles ((Fuchs & Fuchs, 2001, pp. 91-93):

1. A focus on the individual student as the unit of instructional decision making.  Constructivist influences are found in instructional practices. When focusing on the individual student (individually referenced decision making), teachers do not prejudge the efficacy of a particular instructional method.  Judgment can only be made after trying a method to see if it does or does not work for a learner. 
2. Intensive instructional delivery.  Intensive instruction includes, but is not limited to, one-to-one tutoring.  Group lessons can also involve intensive instruction.  Representative of a broader set of instructional features are “(1) high rates of active responding at appropriate levels, (2) careful matching of instruction with the individual student’s skill levels, (3) instructional cues, prompts, and fading to support approximations to correct responding, and (4) detailed task-focused feedback” (p. 92). 
3. Explicit contextualization of skills-based instruction.  Rather than teaching basic-skills in isolation, such skills are explicitly taught situated within a context of application.  For example, students might be explicitly taught four ways that transfer occurs in mathematics: “problems can look different, can ask questions in a different way, can use different vocabulary, and can imbed skills within larger problem-solving contexts” (p. 92).  Further, teachers would make “transparent the connections between knowledge acquisition and knowledge application, rather than leaving the student to discover those connections more incidentally” (p. 93). 

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Assessing Mathematics Learning Needs and Associated Teaching Strategies

According to Kristine Augustyniak, Jacqueline Murphy, and Donna Phillips (2005), the current emphasis on targeted interventions for students makes it important for educators to refine their knowledge of the different learning disabilities and how they might be manifested in children.  The federal government’s current classification system includes reading, language arts, and mathematics as three specific areas of deficit.  The government presumes the disabilities are associated with a central nervous system dysfunction. 

In Psychological Perspectives in Assessing Mathematics Learning Needs, Augustyiak, Murphy, and Phillips (2005) discuss relevant factors in learning mathematics and propose several teaching strategies that may prove helpful for learners with hypothesized primary skill deficits in mathematics.  Their suggested strategies are summarized in Table 1 below. 

Factors in Learning Math

Research in developmental, cognitive, social, and neuro-psychology has shed light on factors related to learning mathematics and the nature of math learning disability (MLD). Typically, students with MLD require over-learning to retain skills required in math. 

Developing numerical skills involves specialized arithmetic language, comprehension of quantity, reasoning, and an ability to convert words (verbal or written) and visual forms into symbols and vice-versa.  A visual-spatial impairment “is often evidenced as problems in discriminating between similar letters, copying shapes and figures, using computerized answer sheets, making sense of graphs and charts, and lining up numbers in math problems.”  Those with spatial acalculia might rotate or omit numbers, misread arithmetic signs, have difficulty with lining up numbers in columns and placing of decimals. (Augustyniak, Murphy, & Phillips, 2005, p. 279).

Cognitive skill development relates to learners’ abilities to perceive, sustain attention, organize, remember, and monitor information such as distinguishing between essential and non-essential details.  However, one should not assume that deficits can be attributed to a specific learning disability.  There is great variability in normal development of those skills.  Math performance might be impeded because the student’s higher order cognitive skills are just underdeveloped.  Thus, an assessment of specific neuropsychological abilities is potentially unreliable.  Yet when paired with assessments of academic skills, both can inform an intervention. (Augustyniak, Murphy, & Phillips, 2005).

Social aspects of learning math are influenced by students’ beliefs about how math is learned (e.g., memorization, only one correct way to solve a problem, quick solutions), beliefs about oneself in relation to math, and beliefs about the social context of math learning and problem solving.  Problems are manifested in emotions and behaviors such as frustration, lack of motivation, and poor problem solving strategies.  A constructivist approach to teaching and learning is recommended that includes making math relevant to real-life situations, hands-on involvement, and exploration within a flexible learning environment. (Augustyniak, Murphy, & Phillips, 2005).

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Teaching Strategies for Hypothesized Primary Skill Deficits in Math

Response to Intervention Resources 

Response to Intervention is a method for identifying students with learning disabilities.  The following will help you to learn more on this process.

National Center for Learning Disabilities, LD News--Response to Intervention Updates: http://www.ncld.org/content/view/1129/389/

U.S. Department of Education, Office of Special Education Programs’ IDEA website: http://idea.ed.gov/explore/home

What You Need to Know about IDEA 2004
Response to Intervention (RTI):
New Ways to Identify Specific Learning Disabilities: http://www.wrightslaw.com/info/rti.index.htm  has numerous articles and links to other web sites on this topic.

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References

Augustyniak, K., Murphy, J., & Phillips, D. (2005). Psychological perspectives in assessing mathematics learning needs. Journal of Instructional Psychology, 32(4), 277-286.

Brodesky, A., Parker, C., Murray, E., & Katzman L. (2002). Accessibility strategies toolkit for mathematics. Newton, MA: Education Development Center, Inc. Available: http://www2.edc.org/accessmath/resources/strategiesToolkit.pdf

Deubel, P. (2003). An investigation of behaviorist and cognitive approaches to instructional multimedia design. Journal of Educational Multimedia and Hypermedia,12(1), 63-90. Available: http://www.ct4me.net/multimedia_design.htm

Fuchs, L., & Fuchs, D. (2001). Principles for the prevention and intervention of mathematics difficulties. Learning Disabilities Research & Practice, 16(2), 85-95.

Kenyon, R. (2000, September). Accommodating math students with learning disabilities.  Focus on Basics, 4(B). Available: http://www.ncsall.net/?id=325 

National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel.  Washington, DC: U.S. Department of Education. Available: http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

Silver, H., Strong, R., & Perini, M. (2007). The strategic teacher: Selecting the right research-based strategy for every lesson.  Alexandria, VA: Association for Supervision and Curriculum Development. Available: http://www.ascd.org 

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See other Math Methodology pages: 

Instruction--Resources,  Assessment and Curriculum: Content and Mapping

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