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Math Methodology

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

 

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). For elementary and middle schools, Gersten, Beckmann, Clarke, Foegen, Marsh, and Witzel (2009) recommend that all students be screened to identify those at risk for potential mathematics difficulties and that interventions be provided to students identified as at risk (p. 6).

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) indicated 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) said 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) discussed 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) presented 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). However, Lynn Fuchs (2008) indicated that six instructional principles must be incorporated at the secondary prevention level of a multi-tier prevention system:

1. Instructional explicitness
2. Instructional design that eases the learning challenge (e.g., via "precise explanations and with the use of carefully sequenced and integrated instruction," beginning with "teaching a set of foundational skills the student can apply across the entire program."
3. A strong conceptual basis for procedures that are taught
4. An emphasis on drill and practice
5. Cumulative review as part of drill and practice
6. Motivators to help students regulate their attention and behavior and to work hard. (para. 1, 4, 5)

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) discussed relevant factors in learning mathematics and proposed 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).  Development of cognitive skills in math is closely aligned with Jean Piaget's theory of cognitive development.  Readers might be interested in David Moursund's (2010) six-level, Piagetian-type, math cognitive development scale found in IAE-pedia.  The scale examines math development from birth to becoming a true mathematician.  You can also learn more about math learning disorders.

As an intervention for developing cognitive skills, educators might consider BrainWare Safari.  This award-winning software program develops 41 cognitive skills in six areas: attention, memory, sensory integration, visual processing, auditory processing, and logic/reasoning.  It has an entertaining and motivating video-game format with a jungle theme. According to its developer, Learning Enhancement Corporation (2009), "By improving students’ underlying mental processing skills, it enables them to be more successful across the curriculum, whether in reading, math or other subject matter. Published research has demonstrated an average of over 4 years of cognitive growth in 12 weeks of using the program. BrainWare Safari benefits all students, not just those with special needs" (para. 1).  This software won the 2010 Distinguished Achievement Award in the category of Special Education from the Association of Educational Publishers.

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

A tiered approach to teaching mathematics is most likely the most effective method to addressing the issues of math difficulties that exist among learners in the same classroom.  According to David Suarez (2007), underperforming students might be bored or overwhelmed by a single approach to learning. "Through tiered instruction, students at different ends of the ability spectrum find success in math class" (p. 60).  His approach consists of thematic units.  Learners choose from green, blue, and black levels of difficulty in both instructional materials and assessments, and can vary their choices from unit to unit.  A green level designates foundational and meets grade level standards for proficiency, blue is intermediate, and black is advanced offering the greatest level of challenge sometimes requiring learners to tackle unfamiliar tasks.  Learners find choices motivational and provide experience taking charge of their own learning.  This tiered instruction has a theoretical foundation in the works of Lev Vygotsky's zone of proximal development, Mihaly Csikszentmihalyi's perspective on how to create joyful concentration, Eric Jensen's work on how stress affects learning, and William Glasser's choice theory.  Suarez maintains a blog, Challenge by Choice, on tiered instruction in math, where he elaborates on this method and offers videos, examples, and results from his implementation with a colleague at the Jakarta International School in Indonesia.

The Tiered Curriculum Project from the Indiana Department of Education further exemplifies this method for K-12.  This project includes examples of tiered lessons for mathematics, science, and language arts differentiated by readiness, interest, and learning styles.  For more on eight steps for creating a tiered lesson, read Tiered Lessons: One Way To Differentiate Mathematics Instruction by Rebecca L. Pierce and Cheryll M. Adams.

 

Table 1: Teaching Strategies for Hypothesized Primary Skill Deficits in Math
Numerical Skills Deficits
Use scaffolding for building computational and conceptual skills, frequent teacher questioning, and student response. Increase exposure to basal math curriculums.	Individualize instruction using small groups. Here teachers can simplify language and instructions for those who need it. Similar students can also make and/or add to their own math dictionaries. These might contain the terms reviewed prior to new lessons for further practice and reinforcement.
Use daily 2-3 minute long timed tests to review and monitor progress. The immediate feedback helps teachers to adjust instruction for the day.	Rather than using traditional worksheets, consider drill and practice using board games, Math Jeopardy, puzzles, dot-to-dots, color by numbers where numbers are obtained by computing math problems.
Visual Spatial Deficits
Provide students with copies of problems to compute that are already written out for them.	Use visual aids so that students receive both auditory and visual reinforcement of concepts.
Scaffold learning of place value and lining up numbers by having students use grid paper, or turning lined paper sideways to create columns.	Use manipulatives.
Underdeveloped Higher Order Cognitive Skills
Review terms before new instruction or testing.	Teach students to highlight key words in problem solving.
Write or illustrate critical information and directions to focus attention on key concepts.	When working on a series of problems, it is helpful to call attention to changes in operations. Students might first highlight each operation in a different color to call attention to those changes.
Use the computer for drill and practice; monitor student performance; preview software for its appropriateness and level of difficulty for the student/	MLD students might need extra time to process information and respond. Rather than just call on them, a private agreed-upon signal between the teacher and student might build confidence. For example, a raised hand with closed fist might mean “I’m thinking and want to participate.” When the hand opens, the teacher would know to call on the student.
Social Cognition—Beliefs about abilities to do math
Make math relevant to real-life. When developing word problems, use familiar names and places.	Use cooperative learning and encourage students to take on different roles within the group.
Have students create their own problems individually or within a group.
Adapted from: Augustyniak, K., Murphy, J., & Phillips, D. (2005). Psychological perspectives in assessing mathematics learning needs. Journal of Instructional Psychology, 32(4), 277-286.

 

As one examines the table 1 above, you observe that using drill and practice and related software is included among those recommendations.  The National Mathematics Advisory Panel (2008) also "recommends that high-quality computer-assisted instruction (CAI) drill and practice, implemented with fidelity, be considered as a useful tool in developing students’ automaticity (i.e., fast, accurate, and effortless performance on computation), freeing working memory so that attention can be directed to the more complicated aspects of complex tasks" (p. 51). Further:

Research has demonstrated that tutorials (i.e., CAI programs, often combined with drill and practice) that are well designed and implemented can has a positive impact on mathematics performance, particularly at the middle and high school levels. CAI tutorials have been used effectively to introduce and teach new subject-matter content. Research suggests that tutorials that are designed to help specific populations meet specific educational goals have a positive impact. However, these studies also suggest several important caveats. Care must be taken to ensure that there is evidence that the software to be used has been shown to increase learning in the specific domain and with students who are similar to those who will use the software. Educators should critically inspect individual software packages and the studies that evaluate them. Furthermore, the requisite support conditions to use the software effectively (sufficient hardware and software; technical support; adequate professional development, planning, and curriculum integration) should be in place, especially in large-scale implementations, to achieve optimal results. (p. 51)

Gersten et al. (2009) made additional recommendations for elementary and middle school students who receive math interventions:

• Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.
• Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
• Interventions should include instruction on solving word problems that is based on common underlying structures.
• Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
• Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
• Monitor the progress of students receiving supplemental instruction and other students who are at risk.
• Include motivational strategies in interventions. (p. 6)

Virginia Department of Education: At-A-Glance" Algebra Readiness Initiative Checklist pertains to their Algebra Readiness Initiative; however, the criteria are appropriate for an intervention program at any level. Look for:

1. Low teacher/student ratio (10 or fewer).

2. The teacher should be actively engaged with students 90% of the time.

3. There are few, if any, interruptions.

4. The teacher uses differentiated instruction.

5. The teacher uses research-based strategies: manipulatives, modeling, scaffolding, questioning techniques, student reflection and writing, direct instruction on how to use calculators, math taught in context, and so on.

6. There are frequent assessments, which are then used to adjust instruction accordingly.

7. Students' work matches their areas of weakness.

8. Highly-structured classroom management: students know where to find materials, directions, and know classroom procedures.

In terms of those frequent assessments, teachers and their school districts might consider adaptive testing software, which can be easily administered several times a school year to determine individual needs and to give a measure of class growth on state standards benchmarks.  Among such products noted by the National Center on Response to Intervention are AIMSweb from Pearson Education, STAR Math and Accelerated Math from Renaissance Learning, and Yearly Progress Pro from CTB/McGraw-Hill.

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Responsiveness to Intervention Resources

Responsiveness to Intervention is typically a three- tiered method for early identification of students who may be at risk for learning disabilities/difficulties. The following will help you to learn more on this process and also provide you with some companies offering RTI software.

American Speech-Language-Hearing Association answers the question: What is Responsiveness-to-Intervention?: http://www.asha.org/Publications/leader/2005/050322/050322b.htm#2

ASCD: http://www.ascd.org/research-a-topic/response-to-intervention-resources.aspx Response to Intervention resources are listed within each other three tiers.

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009, April). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/pdf/practiceguides/rti_math_pg_042109.pdf [Note: This best practice guide from the IES is also available via http://ies.ed.gov/ncee/wwc/publications/practiceguides/  The authors' "goal in this practice guide is to provide suggestions for assessing students’ mathematics abilities and implementing mathematics interventions within an RtI framework, in a way that reflects the best evidence on effective practices in mathematics interventions" (p. 4).  Eight recommendations are included, along with a list of 12 examples of math problems to illustrate concepts.]

Curriculum Associates, Inc.: http://www.curriculumassociates.com/ includes an extensive list of Response to Intervention resources in reading and mathematics.

Intervention Central: http://www.interventioncentral.org/index.php#ideas "is committed to the goal of making quality Response-to-Intervention resources available to educators at no cost. The site was created in 2000 by Jim Wright, a school psychologist and school administrator from Central New York. Visit to check out newly posted academic and behavioral intervention strategies, download publications on effective teaching practices, and use tools that streamline classroom assessment and intervention."

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

National Center on Response to Intervention: http://www.rti4success.org/

PCI Education: http://www.pcieducation.com/lift/ has a line of intensive intervention programs for reading, writing, and math (LIFT) targeted to middle school and high school students. LIFT is designed for struggling students who are performing two or more years below grade level, have learning differences, or are English language learners.

Response to Intervention: http://www.esu1.org/SPED/RtI-interventionmath.html from the Educational Service Unit #1 in Nebraska contains specific math interventions ranging from least intensive to moderate/intensive interventions.

RTI Action Network: http://www.rtinetwork.org/ contains resources for response to intervention for preK, K-5, middle school, high school, parents and families.  Of particular interest related to mathematics are the podcast RTI and Improved Math Achievement featuring David Allsopp and the essay RTI and Math Instruction by Dr. Amanda VanDerHeyden.

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

 

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See other Math Methodology pages:

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