Characteristics of Students' Mathematics Learning Problems
About 23 percent of fourth-grade students and 32 percent of eighth-grade students scored below basic on the 2003 National Assessment of Educational Progress Mathematics Assessments (National Center for Educational Statistics, 2003). Although scores have improved since 1996, when 37 percent of fourth graders and 39 percent of eighth graders scored below basic, between one-fourth and one-third of students could not meet grade-level standards in mathematics on the 2003 assessments.
Those students with mathematics disabilities identified under the Individuals with Disabilities Education Act (IDEA, 2004) typically are classified under the categories of specific learning disabilities, mild to moderate cognitive disabilities, emotional disabilities, language impairments, and other health impairments (such as ADHD), where the disability has been identified as having a significant impact on learning. Students with these identified disabilities may be served anywhere on the continuum of placement options, but most are served in regular classrooms or in limited but intensive pull-out programs.
Other students who perform below standards on state or district mathematics tests sometimes have individual intervention plans depending on district or state policies. The No Child Left Behind Act (NCLB, 2001) made no provisions for meeting the needs of individual students who do not meet standards, other than the option to transfer schools if their overall school does not meet criteria for adequate yearly progress. Students from low-income families may qualify for supplemental after-school or Saturday services under the NCLB provisions for Title I schools. Some of these students are served in remedial mathematics programs funded through a variety of sources and others are served primarily in the regular classroom.
Students who have problems learning mathematics or fail to meet grade-level standards are usually identified between third and fifth grade, much later than those identified for reading problems, and are referred for special education services or other remedial programs. Special education and remedial teachers find that these students’ basic concept and skill development generally one to two years behind their peers upon identification. Even with interventions, the achievement gap for many students with identified disabilities continues to widen. These math achievement problems are usually due to a combination of teaching and student factors including language, cognitive, metacognitive, motor, social and emotional factors, habits of learning, and previous experiences.
- Language problems. Most students with mild disabilities have primary or secondary language problems. A language disorder, according to the American Speech-Language-Hearing Association, is “impaired comprehension and/or use of a spoken, written, and/or other symbol systems” (ASHA, 1993, p. 40). The disorder may involve form, content, or function of the language. Even if a student does not have an identified language disorder, he or she may exhibit language deficiencies related to his or her disability.
In mathematics class, language problems are evident when students have trouble using symbols of math, expressing math concepts to others, and listening to mathematics explanations. Problems also appear with reading or writing word problems and writing and expressing math “sentences.” Language can provide the bridge between the concrete representations of math and the more abstract and symbolic forms. As students advance in math learning, they also use language to think—they manipulate concepts and ideas through language (oral or inner) without having to rely on concrete materials.
Unfortunately, some students have few opportunities to talk about math. Teachers who limit lessons to lecture, demonstration, and worksheets are limiting their students’ language development and related math progress. Students should be responding frequently and discussing math problems and concepts with each other and the teacher. Students whose parents continue the dialog at home will have additional benefits.
- Cognitive factors. Most students with mild to moderate disabilities have cognitive factors that impede learning. These may be perceptual, memory, attention, or reasoning factors. Perception involves taking in information from the environment and processing that information for storage or use. It’s not just seeing the symbols for numbers but seeing and copying them. It’s not just hearing the oral number sequence but hearing it and continuing the sequence. It’s not the seeing or hearing alone, it’s the discrimination and interpretation of visual and auditory input. Perceptual problems show up with difficulties keeping place on a worksheet or within a column of numbers, differentiating numbers or symbols that are close in form, copying shapes or symbols, following directions with algorithms or graphs, recognizing patterns or sequences, and understanding oral directions or drills.
Memory problems can affect long-term, short-term, or active, working aspects. Memory capacities, in the information-processing model, serve to store and retrieve information needed to interact with the environment. Long-term memory is the background knowledge and prior experience to which new information is added in various forms. Inefficient organization and integration of information going to long-term memory will cause problems with retrieval later. Short-term memory is the briefest register of new information, most of which is filtered and discarded. Active working memory is where new information is organized, filtered again, and the destination for previous learning to be retrieved for active use during a learning or problem-solving situation. The student’s processing depth, organization, attention, and integrative abilities affect how well these capacities work (Swanson & Sáez, 2003).
Attention is a regulator for learning. It is concerned with alertness, mental effort, shifting attention, focus, and self-regulation. Students with attention problems either lack organized executive processing abilities or fail to apply efficient strategies and controls. In the mathematics classroom, attention problems are evidenced by difficulty sustaining focused attention during a lesson or while solving a problem, inability to filter out irrelevant parts of problems, and failure to complete work.
Reasoning is a higher-order cognitive ability that is essential for math success. It involves various forms of conceptualization, deductive and inductive thought, working in the abstract, and solving problems. Students who have trouble with tasks involving reasoning have trouble seeing patterns and relationships, use faulty logic, typically accept things at face value rather than questioning and analyzing, and can’t explain the “why” behind a math process.
- Metacognitive factors. Metacognition is an awareness of the skills, strategies, and resources that are needed to perform a task and the ability to use self-regulatory mechanisms, including adjustments, to complete the task (Borkowski & Burke, 1996). Sometimes called “thinking about one’s own thinking,” metacognition is the process involving being aware of and monitoring the use of executive and cognitive strategies. Students with metacognition problems have trouble selecting and using effective learning strategies. They don’t monitor their own use of strategies and have difficulty with generalization across time and setting. For example, a student may have trouble deciding how to solve a non-routine word problem. Even if the student attempts the problem, he doesn’t monitor the process or results. (Does this make sense? How can I change what I attempted?) Further, he can’t draw on experiences with similar problems because they don’t appear similar in his conceptualization of the problem.
- Motor factors. Motor problems with written work are most evident in younger students but even adolescents with no physical disabilities can struggle with number and symbol formation. Motor skills, like perceptual ones, involve more than one process. They may involve memory of the symbol along with its actual formation (visual and motor memories). They may involve visual perception and transfer (copying). Or they may involve integration of fine muscles with task demands. Indicators of motor problems are highly visible: poorly formed symbols, little control of spacing, excessive time for a task, and avoidance of written work.
- Social and emotional factors. Sometimes overlooked in the academic realm, social and emotional factors can cause as many learning problems as cognitive ones. The range of these factors is as diverse as the students served. Some students have trouble with peer or adult relationships, causing problems in cooperative learning settings or seeking assistance. Others have self-concept and self-esteem problems that lower motivation, task persistence, and effort. Impulsive students make careless errors and don’t take the time to understand the deeper concepts and connections. Students with extreme anxiety—either toward mathematics or school in general—tend to avoid the source of their anxiety or perform at much lower levels than their abilities.
- Habits of learning. A combination of environmental, cognitive, social, and emotional factors, habits of learning are formed from an early age but certainly can be modified throughout the lifespan. “Habits of learning” refers to how individuals view and participate in learning, their self-discipline and self-motivation, goal setting, engagement in learning activities, and acceptance of challenges. Habits that could interfere with math learning include avoidance, learned helplessness, impulsivity, little curiosity, poor assignment completion, disinterest, and working for the “right answer” rather than understanding. Even students with high mathematics abilities have habits, such as the drive for perfection, that can interfere with strong concept development and flexible problem solving.
- Previous experiences. A student’s prior knowledge and previous experiences with mathematics are the best predictors of future success. Many of these experiences have been influenced by the factors described above. However, previous instructional experiences also can have a significant impact on achievement. If previous teachers did not explain concepts well, use effective teaching methods, or allow time for mastery and success, students’ mathematics learning will be affected. If the curriculum and materials used weren’t aligned with math standards, learning might be superficial or limited. And if the student wasn’t able to develop the deep concept understanding that comes from good teaching and sound curriculum, his or her math achievement will suffer.
Students who have been served in separate special education settings for part or all of the school day may be affected by factors such as instruction from teachers without specific mathematics training, being “pulled out” of the regular classroom during critical instructional time, or having less than adequate time devoted to mathematics instruction. Students who changed schools frequently may have gaps in learning. Students with few opportunities for learning at home due to poverty or parent background may demonstrate significant delays. Those students whose primary language is not English may experience difficulties with the language demands of mathematics instruction.
Zachary, Kristy, David, and Jessica are young children interested in the world around them. All of these children were involved in excellent preschool programs that offered rich environments with well-trained teachers. One of these children is going to be evaluated for a specific mathematics disability in second grade after spending a frustrating year in first grade unable to add and subtract, write numbers, or draw diagrams like the other first graders. This student’s reading ability is progressing as expected but mathematics has become a source of failure, confusion, and anxiety. Although this child had a strong foundation in preschool, the formal pencil and paper tasks in math class have not been effective for continued progress in concept development. The student evaluation team must plan assessments that will provide information on specific learning problems and plan for more effective interventions.
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