Understanding Math Learning Problems
What Are They?
Students who experience significant problems learning and applying mathematics manifest their math learning problems in a variety of ways. Research indicates that there are a number of reasons these students experience difficulty learning mathematics (Mercer, Jordan, & Miller, 1996; Mercer, Lane, Jordan, Allsopp, & Eisele, 1996; Mercer & Mercer, 1998; Miller & Mercer, 1997.) The following list includes these researchbased math disability characteristics.
Characteristics of Students Who Have Learning Problems
Learned Helplessness  Students who experience continuous failure in math expect to fail. Their lack of confidence compels them to rely on assistance from others to complete tasks such as worksheets. Assistance that only helps the student "get through" the current set of problems or tasks and does not include reteaching the concept/skill, only reinforces the student's belief that he cannot learn math.
Passive Learners  Students who have learning problems often are not "active" learners. They do not actively make connections between what they already know and what they are presently learning. When presented with a problemsolving situation, they do not employ strategies or activate prior knowledge to solve the problem. For example, students may learn that 8 x 4 = 32, but when presented with 8 x 5 = ___, they do not actively connect the process of multiplication to that of repeated addition. They do not think to add eight more to thirtytwo in order to solve the problem. Students that have learning problems often believe that students who are successful in math just know the answers. They do not understand that students who are successful in math are good at employing strategies to solve problems.
Memory Problems  Memory deficits play a significant role in these students' math learning problems. Memory problems are most evident when students demonstrate difficulty remembering their basic addition, subtraction, multiplication, & division facts. Memory deficits also play a significant role when students are solving multistep problems and when problemsolving situations require the use of particular problem solving strategies. A common misconception about the memory problems of these students is that it is an information storage problem; that somehow, these students just never get it stored properly. This belief probably arises because one day the student can do a math task but then the next day they can't. Teachers then reteach the skill only to have the same experience repeated. In contrast to an information storage problem, these memory deficits are often a result of an information retrieval problem. For these students, instruction should include teaching students strategies for accessing and retrieving the information they have stored.
Attention Problems  Math requires a great deal of attention, particularly when multiple steps are involved in the problem solving process. During instruction, students who have attention problems often "miss" important pieces of information. Without these important pieces of information, students have difficulty trying to implement the problem solving process they have just learned. For example, when learning long division, students may miss the "subtract" step in the "divide, multiply, subtract, bring down" long division process. Without subtracting in the proper place, the student will be unable to solve long division problems accurately. Additionally, these students may be unable to focus on the important features that make a mathematical concept distinct. For example, when teaching geometric shapes, these students may attend to features not relevant to identifying shapes. Instead of counting the number of sides to distinguish triangles from rectangles, the student may focus on size or color. Using visual, auditory, tactile (touch), and kinesthetic (movement) cues to highlight the relevant features of a concept is helpful for these students.
Cognitive/Metacognitive Thinking Deficits  Metacognition has to do with students' ability to monitor their learning: 1.) Evaluating whether they are learning; 2.) Employing strategies when needed; 3.) Knowing whether a strategy is successful; and, 4.) Making changes when needed. These are essential skills for any problem solving situation. Because math is problem solving, students who are not metacognitively adept will have great difficulty being successful with mathematics. These students need to be explicitly taught how to be metacognitive learners. Teachers who model this process, who teach students problem solving strategies, who reinforce students' use of these strategies, and who teach students to organize themselves so they can access strategies, will help students who have metacognitive deficits become metacognitive learners.
Low Level of Academic Achievement  Students who experience math failure often lack basic math skills. Because it takes students with math disabilities a longer time to process visual and auditory information than typical learners, they often never have enough time or opportunity to master the foundational concepts/skills that make learning higher level mathematics possible. Providing these students many opportunities to respond to math tasks and providing these practice opportunities in a variety of ways is essential if these students are to ever master the math concepts/skills we teach. Additionally, teachers need to plan periodic review and practice of concepts/skills that students have previously mastered.
Math Anxiety  These students often approach math with trepidation. Because math is difficult for them, "math time" is often an anxietyridden experience. The best cure for math anxiety is success. Providing success starts first with the teacher. By understanding why students are having the difficulties they are having, we are less inclined to place "blame" on the students for their lack of math success. These students already feel they are not capable. The attitude with which we approach these students can be a crucial first step in rectifying the math problems they are having. Providing these students with nonthreatening, riskfree opportunities to learn and practice math skills is greatly encouraged. Celebrating both small and great advances is also important. Last, if we provide instruction that is effective for these students, we will help them learn math, thereby helping them to experience the success they deserve.
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Math Instruction Issues That Impact Students Who Have Math Learning Problems
Although it is very important to understand the learning characteristics of students with math learning problems, it is also important to understand how math instruction/curriculum issues negatively affect these students (Mercer, Jordan, & Miller, 1996; Mercer, Lane, Jordan, Allsopp, & Eisele, 1996; Mercer & Mercer, 1998; Miller & Mercer, 1997). The following list includes these instruction/curriculum issues as well as how they impact the students described above.
Spiraling Curriculum  Within a spiraling curriculum, students are exposed to a number of important math concepts the first year. The next year, students return to those math concepts, expanding on the foundation established the year before. This cycle continues with each successive year. While the purpose of this approach is logical and may be appropriate for students who are average to above average achievers, the spiraling curriculum can be a significant impediment for students who have math learning problems. The primary problem for these students is the limited time that is devoted to each concept. Students who have math learning problems are never able to truly master the concept/skill being taught. For these students, "exposure" to foundational skills is not enough. Without an appropriate number of practice opportunities, these students will only partially acquire the skill. When the concept/skill is revisited the next year, the student is at a great disadvantage because the foundation they are expected to have is incomplete. After several years, the student has not only "not mastered" basic skills, but has also not been able to make the important connections between those basic skills and the higher level math skills being taught as the students moves through the elementary, middle, & secondary grades.
Teaching Understanding/Algorithm Driven Instruction  Although the National Council on Teaching Mathematics (NCTM) strongly encourages teaching mathematical understanding and reasoning, the reality for students with math learning problems is that they spend most of their math time learning and practicing computation procedures. Because of their memory problems, attention problems, and metacognitive deficits, these students have difficulty accurately performing multistep computations. Therefore, instructional emphasis for these students is often placed on procedural accuracy rather than on conceptual understanding. This emphasis on algorithm (procedure) proficiency supersedes emphasis on conceptual understanding. An example of this is the process of multiplication. Students who only are taught the procedure of multiplication through drill and practice often do not really understand what the process represents. For example, consider the relationship of the following two multiplication problems: 2 x 4 = 8 and ½ x ¼ = 1/8. When students are asked why the answer in the first problem is greater than its multipliers but the answer to the second problem is less than each of its multipliers, the students are unable to answer why. They have never really understood that the multiplication sign really means "of" and that "2 x 4 = 8" means two groups of fours objects, while " ½ x ¼ = 1/8" means onehalf of onefourth. Teaching understanding of the math processes as well as teaching the algorithms (procedures) for computing solutions is critical for students with math learning problems.
Teaching to Mastery  As described under "Spiraling Curriculum," students with learning problems need many opportunities to respond to specific math tasks in order to master them. Teaching to mastery requires that both the teacher and the student monitor the student's learning progress on a daily basis. Mastery is indicated only when the student is able to perform a math task at 100% accuracy for at least three consecutive days. In situations where student progress is assessed only by unit tests, it is very difficult to determine whether a student has really mastered the skills covered in that unit. Even if the student performs well on the unit test, a teacher cannot be certain that the student actually has reached mastery. Because of the learning characteristics common for these students, it is possible that the student would not score as well if given the same test the next day. Mastery can only be inferred when the student demonstrates consistent mastery performance over time. Such continuous assessment is rare in math classrooms. When evaluation of student progress occurs only by unit tests and the students with learning problems do not perform well, the teacher is left with a difficult dilemma. Does the teacher take additional class time to reteach the skill, thereby falling behind the mandated curriculum's instructional pace? Conversely, does the teacher instead move on to new material, knowing that these students have not mastered the preceding skills, making it less likely the student will have the prerequisite skills to learn the new information? This nowin situation can be avoided if continuous daily assessment is implemented for these students. It is easier and more time efficient to reteach an individual math skill the same day of initial instruction, or on the following day. Attempting to reteach multiple math skills many days after initial instruction is much more difficult and time consuming. Due to the hierarchical nature of mathematics, if students do not master prerequisite skills, it is likely that they will not master future skills.
Reforms That Are Cyclical in Nature  The cyclical nature of mathematics curriculum/instruction reforms creates changing instructional practices that confuse students with learning problems. Like reforms for reading instruction, math instruction swings from primarily skillsbased emphasis to primarily meaningbased emphasis dependent on the philosophical and political trends of the day. Most students experience at least one of these shifts as they move through grades K to 12. While students who are average to above average achievers are able to manage these changes in instruction, students who have learning problems do not adjust well to such change.
Application of Effective Teaching Practices for Students who have Learning Problems  Research has identified math instructional practices that are effective for students who have learning problems, but these instructional practices are not always implemented in our schools. These instructional practices are described and modeled in this CDROM program. Descriptions also include how the particular characteristics of each instructional strategy complement the learning characteristics of students with learning problems. Guidance is also provided which will help you implement these instructional practices in an organized and systematic way.
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How Does This Information Help Me?
Teachers who understand the learning needs of their students are more empowered to provide the kind of instruction their students need. Knowing why a student is struggling to learn math provides a basis for understanding why particular instructional strategies/approaches are effective for him/her. Each of the instructional strategies included in this program has unique characteristics that positively impact the learning characteristics of students who have math learning problems. As you learn about each strategy, you are encouraged to refer often to the learner characteristics described in this section. While reading about each instructional strategy and then watching a teacher model the strategy, note how the specific instructional characteristics of the strategy complement or "match" the learning characteristics of students with math learning problems. The text descriptions for each instructional strategy found in this manual clarify these relationships. The elaborated video clips in the CDROM also emphasize how the specific characteristics for each instructional strategy positively impact students who have math learning problems.
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