A major component of the child-centered, systematic teaching approach is content. The discipline of mathematics presents many challenges to dissimilar learners. Mathematics has often been termed the “gatekeeper” of success or failure for high school graduation and career success (National Research Council [NRC], 1989). It is essential that “mathematics . . . become a pump rather than filter in the pipeline of American education” (NRC, 1989, p. 7). A lack of sufficient mathematical skill and understanding affects one’s ability to make critically important educational, life, and career decisions.
Students fall below their expected level of mathematics achievement for a variety of reasons. When asked why they were not as successful in learning mathematics, many people reply that they “never understood math,” or “never liked it because it was too abstract and did not relate to them.” These reasons and others can be categorized, in general, as environmental or personal, individualized factors.
Mathematics instruction must provide many opportunities for concept building, relevant challenging questions, problem solving, reasoning, and connections within the curriculum and real-world situations. Students who are taught in a way that relies too heavily on rote memorization isolated from meaning have difficulty recognizing and retaining math concepts and generalizations.
Spiraling the curriculum provides opportunities for learners to deal with content developmentally over time. Concepts can be built upon and related to previous learning throughout the curriculum as students become more proficient and experienced in mathematics. However, it is critical that the same content not be taught year after year, in almost the same manner of delivery. Students who do not “get it” the first time are not likely to “get it” the next several times it is taught in the usual manner. Moreover, underachieving students are frequently assigned repetitious and uninteresting skill-and-drill work each year in order to teach them “the basics.” This type of work often represents a narrow view of mathematical foundations and a low level of expectation of students’ abilities. It limits opportunities to reason and problem solve.
The Gap Between Learner and Subject Matter
When the mathematics content being taught is unconnected to students’ ability level and/or experiences, serious achievement gaps result. This situation may occur if students are absent frequently or transfer to another school during the academic year. A student may find the mathematics curriculum to be more advanced or paced differently than what was being taught in the previous school. Without intervention strategies, students could remain “lost” for the duration of their education.
Too few life experiences, such as trips to neighborhood stores or opportunities to communicate with others about numbers through practical life examples, can make math irrelevant for students. Gaps exist, therefore, not only in the curriculum but between the learner and perceived usefulness of the subject matter.
Personal or Individualized Factors
Locus of Control
Some students believe that their mathematical achievement is mainly attributable to factors beyond their control, such as luck. These students think that if they scored well on a mathematics assignment, they did so only because the content happened to be easy. These students do not attribute their success to understanding or hard work. Their locus is external because they believe achievement is due to factors beyond their control and do not acknowledge that diligence and a positive attitude play a significant role in accomplishment. Students might also believe that failure is related to either the lack of innate mathematical inability or level of intelligence. They view their achievement as accidental and poor progress as inevitable. In doing so, they limit their capacity to study and move ahead (Beck, 2000; Phillips & Gully, 1997).
Some students lack well-developed mental strategies for remembering how to complete algorithmic procedures and combinations of basic facts. However, strategies to improve capacities for remembering facts, formulas, or procedures can be taught. Repetition games such as calling out fact combinations and having students solve them and then repeat those that were called before their turn can help. For example, the teacher would call out “3 X 5 = 15 and a student would respond with “15.” That student would then ask a number question such as “7 - 5" of the group. The responder would reply, “3 X 5 = 15 and 7- 5 = 2.” The game continues as each player calls out a new fact and each responder answers with all the previous combinations and the new answer. Students’ ability to organize their thinking and use it to recall data will affect success throughout the curriculum.
Students may be mentally distracted and have difficulty focusing on multistep problems and procedures. Dealing with long-term projects or a number of variables or pieces of information at one time can interfere with achievement. Effective teachers should use attention getters such as drawings and learning aids. Students who work in pairs can help each other stay on task.
Understanding the Language of Mathematics
Students are confused by words that also have special mathematical meaning, such as “volume,” “yard,” “power,” and “area.” Lack of understanding of mathematical terms such as “divisor,” “factor,” “multiple,” and “denominator” seriously hampers students’ abilities to focus on and understand terms and operations for algorithms and problem solving. Memorizing these terms without meaning and context is not productive.
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