For all students to develop mathematical proficiency, schools should devote a substantial and regular amount of time to mathematics instruction. As an overall guideline, an hour each school day should be devoted to mathematics instruction from kindergarten through eighth grade. This time should be apportioned so that all the strands of mathematical proficiency receive adequate attention. Does working in small groups help students to develop math proficiency?
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Students in groups—sometimes called cooperative groups—of three, four, or five can work on a math task together and thereby increase their proficiency. But if the task does not allow each student to contribute, and if students are not sure what they are supposed to do, precious learning time is wasted. When used appropriately, small groups can both increase achievement and promote positive social interactions among students.
Students placed in lower groups receive fewer opportunities to develop all the strands of math proficiency. Their classes typically have a less demanding curriculum, less capable teachers, and few or no strong peer models. As a result, achievement gaps grow rather than diminish, and socioeconomic, racial, and ethnic disparities widen.
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These observations suggest that the wiser course in elementary and middle school grades is not to group students into classes by ability or achievement. Accelerated classes are not necessary to help high-achieving students learn more; this can be accomplished within mixed groups.
Some countries with impressive math achievement scores do not separate high achievers from low achievers. Effective teaching methods can help all students in mixed-ability classes to develop proficiency, and teachers can be supported to acquire and use these methods. Textbooks and other instructional materials in the United States need to support the learning of all five strands of mathematical proficiency. They should develop the core content of mathematics in a focused way and with continuity within and across grades.
More time should be spent developing fewer topics in each grade, as is done in many countries where mathematics achievement is strong. Instructional materials should incorporate activities and strategies that assist teachers in helping all students become proficient in mathematics, including students low in socioeconomic status, English-language learners, special education students, and students with special interests or talents in mathematics.
In recent years, many states and districts have mandated a variety of assessments to measure the mathematics performance of students. Some of these assessments have serious consequences for students, teachers, and schools, such as determining whether a student will graduate or whether a school will avoid state sanctions. In some cases, the assessment aims at what students have mastered, and all students should be able to demonstrate mastery.
In many other cases, however, the assessment ranks students, schools, or districts; this means that half of those being assessed are necessarily below par.
Such assessments typically do not provide information that can be used to improve instruction. The goal of mathematical proficiency for all requires as one of its first tasks a rethinking of what assessments are measuring. Large-scale comparisons are seldom aligned with the curriculum and often focus on only one or two aspects of.
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Why are students playing with blocks, sticks, and beans in math classes? Reflecting on the use of physical objects—such as blocks, sticks, and beans—can help students to develop understanding by linking their informal knowledge and experiences to school math. Physical materials should not be used simply as tools to calculate answers. Students need to be able to move from using physical objects to finding solutions numerically.
Teachers must provide opportunities for students to make explicit connections between activities with the objects and the math concepts and procedures that the objects are intended to help teach. The math is in the connections, not the objects. Algebra is the gateway to higher math. Proficiency in algebra helps students solidify their proficiency in numbers and integrate their knowledge of math. Algebra provides the concepts and language to move from individual numerical calculations to general relationships.
The study of algebra, however, need not begin with a formal course in the subject.
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The elementary and middle school curriculum can support the development of algebraic ways of thinking and thereby avoid the difficulties many students now experience in making the transition from arithmetic to algebra. The basic ideas of algebra can be learned by the end of middle school if they are taught in ways that draw on and develop all strands of math proficiency.
All assessments need to support the development of mathematical proficiency. They need to measure the five strands of proficiency and their integration. By doing so, they will provide opportunities for students to become proficient rather than taking time away from this goal. This group could recommend how such programs might be modified to promote the goal of proficiency. Like any complex task, effective mathematics teaching must be learned.
Teachers need a special kind of knowledge. To teach mathematics well, they must themselves be proficient in mathematics, at a much deeper level than their students. They also must understand how students develop mathematical proficiency, and they must have a repertoire of teaching practices that can promote proficiency.
Unfortunately, very few teachers have the specialized knowledge needed to teach mathematics in the ways envisioned in this report. Like learning in any profession, learning to teach for mathematical proficiency is a career-long challenge. Acquiring this knowledge and learning how to use it effectively in the classroom will take not only time but resources.
It is not reasonable in the short term to expect all teachers to acquire the knowledge they need to teach for mathematical proficiency.
ICMI Study Volumes (NISS) | International Mathematical Union (IMU)
To further promote effective teaching and learning, mathematics specialists—teachers who have special training and interest in mathematics— should be available in every elementary school. The undergraduate years of teacher training must provide significant and continuing opportunities, linked closely to classroom practice, for prospective teachers to develop the knowledge needed to teach for mathematical proficiency.
People intending to be teachers must continue developing their own mathematical proficiency and learn how to use that proficiency to guide discussions, modify problems, and make decisions about what to pursue in class. They cannot wait until they enter the profession to learn to teach effectively. New tests may be needed, and old tests may need to be changed. Most current math tests, whether standardized achievement tests or classroom quizzes, address only a fraction of math proficiency—usually just the computing strand and simple parts of the understanding and applying strands.
Teachers need tests and other assessment procedures that let them gauge how far students have come along in all five proficiency strands. Furthermore, instead of taking time away from learning, these instruments should allow students simultaneously to build and exhibit their proficiency.
Many people have worked very hard in recent years to change the ways in which mathematics is taught and learned. However, the problem requires greater and different efforts than those made so far.
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Complex systems like school mathematics are what social scientists call overdetermined. A large number of pressures exert forces on these systems, making them remarkably stable and resistant to change. T o argue for the rethinking of curriculum and teaching, This article argues for rethinking curriculum and teaching as two At a time when political interest in mathematics education is at its … download Rethinking The Mathematics Curriculum in pdf read Rethinking The Mathematics Curriculum ebook download B.
In this unique collection, more than 30 articles show how to weave social justice issues throughout the mathematics curriculum, as well as how to integrate mathematics into other curricular areas. Rethinking Mathematics offers teaching ideas, lesson plans, and reflections by practitioners and Rethinking and connecting