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Recommendations for Teaching Mathematics Increase Decrease Teaching Practices Use of manipulative materials Cooperative group work Discussion of mathematics Questioning and making conjectures Justification of thinking Writing about mathematics Problem-solving approach to instruction Content integration Use of calculators and computers Being a facilitator of learning Assessing learning as an integral part of instruction Rote memorization of rules and formulas Single answers and single methods to find answers Use of drill worksheets Repetitive written practice Teaching by telling Teaching computation out of context Stressing memorization Testing for grades only Being the dispenser of knowledge Rote practice Mathematics as Problem Solving Word problems with a variety of structures and solution paths Everyday problems and applications Problem-solving strategies Open-ended problems and extended problem solving projects Investigating and formulating questions from problem situations Use of cue words to determine operation to be used Practicing routine, one-step problems Practicing problems categorized by types Mathematics as Communication Discussing mathematics Reading mathematics Writing mathematics Listening to mathematical ideas Doing fill-in-the blank worksheets Answering questions that need only yes or no responses Answering questions that need only numerical responses Mathematics as Reasoning Drawing logical conclusions Justifying answers and solution processes Reasoning inductively and deductively Relying on authorities (teacher, answer key) Mathematical Connections Connecting mathematics to other subjects and to the real world Connecting topics within mathematics Applying mathematics Learning isolated topics Developing skills out of context Numbers/Operations/Computation Developing number and operation sense Understanding the meaning of key concepts such as: place value, fractions, decimals, ratios, proportions, and percents Various estimation strategies Thinking strategies for basic facts Using calculators for complex calculation Early use of symbolic notation Complex and tedious paper and pencil computations Memorizing rules and procedures without understanding Geometry/Measurement Developing spatial sense Actual measuring and the concepts related to units of measure Using geometry in problem solving Memorizing facts and relationships Memorizing equivalencies between units of measure Memorizing geometric formulas Statistics/Probability Collection and organization of data Using statistical methods to describe, analyze, evaluate, and make decisions formulas Memorizing Patterns/Functions/Algebra Pattern recognition and description Identifying and using functional relationships Developing and using tables, graphs, and rules to describe situations Using variables to express relationships Manipulating symbols Memorizing procedures and drilling Evaluation Having assessment be an integral part of teaching Focusing on a broad range of mathematical tasks and taking a holistic view of mathematics Developing problem situations that require applications of a number of mathematical ideas Using multiple assessment techniques, including written, oral, and demonstration formats Having assessment be simply counting correct answers on tests for the sole purpose of assigning grades Focusing on a large number of specific and isolated skills Using exercises or word problems requiring only one or two skills Using only written tests "Evidence from many sources shows that the least effective mode for mathematics learning is the one that prevails in most of America's classrooms: lecturing and listening . . . Students simply do not retain for long what they learn by imitiation from lectures, worksheets, or routine homework . . . Research on learning shows that students actually construct their own understanding based on new experiences . . . Mathematics becomes useful to a student only when it has been developed through a personal intellectual engagement that creates new understanding. Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn." --Everybody Counts: A Report to the Nation on the Future of Mathematics Education. National Research Council. Washington, D.C.: National Academy Press, 1989. RECOMMENDATIONS FROM: National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Commission on Standards for School Mathematics, Reston, VA. SUGGESTED READINGS: Cooney, Thomas J. (ed.). (1990) Teaching and Learning Mathematics in the 1990's. Reston, VA: National Council of Teachers of Mathematics. Post, Thomas R. (ed.). (1988). Teaching Mathematics in Grades K-8: Research Based Methods. Boston: Allyn & Bacon. Trafton, Paul R. (ed.). (1989). New Directions for Elementary School Mathematics. Reston, VA: National Council of Teachers of Mathematics. [email protected] Copyright © North Central Regional Educational Laboratory. All rights reserved. Disclaimer and copyright information.