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54 Reshaping School Mathematics Chambers, Donald L. A Guide to Curriculum Planning in Mathematics. Madison: Wisconsin Department of Public Instruction, 1 986. Charles, Randall 1., and Silver, Edward A. (Eds.~. The Teaching and Assessing of Mathematical Problem Solving. Reston, Va.: Natior~al Council of Teachers of Mathematics, 1988. Clement, J. "Quantitative Problem Solving Processes in Children." Ed.D. Dissertation, University of Massachusetts, Amherst, May 1977. Clement, J.; Lockhead, J.; and Monk, G. "Translation Difficulties in Learning Mathematics." American Mathematical Monthly, 88 (1 979), 4, Collins, A.; Brown, J.S.; and Newman, S. "The New Apprenticeship: Teaching Students the Craft of Reading, Writing, and Mathematics." In D. Klahr (Ed.~. Cognition and Instruction: Issues and Agendas. Hillsdale, NiJ.: Lawrence Erlbaum Associates, 1989. Conference Board of the Mathematical Sciences. "The Mathematical Sciences Curriculum K-12: What Is Still Fundamental and What Is Not." In Educating Americans for the 21st Century: Source Materials. National Science Board Commission on Precollege Education in Mathematics, Science, and Technology. Washington, D.C.: National Science Foundation, 1983, 1-23. Crosswhite, F. Joe; Dossey, John A.; Swafford, Jane O.; McKnight, Curtis C.; Cooney, Thomas J.; Downs, Floyd L.; Grouws, Douglas A.; and Weinzweig, All. Second International Mathematics Study: Detailed Report for the United States. Champaign, 111.: Stipes Publishing Company, 1986. Davis, Robert B. Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, N.J.: Ablex Publishing, 1984. Denham, Walter F., and O'Malley, Edward T. (Eds.~. Mathematics Framework for California Public Schools, Kindergarten Through Grade Twelve. Sacramento, Calif: California State Department of Education, 1985. Dessart, Donald J., and Suydam, Marilyn N. Classroom Ideas from Research on Secondary School Mathematics. Reston, Va.: National Council of Teachers of Mathematics, 1983. Dossey, John A.; Mullis, Ina V.S.; Lindquist, Mary M.; and Chambers, Donald L. The Mathematics Report Card: Are We Measuring Up? Trends and Achievement Based on the 1986 National Assessment of Educational Progress (NAEP). Princeton, N.J.: Educational Testing Service, 1988. Dugdale, Sharon. "Green Globs: A Microcomputer Application for Graphing of Equations." Mathematics Teacher, 75:3 (March 1982), 208-2 1 4. Easley, J., and Easley, E. "Ichinensei: The Introduction to Quantity in a Japanese Elementary School." Technical Report, University of Illinois, 1982. Enloe, W., and Lewin, P. "The Cooperative Spirit ir~ Japanese Primary Education." Educational Forum, 51:3 (Spring 1987), 233-247. Erlwanger, S.H. "Case Studies of Children's Conceptions of Mathematics." Ph.D. Dissertation, University of Illinois, Champaign Urbana, 1974.

55 A Philosophy and Framework Fawcett, Harold P. The Nature of Proof. Thirteenth NCTM Yearbook. New York: Columbia University Teacher's College, Bureau of Publications, 1938. Fey, James T. "Mathematics Teaching Today: Perspectives From Three National Surveys." Mathematics Teacher, 72:7 (October 1979), 490- 504. Fey, James T. (Ed.~. Computing and Mathematics: The Impact on Secondary School Curricula. Reston, Va.: National Council of Teachers of Mathematics, 1984. Fuson, Karen C.; Stigler, James W.; and Bartsch, Karen. "Grade Placement of Addition and Subtraction Topics in Japan, Mainland China, the Soviet Union, Taiwan, and the United States." Journal for Research in Mathematics Education, 19 (November 1988), 449-456. Gelman, R., and Gallistal, C. The Child's Understanding of Number. Cambridge, Mass.: Harvard University Press, 1978. Ginsburg, H.P. Children's Arithmetic: How They Learn It & How You Teach It. New York: Van Nostrand Reinhold, 1977. Ginsburg, H.P. (Ed.~. The Development of Mathematical Thinking. New York: Academic Press, 1983. Grouws, Douglas A.; Cooney, Thomas J.; and Jones, Douglas (Eds.~. Perspectives on Research on Effective Mathematics Teaching. Reston, Va.: National Council of Teachers of Mathematics, 1988. Hansen, Viggo P. (Ed.~. Computers in Mathematics Education. 1984 NCTM Yearbook. Reston, Va.: National Council of Teachers of Mathematics, 1984. Hatano, G. "Learning to Add and Subtract: A Japanese Perspective." In T. Carpenter, J. Moser, and T. Romberg (Eds). Addition & Subtraction: A Cognitive Perspective. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1982. Held, M.K. "Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool." Journal for Research in Mathematics Education, 19:1 (January 1988), 3-25. Held, M.K., and Kunkle, D. "Computer Generated Tables: Tools for Concept Development in Elementary Algebra." In The Ideas of Algebra, K-12. 1988 NCTM Yearbook. Reston, Va.: National Council of Teachers of Mathematics, 1988, 170-177. Hembree, R., and Dessart, D. "Effects of Hand-held Calculators in Precollege Mathematics Education: A Meta-analysis." Journal for Research in Mathematics Education, 17:2 (1986), 83-89. Hiebert, J. (Ed.~. Conceptual and Procedural Know/edge: The Case of Mathematics Hillsdale, N.J.: Lawrence Erlbaum Associates, 1986. Howson, Geoffrey, and Kahane, J.-P. (Ecis.~. The Influence of Computers and Informatics on Mathematics and Its Teaching. Interrogational Commission on Mathematical Instruction Study Series. Cambridge, Mass.: Cambridge University Press, 1986. Johnston, William B., and Packer, Arnold ~E. (Eds.~. Workforce 2000: Work and Workers for the Twenly-First Century. Indianapolis, Ind. Hudson Institute, 1987. Krulik, Stephen (Ed.~. Problem Solving in School Mathematics. 1980 NCTM Yearbook. Reston, Va.: National Council of Teachers of Mathematics, 1980.

56 Reshaping School Mathematics Lapointe, Archie E., Mead, Nancy A.; and Phillips, Gary W. A World of Differences: An International Assessment of Science and Mathematics. Princeton, N.J.: Educational Testing Service, 1989. Love, Jean; Smith, Steve; and Butler, Michael. "Problem Solving as Everyday Practice." In Randall 1. Charles and Edward A. Silver (Eds.~. The Teaching and Assessing of Mathematical Problem Solving. Reston, Va.: National Council of Teachers of Mathematics, 1988, 61-81. Lesh, Richard. "The Evolution of Problem Representations in the Presence of Powerful Conceptual Amplifiers." In C. Janvier (Ed.~. Problems of Representation in the Teaching and Learning of Mathematics. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1987, 1 97-206. Linn, Marcia C. (add. "Establishing a Research Base for Science Education: Challenges, Trends and Recommendations." Report of a National Conference held January 16-19, 1986, University of California, Berkeley, May 1986. Mason, J.; Burton, L.; and Stacey, K. Thinking Mathematically. Reading, Mass.: Addison-Wesley, 1982. Mathematical Sciences Education Board, National Research Council. The Teacher of Mathematics: Issues for Today and Tomorrow. Washington, D.C. National Academy Press, 1987. Maurer, Stephen B. "New Knowledge About Errors and New Views About Learners: What They Mean to Educators and More Educators Would Like to Know." In A. Schoenfeld (Ed.), Cognitive Science and Mathematics Education. Hillsdale, N.J .: Lawrence Erlbaum Associates, 1987, 165- 187. McKnight, Curlis C.; Crosswhite, F. Joe; Dossey, John A.; Kiter, Edward; Swafford, Jane O.; Travers, Kenneth J.; and Cooney, Thomas J. The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective. Champaign, 111.: Stipes Publishing Company, 1987. Mestre, Jose. "Why Should Mathematics and Science Teachers Be Interested in Cognitive Research Findings?" Academic Connections, The College Board (1987), 3-5, 8-11. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: National Council of Teachers of Mathematics, 1989. National Research Council: Committee on Research in Mathematics, Science, and Technology Education. Mathematics, Science, and Technology Education: A Research Agenda. Washington, D.C.: National Academy Press, 1985. National Research Council. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D,C.: National Academy Press, 1989. Noddings, Nell "Preparing Teachers to Teach Mathematical Problem Solving." In Randall 1. Charles and Edward A. Silver (Eds.~. The Teaching and Assessing of Mathematical Problem Solving. Reston, Va.: National Council of Teachers of Mathematics, 1988, 244-258. Oaxaca, Jaime, and Reynolds, Ann W. Changing America: The New Face of Science and Engineering. Washington, D.C.: Task Force on Women, Minorities, and the Handicapped in Science and Technology, September 1988.

57 A Philosophy and Framework Office of Technology Assessment. Educating Scientists and Engineers, Grade School to Grad School. Washington, D.C.: Office of Technology Assessment, 1988. Palmiter, J.R. "The Impact of a Computer Algebra System on College Calculus." Dissertation, Ohio State University, November 1986. Papert, Seymour. Mindstorms: Children, Computers, and Powerful Ideas. New York: Basic Books, 1980. Papert, Seymour. "Information Technology and Education: Computer Criticism vs. Technocentric Thinking." Educational Researcher, 16:1 (1 987). Pea, Roy D. "Cognitive Technologies for Mathematics Education," In Alan H. Schoenfeld (Ed.~. Cognitive Science and Mathematics Education. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1987a, 89- 122. Pea, Roy D. "The Aims of Software Criticism: Reply to Professor Papert." Educational Researcher, 16:5 (1 987b), 4-8. Peterson, P.L. "Teaching for Higher-order Thinking in Mathematics: The Challenge for the Next Decade." In Douglas A. Grouws, Thomas J. Cooney, and Douglas Jones (Eds.~. Perspectives on Research on Effective Mathematics Teaching, Volume 7. Reston, Va.: National Council of Teachers of Mathematics, 1988, 2-26. Peterson, P.L., and Carpenter, T.L. (Eds.~. "Learning Through Instruction: The Study of Students' Thinking During Instruction in Mathematics." A Special Issue of the Educational Psychologist, 1989. Piaget, J. Ou va L'education? International Commission on the Development of Education, UNESCO (1948~. English Translation. New York: Grossman, 1973. Piaget, J. The Construction of Reality in the Child. New York: Basic Books, 1954. Polya, G. How to Solve It. Princeton, N.J.: Princeton University Press, 1 945. Resnick, Lauren B. "Task Analysis in Instructional Design: Some Cases from Mathematics." In D. Klahr (Ed.). Cognition and Instruction: Issues and Agendas. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1976, 51-80. Resnick, Lauren B. "Mathematics and Science Learning: A New Conception." Science, 220 (April 29, 1983), 477-478. Resnick, Lauren B. Education and Learning to Think. Committee on Mathematics, Science, and Technology Education, Commission on Behavioral and Social Sciences and Education, National Research Council. Washington, D.C.: National Academy Press, 1987. Resnick, Lauren B. "Treating Mathematics as an III-Structured Discipline." In Randall 1. Charles and Edward A. Silver (Eds.). The Teaching and Assessing of Mathematical Problem Solving. Reston, Va.: National Council of Teachers of Mathematics, 1988, 32-60. Reusser, Kurt. "Problem-Solving Beyond the Logic of Things: Textual and Contextual Effects on Understanding and Solving Word Problems." Paper presented at the annual meeting of the American Educational Research Association, San Francisco, Calif., April 1986.

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