You re going to be sorry you ever asked

You’re going to be sorry you ever asked that.

• Better define “early number concepts”. • Compare and contrast early number concept theories. • Describe how early number concepts impacts your teaching and your students. • Have some good, CLEAN fun while learning.

Innate Ability Conceptual Knowledge Versus Experience Procedural Knowledge

Only people ignorant of history and too shallow to understand what an enormous task it is to create anything could believe that children can reach any serious understanding of even basic concepts by the faddish methods of “discovery” and “creativity” being used in public school classrooms… Most of us will never discover or create anything significant that was not understood. – Thomas Sowell, a senior fellow at the Hoover Institute, Stanford University

For instance, some critics have claimed that instruction based on the NCTM standards is “fuzzy math. ” It is not clear, though, why encouraging students to understand mathematics, use this understanding to invent solution procedures for challenging problems, justify and defend their procedures and solutions, and critically analyze others’ procedures and solutions is fuzzy math. – Arthur J. Baroody, Professor, University of Illnois at Urbana-Champagin

PART 1 PART 2 Development of Number Concepts Counting Magnitude of Numbers One to One Correspondence Collections Interlude 1: Let the Games Begin! Number Words Operations (+, -, *) Commutative Property Equality Interlude 2: More Games!! PART 3: Advanced Early Number Concepts – Quaternions!!!

Eliz had 8 cookies. She ate 3 of them. How many cookies does Eliz have left? Eliz has 3 dollars to buy cookies. How many more dollars does she need to earn to have 8 dollars? Eliz has 3 dollars. Tom has 8 dollars. How many more dollars does Tom have than Eliz?

1. What proportion of the time do you automatically remember the answers for simple multiplication/addition/subtraction problems? 2. Some people use solutions called derived facts where they use a problem they know how to figure out the answer for another problem. How often do you use derived fact solutions? 3. Some people use counting to solve addition problems. How often do you use counting to solve addition problems? 4. Overall, please estimate what proportion of the time you solve simple multiplication/addition/subtraction problems in ways other than by automatically remembering the answers?

1 8 3 6 + 3 5 9 7 7 3 2 8 3 4 + 1 9 6 3 3 7 0 8 1 5 2 7 + 1 9 4 6 6 1 9 1 1 2 4 8 + 2 5 7 9 9 9

4 6 5 2 – 1 9 6 8 3 3 1 6 5 2 8 6 – 1 6 2 9 4 0 6 0 4 6 5 2 – 1 9 6 8 3 7 9 4

At Ideal University, there are six times as many students as professors. Write an equation to represent this situation using S to indicate the number of students and P to indicate the number of professors.

Born: May 19, 1895 Died: May 28, 1977 Allegheny College A. B. 1917 University of Chicago A. M. 1923 University of Chicago Ph. D. 1926 University of Illinois Cornell University of Michigan George Peabody College for Teachers Northwestern University Duke University 1930 – 1949 University of California at Berkeley 1950 – 1961

Emeritus Ph. D. University of Wisconsin, 1971 Teacher Education Building Rm: 476 A 225 N. Mills Street Madison, Wisconsin 53706 Phone: (608) 263 -4266 Email: tpcarpen@wisc. edu Website: http: //www. education. wisc. edu/ci/mathed/carpenter/

Professor Curriculum & Instruction 310 Education Building 1310 S. 6 th St. MC 708 baroody@uiuc. edu 217 333 -4791 Degrees Ph. D. , Educational and Developmental Psychology, Cornell University, 1979 B. S. , Science Education, Cornell University, 1969

UC Berkeley 4315 Tolman Hall Berkeley, CA Office: (510)643 -6627 saxe@socrates. berkeley. edu EDUCATION 1976 -1977 Post-doctoral Trainee, Children's Hospital Medical Center (Harvard University Medical School) and Boston Veteran's Administration Hospital, Boston, Massachusetts. Atypical cognitive development. 1976 Ph. D. , University of California, Berkeley. Psychology. 1970 B. A. , University of California, Berkeley. Psychology

Professor Department of Psychology Yale University 2 Hillhouse Ave Box 208205 New Haven, CT 06520 -8205 EDUCATION Ph. D. , 1990, Massachusetts Institute of Technology B. A. , 1985, Mc. Gill University (Montreal)

About. com edited by Deb Russell Baroody, A. J. (1984). More precisely defining and measuring the orderirrelevance principle. Journal of Experimental Child Psychology, 38, 33 -41. Baroody, Arhtur Teaching Children Mathematics / August 2006, NCTM Baroody, Arthur J. , Tiilikainen, Sirpa H. , & Wilkins, Jesse L. M. “Additive Commutativity. ” Development of Arithmetic Concepts and Skills (2002). PP. 127 -150. Baroody, Arthur & Benson, Alexis. Early Number Instruction. Teaching Children Mathematics 2001 NCTM. Butterworth, Brian, Marchesini, Moemi, & Girelli, Luisa. “Dynamic Reorganization of Combinations. ” Development of Arithmetic Concepts and Skills (2002). PP. 189 -200.

Carpenter, Thomas P. , Elizabeth Fenneman, Megan Loef Franke, Linda Levi, and Susan B. Empson, Children's Mathematics: Cognitively Guided Instruction, The National Council of Teachers of Mathematics, Inc. , 1999, Heinemann: Portsmouth, NH. Cowan, Richard. “Knowledge of Addition. ” Development of Arithmetic Concepts and Skills (2002). PP. 35 -74. Cowan, R. , & Renton, M. (1996). Do they know what they are doing? Children’s use of economical addition strategies and knowledge of commutativity. Educational Psychology, 16, 409 -422. Donlan, C. (1998). The development of mathematical skills: Studies in developmental psychology. London: Psychology Press Fuson, Karen C. & Burghardt, Birch H. “Multidigit Addition and Subtraction. ” Development of Arithmetic Concepts and Skills (2002). PP. 245 -301 Fuson, K. C. (1988). Children’s counting and concept of number. New York: Springer. Verlag. Gelman, R. (1991). Epigenetic foundations of knowledge structures: Initial and transcendent constructions. In S. Carey & R. Gelman (Eds. ), The Epigenesis o f Mind: Essays on biology and cognition (pp. 293 -322). Hillsdale, NJ: Erlbaum.

Gelman, R. (1978). Counting in the preschooler: What does and does not develop. In R. S. Siegler (Ed. ), Children’s Thinking, what develops? (pp. 213 -241). Hillsdale, NJ: Lawrence Erlbaum. Kilpatrick, Jeremy & Weaver, J. Fred. “The Place of William A. Brownell Place in Mathematics Education. ” Journal for Research in Mathematics Education. Vol 8 No. 5 (Nov 1977). PP. 382 -384 Hiebert, J, & Wearne, D. (1992). Links between teaching and learning place value with understanding in the first grade. Journal for Research in Mathematics Education, 23, 68, 122. How People Learn: Brain, Mind, Experience, and School. Commission on Behavioral and Social Sciences and Education, National Research Council Le. Fevre, Jo-Anne. “Multiple Procedures in Adults. ” Development of Arithmetic Concepts and Skills (2002). PP. 203 -220. Miura, Irene T. & Okamoto, Yukari. “Language Supports. ” Development of Arithmetic Concepts and Skills (2002). PP. 230 -236. Mix, K. (2002). The construction of number concepts. Cognitive Development, 17, 1345 -1363.

Piaget, J. (1952). The child’s conception of number. London: Routledge and Kegan Paul. Simon, T. J. , & Hespos, S. J. , & Rochat, P. (1995). Do infants understand simple arithmetic? A replication of Wynn (1992). Cognitive Development, 10, 253 -269. Starkey P. , & Cooper, Jr. , R. G. (1980). Perception of numbers by human infants. Science, 210, 1033 -1035. Wynn, K. (1996). Infants’ individuation and enumeration of sequential actions. Psychological Science, 7, 164 -169 Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749 -750. Wynn, K. (1990). Children’s understanding of counting. Cognition, 36, 155 -193.