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"For the great majority of mathematics educators, multiplication is only a faster way of doing repeated addition, and most of them introduce multiplication in third grade. The purpose of this article is to show that (a) there is a clear difference between repeated addition (like 4 + 4 + 4) and multiplication (like 3 x 4) and that (b) many third graders do not understand multiplication because they are not yet able to think multiplicatively. The article will conclude with implications for the classroom, including a diagnostic task to assess children’s ability to think multiplicatively."

"Piaget studied the nature of human knowledge scientifically and made a fundamental distinction among three kinds of knowledge–physical knowledge, social (conventional) knowledge and logico-mathematical knowledge. This distinction enables us to view arithmetic in a new way and to teach it very differently from traditional instruction. In this paper, I will first clarify the nature of logico-mathematical knowledge. I will then show that a new theory about the origin and construction of mathematics leads to educational goals, classroom practices and an approach to evaluation that are very different from traditional ones. In the final part of this paper, I will explain why, in my opinion, traditional instruction is not only unnecessary but also harmful to children's development of numerical reasoning."

"A more complete account of this study entitled “The Harmful Effects of Algorithms in Grades 1-4” can be found in Chapter 17 of the l998 NCTM Yearbook (Kamii & Dominick, 1998). The National Council of Teachers of Mathematics holds the copyright to this chapter and permitted the reprinting only of the tables for the present article."

"Three hundred and eighty-three children in grades 1–5 were individually interviewed to find out at when they construct unit iteration out of transitive reasoning as described by Piaget, Inhelder, and Szeminska (1960). The results indicated that most children (72%) construct transitive reasoning by second grade and that most (76%) construct unit iteration out of transitive reasoning by fourth grade. The article explains why traditional instruction produces the poor results revealed by the National Assessment of Educational Progress. It also suggests a better approach to the teaching of measurement that presents problems and encourages children to modify their ways of thinking."

For more information visit Wiley Online Library. 

http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1997.tb17354.x/abstract

"One hundred twenty children in kindergarten and grades 2, 4, and 6 were individually interviewed with five Piagetian tasks to determine the grade level at which most have constructed transitive reasoning, unit iteration, and the conservation of speed. The responses were categorized as “successful,”“unsuccessful,” or “transitional.” By combining the “successful” and “transitional” categories, it was found that the children reasoned transitively by second grade (70.0%) and demonstrated unit iteration and conservation of speed by sixth grade (70.0% and 83.3%, respectively). It was concluded that the construction of the logic necessary to make sense of the measurement of time is generally not complete before sixth grade."

For more information visit Wiley Online Library.

http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.2001.tb18015.x/abstract 

"A total of 257 children in grades 2–5 were individually interviewed to find the grade level at which they demonstrated transitive reasoning and unit iteration in the measurement of volume. In the transitivity task, the children were asked if a larger, empty container could be used to compare the quantity of popcorn kernels (about 350 cc) in two containers that looked very different. The unit-iteration task was similar except that children were asked if a small cup could be used to compare similar quantities of rice in two containers. It was found that a majority of children (51%) demonstrated transitive reasoning by third grade and that a majority (56%) demonstrated unit iteration by fourth grade. A conclusion reached is that the standard of the National Council of Teachers of Mathematics (2000) expecting children to understand units of measurement by grade 2 is unrealistic. Better principles of teaching are also suggested to encourage children to think logically."

For more information visit Wiley Online Library.

http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.2001.tb17969.x/abstract 

"The people who set goals for education seldom take into account scientific knowledge about how children acquire knowledge and moral values."

PHI DELTA KAPPAN holds the copyright for this article. PHI DELTA KAPPAN needs to be contacted to use the material for anything except personal study.

Math games with playing cards for children in grades K-3. 

NAEYC holds the copyright for this article. 

NAEYC needs to be contacted to use the material for anything except personal study. 

For more information, visit: https://www.naeyc.org/resources/permissions

NAEYC holds the copyright for this article. 

NAEYC needs to be contacted to use the material for anything except personal study. 

For more information, visit: https://www.naeyc.org/resources/permissions

NAEYC holds the copyright for this article. 

NAEYC needs to be contacted to use the material for anything except personal study. 

For more information, visit: https://www.naeyc.org/resources/permissions

The Association for Childhood Education International holds the copyright for this article. 

The Association for Childhood Education International needs to be contacted to use the material for anything except personal study.