Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infinitely divisible in the sense that they can be repeatedly divided without ever getting to zero. Their conceptualizing number as infinitely divisible was strongly related to their having a model of fraction notation based on division and to their successful judgment of the relative magnitudes of fractions and decimals. In addition, their understanding number as infinitely divisible was strongly related to their understanding physical quantities as infinitely divisible. These results support a conceptual change account of knowledge acquisition, involving two-way mappings between the domains of number and physical quantity.

[1]  D. Tirosh,et al.  Intuitive rules in science and mathematics: the case of ‘more of A ‐‐ more of B’ , 1996 .

[2]  Robert P. Craig.  The Child's Construction of Space and Time. , 1981 .

[3]  S. Carey,et al.  The Epigenesis of mind : essays on biology and cognition , 1991 .

[4]  S. Carey.  Knowledge Acquisition: Enrichment or Conceptual Change? , 1991 .

[5]  Kelly S. Mix,et al.  Early fraction calculation ability. , 1999, Developmental psychology.

[6]  S. Carey,et al.  On differentiation: A case study of the development of the concepts of size, weight, and density , 1985, Cognition.

[7]  J. Piaget,et al.  Child's Conception Of Geometry , 1960 .

[8]  Thomas A. Romberg,et al.  Rational numbers : an integration of research , 1993 .

[9]  John Monaghan.  Young Peoples' Ideas of Infinity , 2001 .

[10]  Lauren B. Resnick,et al.  Protoquantitative origins of ratio reasoning. , 1993 .

[11]  Dina Tirosh,et al.  The intuition of infinity , 1979 .

[12]  Carol L. Smith,et al.  Teaching for Understanding: A Study of Students' Preinstruction Theories of Matter and a Comparison of the Effectiveness of Two Approaches to Teaching About Matter and Density , 1997 .

[13]  J. Hiebert.  Conceptual and procedural knowledge : the case of mathematics , 1987 .

[14]  F. Keil.  Concepts, Kinds, and Cognitive Development , 1989 .

[15]  UNDERSTANDING ZERO AND INFINITY IN THE EARLY SCHOOL YEARS , 1983 .

[16]  Joseph Nussbaum,et al.  Junior high school pupils' understanding of the particulate nature of matter: An interview study , 1978 .

[17]  Jere Confrey,et al.  The Development of multiplicative reasoning in the learning of mathematics , 1995 .

[18]  Thomas P. Carpenter.  Calculators in Testing Situations: Results and Implications from National Assessment. , 1981 .

[19]  Arthur B. Markman,et al.  Analogical Reasoning and Conceptual Change: A Case Study of Johannes Kepler , 1997 .

[20]  P. Kitcher.  The Child as Parent of The Scientist , 1988 .

[21]  Rochel Gelman,et al.  Early understandings of numbers: paths or barriers to the construction of new understandings? , 1998 .

[22]  J. Moss.  Developing Children's Understanding of the Rational Numbers: A New Model and an Experimental Curriculum , 1999 .

[23]  T. Kuhn,et al.  The essential tension : selected studies in scientific tradition and change , 1977 .

[24]  D. Kerslake,et al.  Fractions: Children's Strategies and Errors. A Report of the Strategies and Errors in Secondary Mathematics Project. , 1986 .

[25]  H. Wellman,et al.  Insides and essences: Early understandings of the non-obvious , 1991, Cognition.

[26]  C. Gallistel,et al.  The Child's Understanding of Number , 1979 .

[27]  Ruth Stavy,et al.  How Students Mis/Understand Science and Mathematics: Intuitive Rules (Ways of Knowing in Science Series) , 2000 .

[28]  Ipke Wachsmuth,et al.  IDENTIFYING FRACTIONS ON NUMBER LINES , 1988 .

[29]  B. Rittle-Johnson,et al.  Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process. , 2001 .

[30]  P. Nesher,et al.  Shifts in reasoning , 1986 .

[31]  Carol L. Smith,et al.  Linking phenomena with competing underlying models: A software tool for introducing students to the particulate model of matter , 2003 .

[32]  E. Fischbein.  The Autonomy of Mental Models. , 1990 .

[33]  D. Tirosh,et al.  Intuitive rules in science and mathematics: the case of ‘Everything can be divided by two’ , 1996 .

[34]  C. W. Anderson,et al.  Changing middle school students' conceptions of matter and molecules , 1990 .

[35]  R. Ackermann.  Minnesota Studies in the Philosophy of Science , 1975 .

[36]  K. Fuson.  Children's Counting and Concepts of Number , 1987 .

[37]  Richard Lesh,et al.  ORDER AND EQUIVALENCE OF RATIONAL NUMBERS: A CLINICAL TEACHING EXPERIMENT , 1984 .