Students' Coordination of Geometric Reasoning and Measuring Strategies on a Fixed Perimeter Task: Developing Mathematical Understanding of Linear Measurement

This article examines students' development of levels of understanding for measurement by describing the coordination of geometric reasoning with measurement and numerical strategies. In analyzing the reasoning and argumentation of 38 Grade 2 through Grade 10 students on linear measure tasks, we found support for the application and elaboration of our previously established categorization of children's length measurement levels: (1) guessing of length values by nai've visual observation, (2) making inconsistent, uncoordinated reference to markers as units, and (3) using consistent and coordinated identification of units. We elaborated two of these categories. Observations supported sublevel distinctions between inconsistent identification (2a) and consistent yet only partially coordinated identification of units (2b). Evidence also supported a distinction between static (3a) and dynamic (3b) ways of coordinating length; we distinguish integrated abstraction (3b) from nonintegrated

[1]  Ron Tzur.  An Integrated Study of Children's Construction of Improper Fractions and the Teacher's Role in Promoting That Learning , 1999 .

[2]  Catherine Sophian.  Learning about "One" : Units as a Cornerstone for Head Start Mathematics , 2003 .

[3]  D. Schifter,et al.  Interpreting the Standards: Translating Principles into Practice. , 1997 .

[4]  Annette Karmiloff-Smith,et al.  Children's problem solving , 1984 .

[5]  Julie Sarama,et al.  Logo and geometry , 2001 .

[6]  Constance Kamii,et al.  Measurement of Length: The Need for a Better Approach to Teaching , 1997 .

[7]  K. VanLehn,et al.  Why Do Only Some Events Cause Learning During Human Tutoring? , 2003 .

[8]  Leslie P. Steffe,et al.  Operations that generate quantity , 1991 .

[9]  C. Hoyles,et al.  Visual and Symbolic Reasoning in Mathematics: Making Connections with Computers? , 1999 .

[10]  Lauren B. Resnick,et al.  Shared cognition: Thinking as social practice , 1991, Perspectives on socially shared cognition.

[11]  K. Hart,et al.  Children's understanding of mathematics: 11-16 , 1981 .

[12]  Kathleen Hart,et al.  Children's Understanding of Mathematics , 1989 .

[13]  Paul Light,et al.  Tools for Thought: The Measurement of Length and Area. , 1993 .

[14]  Douglas H. Clements.  Teaching Length Measurement: Research Challenges , 1999 .

[15]  Paul Cobb,et al.  Supporting Students' Development of Measuring Conceptions: Analyzing Students' Learning in Social Context , 2003 .

[16]  Lynne Outhred,et al.  Young Children's Intuitive Understanding of Rectangular Area Measurement. , 2000 .

[17]  Tami S. Martin,et al.  Assessing High School Students’ Understanding of Geometric Proof , 2004 .

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

[19]  P. M. Hiele.  Structure and Insight: A Theory of Mathematics Education , 1985 .

[20]  C. Kamii.  Why Is the Use of a Ruler So Hard , 1995 .

[21]  James Hiebert,et al.  Cognitive Development and Learning Linear Measurement , 1981 .

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

[23]  G. Boulton‐Lewis,et al.  An analysis of young children's strategies and use of devices for length measurement☆ , 1996 .

[24]  Douglas H. Clements,et al.  Rethinking "Concrete" Manipulatives , 1996 .

[25]  R. Marrades,et al.  Proofs produced by secondary school students learning geometry in a dynamic computer environment , 2000 .

[26]  Michael Mitchelmore,et al.  Children's informal knowledge of physical angle situations , 1997 .

[27]  Thomas P. Carpenter,et al.  The Development of the Concept of a Standard Unit of Measure in Young Children. , 1976 .

[28]  Using geometrical models in a process of reflective thinking in learning and teaching mathematics , 1990 .

[29]  Norma C. Presmeg,et al.  Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics , 1992 .

[30]  Vicky L. Kouba.  Results of the Fourth NAEP Assessment of Mathematics: Measurement, Geometry, Data Interpretation, Attitudes, and Other Topics. , 1988 .

[31]  S. Senk.  How Well Do Students Write Geometry Proofs , 1985 .

[32]  Leslie P. Steffe,et al.  Children's Thinking in Measurement Situations. , 1976 .

[33]  Paul Cobb,et al.  Developing Mathematical Reasoning within the Context of Measurement , 1999 .

[34]  James Hiebert,et al.  Why Do Some Children Have Trouble Learning Measurement Concepts , 1984 .

[35]  Luciano Meira,et al.  The Microevolution of Mathematical Representations in Children's Activity , 1995 .

[36]  Elana Joram,et al.  Benchmarks as Tools for Developing Measurement Sense , 2003 .

[37]  C. Kynigos.  Children's inductive thinking during intrinsic and Euclidean Geometrical activities in a computer programming environment , 1993 .

[38]  Celia Hoyles,et al.  Abstraction in expertise: a study of nurses' conceptions of concentration , 2002 .

[39]  Gila Hanna,et al.  Proof, Explanation and Exploration: An Overview , 2000 .

[40]  Douglas H. Clements,et al.  Students' Spatial Structuring of 2D Arrays of Squares , 1998 .

[41]  Douglas H. Clements,et al.  Quantifying Path Length: Fourth-Grade Children's Developing Abstractions for Linear Measurement , 2003 .

[42]  E. Fischbein.  The theory of figural concepts , 1993 .

[43]  Douglas H. Clements,et al.  Geometry and spatial reasoning. , 1992 .

[44]  Richard Lehrer,et al.  Research into Practice: Developing Spatial Sense through Area Measurement. , 1996 .

[45]  J. Shaughnessy,et al.  CHARACTERIZING THE VAN HIELE LEVELS OF DEVELOPMENT IN GEOMETRY , 1986 .