Karen C. Fuson

Children's counting and concepts of number

I Number Words.- 1 Introduction and Overview of Different Uses of Number Words.- 2 The Number-Word Sequence: An Overview of Its Acquisition and Elaboration.- II Correspondence Errors in Counting Objects.- 3 Correspondence Errors in Childrens Counting.- 4 Effects of Object Arrangement on Counting Correspondence Errors and on the Indicating Act.- 5 Effects of Object Variables and Age of Counter on Correspondence Errors Made When Counting Objects in Rows.- 6 Correspondence Errors in Childrens Counting: A Summary.- III Concepts of Cardinality.- 7 Childrens Early Knowledge About Relationships Between Counting and Cardinality.- 8 Later Conceptual Relationships Between Counting and Cardinality: Addition and Subtraction of Cardinal Numbers.- 9 Uses of Counting and Matching in Cardinal Equivalence Situations: Equivalence and Order Relations on Cardinal Numbers.- IV Number Words, Counting, and Cardinality: The Increasing Integration of Sequence, Count, and Cardinal Meanings.- 10 Early Relationships Among Sequence Number Words, Counting Correspondence, and Cardinality.- 11 An Overview of Changes in Childrens Number Word Concepts from Age 2 Through 8.- References.- Author Index.

Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics

We argue that reform in curriculum and instruction should be based on allowing students to problematize the subject. Rather than mastering skills and applying them, students should be engaged in resolving problems. In mathematics, this principle fits under the umbrella of problem solving, but our interpretation is different from many problem-solving approaches. We first note that the history of problem solving in the curriculum has been infused with a distinction between acquiring knowledge and applying it. We then propose our alternative principle by building on John Dewey’s idea of “reflective inquiry,” argue that such an approach would facilitate students’ understanding, and compare our proposal with other views on the role of problem solving in the curriculum. We close by considering several common dichotomies that take on a different meaning from this perspective

Describing Levels and Components of a Math-Talk Learning Community.

The transformation to reform mathematics teaching is a daunting task. It is often unclear to teachers what such a classroom would really look like, let alone how to get there. This article addresses this question: How does a teacher, along with her students, go about establishing the sort of classroom community that can enact reform mathematics practices? An intensive year-long case study of one teacher was undertaken in an urban elementary classroom with Latino children. Data analysis generated developmental trajectories for teacher and student learning that describe the building of a math-talk learning community—a community in which individuals assist one another’s learning of mathematics by engaging in meaningful mathematical discourse. The developmental trajectories in the Math-Talk Learning Community framework are (a) questioning, (b) explaining mathematical thinking, (c) sources of mathematical ideas, and (d) responsibility for learning.

The Acquisition and Elaboration of the Number Word Sequence

In this chapter we describe children’s acquisition and elaboration of the sequence of counting words from its beginnings around age two up to its general extension to the base ten system notions beyond one hundred (around age eight). This development occurs, in our view, in two distinct, though overlapping, phases: an initial acquisition phase of learning the conventional sequence of number words and an elaboration phase, during which this sequence is decomposed into separate words and relations upon these pieces and words are established. During acquisition, the sequence begins to be used for counting objects. Near the end of the elaborative phase, the words in the sequence themselves become items which are counted for arithmetic and relational purposes.

Advancing Children's Mathematical Thinking in Everyday Mathematics Classrooms

In this article we present and describe a pedagogical framework that supports childrens development of conceptual understanding of mathematics. The framework for Advancing Childrens Thinking (ACT) was synthesized from an in-depth analysis of observed and reported data from 1 skillful 1st-grade teacher using the Everyday Mathematics (EM) curriculum. The ACT framework comprises 3 components: Eliciting Childrens Solution Methods, Supporting Childrens Conceptual Understanding, and Extending Childrens Mathematical Thinking. The framework guided a cross-teacher analysis over 5 additional EM 1st-grade teachers. This comparison indicated that teachers often supported childrens mathematical thinking but less often elicited or extended childrens thinking. The ACT framework can contribute to educational research, teacher education, and the design of mathematics curricula.

Achievement Results for Second and Third Graders Using the Standards-Based Curriculum Everyday Mathematics

Students using Everyday Mathematics (EM), developed to incorporate ideas from the NCTM Standards, were at normative U.S. levels on multidigit addition and subtraction symbolic computation on traditional, reform-based, and EM-specific test items. Heterogeneous EM 2nd graders scored higher than middleto upper-middle-class U.S. traditional students on 2 number sense items, matched them on others, and were equivalent to a middle-class Japanese group. On a computation test, the EM 2nd graders outperformed the U.S. traditional students on 3 items involving 3-digit numbers and were outperformed on the 6 most difficult test items by the Japanese children. EM 3rd graders outscored traditional U.S. students on place value and numeration, reasoning, geometry, data, and number-story items.

Children's Knowledge of Teen Quantities as Tens and Ones; Comparisons of Chinese, British, and American Kindergartners.

Three studies were conducted to examine the effects of individual differences and language differences on childrens understanding of teen quantities (11 < n ≤ 19) as counted cardinal tens and ones (embedded-ten cardinal understanding). At age 4, most Chinese children, using named-ten number words (e.g., 12 is said as ten two), did not show such understanding on a task in which y quantities were added to 10 quantities. At age 5, half the children of average or above intelligence who had high rote-counting sequences (M = 90) did show such understanding; those with lower rote-counting sequences did not. English-speaking 5-year-old children in England and in the United States, whose teen words obfuscate the tens and ones, showed no evidence of understanding teen quantities as cardinal tens and ones.

Roles of representation and verbalization in the teaching of multi-digit addition and subtraction

First- and second-grade children were successfully taught symbolic multi-digit addition and subtraction procedures by first doing the procedures with a physical embodiment of the first four places of the base-ten system. Relationships between the embodiment and the numerical symbols were supported by close linking of the physical and the symbolic procedures and by the use of base-ten, named-value (standard English), and embodiment-name words. Most children successfully extended the procedures to ten-digit symbolic problems done without the embodiment. For many children who made procedural errors on delayed tests, the mental representation of the procedure with the physical embodiment was strong enough for them to use it to selfcorrect their symbolic procedure. The physical embodiment also seems to support the learning of various other concepts related to place value.RésuméDes enfants de CP et CE1 ont appris les procédures d’addition et de soustraction à plusieurs chiffres en commençant par des manipulations physiques de différents blocs (cubes et buchettes de différentes formes et tailles) représentant les quatre premières places du système de base dix. Les relations entre la manipulation physique et celle des symboles numériques étaient assurées par l’utilisation conjointe des deux activités et la mise en correspondance des termes utilisés dans l’une et l’autre (nom des chiffres, nom des places en base dix, nom des objets représentant ces places). La plupart des enfants sont parvenus à généraliser leurs procédures de calcul à des opérations comportant dix chiffres et cela sans manipulations physiques. Pour la plupart des enfants ayant commis des erreurs aux épreuves de généralisation, la persistance de la représentation mentale des procédés de manipulation physique, constitua une aide efficace pour l’auto-correction de leurs erreurs de calcul symbolique. La concrétisation utilisée lors de la manipulation semble également favoriser l’apprentissage de nombreuses autres notions relatives à la valeur des places occupées par les chiffres dans les nombres sujets à des opérations arithmétiques.

Chapter 2 Relationships Children Construct Among English Number Words, Multiunit Base-Ten Blocks, And Written Multidigit Addition

Publisher Summary Arithmetic has arisen in many different cultures as a way to solve problems concerning the quantitative aspects of real world situations. These quantitative aspects are described by words and, in many cultures, by marks that are written on some surface. In traditional cultures, children learn arithmetic by observing and eventually using the quantitative words and written marks in their situations. In modern cultures, however, children are taught the arithmetic of single-digit whole numbers, multi-digit whole numbers, integers (negative numbers), decimal fractions, and rational numbers. In much of this teaching, children do not learn to talk and write about the quantitative aspects of real world situations, but rather stay within the arithmetic marks world and memorize the sequences of written marks steps (routines) to accomplish each operation for each kind of number. For too many children, this approach results in a verbal superstructure of hierarchical routines unrelated to anything. As a result, there is massive interference among the routines. This chapter presents an empirical investigation in the domain of multi-digit addition and subtraction.

Effects of collection terms on class-inclusion and on number tasks

Abstract Ten experiments examined the effects on childrens performance of using collection (e.g., “army”) versus class (e.g., “soldiers”) terms to describe sets of familiar objects. In seven experiments using number tasks (conservation of number and two other equivalence tasks) the facilitative effect of collection terms reported by Markman (1979) was not found. However, in experiments with a class-inclusion task, performance was better with collection than with class terms, replicating the original labeling effect reported by Markman (1973) and Markman and Seibert (1976) . Additional data supported the interpretation that collection terms facilitate class-inclusion performance because they help the child represent the class-inclusion situation as a combine object situation in which two sets are put together to form a combined aggregate set. Formation of a mental combined set is not required by the number tasks, where both sets are always available in their entirety.