Carol L. Smith

On differentiation: A case study of the development of the concepts of size, weight, and density

Abstract This paper presents a case study of 3- to 9-year-old childrens concepts of size, weight, density, matter, and material kind. Our goal was to examine two claims: (1) that individual concepts undergo differentiation during development; and (2) that young childrens concepts are embedded in theory-like structures. To make progress on the first issue, we needed to specify in representational terms what an undifferentiated concept is like and in what sense this undifferentiated concept is a parent of the more differentiated concepts. Our strategy was to use a model of conceptual differentiation suggested by the history of science to guide our search for evidence. In this model, undifferentiated concepts, like differentiated concepts, can be analyzed in terms of their component properties, features, or dimensions. The key difference is that an undifferentiated concept unites certain components which will subsequently be analyzed as components of distinct concepts, and that the undifferentiated concept is embedded in a different theoretical structure from the differentiated concepts. In our study, the same group of 78 children (18 3-year-olds, 18 4-year-olds, 18 5-year-olds, 12 6–7-year-olds, and 12 8–9-year-olds) were given a range of tasks probing their understanding of size, weight, and density; a subgroup of these children were given additional tasks probing their concepts of matter and material kind. We found that young children had a theoretical system which included distinct concepts of size, weight, and material kind and were beginning to form generalizations relating these concepts (e.g., size is crudely correlated with weight, steel objects are typically heavy). The core of their weight concept was felt weight, with density absent from their conceptual system; material kinds were defined in terms of properties which characterize large scale chunks of stuff. Slightly older children (5–7-year-olds) had made modifications to their concepts of weight and material kind. At these ages, their concept of weight now contained both the properties heavy and heavy for size (which we take as evidence of their having an undifferentiated weight/density concept) and they were coming to see weight differences as important in distinguishing whether large-scale objects were made of the same kind of stuff. However, the core of their weight concept was still felt weight and material kinds were still defined in terms of properties of large scale objects. Finally, still older children (8–9-year-olds) had a theoretical system in which weight and density were articulated as distinct concepts, material kinds were reconceptualized as the fundamental constituents of objects, and weight was seen as a fundamental property of matter. We conclude that childrens concepts of weight and density do differentiate in development and that it does make sense to view childrens concepts in the context of theory-like structures.

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 childrens conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Childrens 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.

Quantifiers and Question Answering in Young Children.

Two hypotheses were tested about how young children answer questions with the quantifiers all and some: (a) that children use syntactic cues in determining which noun phrase is quantified, and (b) that children evaluate a some-statement as part of evaluating an all-statement. To test these hypotheses, the same group of 60 4- to 7-year-olds were asked four contrasting types of quantitative questions. The results indicated that children can use syntactic cues under some presentation conditions. However, there was no evidence for an asymmetry between the all-and some-questions. A model of how young children might answer quantitative questions was then considered.

Children's understanding of natural language hierarchies

Abstract Three experiments examined young childrens ability to evaluate the relationship between concepts as one of inclusion. In Experiment 1, the same group of 4-to 7-year-old children were given three tasks: one in which they made judgments about whether “all” or “some” members of a category were included in another category, and two tasks in which they made inferences based on knowledge of inclusion relations. The majority of children succeeded on at least one of the tasks, thereby implying that they could evaluate inclusion relations. Two further experiments provided more evidence for this hypothesis: Experiment 2 confirmed that nursery children could answer quantitative questions about conceptual interrelationts; Experiment 3 demonstrated that nursery schoolers could solve certain inference problems involving the construction and evaluation of hierarchies.

Conceptually Enhanced Simulations: A Computer Tool for Science Teaching

In this paper, we consider a way computer simulations can be used to address the problem of teaching for conceptual change and understanding. After identifying three levels of understanding of a natural phenomenon (concrete, conceptual, and metaconceptual) that need to be addressed in school science, and classifying computer model systems and simulations more generally in terms of the design choices facing the programmer, we argue that there are ways to design computer simulations that can make them more powerful than laboratory models. In particular, computer simulations that provide an explicit representation for a set of interrelated concepts allow students to perceive what cannot be directly observed in laboratory experiments: representations for the concepts and ideas used for interpreting the experiment. Further, by embedding the relevant physical laws directly into the program code, these simulations allow for genuine discoveries. We describe how we applied these ideas in developing a computer simulation for a particular set of purposes: to help students grasp the distinction between mass and density and to understand the phenomenon of flotation in terms of these concepts. Finally, we reflect on the kinds of activities such conceptually enhanced simulations allow that may be important in bringing about the desired conceptual change.

Learning Progressions as Tools For Curriculum Development

Cognitively-based curricula may take into account research on students’ difficulties with a particular topic (e.g., weight and density, inertia, the role of environment in natural selection) or domain-general learning principles (e.g., the importance of revisiting basic ideas across grades). Learning progressions (LPs) integrate and enrich those approaches by organizing students’ beliefs around core ideas in that domain, giving a rich characterization of what makes students’ initial ideas profoundly different from those of scientists, and specifying how to revisit those ideas within and across grades so that young children’s ideas can be progressively elaborated on and reconceptualized toward genuine scientific understanding.

REJOINDER: Response to Commentaries

In responding to the thoughtful commentaries on our article, we focus on two important issues raised by most of the reviewers: (a) the curricular dependence of the proposed learning progression and the daunting (twin) challenges of bringing about changes in curriculum and assessment and consolidating and validating the proposed learning progression, and (b) the need for further research and development of a wide variety of sorts. We agree they are major challenges but see them as opportunities as well: opportunities to use research on children’s learning to inform curriculum and assessment much more than is currently the case.