Arthur J. Baroody Children's counting-based knowledge of number and arithmetic builds on their nonverbal knowledge in these domains. The nature of this pre-counting phase and how children make the transition to the counting phase, however, are not clear. According to the mental model proposed by Janellan Huttenlocher and colleagues, children initially represent even small collections of 1 to 4 items inexactly, not precisely as much current theorizing suggests. With Transition 1, children develop the ability to represent collections exactly but nonverbally. This and the development of counting permit Transition 2 to an exact, verbally based representation of number. Whereas the mental model focuses on how number is represented, Lauren Resnick's developmental model focuses on what is represented. According to this model, mathematical thinking evolves from concrete (context-bound) to abstract (general). In the first phase, children form a nonverbal understanding of uncounted quantities (engage in protoquantitive reasoning). In the second phase, they construct understandings of counted collections (become capable of quantities-level reasoning). In the third phase, children construct knowledge and can reason about specific numbers in the absence of actual collections (numbers-level thinking). In the fourth phase, they discover numerical or arithmetic relations and can reason with and about generalities (abstract l-level reasoning). The proposed project will entail evaluating a model that integrates the two models discussed above. According to this integrated model, children may pass through three subphases between protoquantitative- and quantities-level thinking. After Transition 1, they may first reason sensibly but imprecisely (qualitatively) about exact representations of number (subphase 1) and then reason precisely (quantitatively) about them (subphase 2). After children learn number names but before they can enumerate collections (can use counting to determine the number of items in a collection), they may be able to reason quantitatively about exact representations of collections and attach number labels to them (subphase 3). Although the integrated model is consistent with much existing research, including that which suggests the rapid recognition of number without counting is a basis for quantitative-level thinking, specific implications of the model need to be tested. Studies 1 and 2 will involve evaluating predictions that follow from the integrated model about how the development of an exact nonverbal representation of number, verbal counting, and simple addition/subtraction are inter-related. For example, according to this model, pre-counting children in subphase 1 should be to nonverbally create a matching collection for one previously seen but now hidden. Whereas, these children should can only estimate the effects of addition or subtraction on a collection, subphase 2 pre-counters can mentally determine small sums and differences accurately, and subphase 3 pre-counters can further identify such results by verbally labeling it with a number. Study 1 will entail combining a cross-sectional design, questionnaire-based interviews of children's parents or teachers (to provide context and observational data), and repeated re-testing (to examine the learning effects often induced by testing young children). Study 2 will consist of long-term case studies based on naturalistic observations and microgenetic methods (repeatedly administering selected tasks in a specific manner at a prescribed interval, particularly during a developmental transition phase). Using a combination of methods should provide richer data on number and arithmetic development than using any single method. Understanding children's performance in a microgenetic study, for instance, can be significantly improved by a detailed knowledge of their developmental readiness and performance in their natural environment (as documented by naturalistic data). In return, the focused nature of a microgenetic study can provide naturalistic observations with a clear direction (e.g., direct attention to key behaviors or patterns of behavior). Four other studies will involve examining the early development (nonverbal understanding) of the following key number and arithmetic concepts: part-whole knowledge (e.g., a whole is larger than any single part), additive composition (e.g., a sum is larger than either part), the inverse principle (addition of a certain amount is undone by the subtraction of the same amount), and additive commutativity (the order in which two collections are combined does not affect the outcome). The proposed project should have important theoretical, methodological, and practical value. Directly testing the implications of the integrated model should lead to a better understanding about the origins of number and operation sense and how it evolves. The integrated methodologies and task-specific tests developed should be useful in investigating pre-counting mathematical knowledge, exploring the transition to counting-based knowledge, and gauging the effects of the former on the latter. The more powerful developmental framework and assessment measures should be useful for those planning, developing, or implementing early childhood mathematics programs.
