Mathematical Cognition
In general, I am interested in understanding the knowledge, processes, and underlying mechanisms that both emerge from and support mathematical thinking in the child, especially when such thought involves external representations.
External Representations in Mathematical Problem Solving and Reasoning
External representations such as gestures, inscriptions, diagrams, manipulative, and multi-media environments figure prominently in mathematics. Acquiring mathematical knowledge and skills for problem solving and reasoning is tightly constrained by (a) the knowledge of task-relevant ideas in mathematics, (b) the semantic properties of external representations such as inference affordances, schema abstraction (e.g., Koedinger & Anderson, 1990), and computational offloading (e.g., Larkin & Simon, 1987), and (c) the schematic knowledge of the structural configuration of an external representation such as understanding that matrices are good for dealing with relationship between two variables and hierarchies work well for classifying super and subordinate class relations (e.g., Novick, Hurley, & Francis, 1999).
A person's ability to express what they know and understand about say, multiplicative relations, is influenced by his or her ability to detect those relations in the working environment or, task domain. Often, the task domain can be operationalized in terms of the verbal and the non-verbal information structures available to the reasoner. In order to think productively with these forms of information the reasoner must be knowledgeable in both information systems. Variation in the configuration and properties of the task have been found to influence computational demands, which in turn, impact representation, processing, and performance. Therefore, it is informative to understand mathematical thought as a function of the interaction between knowledge and physical information in the environment including external representations.
Application of Cognitive Science to the Design of Cognitive Technology
One practical goal of my research is to inform the design of cognitive technologies and artifacts (including school and home environments which can be engineered as cognitive systems) to support thinking and learning, especially in mathematics and science. The design of cognitive technologies in math and science should be based on a deep understanding about how people learn, especially the interplay of knowledge and action. By examining how mathematical thinking is moderated by domain-specific and schematic knowledge of external representations, individual and social actions, I aim to advance the way that children think mathematically and scientifically, particularly children from communities that are traditionally marginalized in these domains.
External Representations and Representing
An external representation (ER) is a physical configuration of information or a material artifact that, to the user, has symbolic properties. An ER that does not represent anything is a mere material mass. ERs may be partially or completely symbolic to the user. ERs may be either presented to the user such as when one encounters a diagram in the Weather Section of the newspaper or constructed as is often the case when a person designs something such as a floor plan or a solution to a novel math problem. The difference between presented and constructed ER use is that the former involves the interpretation of matters in the physical world whereas the latter involves constructing matter in the physical world. Whether an ER is presented or constructed impacts the nature of representing.
Representing refers to acts of interpretation and construction. Whereas most research on representing focuses on how domain-specific knowledge or experience (e.g., grade, expertise) impact ER performance, very little research examines the impact that knowledge about the representational functions of an artifact has on performance. I am interested in three types of "ER function" knowledge: applicability, mapping, and optimality. Given a task domain, applicability refers to the user's capacity to select the ER that matches the task demands. Mapping refers to the adequacy with which the user can relate physical aspects of the ER to symbolic referents either in the task or in the actions imposed on the task by the user. Finally, optimality refers to the user's potential to select the best ER for a given task domain. My recent work around knowledge of ER function builds on more recent theoretical and empirical work on meta-representation (diSessa, 2002, 2004; diSessa & Sherin, 2000; Sherin, 2000), diagram literacy (Diezmann & English, 2001), abstract or schematic knowledge (e.g., Novick, 2006; Uesaka & Manalo, 2006) and progressive symbolization (Enyedy, 2005).
External Representations and Representing
An external representation (ER) is a physical configuration of information or a material artifact that, to the user, has symbolic properties. An ER that does not represent anything is a mere material mass. ERs may be partially or completely symbolic to the user. ERs may be either presented to the user such as when one encounters a diagram in the Weather Section of the newspaper or constructed as is often the case when a person designs something such as a floor plan or a solution to a novel math problem. The difference between presented and constructed ER use is that the former involves the interpretation of matters in the physical world whereas the latter involves constructing matter in the physical world. Whether an ER is presented or constructed impacts the nature of representing.
Representing refers to acts of interpretation and construction. Whereas most research on representing focuses on how domain-specific knowledge or experience (e.g., grade, expertise) impact ER performance, very little research examines the impact that knowledge about the representational functions of an artifact has on performance. I am interested in three types of "ER function" knowledge: applicability, mapping, and optimality. Given a task domain, applicability refers to the user's capacity to select the ER that matches the task demands. Mapping refers to the adequacy with which the user can relate physical aspects of the ER to symbolic referents either in the task or in the actions imposed on the task by the user. Finally, optimality refers to the user's potential to select the best ER for a given task domain. My recent work around knowledge of ER function builds on more recent theoretical and empirical work on meta-representation (diSessa, 2002, 2004; diSessa & Sherin, 2000; Sherin, 2000), diagram literacy (Diezmann & English, 2001), abstract or schematic knowledge (e.g., Novick, 2006; Uesaka & Manalo, 2006) and progressive symbolization (Enyedy, 2005).