You’ll see what you mean: Students encode equations based on their knowledge of arithmetic

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth-grade students solved and explained a set of typical addition problems (e.g., ) and mathematical equivalence problems (e.g., or ). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems of the form overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students’ encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank-final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank-final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding.     

Do details bug you? Effects of perceptual richness in learning about biological change

People often have difficulty in understanding processes of biological change, and they typically reject drastic life cycle changes such as metamorphosis, except for animals with which they are familiar. Even after a lesson about metamorphosis, people often do not generalize to animals not seen during the lesson. This might be partially due to the perceptual richness of the diagrams typically used during lessons on metamorphosis, which serves to emphasize the individual animal rather than a class of animals. In two studies, we examined whether the perceptual richness of a diagram influences adults' learning and transfer of knowledge about metamorphosis. One study was conducted in a laboratory setting, and the other was online. In both studies, adults who saw the bland diagram during the lesson accurately transferred more than adults who saw the rich diagram during the lesson.     

Understanding and using principles of arithmetic: operations involving negative numbers

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers ( Dixon, Deets, & Bangert, 2001 ). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation. Participants ( N = 44) completed two tasks. In the Evaluation task, participants rated how well sets of equations were solved. Some sets violated principles of arithmetic and others did not. Participants rated non-violation sets higher than violation sets for two different principles for subtraction with a negative number. In the Word Problem task, participants read word problems and set up equations that could be used to solve them. Participants who displayed greater knowledge of principles of arithmetic with a negative number were more likely to set up equations that involved negative numbers. Thus, participants' knowledge of arithmetic principles was related to their problem representations.     

Spontaneous gestures influence strategy choices in problem solving

Do gestures merely reflect problem-solving processes, or do they play a functional role in problem solving? We hypothesized that gestures highlight and structure perceptual-motor information, and thereby make such information more likely to be used in problem solving. Participants in two experiments solved problems requiring the prediction of gear movement, either with gesture allowed or with gesture prohibited. Such problems can be correctly solved using either a perceptual-motor strategy (simulation of gear movements) or an abstract strategy (the parity strategy). Participants in the gesture-allowed condition were more likely to use perceptual-motor strategies than were participants in the gesture-prohibited condition. Gesture promoted use of perceptual-motor strategies both for participants who talked aloud while solving the problems (Experiment 1) and for participants who solved the problems silently (Experiment 2). Thus, spontaneous gestures influence strategy choices in problem solving.     

Property content guides children’s memory for social learning episodes

How do children’s interpretations of the generality of learning episodes affect what they encode? In the present studies, we investigated the hypothesis that children encode distinct aspects of learning episodes containing generalizable and non-generalizable properties. Two studies with preschool (N = 50) and young school-aged children (N = 49) reveal that their encoding is contingent on the generalizability of the property they are learning. Children remembered generalizable properties (e.g., morphological or normative properties) more than non-generalizable properties (e.g., historical events or preferences). Conversely, they remembered category exemplars associated with non-generalizable properties more than category exemplars associated with generalizable properties. The findings highlight the utility of remembering distinct aspects of social learning episodes for children’s future generalization.     

Conducting Research in Schools: A Practical Guide

Cognitive development unfolds in many contexts, and one of the most important of these contexts is school. Thus, understanding the school context is critical for understanding development. This article discusses some of the reasons why cognitive developmental researchers might wish to conduct research in schools, describes how to get started conducting research in schools, and offers advice to help make school-based research proceed more smoothly.     

Children's Acquisition of Arithmetic Principles: The Role of Experience

The current study investigated how young learners' experiences with arithmetic equations can lead to learning of an arithmetic principle. The focus was elementary school children's acquisition of the Relation to Operands principle for subtraction (i.e., for natural numbers, the difference must be less than the minuend). In Experiment 1, children who viewed incorrect, principle-consistent equations and those who viewed a mix of incorrect, principle-consistent and principle-violation equations both showed gains in principle knowledge. However, children who viewed only principle-consistent equations did not. We hypothesized that improvements were due in part to improved encoding of relative magnitudes. In Experiment 2, children who practiced comparing numerical magnitudes increased their knowledge of the principle. Thus, experience that highlights the encoding of relative magnitude facilitates principle learning. This work shows that exposure to certain types of arithmetic equations can facilitate the learning of arithmetic principles, a fundamental aspect of early mathematical development.     

Making Connections in Math: Activating a Prior Knowledge Analogue Matters for Learning

This study investigated analogical transfer of conceptual structure from a prior-knowledge domain to support learning in a new domain of mathematics: division by fractions. Before a procedural lesson on division by fractions, fifth and sixth graders practiced with a surface analogue (other operations on fractions) or a structural analogue (whole number division). During the lesson, half of the children were also asked to link the prior-knowledge analogue they had practiced to fraction division. As expected, participants learned the taught procedure for fraction division equally well, regardless of condition. However, among those who were not asked to link during the lesson, participants who practiced with the structurally similar analogue gained more conceptual knowledge of fraction division than did those who practiced with the surface-similar analogue. There was no difference in conceptual learning between the two groups of participants who were asked to link; both groups performed less well than did participants who practiced with the structural analogue and were not asked to link. These findings suggest that learning is supported by activating a conceptually relevant prior-knowledge analogue. However, unguided linking to previously learned problems may result in negative transfer and misconceptions about the structure of the target domain. This experiment has practical implications for mathematics instruction and curricular sequencing.     

Learning about the equal sign: Does comparing with inequality symbols help?

This study investigated whether instruction that involves comparing the equal sign with other relational symbols is more effective at imparting a relational interpretation of the equal sign than instruction about the equal sign alone. Third- and fourth-grade students in a comparing symbols group learned about the greater than, less than, and equal signs and had the opportunity to compare the inequality symbols with the equal sign. Students in an equal sign group learned about the equal sign only. A third group of students served as a control group. Three aspects of students’ knowledge were assessed before and after the lesson: (a) conceptual understanding of the equal sign, (b) equation encoding, and (c) problem solving. Students in the comparing symbols group showed greater gains in conceptual understanding from pretest to posttest than students in the other two groups, and students in the comparing symbols group also scored higher on a posttest that assessed knowledge about inequality symbols and inequality problem solving. Thus, they learned about three symbols in the same amount of time as other students learned about the equal sign alone or not at all. Therefore, an instructional approach involving comparison can be an effective tool for learning about concepts in mathematics.