Feedback influences children's reasoning about math equivalence: A meta-analytic review

Cross-notation knowledge of rational numbers predicts fraction arithmetic

BACKGROUND

Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered.

AIMS

To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally.

SAMPLE

Three hundred and fifty-four Chinese children (Mage  = 112.1 months).

METHODS

During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division.

RESULTS AND CONCLUSIONS

Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.

Strategy adoption depends on characteristics of the instruction, learner, and strategy

Why do people change their strategies for solving problems? In this research, we tested whether negative feedback and the context in which learners encounter a strategy influence their likelihood of adopting that strategy. In particular, we examined whether strategy adoption varied when learners were exposed to a target strategy in isolation, in conjunction with their own current strategy, and in conjunction with another novel strategy. We also investigated the roles of individual differences, including learners’ need for cognition and their confidence in their current strategies. In Study 1, undergraduate participants who encountered a target strategy in isolation were more likely to adopt it than participants who encountered it in the context of their own current strategy. Negative feedback, low confidence, and high need for cognition also predicted greater adoption. In Study 2, we examined whether rates of strategy adoption depended on the target strategy itself. Indeed, participants were more likely to adopt one strategy than the other, and the effects of feedback also varied across strategies. Individual differences—need for cognition and confidence—also influenced patterns of strategy adoption. These results suggest that strategy adoption depends on the confluence of many factors, including the context in which a target strategy is introduced, characteristics of the learner, and characteristics of the strategy itself.

Understanding Strategy Change: Contextual, Individual, and Metacognitive Factors

Learning, development, and response to instruction often involve changes in the strategies that learners use to solve problems. In this chapter, our focus is on mathematical problem solving in both children and adults. We offer a selective review of research on three classes of factors that may influence processes of strategy change in mathematical problem solving: contextual factors, individual factors, and metacognitive factors. Contextual factors involve information that learners encounter in the learning context, such as feedback about prior strategies and examples of alternative strategies. Individual factors involve the abilities, dispositions, and knowledge that learners bring to the learning context. Metacognitive factors involve knowledge about strategies and factors that affect the application of strategies-including perceptions of problem difficulty, confidence in the strategies one already knows, and judgments about the qualities of alternative strategies. These factors operate both independently and in combination to influence learners' behavior. Therefore, we argue that scientific progress in understanding strategy change will require comprehensive conceptual models that specify how different factors come together to explain behavior. We discuss several such models, including vulnerability-trigger models, cumulative risk models, and dynamic systems models. Research guided by such models will contribute to greater progress in understanding processes of strategy use and strategy change.