Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

ADOLESCENT DAUCHTER/MIDLIFE MOTHER DYAD A New Look at Separation and Self-Definition

The mother-daughter relationship is seen as complex, unique, and emotionally charged, with adolescence being the most difficult period for this dyad. Yet much of the literature has highlighted the struggle of adolescents, while underemphasizing that many of their mothers themselves are confronting their own developmental issues at this time. In this article, the developmental tasks and dyadic issues of adolescent daughters (aged 13–19) and their midlife mothers (aged 35–55) are examined from a psychodynamic perspective. The authors posit that both mother and daughter are experiencing a lifecycle crisis of separation and self-definition. Drawing from psychoanalytic and lifespan developmental literature as well as their own clinical experience, the authors propose that the current generation of mothers and daughters requires a new look at old models, parts of which may still be relevant and parts of which need to be revised or dropped. The issues of competition and rebellion as they relate to separation and self-definition are also addressed.