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In summary the findings of this study confirm the need to incorporatein teaching methods those procedures that facilitate students’ problemsolvingperformance by encouraging them to make full use of their availablecognitive and metacognitive resources. This involves the provision oflearning situations that give experience in knowing how to make conscioususe of available metacognitive information in order to predict future events,and to plan appropriate responses in order to reach a problem solution. Anastute choice of classroom problems with associated follow-up time for reflectionand discussion of both appropriate and inappropriate approaches tosolutions is one necessary inclusion. Such reflection and discussion shouldinclude the discussion of metacognitive activities that were (or should havebeen) used. It is crucial that students enhance their abilities to focus on theselection of relevant information, the construction of a feasible model, andAPPLYING MATHEMATICS AND METACOGNITION 187the testing of that model, as well as the employment of monitoring andverification strategies.In essence, metacognition concerns decision making and decision makinginvolves using both strategic and temporal resources to determine theappropriate response. Both the speed and accuracy of this process of responseselection is influenced and linked to the number of decisions tobe made, the number of options to be selected from, the total time perceivedby the student to be available for decision making and the perceivedtime-cost associated with incorrect decisions. As has been observed inthis investigation, students’ decision making about temporal resources iscrucial to their success and students need sufficient practice in problemsolving during an extended time interval in order to develop appropriatemechanisms for dealing with efficient use of time.Finally, two further general observations follow from the outcomes ofthe research.Firstly, the results indicate that a symbiotic relationship exists betweenthe cognitive, metacognitive, and affective elements of student problemsolvingactivity. To attempt to treat affect as a separate issue from cognitionwould be to go against all the evidence this study has provided. Teachingapproaches are needed through which both can be made overt andinteracting components of learning contexts.Secondly, nothing in the research suggests that problem-solving willbe enhanced by a focus on so-called generic skills. We have noted thatteaching which encourages compartmentalised knowledge and stereotypedexpectations works against the development of problem-solving expertisewhich is consistent with studies elsewhere. However, equally, studentswere defeated through lack of conceptual understanding and automatedfacility with essential mathematical skills. More knowledge, rather thanless is required, but it must be introduced and applied in problem contextsin which the various metacognitive strategies are consciously brought tobear.
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