On recognizing and formulating mathematical problems

When mathematics is used to help people cope with real-life situations, a three-stage intellectual process is involved. First, a person becomes aware of a problem-situation which stimulates him to generate a problem-statement, a verbal story-problem. This may be in writing, expressed orally, or merely thought and evidenced by other behavior. Second, he transforms the verbal problem-statement into a mathematical formulation. Third, he analyzes this mathematically stated problem into subproblems to which the solution is more immediate.We propose an operational instrument for assessing how a person moves from the first to the second and third stage. This involves coding questions asked, actions taken, hesitancy and latency. Results of using this procedure in experimental situations were used to determine reliability. We also propose and compare methods for effecting and accelerating motion from the first to the third stage. These involve operant conditioning, need arousal cues, controlled verbal instructions. Experience with one such technique to teach problem-formulation in actual teaching practice is reported.The theoretical basis for these experiments is developed by showing how to program a computer to go through these stages for a reasonably large and interesting class of problem-situations to which mathematics is applicable.This research was partially supported by an Office of Education Grant OEG5-72-0050(509).     

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Ohne Zusammenfassung     

Effects of age at school entry (ASE) on the development of non-cognitive skills: Evidence from psychometric data

We identify effects of age at school entry (ASE) on the development of child temperament. Our analysis is based on psychometric measures from a longitudinal cohort study of children in the Rhine-Neckar region in central Germany. In children with a higher ASE due to a birthday late in the year, we find more favorable outcomes with respect to several temperamental dimensions: these children are more persistent and less often hyperactive. The findings are robust if we control for the respective temperamental dimension before entering school. We also show that the ASE effect on persistence is stable over time by comparing the children at age eight and age eleven, after the children have entered Germany's segregated secondary-school tracks. At age eleven, we additionally find significant ASE effects on adaptability to change. Overall, the results point to a high degree of malleability in the considered non-cognitive skills after school entrance. By contrast, we do not find a significant effect of ASE on cognitive skills as measured by IQ.     

Parental resources, parental stress, and socioemotional development of deaf and hard of hearing children

In recent years, empowerment and resource orientation have become vital guidelines for many of the sciences. For the field of deaf education, it is also highly important to look carefully at these guidelines if we are to acquire a better understanding as regards both the situation of the parents involved and the development of the deaf and hard of hearing children themselves. A resource-oriented approach to deaf education has therefore proved especially helpful. If both the theoretical and practical aspects of educating deaf and hard of hearing children are to benefit, research on parental experience with deafness and research on the socioemotional development of the children must always be combined and studied in the context of resource availability. In a study of 213 mothers and 213 fathers of deaf and hard of hearing children, we used an array of different questionnaires (PSI, SDQ, SOC, F-SozU, etc.) to examine the correlation between parental resources, sociodemographic variables, parental stress experience, and child socioemotional problems by way of a path analysis model. The results show that high parental stress is associated with frequent socioemotional problems in the children, thus emphasizing the importance of a resource-oriented consulting and support strategy in early intervention, because parental access to personal and social resources is associated with significantly lower stress experience. Child development seems to profit enormously from a resource-oriented support concept. In addition, the results confirm two earlier findings: parents with additionally handicapped children are especially stressed and the child's communicative competence makes for a more sound prediction than its linguistic medium (spoken language or sign). The path models for mothers and fathers agree in all essential factors. The results are discussed with a view to their meaning for pedagogical practice, and recommendations for further research are given (longitudinal data, more representative samples, cochlear implant).     

Prediction of the P-leaching potential of arable soils in areas with high livestock densities

Due to long-term positive P-balances many surface soils in areas with high livestock density in Germany are oversupplied with available P, creating a potential for vertical P losses by leaching. In extensive studies to characterize the endangering of ground water to P pollution by chemical soil parameters it is shown that the available P content and the P concentration of the soil solution in the deeper soil layers, as indicators of the P-leaching potential, cannot be satisfactorily predicted from the available P content of the topsoils. The P equilibrium concentration in the soil solution directly above ground water table or the pipe drainage system highly depends on the relative saturation of the P-sorption capacity in this layer. A saturation index of 〈20% normally corresponds with P equilibrium concentrations of〈0.2 mg P/L. Phytoremediation may reduce the P leaching potential of P-enriched soils only over a very long period.     

Value orientations as determinants and outcomes of conflicts between on-task and off-task actions in the classroom

Off-task behavior in the classroom was conceptualized as a manifestation of students pursuing goals they bring into the classroom aside from achievement goals. Regulation during on-task and off-task behavior in action conflict scenarios was elaborated on using the constructs motivational interference and flow. It was argued that achievement and well-being values that students hold can be determinants as well as outcomes of these conflict experiences. Data from 697 students (mean age 13.43) was collected at two time points within one school year. Results supported a reciprocal model in which value orientations (t1) lead to conflict variables (t2) as well as conflict variables (t1) to value orientations (t2).