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Abstract
The goal of this study was to investigate high school mathematics teachers instructionalpractices with contextual problems and their notions of terminology associated with suchproblems. The study was motivated by the literature on research and teaching that calls for theincorporation of contextual problems in high school mathematics. Teaching with such problemsis not easy. Yet little research has focused on teachers practices with these problems. Inaddition, the increased inclusion of contextual problems in the secondary mathematicscurriculum has introduced terminology associated with such problems. Three suchterms mathematics in context, applications of mathematics, and mathematical modeling havebeen defined and redefined in the literature, and their meaning has been confounded by disputesabout what constitutes the real world and reality. As a result, mathematics educators andteachers alike have somewhat ambiguous notions about these concepts.Six teachers who taught with contextual problems on a near-daily basis in two schools inthe southeastern United States agreed to participate in this study. All teachers were interviewedconcerning their notions of terminology associated with contextual problems and their practiceswith such problems. Three of them were also observed teaching lessons that incorporatedcontextual problems and interviewed about those lessons. A grounded theory approach andconstant comparative analysis were applied to the data.The results of this study highlighted that these teachers notions of terminology varied,but that they defined and differentiated the terms along a number of dimensions including thedegree of reality and the role and complexity of the mathematics in the problem. The data alsoshowed that three features of the teachers instruction with contextual problems were importantin shaping the lessons how they (a) adapted and used problems from other sources, (b) helpedthe students formulate the problem, and (c) balanced time and attention to the context and themathematics. Four conditions were identified that enabled the teachers to do this work:technology, commitment, community support, and beliefs. The results of this study haveimplications for the preparation and support of high school teachers who incorporate contextualproblems in their teaching, and for mathematics educators who engage in scholarly writing onthis subject.