Virtual School Students’ Reactions to Undertanding Tasks: The Case of Local Extrema Point

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The main aim this study is to examine the understanding of the concept of a local extreme point of differentiable function. Three research questions rose: (a) To what extent do students who study 5 mathematics units in virtual school understand the concept extremum point? (b) To what extent does the concept name (i.e. “extremum point”) influence the mathematical understanding of the concept upon students who study 5 mathematics units in the virtual school? (c) To what extent do students who study 5 mathematics units in the virtual school understand the concept maximum point? To answer the research questions we used qualitative research methods based on semi-structured interviews. The results of the study showed that: All the four interviewers know to find correctly the stationary points, but they don’t distinguish between a stationary point and an Extremum point. All the interviewers mention that the two points are Extremum points instead of stationary points. We have to emphasize that mathematically, a stationary point not necessarily an extremum point. In addition, one of the interviewers confuses an extremum point and a maximum point because of the daily meaning of “Extremum” in Arabic languages. He also reveals Pseudo-Conceptual Behavior; he used sentences like “the tangent equal to 0” instead of the “slope of the tangent equal zero”. It turns out also, that while three out of four interviewers determined the type of the extremum points by using the acceptable value table method correctly, the fourth determines the type of the extremum points by one point and by using the “odd function” characteristic, correctly. According to the requirement to find the extremum point the results showed that two of the four interviewers reveal pseudo-conceptual mode of thinking according to the maximum point concept. They use “slope at the extremum points equal 0”. One of them even used concepts: increasing, decreasing, positive, negative meaninglessly, without clarifying who is positive (negative) or who is increasing (decreasing). One of the important but interesting results is the reaction of the interviewers to the definitions of the maximum point concept. The results showed that 3 students out of 4 gave correct mathematical definition, but; the 4th student gave meaningless definition. We crossed his meaningless answer with his answers to the other questions during the interviews; it turned out that his answers to most of the questions revealed pseudo-conceptual mode of thinking.