Identifying Challenges within Transition Phases of Mathematical Modeling Activities at Year 9

Abstract

The Galbraith, Stillman, Brown, and Edwards Framework (2007) for identifying blockages hindering progress in transitions in the modeling process is applied to a modeling task undertaken by 21 Year 9 students. The Framework identified where challenges occurred; but, because some blockages proved to be more robust than others, another construct “level of intensity” was added. The blockages described here occurred during the formulation phase of the modeling cycle. We infer that blockages induced by lack of reflection, or by incorrect or incomplete knowledge, are different in nature and cognitive demand from those involving the revision of mental schemas (i.e., cognitive dissonance). The nature and intensity of the blockage have consequences for teacher intervention and task implementation.

Appendix: A Possible solution

An outline of essential steps in the solution follows. Table 33.1 shows calculations obtained using the LIST facility of a TI-83 Plus graphing calculator. Calculations are shown for positions of the goal shooter at (typical) distances from the goal line of between 2 and 30 m; along a run line that is 20 m from the near goalpost (see Fig. 33.6). Width of goalmouth is 7.32 m. (The students encompassed more calculations than these, increasing the distance along the run line beyond 26 m, and the lateral position of the run line varied for each group.) The maximum angle and its reference points are highlighted in the table, which was generated by the LIST facility of the calculator (following hand calculations to establish a method).

Fig. 33.6

Angle (α) to be maximised, and function for (α) passing through scatter plot for run line 20 m from near goal post

Table 33.1 Sample calculations from a typical solution to Shot at Goal

A graph (Fig. 33.6), showing angle against distance along the run line is drawn, using the graph plotting facility of the calculator. Additional points near the maximum can then be calculated, to provide a numerical approach to the optimum position (8.90 ° at 24 m) from the goal line – a suitable approach for early or middle secondary students – or an algebraic model can be constructed and the maximum found using the graphing calculator operations. An interpretive statement of advice to an attacking player as strategy adviser could be “your optimal angle will be at 24 m but shooting in the range 30–18 m would be fine. The closer you are towards the goal with a clear shot in this range is best as the further away when you shoot the more chance your shot will be intercepted”.