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Evidence of implemented anticipation in mathematising by beginning modellers

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Abstract

Data from open modelling sessions for year 10 and 11 students at an extracurricular modelling event and from a year 9 class participating in a programme of structured modelling of real situations were analysed for evidence of Niss’s theoretical construct, implemented anticipation, during mathematisation. Evidence was found for all three proposed aspects. With respect to Niss’ s enablers of ideal mathematisation explaining unsuccessful mathematisations, flaws in the modelling of the year 10–11 students were related to the required mathematics being beyond the knowledge of the group members or poor choice of the particular mathematics to use in the modelling context; whilst unsuccessful attempts at mathematisations in the year 9 class were related to inability to use relevant mathematical knowledge in the modelling context. The necessity of these enablers as requisites for modelling, particularly in a classroom context, needs further investigation.

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Notes

  1. All references to Niss refer to Mogens Niss except where explicitly stated otherwise.

  2. Martin Niss (2012) alludes to a similar need for further research into mathematisation processes in the context of viewing real world problem solving as modelling in Physics.

  3. Student names used throughout this article are pseudonyms.

  4. In the Forum, Singaporean students with their Australian hosts (including our focus group) visited tourist theme park, Movie World, to collect data using various technological tools about amusement rides such as the Batwing Spaceshot Ride. Students posed their own problems about the rides and tried to solve these using mathematical modelling [Video, Day 3 Forum].

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Acknowledgments

Some data used here were collected by the authors as part of the RITEMATHS project (an Australian Research Council funded linkage project—LP0453701). The year 9 task was developed from a task originally designed by Ian Edwards, Luther College.

Author information

Authors and Affiliations

  1. Australian Catholic University (Ballarat), PO Box 650, Ballarat, VIC, 3350, Australia

    Gloria Stillman

  2. Australian Catholic University (Melbourne), 115 Victoria Pde, Fitzroy, VIC, 3065, Australia

    Jill P. Brown

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  1. Gloria Stillman
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Correspondence to Gloria Stillman.

Additional information

This article is an extended version of the following conference paper including further data from a different context and further analysis:

Stillman, G. and Brown, J. (2012). Empirical evidence for Niss’ implemented anticipation in mathematising realistic situations. In J. Dindyal, L. P. Cheng and S. F. Ng (Eds.), Mathematics education: expanding horizons. Proceedings of the 35th Annual Conference of the Mathematics Education Research Group of Australasia (MERGA), Singapore (Vol. 2, pp. 682–689). Adelaide: MERGA.

Appendices

Appendices

Appendix 1: solution elements to Shot on Goal

An outline of essential steps in the solution for a run line 18 m from the near goal post follows. Table 6 shows calculations obtained using the LIST facility of a TI-83 Plus graphing calculator. Calculations are shown for positions of the goal shooter at distances from the goal line of between 14 and 28 m along the run line (see Fig. 5). Width of goalmouth is 7.32 m. (the students carried out calculations from 1–30 m) The maximum angle is highlighted in the table, which was generated by the LIST facility of the calculator, following hand calculations to establish a method.

Table 6 Sample calculations from a typical solution to Shot on Goal
Full size table

A graph (Fig. 8) 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 (9.73° at 21.35 m from the goal line) or an algebraic model, y = tan − 1(25.32/x) − tan − 1(18/x) can be constructed and the maximum found using graphing calculator operations.

Fig. 8
figure 8

Angle (α) to be maximised

Full size image

Appendix 2: selected interview questions

Q10.1 Could you have written an algebraic model for your answer to Task 11?

Q10.2 One group [Group 6] said that the position of the spot for the maximum shot on goal was 3 m more than the distance of the run line from the goal post. What type of mathematical relationship is this?

Q10.2.2 Draw me a graph to show it.

Q10.2.3 How could a coach of a defending player use this information?

Q11.1 Do you like doing challenging tasks like this in maths? Can you elaborate on that? [Prompt: What makes it challenging/not challenging?]

Q11.1.1 What types of maths tasks do you prefer?

Q11.2 What is the purpose of tasks such as Cunning Running and Shot on Goal?

Q11.3 Do you like the fact these tasks are set in a real world context?

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Stillman, G., Brown, J.P. Evidence of implemented anticipation in mathematising by beginning modellers. Math Ed Res J 26, 763–789 (2014). https://doi.org/10.1007/s13394-014-0119-6

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