«Mathematical applications and modelling of real world situations are receiving increased emphasis in several curricula internationally at the moment ...»
Empirical Evidence for Niss’ Implemented Anticipation in
Mathematising Realistic Situations
Gloria Stillman Jill P. Brown
Australian Catholic University (Ballarat) Australian Catholic University (Melbourne)
Mathematisation of realistic situations is an on-going focus of research. Classroom data from
a Year 9 class participating in a program of structured modelling of real situations was analysed for evidence of Niss’s theoretical construct, implemented anticipation, during mathematisation. Evidence was found for two of three proposed aspects. In addition, unsuccessful attempts at mathematisations were related in this study to inability to use relevant mathematical knowledge in the modelling context rather than lack of mathematical knowledge, an application oriented view of mathematics or persistence.
Mathematical applications and modelling of real world situations are receiving increased emphasis in several curricula internationally at the moment (e.g., Ministry of Education, 2006). The teaching and learning of applications and modelling has been the subject of on- going research for many years (Blum et al., 2007; Kaiser et al., 2011; Niss, 2001). Two of the areas receiving on-going attention have been the mathematisation (i.e., translation into mathematics) of the idealised problem formulated from the real situation and the reverse process, de-mathematisation (i.e., interpretation of mathematical outputs of modelling in terms of the real situation). Recently, Niss (2010) has added to the theoretical models informing research in these areas.
The notion of mathematising has been promoted by the mathematics frameworks for PISA 2003, 2006, 2009 and 2012 assessing the mathematical literacy of 15 year olds roughly at the end of compulsory schooling. In the earlier frameworks what seemed to be presented as mathematisation was the entire mathematical modelling cycle (see OECD, 2009, p. 105). The mathematical modelling cycle is described as a key feature of the draft PISA 2012 mathematics framework (OECD, 2010) but mathematising is less prominent. It is one of seven fundamental mathematical capabilities that underpin the framework.
Mathematising is taken to mean the fundamental mathematical activities that are involved in “transforming a problem defined in the real world to a strictly mathematical form…or interpreting or evaluating a mathematical outcome or a mathematical model in relation to the original problem” (OECD, 2010, p. 18). The latter would be termed “de- mathematisation” by Niss (2010). Mathematising is thus seen in PISA as one of the cognitive capabilities that can be learnt through schooling so as to enable students to understand and engage with the world in a mathematical manner. The purpose of this paper is to demonstrate whether or not there is empirical evidence for one of the main explanatory constructs of Niss’s model of mathematisation processes (Niss, 2010, p. 57).
Theoretical Framework Researchers in the area of applications and modelling often use diagrams of the “socalled” modelling cycle to discuss what appears to be happening at a task and mental level during modelling. These are mere simplifications but they are a useful means of communicating amongst international researchers. The diagram used by Niss (2010) is reproduced in Figure 1. This shows two quite disparate domains in this particular representation of modelling: the extra-mathematical domain (i.e., the real world situation In J. Dindyal, L. P. Cheng & S. F. Ng (Eds.), Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia). Singapore: MERGA.
© Mathematics Education Research Group of Australasia Inc. 2012 and the modeller’s idealisation of this) and the mathematical domain. Others such as Stillman (1998) argue for the importance, particularly in the context of schooling, of also including in these diagrams a representation of the blending of the real world and mathematical world. Idealisation occurs through making assumptions and identifying elements in the situation which are of interest which are then formulated, that is, specified into a problem statement which may take the form of a question. The mathematical domain includes the mathematical model that has been made of the situation, and mathematical artefacts (such as graphs and tables) that might be used in solving the mathematical model.
The idealised situation is mathematised through a process of translation into mathematics and similarly mathematical outputs need to be de-mathematised, that is, interpreted in terms of the idealised situation and the real situation which was the launching point for the modelling in the first place.
In order to produce a theoretical model of the mathematisation process, Niss (2010) introduces the construct, “implemented anticipation” (p. 54). Successful mathematisation, according to Niss, involves anticipating what will be useful mathematically in subsequent steps of the cycle and implementing that anticipation in decision making and carrying out actions. Firstly, the idealisation of the real situation from the extra-mathematical domain involves implementing decisions about what elements or features are essential as well as posing any related question or statement of the problem in light of their anticipated usefulness in mathematising. Secondly, when mathematising this formulation of the problem situation the modeller needs to do this by anticipating mathematical representations and questions that, from previous experience, have been successful when put to similar use.
Thirdly, when anticipating these mathematical representations, the modeller has to be cognisant of the utility of the selected mathematisation and the resulting model in future solution processes to provide mathematical answers to the mathematical questions posed by the mathematisation. This involves anticipating mathematical procedures and strategies to be used in problem solving after mathematisation is complete. Thus successful implemented anticipation involves a three step foreshadowing process.
To be able to successfully use implemented anticipation in mathematising a real or realistic situation modellers need to: (1) possess relevant mathematical knowledge, (2) be capable of using this when modelling, (3) believe a valid use of mathematics is modelling real phenomena, and (4) have persistence and confidence in their mathematical capabilities (Niss, 2010, p. 57). Clearly, this is a challenging process. It is reasonable to expect that modellers, especially new modellers, would experience this challenge and have difficulties related to the three foreshadowing aspects of implemented anticipation. These difficulties 681 might be explained by these four requisites. Both successful and unsuccessful attempts at modelling or applying mathematical knowledge to real situations are opportunities for developing deeper “metaknowlege about modelling and mathematisation, in particular” (Schaap, Vos, & Goedhart, 2011, p. 145) and thus should be the foci of any study of mathematisation.
Researchers (e.g., Galbraith & Stillman, 2006; Schaap et al., 2011; Stillman, Brown, & Galbraith, 2010) have found evidence of beginning modellers in secondary schools having difficulties with mathematising because of impeding formulations of the problem statement.
However, no one to date has attempted to use Niss’s model in analysing classroom data. In this paper we will attempt to use the model as the basis for our analysis of data from a year 9 class of beginning modellers who had participated in a program of quite structured modelling over one year. To operationalise the mathematising construct for research purposes we take as its starting point the formulated statement of the problem situation. This may or may not be formulated as a question. The end point of mathematising will be the mathematical model.
The Study As part of the RITEMATHS project, a series of three modelling tasks were used in a class of 21 Year 9 students. The data used in this paper relate to the implementation of the last of these tasks, Shot On Goal. Names used are pseudonyms. In brief, the task uses a soccer context (see Figure 2) where the modelling problem involves optimising a position for an attacking player to attempt a shot on goal whilst running parallel to the sideline (for details, see Stillman et al., 2010). The teacher anticipated that students might have difficulty with task formulation so he used a nearby soccer field to provide an outdoor demonstration at the beginning of the lesson sequence. The teacher used a rope parallel to the sideline (i.e., perpendicular to the goal line) as a run line for several students to run down and stop when they had the best shot on goal. One student then stayed on this run line, marking the average of their estimates. The process was repeated for a run line closer to the goal. Students then discussed the effect on the average position for the best shot before returning to the classroom to begin the task. They worked in 7 groups of 2 to 4 students. The teacher allocated each group a particular distance for their run line from the near goal post.
Figure 2. Student diagram of Shot on Goal
The main part of the task (Tasks 1 to 10) offered structured scaffolding. Two questions at the end included no details of how to mathematise or approach them mathematically (see Figure 3). Fifteen students attempted part or all of these. Tasks 11 and 12 are the focus here.
The task was implemented over three lessons on consecutive days, and these final tasks were 682 attempted during the third lesson. Data collected which is relevant to the focus of this paper consisted of transcriptions of 3 video-recordings and a further audio-recording of groups working on the tasks, individual task scripts from 15 students and interview responses of nine of these students (see Appendix for example questions).
TASK 11 – CHANGING THE RUN LINE Investigate whether the position of the spot for the maximum shot on goal changes as you move closer or further away from the near post. [Collect data from other students results to help you see if there are any patterns in the position of spots for the maximum angle.] What does the relationship between position of the spot for the maximum shot on goal and the distance of the run line from the near post reveal?
TASK 12 – CHANGING THE RULES Soccer is often a low scoring game. Some have suggested that it would be a better game if the attackers had more chance of scoring, so the width of the goal mouth should be increased. Others claim it would be a more skilful game if the goal keeper was given more of a chance to stop goals by reducing the width of the goal mouth. Investigate what effect changing the width of the goal mouth would have on the position of the maximum shot on goal for the run lines and give your recommendation.
Figure 3. Tasks for mathematising at end of Shot on Goal
The research questions addressed in this paper are:
1. Is there evidence for the existence of Niss’s implemented anticipation in mathematisations occurring in the classroom?
2. Do Niss’s four requisites explain unsuccessful mathematisations?
To analyse the data student responses to Tasks 11 and 12 were classified as (a) mathematisations showing (i) successful or (ii) unsuccessful implemented anticipation (b) qualitative statements (i) identifying a relationship between relevant variables or (ii) identifying variables but not supported, in either case, by any mathematical objects or representations and (c) incomplete as only raw data with no translation into mathematics or interpretations recorded. When a(ii), (b) or (c) classifications were given, interview data and video and audiotape data were scrutinised in detail for explanations. These were then compared to Niss’s four requisites for successful implemented anticipation.
Results and Analysis Fifteen students from 6 different groups attempted Task 11. Group 5 was a long way behind other groups because of difficulties with formulation of the task (see Stillman, et al., 2010). Two students from Group 7 were also behind because of difficulties they were experiencing using their graphing calculator. These students did not attempt tasks 11 or 12.
Ned from Group 6 recorded his results of collecting data for Task 11 in a partially ordered table which he called a “commentary table”. He correctly identified the three relevant variables “run line to goal post”, “dist along RL”, and “Angle” (“Max angle” used in recording the raw data). There was an error in the data which Ned and Len created themselves and recognised at the time of data collection as they decided Group 4’s angle for a 14 m run line distance maximised at 18 m as the data were recorded to only one decimal place and thus the angle was the same for 16 to 19 m. They did not correct this datum as they felt the rest of the data supported their conclusion.
After tabulating the data in this fashion from Len’s systematically organised two column
recording of the data and scrutinising it for less than 15 seconds, Ned exclaimed:
Ned: Check it out. The distance along the run line you have to be is, with the exception of-f-f 14, ah, is 3 more than the distance, what the distance is from the run line to the goal post. [Video, Group 6] 683 This was correct. In recording his interpretation of the data Len added: “It also proves our theory that the closer you come to goal, the closer you have to be on your run line to achieve maximum angle.” Neither student recorded this symbolically. However, when interviewed Len, wrote “y = x + 3 where maximum angle is = y and distance from near post = x” in response to being asked to write his answer algebraically. The group’s work was classified as successful implemented anticipation (ai).
Ozzie and Jaz of Group 2 used correctly ordered tables with Jaz also identifying and
labelling the variables in the columns correctly on his script. Ozzie concluded: