Exemplifying a model-eliciting task for primary school pupils

Mathematical modelling is a field that is gaining prominence recently in mathematics educaiton research and has generated interests in schools as well.  In Singapore, modelling and applications are included as process componens in revised 2007 curriculum document (MOE, 2007) as keeping to reform efforst. In Indonesia, efforts to place stronger emphasis on connecting school mathematics with real-world contexts and applications have started in Indonesian primary schools with the Pendidikan Matematika Realistik Indonesia (PMRI) movement a decade ago (Sembiring, Hoogland, Dolk, 2010). Amidst others, modeling activities are gradually introduced in Singapore and Indonesian schools to demonstrte the relevance of school mathematics with real-world problems. However, on order for it to find a place in the mathematics classroom, ther eis a need for teacher-practitioners to know what mathematical modelling and what a modelling task is. This paper sets out to exemplify a model-eliciting task that has been designed and used in both a Singapore and Indonesian mathematics classroom. Mathematical modelling, the features of a model-eliciting task, and its potential and advice on implementation are discussed.


Introduction
Research have found that modelling activities help children in the promotion of important mathematical reasoning processes such as constructing, explaining, justifying, predicting, conjecturing, and representing (Chan, 2008;English & Watters, 2005;Lehrer & Schauble, 2000); reasoning aspects that are valued as a powerful way to accomplishing learning with understanding. Furthermore, studies have shown that children have been able to manage complex mathematical constructs irrespective of their academic mathematics achievement . What is it about mathematical modelling that is asserted to be a promising reform effort in the mathematics classroom? This paper introduces mathematical modelling from a Models-and-Modelling Perspective (MMP) and shares the features of a modelling task (both from a Singapore and Indonesian context) towards understanding the potential of using such tasks. It will also discuss the potentials of the task and suggest teacher scaffolding strategies for task implementation.

Mathematical Modelling
There are various conceptions as to what mathematical modelling is just as there are various diagrammatic representations of the modelling process found in Key words: Mathematical Modelling, Singapore, Indonesia, Model-Eliciting Task, Primary Mathematics, Task Scaffolding literature (e.g., Galbraith, Goos, Renshaw, & Geiger, 2003;Lesh & Zawojewski, 2007;Stillman, 1998). The common stance though is that mathematical modelling begins with a real-world problem or situation and the engagement process results in the representation of such problems in mathematical terms towards finding solutions to the problems. In this paper, we adopt a Models-and-Modelling Perspective  where the modelling task used is termed as a Model-Eliciting Task.
Based on the MMP, students develop internal conceptual models that are powerful but are under-utilized unless they are expressed externally through some representational media as they complete the modelling task. The external projection of the models can be expressed spoken language, written symbols, graphs, diagrams, and experiencebased metaphors. These representations are continually tested and revised as students aim to reach workable solutions. In this sense, the MMP focuses on students' representational fluency through the flexible use of mathematical ideas when students make mathematics descriptions of the problem context and data (Doerr & English, 2003;.
The development of models takes place during the modelling process and is seen as involving mathematizing reality as contrasted with realizing mathematics . The mathematical modelling process involves several stages.

Features of a Model-Eliciting Task
The mathematical modelling endeavour (involving a model-eliciting task) is commonly termed as a model-eliciting activity. During the modelling process, rich mathematical discourse and reasoning are made manifest during when students develop models. This is facilitated by the task when students confront it. As such, the design of the modelling task needs to ensure that the mathematics embedded can be surfaced when students are deeply engaged in it. Lesh, Cramer, Doerr, Post, and Zawojewski (2003) asserted a model-eliciting task is designed based on six instructional design principles: (1) the reality principle (i.e., elicits sense-making and extension of prior knowledge); (2) the model construction principle (i.e., warrants the need to develop a mathematically significant construct); (3) the self-evaluation principle (i.e., requires self-assessment); (4) the construct documentation principle (i.e., requires students to make visible their thinking); (5) the construct generalization principle (i.e., sharable, adaptable, reusable in other similar situations); and (6) the simplicity principle (i.e., the simplicities of the problem-solving situation). Thus the construct 'model-eliciting' circumscribes a problem-solving situation, its mathematical structure, the problem-solving processes and the mathematical models generated that are invoked by the problematic situation. The features of a modeleliciting task thus are what the principles aim to promote. The modelling tasks were set as a group work for students to complete over three hours on three consecutive mathematics lessons. This offered a good platform for students to develop their mathematical communications skills during group and whole class discussion (English, 2010b). Students were required to justify their solutions and check the reasonableness of their solution. It was also ideal that students revisit their assumptions and the conditions set in the task in order to check if the solution needed further refinement. However, numerous studies on mathematical modelling have identified that validating results and re-interpreting the results in the real-world context proved to be challenging within the time constraints of classroom implementation (English, 2010a;Galbraith & Stillman, 2006). The teachers involved in the implementation of the two tasks in the Singapore and Indonesian classrooms tried their best in helping their students refine the models within the constraints.

Suggestions to Task Scaffolding
This section will propose some teacher scaffolding suggestions for the implementation of the Bus Route Task which may apply across the Singapore and Indonesian classroom contexts. Firstly, it is important that the teacher sets aside time for students to discuss their interpretations of the context provided in the problem situation. For instance, students should be encouraged to brainstorm on what "efficient" means within the given context and how they would determine "efficiency" through use of mathematical measurements. At this stage, it is plausible that students would be able to suggest efficiency based on time, speed, and cost of travelling, well within their mathematical pre-requisites of the task. The scaffolding strategies of the teacher would be concentrating on questioning for ideas, exploration of possible choice of mathematical concepts and skills useful for the task, and helping students determine their next stage in the problem solving process. Students need to be reassured that multiple interpretations of "efficiency" are recognised and differing solution pathways for mathematical diversity are thus encouraged. They should also be allowed to put forth questions about task for clarifications.
Secondly, it is also important to help students understand the role of assumptions, conditions, and variables within the task. Assumptions are made about the context of the task (e.g., the bus travels at a constant speed, regular traffic conditions) so as to narrow down the focus for choice of appropriate mathematical approach. For example, it may be logical for students to assume regular traffic conditions (i.e., no vehicle break-downs, traffic lights are working) because data on traffic conditions vary day-to-day and it may not be readily available for the feasibility of classroom implementation. Conditions of the task need to be articulated and recorded because it would be necessary for students to decide on the parameters in which they would like to work within in order to develop a "mathematically valid" model as a proposed solution to the problem. For instance, one of the conditions in the Bus Route Task (Singapore version) where students can work with was to discuss the problem situation with the peak hour (i.e., 06 30 -07 30) of travelling stipulated, based on the starting time of their school. This helps in decision making about the frequency of bus services, along with other real-world considerations of day-to-day bus journeys. Variables of the tasks include the ways in which efficiency is measured (e.g., time, speed, cost). Students can be prompted to explore how these could be determined from the given information (i.e., map) and whether more information needs to be collected for more accurate calculations. Furthermore, these variables are also related to each other. Students can work on how to develop a more sophisticated mathematical model holistically connecting the variables to help determine the most efficient route from the given three routes.
Last and not least, model refinement requires continue evaluation of the mathematisation process and critique of initial models based on the validity and applicability to the given situation. It is crucial for students to be encouraged to reflect and review their initial models in several ways. For one, the appropriateness of assumptions made, conditions set, variables chosen for exploration. These form the foundation of the mathematical model or representation of the students' arguments and reasoning for their choice of the most efficient bus route. Students have to decide if the final decision made is a logical one, based on the above. Another way where students could review their initial model is to check their mathematical calculations for accuracy and reasonableness, whether these calculations present substantial support for their decision. Finally, students can reflect on their models to determine if it meets the requirements of the task. They need to recognise the possible limitations of their models for applications across other similar contexts (e.g., the most efficient train route) based on the parameters they had initially chosen to work with.

Conclusion
This paper serves three purposes to: (a) introduce the Model-Modelling Perspective (MMP) in mathematical modelling from a practice-oriented point of view; (b) exemplify a model-eliciting task (The Bus Route Task); and (c) discuss the potentials of the task and teacher scaffolding strategies. However, the implementation of modelling activities in classrooms can be challenging (Ng, 2011). Teachers who are used to prescriptive teaching approaches would need to explore other scaffolding approaches to more student-centred focus, encouraging multiple interpretations and solution pathways. Teacher beliefs about mathematics and how mathematics should be taught and applied may be challenged as mathematical models can take many forms; tables, graphs, and drawings, amidst abstract algebraic representations and calculations. In view of bridging the gap between the potentials of mathematical modelling and the recognition of these potentials by teachers, Ng (2011) recommends building teacher repertoire in two specific areas such as teacher questioning techniques during scaffolding and facilitating a conducive modelling climate in the classroom. The former recommendation has been elaborated in detail above based on the Bus Route Task. The latter recommendation is just as crucial for both teachers and students engaged in mathematical modelling as a positive climate which encourages inquiry, discussion, and fruitful mathematical arguments enhances sophisticated mathematical thinking.