The seventh stage of the modelling process:
reflection and mathematical modelling

‘Problem-based learning places emphasis on what is needed, on the ability to gain propositional knowledge as required, and to put it to the most valuable use in a given situation. It does not, therefore, deny the importance of ‘content’ - but it does deny that content is best acquired in the abstract, in vast quantities, and memorized in a purely propositional form, to be brought out and ‘applied’ (much) later to problems. Problem-based learning requires a much greater integration of knowing that with knowing how’ (Margetson 1991, p. 44).

Mathematical modelling is a problem-based learning approach. The modelling cycle consists of seven stages (Figure 1). Briefly the stages are: one, consider an ill-structured general problem (with little specific data); two, make assumptions and decide what data are needed; three, formulate the mathematical problem; four, solve the mathematical problem; five, interpret the solution; six, verify the model and consult the experts for better models; and seven, examine your thinking, make predictions, provide explanations and produce a report. Modelling, however, is not as clean as this seems and it is not merely a one-directional progression through the seven stages. In reality it is often a reiterative process of considering the problem, adjusting the assumptions, hypothesis testing, evaluation and so on, which is very similar to the processes involved in metacognition. Figure 1 attempts to indicate this by using multiple paths.

A metacognitive view of learning emphasizes reflective thinking and involves analyzing what you are doing by monitoring and evaluating your learning experiences. It includes looking back over your learning process and comparing it to other people’s thinking and work. It is an intentional exploration and discovery of learning and it is also a combination of both individual active reflection and the ‘social constructivist’s’ (Dengate & Lerman 1995, p. 31) emphasis upon collective reflection through group interactions. A further elaboration of reflection as a form of metacognition can be found elsewhere (Flavell 1987) so I will list some of the main components needed for enhancing reflective thinking. There are four of interest to us: the opportunity for students to decide what they need to learn; the opportunity for students to decide how they should learn; the opportunity for students to decide how self-assessment should be conducted; and the opportunity for individual and social reflection. The modelling process is an excellent instrument for fostering all four components.

So let us examine how these ingredients can be mixed. Before beginning the modelling process, it is a good strategy to organize the class into groups, assign roles to the members, and set classroom and group rules. Mathematics teachers these days are well versed in dividing a class into groups and allowing them to work together. It is a recognition that the effort to communicate is one of the most powerful motivations in sorting out thinking and understanding. There are many ways of forming groups (friendship, mixed ability, random, teacher assigned, etc.), and assigning tasks, and most teachers have a preferred method. An example is included below and students are permitted to assume two roles if there is a group of three (for example, recorder and presenter can be combined). The students will also need a copy of the stages of the modelling process, as this helps them organize their thinking and their report. It is also necessary to point out that during the modelling cycle there will be times when the class will work in small groups, and other times as a whole class.

A set of rules to be followed by all students in the classroom during groupwork

Roles for table group members

Equipment monitor - When directed by the teacher you are responsible for the collection and return of all materials used in the lesson. You are also to supervise the cleaning up, and ensure that every member of the group does his/her share of this work. (Note that the class mathematics textbook is regarded as a resource.)

Question officer - It is your job to ask questions of the teacher, as no other member of the group may directly approach the teacher. At least two people in your group must agree upon the question before you may ask it. In some lessons you may be limited to a maximum of five questions, so think before asking. (Note that this may also include asking the teacher to explain a particular mathematical model.)

Recorder - It is your job to keep track (on paper) of the work of the group. Sometimes you will keep just a brief diary of events, or complete an impact report, or at other times you may prepare a group report. (Note that the recorder will be provided with a sheet setting out what is needed.)

Presenter - It is your job to speak on behalf of your group. You may be called upon to work with the recorder when a report is needed. It is also your job to monitor the noise level of your group.

The modelling process

Now the modelling process can begin, where the classroom dynamic at each stage consists of the groups initially discussing and recording their thinking and then reporting back to the class, comparing their thinking with the others, and then making the necessary adjustments. The teacher’s role at all stages but one is to facilitate this classroom dynamic. Only in stage three does the teacher intervene. How this intervention is carried out has been discussed elsewhere (White 1994, 1995a). However, this raises an interesting point of debate as some teachers prefer to have previously taught the desired mathematical model, whereas others prefer to leave this to the decision of the group. The first choice allows the students to apply a newly acquired tool to an appropriate problem situation. In the second choice the groups may consult textbooks or other resources and discuss how to find and apply the relevant mathematics, or they may ask for the teacher’s assistance. The teacher then may teach to a subset of the groups who requested assistance. Not very efficient says the first group, but very powerful and effective replies the second.

Leaving this issue, one of the obvious benefits of the continual reflection and reporting at each modelling stage lies in the ease of constructing a final report in stage seven of the modelling process. Another advantage for teachers is the ready access to the progress of the students via their modelling folders.

Stage 1: Real-world problem

Focusing question: What is the problem asking us to find?

The complexity of many real-life problems found in textbooks often scares both the teacher and student. There are ways of reducing this complexity and also avoiding the danger of making the problem too trivial, or too far removed from ‘real life’. Most textbook problems need to be rewritten so that the problem statement is very general and free of as much data as possible, because later stages of the modelling process will consider and gather what is needed. The example we will discuss has been adapted from the excellent AAMT 1994 Mathematics Week publication by Henry & McAuliffe (1994) and is titled Staying Wet.

Problem: A family living in Adelaide needs a regular supply of water to supplement its mains supply. You are asked to investigate and prepare a report on how much water can be drawn from a house rainwater tank without it running dry?

There is no actual data in this question. Divide the class into groups and allow them to discuss the problem. In answering the focus question, students should be able to answer the question: ‘The answer will be in what units?’

Stage 2: Consider the variables and make assumptions

Focusing question: What are all the things that are important to this problem and what extra information do we need?

I regard this as the most valuable part of the process and it should not be rushed. It consists of listing all the variables involved and then trying to simplify or modify this list. In this process it becomes obvious that there is a need to obtain certain information which will constitute the initial conditions of the problem. Obtaining the information can be left to the class as library research. However, the teacher may use this stage as a way of controlling the direction of the investigation, as the choice of data can greatly influence the choice of model (ask any politician). The teacher leads the class discussion wherein each variable is considered and is either regarded as being important to the model and more information is needed, or is discarded because it is already included in another variable or has been controlled. This evaluation process is really a way of reducing the complexity of the problem. Average rates can also help to overcome a host of seasonal variables but can cause your model to be less sensitive. The following list could be applied to all the variables:

Example

Initial information

The family occupies a house with a roof area of 195 m².

The following figures give the mean monthly rainfall for the City of Adelaide.

There is also a choice of tanks:

Also we have 1m³ = 1 000 000 cm³

= 1 000 litres,

and ¼ = 3.14.

Stage 3: Formulate mathematical problem

Focusing questions: How do we find the answer to our problem? What calculations do we need to make?

The choice of mathematical model will depend on the approach used by the teacher and the demands of the syllabus. If the class decides on a model which does not match the wishes of the teacher, then the teacher has a choice either to intervene (a structured approach) or to delay until the completion of one cycle (an open approach) (see White 1995b). This is not a major consideration for this present problem.

Stage 3: Model for the roof is a rectangular prism, and the tank is a cylinder. Calculations needed are:

Also it was decided to use a spreadsheet to assist this process.

Model: Solve mathematical problem

Focusing question: What are the answers to our problem?

This stage describes what is found in most classrooms, with the students applying some procedure to given data.

A spreadsheet (Table 2) is set up to investigate these calculations. Columns are used to indicate the month, the rainfall (mm) and water collected (litres).

Calculations needed: The water gathered each month.

For one month, Jan - Volume = surface area x depth

= 195 x 0.02

= 3.9 m³

= 3900 litres

(S.S. eqn: = 195*B5 Remember that the equation must begin with an equals sign. Type the equation into cell C5, then highlight cells C5 to C16 and choose the fill-down command.

Calculations needed: The volume of each tank in litres.

Vol. of tank 1 = 3.14*1000*B22^2*C22 litres

Enter in cell D22, highlight cells D22 to D24 and fill down.

Calculations needed: The volume for each tank that can be withdrawn each month. This can be done visually or using spreadsheet equations.

Sample solutions are also shown in Table 2.

If you examine the modelling cycle in Figure 1 you will notice that the modelling process may mean a return to the initial assumptions in order to modify the problem being considered. You may wish to alter some of the assumptions, such as the water being withdrawn at the end of each month. It is also here that a lot of other investigations spring to mind. For example:

If you decide to make alterations, then you need to return to Stage 2.

Stage 5: Interpret the solution

Focusing question: What do our answers mean?

After obtaining their solutions, the students are then directed back to the problem. They must check to ensure that they have answered the problem within the assumptions they have made. Interpretations made should make explicit these assumptions and initial conditions. This is an important step in helping students realize that solutions to problems are very constrained by the context and not easily transferable to other situations.

Stage 6: Verify the model

Focusing questions: What is wrong with the way we obtained our answer? Can you suggest any improve-ments?

This is also an important stage where the strengths and weaknesses of their model are discussed. We are reflecting upon the mathematics that has been used. The statement that ‘All models are wrong, but some are useful’ is an important reminder of the dangers of oversimplification and of ignoring any underlying assumptions.

Models should be evaluated in terms of the variables used and, more importantly, those omitted. For example, our assumption of a flat roof and nil water loss is an unusual constraint. However, either we may leave it for others to make comparisons with our model, or we may increase the variables we wish to consider and then repeat the process in order to improve our initial model. The class can consider different models to the flat roof and may consider different rainfall or tank values. This is extremely easy to do because it merely involves minor adjustments to the spreadsheet. The students are able to experience the power of their mathematics in making predictions and improving their model. This is also where the students are able to discuss their mathematical understandings and thus it provides the teacher with an opportunity to introduce more sophisticated mathematics.

Stage 7: Produce report, explain, predict, etc.

Focusing question: How did we think about and solve this problem?

This is a valuable part of the process, as students need experience in using language to express mathematical ideas. It is here that we reflect on the quality of the students’ thinking. It should include documentation of the students’ progress through the stages of the cycle, as well as their final predictions and answers. The structure of the modelling process provides a good organizing device for their report.

Conclusion

The purpose of this paper was to emphasize the importance of planning and working. I also tried to highlight the role of the teacher in providing opportunities for reflective thinking within the modelling cycle. This is achieved by providing a classroom dynamic whereby at each stage the students have the opportunity individually to reflect upon and record their thinking; then to discuss, compare and modify their thinking while in groups; and finally to report back to the class. This classroom dynamic, together with the structure of the modelling cycle and diary sheets provides the opportunity for:

Students can only learn mathematical modelling by participating in the experience of constructing models and in the struggle to perfect these models. While this requires the teacher to allow the students to attempt, to fail, to discuss, and to test their thinking, it also requires the teacher to ensure that the students have time to reflect upon their thinking, which is the essential metacognitive ingredient. The role of the teacher is to provide the opportunity and the structure for this reflection and learning to take place, and to guide the students through the modelling process by allowing them freedom within each stage for both active individual reflection and collective reflection.

References