Why do we do modelling

A second example from documenta in Kassel: During documenta everything is more expensive in downtown Kassel. A Hercules T-shirt, for instance, as a Kassel souvenir, costs The second question is: Is it worthwhile to drive to dez in order to buy this T - shirt there? Step 5 : We interpret this mathematical result in the real world: It is indeed by 1.

Step 6 : We validate our result: Does it really make sense to drive 10 km in order to save 1. What about the risk of an accident or the air pollution caused by our trip?

So perhaps we will refine our model and start again, or we will simply decide against that simple mathematical solution. Step 7 : In the end, we write down the whole solution. All these schemas have their specific strength and weaknesses, depending on the respective purposes.

For cognitive analyses, this seven-step-model seems particularly helpful. It is a blend of models from applied mathematics Pollak ; Burghes , linguistics Kintsch and Greeno and cognitive psychology Staub and Reusser Here comes some theory. The topic of this paper is the teaching and learning of mathematics in the context of relations between mathematic and the extra-mathematical world.

The process of solving real world problems by means of mathematics can, from a cognitive point of view, be described by the schema from Fig. If need be, one has to go round the loop several times.

A key concept here is the concept of a model. A mathematical model is a deliberately simplified and formalized image of some part of the real world, formally speaking: a triple D, M, f consisting of a domain D of the real world, a subset M of the mathematical world and a mapping from D to M Niss et al. Modelling competency in a comprehensive sense means the ability to construct and to use or apply mathematical models by carrying out appropriate steps as well as to analyse or to compare given models Blum et al.

It is this comprehensive idea of modelling that will be used in the following. In large parts, PISA items require some modelling in a broad sense. The correct solution is C. Several studies have shown that each step in the modelling process see Fig.

Many students get stuck already here. This is not only or even not primarily a cognitive deficiency. For, many students around the world have learned, as part of the hidden curriculum, that they can survive without the effort of careful reading and understanding given contextual tasks.

This strategy even becomes more popular with age, and in the school context it may indeed make a lot of sense to follow this strategy in order to pass tests and to survive.

This is empirically well documented, in very many countries. Here is a well-known example Verschaffel et al. Each army bus can hold 36 soldiers. How many busses are needed?

Another example of a calculation without imagining the situation clearly is:. The popular answer is 60 min. In particular, learners are afraid of making assumptions by themselves. The answer Another well-documented observation is that students normally do not have strategies available for solving real world problems.

More generally, students usually do not reflect upon their activities and, closely related to that, are not able to transfer their knowledge and skills from one context or task to a different context or task, even if there are structural similarities. The question is whether it is worthwhile for a certain Mrs. Stone to drive from her hometown Trier across the nearby border of Luxemburg, where the gas is cheaper, in order to fill up her car there. In the following test, the students had to solve very similar tasks, among others whether it is worthwhile to drive to a nearby strawberry field in order to pick the berries for a cake instead of buying them in a supermarket, or whether it is worthwhile to use cloth-diapers instead of disposable ones.

For many students, these were totally new challenges, now about strawberries and diapers instead of cars. The PISA study also demonstrates every three years how difficult it is for year-olds to transfer their school knowledge to real world problem situations. This is particularly relevant for learning in the field of relations between the real world and mathematics DeCorte et al. Much more research is necessary into how and how far the desired transfer can be achieved. I will come back to this aspect in parts 5 and 7.

The basis for that are general educational goals such as the ability to take part in social life as an independent and responsible citizen. We can see a certain duality here Niss et al. Niss or epistemologically rich examples that shed some light on mathematics as a science including ethno-mathematical examples ; in both cases, the role of mathematics and its relations to the real world must be made more conscious;. So, examples are not good or bad per se, it depends on their purpose.

For each perspective, there is a certain model of the modelling process that is best suitable for that purpose. There is no space here to elaborate more on this. It is important to offer various aspects of sense since learners will react differently, also according to their beliefs about and attitudes towards mathematics. Back from theory to practice. In the first few parts of this paper, the focus was on learning. It is clear that all aims and purposes can only be reached by high - quality teaching.

Applications and modelling are important, and learning applications and modelling is demanding. This implies that there have to be particularly big efforts to make applications and modelling accessible for learners.

In fact, there are such efforts in many countries around the world. However, in everyday mathematics teaching practice in most countries, there is still relatively few modelling. Applications in the classroom still occur mostly in the context of dressed-up word problems. We have been deploring this gap between the educational debate and classroom practice for decades. Why do we still have this gap? The main reason is that teaching applications and modelling is demanding, too Freudenthal ; Pollak ; DeLange ; Burkhardt ; Ikeda Also the teachers have to have various competencies available, mathematical and extra-mathematical knowledge, ideas for tasks and for teaching as well as appropriate beliefs.

Instruction becomes more open and assessment becomes more complex. This is the main barrier for applications and modelling. What can we do to improve the situation? What do we know empirically about effective teaching of applications and modelling according to those various aims and purposes? Generally speaking, the well-known findings on quality mathematics teaching of mathematics hold, of course, also for teaching mathematics in the context of relations to the real world.

This seems self-evident but is ignored in classrooms around the world every day a million times. A necessary condition is an effective and learner - oriented classroom management see, e.

For modelling, group work is particularly suitable Ikeda and Stephens The group is not only a social but also a cognitive environment co-constructive group work; see Reusser This is not a matter of surface structures such as whole-class teaching versus group work versus individualized teaching, which may be dependent on cultural backgrounds.

What only counts is that learners are cognitively active Schoenfeld We have to distinguish carefully here between students working independently with teacher support, on the one hand, and, on the other hand, students working on their own, alone. I will come back to this aspect in part 6 of this paper. Learners have to be activated not only cognitively but also meta - cognitively. All activities ought to be accompanied by reflections and ought to be reflected in retrospective, with the aim to advance appropriate learning strategies.

Again this is not a matter of lesson surface structures. I will elaborate more on this aspect in part 7 of this paper. There has to be a broad variety of suitable examples as the substance of mathematics lessons since we cannot expect any mystical transfer from one example or context to another. In particular, there has to be a well-aimed variation of real world contexts as well as of mathematical contexts and topics. As I have said in part 4, different kinds of examples may serve different purposes and authenticity is not always required.

Materials from the modelling weeks in various cities, in Germany, Singapore or Queensland. Teachers ought to encourage individual solutions of modelling tasks. There are several reasons for encouraging multiple solutions Schoenfeld ; Hiebert and Carpenter ; Krainer ; Neubrand ; Rittle-Johnson and Star ; Tsamir et al. In the current project MultiMa Schukajlow and Krug , two independency-oriented teaching units with modelling tasks are compared where in one unit students are explicitly required to produce multiple solutions.

It turned out that those students who developed several solutions had higher learning gains. Necessary and not at all out-of-date are permanent integrated repeating and intelligent practising. It is also important to have a permanent balance between focussing on sub-competencies of modelling and focussing on modelling competency as a whole. It is an open research question what such a balance would look like.

What would be needed is a competency development model for modelling, theoretically sound and empirically well-founded, or several such models. This is a big deficit in research. Not only teaching but also assessment has to reflect the aims of applications and modelling appropriately.

Quality criteria such as variation of methods are relevant here, too Haines and Crouch ; Izard et al. One method is, of course, to work with tests. Some items measure certain sub-competencies and all items measure a general competency. It is important to care for a parallel development of competencies and appropriate beliefs and attitudes. Taking into account the remarkable stability of beliefs and attitudes, this also requires long-term learning processes.

What we need here are much more controlled studies into the effects of digital technologies on modelling competency development. The best message comes last. Several case studies have shown that mathematical modelling can in fact be learned by secondary school students supposed there is quality teaching a. However, much more research is needed, especially small-scale studies using a mixture of qualitative and quantitative methods. In closing part 5, I would like to emphasise that all these efforts will not be sufficient to assign applications and modelling its proper place in curricula and classrooms and to ensure effective and sustainable learning.

The implementation of applications and modelling has to take place systemically , with all system components collaborating closely: curricula, standards, instruction, assessment and evaluation, and teacher education. I cannot elaborate more on this aspect. Such an intervention allows students to continue their work without losing their independence—in the Vygotski terminology: an intervention in the Zone of Proximal Development.

Adaptive interventions can be regarded as a special case of scaffolding Smit et al. At the very least they require some form of testing so that a model can maintain its correspondence with reality. As towns and cities expand and contract, a road map must be changed to reflect the new situation. In the worst case, a change in scope necessitates a whole new model.

For example, if there was a need to reflect the current status of roads and the traffic on them, a simple road map is inadequate since we would want to show, amongst other things, the changes in traffic density over time.

Unfortunately, not all models of a particular type share the same notation, often because they originate from different sources.

For example, different publishers will have different ways of constructing and presenting road maps. When developing a product, a variety of models are likely to be constructed. It is unrealistic to expect to put everything into just one model. Too much detail in a model can only be a distraction.

It would be hard to use such a model as an aid to communication. Each model is used to illustrate a different point of view. For example, there are two different kinds of views that modellers often distinguish:. To Jackson, a model refers to the machine what the software does when it executes on a computer , which embodies a simulation of the real thing; it is a description of a domain that part of reality that you are interested in.

A model and the domain are different so he draws attention to the difference between a description that is true in both the machine and the domain, a description that is true only of the domain, and another description that is true only of the machine.

It is important to be clear whether you are talking about reality, the machine, or both. Making the decision to study can be a big step, which is why you'll want a trusted University.

Take a look at all Open University courses. If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module , or a module which allows you to count your previous learning towards an Open University qualification.

Read our guide on Where to take your learning next for more information. Remove too quickly and students will not be ready and will miss out vital steps. Remove too slowly and you might cause learned helplessness, which occurs when students become too reliant on the scaffold and struggle to work independently. Lack of reflection.

Useful questions include: What worked well? What did you find hard? How would you approach it differently next time? Modelling is no substitute for knowledge. This is especially the case when modelling writing.

Students always need a handle on the subject they are writing about before attempting more difficult writing tasks. We have found the I-We-You approach to be a useful way into modelling with many teachers. We urge you to try it out for yourself. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.

Notify me of new comments via email. Notify me of new posts via email. Email Address:. Modelling is a technique that allows educators to teach more efficiently and ensure that their students are making the most of their time in class.

Modelling is an instructional strategy. This strategy includes demonstrating the desired skill or behaviour. Research suggests that modelling fosters positive student outcomes.

It is an interactive process that, through structured guided practice, makes concepts more accessible. It can encourage students to become more reflective, and has been shown to increase on-task behaviour, student engagement and achievement.

Modelling a thought process, rather than an action, can be a little complex. Explaining to students how you came to a decision, based on various mental processes, can be challenging. However, sometimes if you share your own experiences that helped aid your own understanding of the concept, it can often offer your students an authentic insight into the development of that skill. Learning becomes a much smoother process after teaching your students efficient ways to build on the quality of their thinking.