Task Competency: For Your Instructional Flexibility

Abstract

Tasks are the core of mathematics lessons. The selection and the quality of tasks for lessons are essential for mathematical understanding, for promoting students’ mathematical practices and competencies, and can be the basis for structuring lessons and using several teaching methods. This chapter will be a challenge for you – as you can imagine by reading the statement from Christina, a pre-service teacher in my modeling course. Developing mathematical tasks for to cover certain mathematical topics is not trivial at all and you need time to do this. Due to the many demands teachers have to deal every day in school, there is hardly any space for creating modeling problems for students. Thus there is a great need for good and high quality teaching materials including mathematical tasks and especially modeling problems. With the support of these resources, you would be able to incorporate modeling into your everyday teaching. Particularly since there is a lot of work in school every day, it is important that in-service teachers can at least have time during a modeling workshop, and pre-service teachers in a university seminar, to create a modeling problem in a group. Additionally, they learn to classify the different solution steps according to the phases of the modeling cycle, and to reflect about possible difficulties their students could have while modeling. At the beginning of task development, the participants, even if they are well experienced teachers, need time to think about an appropriate context for an age group, about the complexity of the task in connection with the working time and also about further materials the students need to model the problem. As an alternative to developing a new modeling problem, the transformation of a given mathematical task (for example from a school book) into a modeling problem is another way to understand what characterizes a modeling problem or distinguishes it from “normal” mathematical problems. Lastly, when doing this activity the difference between mathematical problem solving and mathematical modeling becomes obvious. Lesh and Zawojewski (Problem solving and modeling. In: Lester F Jr (ed) Second handbook of research on mathematics teaching and Learning. Information Age, Greenwich, pp 763–804, 2007) talked about the difference between problem solving and mathematical modeling. Lesh and Doerr (Beyond constructivism: a models and modeling perspective on mathematics problem solving, learning, and teaching. Lawrence Erlbaum Associates, Mahwah, 2003) also made it clear that every modeling problem is also a “problem” in the sense that you can’t solve it with known algorithms (e.g. Schoenfeld, Mathematical thinking and problem solving. Erlbaum, Hillsdale, 1994) but that you need strategies to do it. Mathematical problems can be strongly focused on inner-mathematical aspects, but can also have a context. Modeling problems are defined as real life questions coming from the extra-mathematical world. So we can’t speak about mathematical modeling problems without a real context. Additional criteria necessary for modeling problems, are shown later on.