Abstract
We interpret problem posing not as an end in itself, but as a means to add quality to students’ process of learning content. Our basic tenet is that all along students know the purpose(s) of what they are doing. This condition is not easily and not often satisfied in education, as we illustrate with some attempts of other researchers to incorporate mathematical problem-posing activities in instruction. The emphasis of our approach lies on providing students with content-specific motives and on soliciting seeds in their existing ideas, in such a way that they are willing and able to extend their knowledge and skills in the direction intended by the course designer. This requires a detailed outlining of teaching–learning activities that support and build on each other. We illustrate and support our theoretical argument with results from two design-based studies concerning the topics of radioactivity and calculus.
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Notes
- 1.
Klaassen’s research was carried out in the late 1980s and early 1990s. At that time, Dutch students all knew about the accident that had happened just a few years earlier (in 1986) with a nuclear power plant in Chernobyl. Also in the Netherlands the accident had consequences. For example, fresh products such as milk and spinach had become radioactive and had to be withdrawn from the market. Our alternative approach also draws heavily on students’ familiarity with the Chernobyl accident.
- 2.
We do not mean to suggest that terms such as “affector” or “instrument” are used by students. It is we who use these terms to talk about their ideas.
- 3.
Here, we have a first major example where a reason is induced in students for what they are going to do next. It is not of a general nature, such as: we are going to do this, because we want to please the teacher, get a good grade, or stay out of trouble. Instead the reason directly and specifically concerns the topical content: in order to reach mutual agreement and secure knowledge about safety measures and applications of radioactivity, we first of all need an objective criterion of telling when something is radioactive, and that is what we are going to find out now. Because this reason is specifically directed at topical content, we call it content specific or content directed.
- 4.
Apart from some weak radioactive sources, also a small X-ray machine was present in the classroom.
- 5.
This is a second major example, where a reason is induced in students for what they are going to do next. This reason is content specific: we do not yet know how to make something radioactive, but clearly this is at least one thing we need to know in order to properly understand safety measures and applications of radioactivity. So what we are going to do next is find out why all of our proposals did not work and how something can be made radioactive.
- 6.
This is a third major example, where a reason is induced in students for what they are going to do next. This reason is content specific and of a theoretical rather than practical nature: we now know a lot about safety measures and applications, but some questions are left open, especially concerning the interaction of radiation with matter and living tissue; we are going to find out more about that now. The theoretical questions invite an account of what radiation does in terms of what radiation is.
- 7.
It was not expected that students’ theoretical questions would provide a basis that was strong enough to support the introduction of full-fledged nuclear models. The bottom arrow in Figure 10.5 is drawn dotted because it represents only a weak content-directed reason suggested by students. A rather detailed nuclear model was only included to meet the requirements of the then examination program.
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We would like to thank Carolina Kühn, two anonymous reviewers, and the editors of this book, for their help in improving previous versions of this chapter.
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Klaassen, K., Doorman, M. (2015). Problem Posing as Providing Students with Content-Specific Motives. In: Singer, F., F. Ellerton, N., Cai, J. (eds) Mathematical Problem Posing. Research in Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6258-3_10
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