Acknowledgements

In our study there were challenging proposals which not always were possible to be solved by the prospective teachers. This fact is in accordance with Silver and collaborators, who remarked that "… mathematicians certainly pose mathematical problems or conjectures that they are not certain they can solve (e.g., Goldbach's Conjecture), and research with adult subjects has found that they often pose mathematical problems that they could not solve on their own" [27]. This situation takes place in Example 4, where the inverse reformulation was solved only because the

Taking into account that the participants are prospective teachers, it is important for them to be able to create new problems to work on their classes. Moreover, as Silver noted "Problem posing has figured prominently in some inquiry-oriented instruction that has freed students and teachers from the textbook as the main source of wisdom and problems in a school mathematics course" [23]. For this purpose, Kilpatrick [22] argued that one of the basic cognitive processes involved in problem posing is association: "[Because] knowledge is represented as a network of associated ideas, that network can be used to generate problems by taking a concept node in the network and raising questions about its associates" (p. 136). In our research, unexpected association of ideas was found in several proposals, like in Example 1 where different ideas from calculus and trigonometry were combined in

Finally, motivation is a very important issue which was also relevant in this experience. It deserves to be mentioned that some of the prospective teachers pose up to three different reformulations of the given direct problem, which can be considered as a result of a motivating activity. This fact was also observed by Winograd who reported that students in his study appeared to be highly motivated to pose problems that their classmates would find interesting or difficult [40].

The first immediate conclusion is that the results of both experiences—carried

In 2017 the prospective teachers imitated previous examples provided for the

On one hand, in 2019 the previous examples were very simple and concerned other mathematics topics like proportions, arithmetic and geometric sequences, and solving for unknown sides and angles in right triangles. On the other hand, the prospective teachers were asked to solve the direct problem before proposing their own reformulation. So, in this new fieldwork, their experience was more involved

reformulations. It can be observed that this fact led the proposals in many different ways. For instance, they gave a formula in the reformulated problem and asked for an interpretation, a sketch of the corresponding region, or another way to get the

Other important difference is the use (or not) of external variables, which can be physical (time and velocity), chemical (amount of fertilizer and herbicide), economical (cost of a fence), or biological variables (like kilograms of grass per day). Those variables were widely used in 2017; however, in 2019 they only appear in a

first problem, i.e., the filling of a swimming pool. When they were asked to reformulate the second one (the sheep problem), there were no examples that can be imitated, but they followed the same ideas that they used in their proposal for the swimming pool, like inverting the function, changing the geometry, or including

in the mathematical solution of the direct problem than other inverse

problem was unwittingly simplified.

Theorizing STEM Education in the 21st Century

a very creative proposition.

out in 2017 and 2019—are absolutely different.

6. Conclusions

obstacles, among others.

same result without using integrals.

40

few cases, like in the Example 2, in Section 4.

The authors wish to thank Marjorie Chaves for her valuable contribution to this work.

This work has been possible thanks to the collaboration of Professors Juan Francisco Ruiz-Hidalgo and Antonio Moreno Verdejo, from the Master's Degree in Teaching Secondary Education of the University of Granada.

The research was carried out with the support of the research project "Professional Competence of teachers in initial training and STEM Education" (PGC2018-095765-B-100) of the National R+D+I Plan and of the Research Group FQM-193: Didactics of Mathematics, Numerical Thought, of the Andalusian Plan for Research, Development and Innovation (PAIDI).

### Author details

Victor Martinez-Luaces\*, José Antonio Fernández-Plaza and Luis Rico Department of Didactics of Mathematics, University of Granada, Spain

\*Address all correspondence to: victorml@correo.ugr.es

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
