5. Discussion

Problem posing has an important role to play in the STEM classroom. For instance, Beal and Cohen [36] suggested "… significant changes to the existing model of education, in which students would move from passive consumers of educational resources that have been developed by others to creators of rich, innovative and authentic STEM content…".

Moreover, problem posing has a positive influence on students' ability to solve word problems and provided a chance to gain insight into students' understanding of mathematical concepts and processes.

Some researchers have found evidences that students' experience with problem posing enhances their perception of the subject; causes excitement and motivation; improves students' thinking, problem solving skills, attitudes, and confidence in mathematics; and contributes to a broader understanding of mathematical concepts [37].

Some of these capabilities were observed in our fieldwork with prospective teachers, particularly in the selected examples, where some responses were very creative. This is an important aspect not only for teaching but also as a prominent feature of mathematical activity. As Poincaré said "Mathematicians may solve some problems that have been posed for them by others or may work on problems that have been identified as important problems in the literature, but it is more common for them to formulate their own problems, based on their personal experience and interests" [38]. In the same way, Hadamard [39] identified the ability to find key research questions as an indicator of exceptional mathematical talent. For instance, a challenging question appears in the solution of Example 2 of the previous section, where the arc length is obtained as the derivative of the area accessible to the sheep. The question about the accuracy of this procedure led us to an interesting research question whose answer is not trivial (see [35] for a general discussion).

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 problem was unwittingly simplified.

Another remarkable observation is that the proposals corresponding to 2019 are usually more challenging from a pure mathematical viewpoint. For instance, they ask for different types of solutions (analytical, geometrical, etc.), and they need the analysis of monotony, existence of a pre-image, solving nonlinear equations, etc. They also ask for more conceptual issues, like identify a region or give a meaning to one or more given variables in a certain formula, among other options that were

It can also be observed that the proposals in year 2017 were more practical, i.e., hands-on problems more involved with other disciplines and more connected with the reality and its mathematical modeling, whereas in 2019 they are more concep-

As a general conclusion, it seems that the prospective teachers tend to propose the reformulations based on their own recent experiences. If these experiences consist in working with previous examples, they try to imitate them, and if their experience consists mainly in solving the direct problem, they tend to use this solution—or the process that led to it—as the main input for problem posing.

Finally, it is difficult to say that one of these experiences yielded better results

predominated, while in the other one, different characteristics were observed. As a consequence, the resulting proposals more than antagonistic can be regarded as

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

The research was carried out with the support of the research project "Profes-

(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

© 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,

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.

for Research, Development and Innovation (PAIDI).

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

provided the original work is properly cited.

sional Competence of teachers in initial training and STEM Education"

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

than the other. In fact, in one of these experiences, certain characteristics

absolutely unusual in 2017.

truly complementary.

Acknowledgements

Author details

41

work.

tual, mathematically challenging, and self-contained.

Inverse Modeling Problems in Task Enrichment for STEM Courses

DOI: http://dx.doi.org/10.5772/intechopen.89109

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 a very creative proposition.

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].

### 6. Conclusions

The first immediate conclusion is that the results of both experiences—carried out in 2017 and 2019—are absolutely different.

In 2017 the prospective teachers imitated previous examples provided for the 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 obstacles, among others.

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 in the mathematical solution of the direct problem than other inverse 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 same result without using integrals.

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

Inverse Modeling Problems in Task Enrichment for STEM Courses DOI: http://dx.doi.org/10.5772/intechopen.89109

Another remarkable observation is that the proposals corresponding to 2019 are usually more challenging from a pure mathematical viewpoint. For instance, they ask for different types of solutions (analytical, geometrical, etc.), and they need the analysis of monotony, existence of a pre-image, solving nonlinear equations, etc. They also ask for more conceptual issues, like identify a region or give a meaning to one or more given variables in a certain formula, among other options that were absolutely unusual in 2017.

It can also be observed that the proposals in year 2017 were more practical, i.e., hands-on problems more involved with other disciplines and more connected with the reality and its mathematical modeling, whereas in 2019 they are more conceptual, mathematically challenging, and self-contained.

As a general conclusion, it seems that the prospective teachers tend to propose the reformulations based on their own recent experiences. If these experiences consist in working with previous examples, they try to imitate them, and if their experience consists mainly in solving the direct problem, they tend to use this solution—or the process that led to it—as the main input for problem posing.

Finally, it is difficult to say that one of these experiences yielded better results than the other. In fact, in one of these experiences, certain characteristics predominated, while in the other one, different characteristics were observed. As a consequence, the resulting proposals more than antagonistic can be regarded as truly complementary.
