Abstract

Problem solving is considered as one of the most important topics in STEM (science, technology, engineering, and mathematics) education, and this is especially relevant when problems require modeling skills in order to be solved. Also, it should be noted that in many branches of science and technology, typical problems are posed in an inverse form. Then, combining both characteristics, the so-called inverse modeling problems deserve to be studied deeply, particularly in their potential for task enrichment. For those reasons, since 2016, a research project was carried out, by using inverse modeling problems to develop prospective teacher's task enrichment skills. The results of this experience that took place in 2017 showed nine different groups of proposals where only few participants were very creative, whereas many others posed trivial problems or simply imitated examples analyzed previously. After that, a new research design was proposed during 2018 and implemented during the first months of 2019, with the aim of avoiding—or at least attenuating—those difficulties observed in the previous fieldwork. The new results showed interesting differences and few similarities when compared with the other experience. In this chapter, both experiences are analyzed, and lastly, findings and final conclusions are reported.

Keywords: inverse problems, task enrichment, prospective teachers, analysis of solution strategies, sketches, mathematical modeling

#### 1. Introduction

In the last decade, STEM education has become an important topic, deeply analyzed by several authors, particularly in North America and Europe [1–6].

It must be remarked that the conjunction of the subjects to which the STEM education refers is not arbitrary. Sciences provide a context for reflection, organization, and action. They propose problems and questions that invite exploration and discovery and provide criteria to classify and organize the natural environment, thus allowing us to deepen into its richness and complexity. Technology and engineering offers tools and techniques that make the construction of models and artifacts that resolve conflicts or minimize impacts easier. Mathematics provides a mode of expression and representation and a set of notions and skills that allow interpreting and modeling the environment, providing strategies to invent and solve problems and promote logical and critical thinking. As a consequence, STEM

education permits the students to understand the world and interact with it in a critical, constructive, and efficient manner.

others, represents some of the best known examples. Under the common denomination of "problem posing," these authors include the formulation of new problems and/or the reformulation of problems previously proposed, in a certain format that

A particular case worthy of study occurs when students pose a new problem during the resolution of one of greater complexity [27]. This situation can already be seen in the work of Polya [28] that proposes, as a possible strategy, the approach of the problem in a different way or the establishment of variants, discarding some

In works done by other researchers, the formulation of problems does not have to be linked to the resolution of a specific problem. For example, in some cases the invention of problems is proposed starting from a certain situation or experience

Another option is to combine the two previous approaches and ask students to solve a problem after changing a condition or the final question of the problem,

Other researchers such as Brown and Walter [20, 21] propose a strategy to raise

Stoyanova [29] identifies three possible ways in terms of the formulation and invention of problems: free situations and semi-structured and structured situations. In the first of the aforementioned, there are no restrictions on the invention of problems. In the semi-structured, the problem-based approach is proposed, based on any experience or quantitative information. Lastly, in the structured situations, a

In our research in Granada, the participants are given a direct problem, which should be reformulated in the form of an inverse problem. Therefore, this can be considered as

According to Groetsch's [15, 16] ideas, the process of solving a direct problem

In contrast, inverse problems may have multiple solutions or simply no solutions, thus making them more interesting though consequently more difficult [30]. In essence there are two types of inverse problems; firstly, the causation problem, where the procedure is well-known and the question is concerned with the necessary data in order to obtain a given result. This situation is schematized in Figure 2. The other inverse problem found is the specification problem, where data and result are given and the question is concerned with which procedure can let reach the desired result (output) with the chosen data (input). This process is schema-

Both of these problems are common in the experimental sciences and real-life

new problems that they call "What if not?" consisting in changing conditions,

restrictions, etc. of a certain problem and then generating a new one.

certain given problem is reformulated or some condition of it is changed.

a structured situation, following the classification given by Stoyanova [29].

can be more or less structured [23–26].

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

Inverse Modeling Problems in Task Enrichment for STEM Courses

thereby creating a new problem [23].

of its conditions.

2.2 Inverse problems

tized in Figure 3.

Figure 1.

31

Scheme for direct problems.

can be schematized as in Figure 1.

situations, as noted in previous research [31, 32].

[23, 24].

The natural link that exists between mathematics and science—which is at the core of STEM education—establishes important challenges for mathematics and science teachers. In particular, the mathematics teacher should know precisely the meaning of mathematical contents, identify the needs of students, diagnose learning problems, and prepare proposals for intervention and instruction for their approach and resolution. The abovementioned is important in all cases of mathematics teaching but especially important when working with STEM students, due to the strong bond between mathematics, science, and technology [7–9].

Besides, for future teachers to carry out these teaching strategies, it is necessary to look for significant situations in which the mathematical and scientific contents acquire meaning, for which it is essential to deepen their meanings (performing the semantic analysis, according to the method of analysis of content), as well as cognitive aspects (plausible expectations, learning stages, limitations, and opportunities, which constitute cognitive analysis) and instructive aspects. Therefore, the didactic analysis [10–13] becomes an important tool for the teacher to carry out teaching strategies that promote the development of the STEM competence of the students.

For these reasons, educational research must respond to the training needs of university students who are going to be teachers in the coming years in order to promote favorable attitudes toward sciences and mathematics.

One of the challenges consists in developing prospective teacher's task enrichment skills, [14] and for this purpose, inverse problems [15, 16] are especially relevant since in many branches of science and technology, typical problems are posed in an inverse form. In previous works we analyzed the particular cases when modeling skills are combined with inverse problems, and we called them inverse modeling problems [17, 18].

In this chapter we consider inverse modeling problems, focusing on their posing for task enrichment purposes. We describe our research carried out during the last 4 years, when working with prospective teachers at the University of Granada, Spain (UGR), and some of our most recent findings are reported and discussed in the following sections.
