2. Theoretical framework

In our research relatively simple problems are proposed to prospective teachers. In all of them, only fundamental concepts of calculus, linear algebra, and geometry are necessary to be considered. The idea is to analyze easy problems, susceptible of being reformulated in the form of an inverse problem by prospective teachers.

It is expected that the reformulations raised by the participants will be richer than the original and will favor a teaching-learning process based more on exploration than repetition of procedures. As Lester and Cai [19] observed: "…teachers can develop worthwhile mathematical tasks by simply modifying problems from the textbooks" (p. 124).

The latter links the work to be done with a traditional area of research in mathematics education, as is the case of problem posing, the first subsection of our theoretical framework. In the end, the other two subsections are devoted to inverse problems and mathematical modeling.

#### 2.1 Problem posing

There is a long tradition in the literature in English regarding problem-solving research, and the work of Brown and Walter [20, 21] and Kilpatrick [22], among

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

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

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 of its conditions.

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 [23, 24].

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, thereby creating a new problem [23].

Other researchers such as Brown and Walter [20, 21] propose a strategy to raise new problems that they call "What if not?" consisting in changing conditions, restrictions, etc. of a certain problem and then generating a new one.

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

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

#### 2.2 Inverse problems

education permits the students to understand the world and interact with it in a

the strong bond between mathematics, science, and technology [7–9].

promote favorable attitudes toward sciences and mathematics.

modeling problems [17, 18].

2. Theoretical framework

the following sections.

textbooks" (p. 124).

2.1 Problem posing

30

problems and mathematical modeling.

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

Besides, for future teachers to carry out these teaching strategies, it is necessary to

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

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

In our research relatively simple problems are proposed to prospective teachers. In all of them, only fundamental concepts of calculus, linear algebra, and geometry are necessary to be considered. The idea is to analyze easy problems, susceptible of being reformulated in the form of an inverse problem by prospective teachers. It is expected that the reformulations raised by the participants will be richer than the original and will favor a teaching-learning process based more on exploration than repetition of procedures. As Lester and Cai [19] observed: "…teachers can develop worthwhile mathematical tasks by simply modifying problems from the

The latter links the work to be done with a traditional area of research in mathematics education, as is the case of problem posing, the first subsection of our theoretical framework. In the end, the other two subsections are devoted to inverse

There is a long tradition in the literature in English regarding problem-solving research, and the work of Brown and Walter [20, 21] and Kilpatrick [22], among

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

critical, constructive, and efficient manner.

Theorizing STEM Education in the 21st Century

According to Groetsch's [15, 16] ideas, the process of solving a direct problem can be schematized as in Figure 1.

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 schematized in Figure 3.

Both of these problems are common in the experimental sciences and real-life situations, as noted in previous research [31, 32].

Figure 1. Scheme for direct problems.

The productions of the prospective teachers of both groups underwent a first

Then, in a second work session, showing these reformulations, they were given a brief explanation about direct problems and inverse problems. At the end, prospec-

Unlike what happened with the problem of the pool, in this case the participants were asked to reformulate the problem in an inverse manner for task enrichment

When those prospective teachers worked with the sheep problem, nine different groups of inverse problems were identified—some of them including up to four variants—and in almost all cases the participants added to their proposal the corresponding task analysis. The productions and the most creative reformulations were analyzed in a previous book chapter [14]. Brief descriptions of the nine groups

1.Reformulations based on the inverse function, asking the length of the rope

3. Inverse problem asking the location of the peg at which the sheep is tied

in which the accessible area without intersection is maximum

5.Optimization problem, including two sheep and asking for the length of rope

6. Sequential inverse problem, in which, from a given R0 given and an accessible area A0, the student has to define a sequence of Rn, in which, the area between Rn-1 and Rn, that is to say, An An-1 is A0, and find out the value of n such that

7. Incremental problem, in which, given a length of rope, the student asks the increment in the length in order to increase the accessible area by a certain

8.Dynamic problem, in which the student includes new magnitudes, such as

9.Equivalent area problem, given different locations of the peg, in which the student asks the length of rope such that the accessible area remains invariant

After that experience, it was observed that several prospective teachers were particularly creative in both the reformulation itself and in the tasks enrichment; however, the vast majority opted for a standard approach and, in some cases, for a

For these reasons, 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. As an example,

analysis, and among all the reformulations presented, three of them were highlighted and selected since they had been posed spontaneously in an inverse form. They were particularly interesting, one of them was proposed by a participant

from group A, and the other two were proposed by members of group B.

tive teachers proposed a new direct problem: the sheep problem.

Inverse Modeling Problems in Task Enrichment for STEM Courses

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

purposes.

of inverse problems are the following:

4. Inverse problem asking the side of the square

it is not possible to find the corresponding Rn

given the ratio of area

2.Trivial reformulations

percentage

trivialization of the problem.

speed

33

Figure 4.

Figure 3.

Figure 2.

Comparison scheme between modeling and application problems.

#### 2.3 Mathematical modeling

In the preliminary discussion document to the International Commission on Mathematical Instruction (ICMI) Study 14 [33], the term "modeling" focuses on the direction that goes from the real world toward mathematics, whereas the term "applications" implies the opposite direction. In addition, the term "modeling" emphasizes the process that is taking place, while the word "applications" stresses the object involved, particularly real-life cases that are susceptible to mathematical manipulation. Taking into account these ideas, we arrive at the following schema (Figure 4).

An extended discussion about modeling and application problems in our previous research can be found in papers [31, 34].

#### 3. Our previous experiences at UGR master courses

In the University of Granada, the research was designed to work with one or more groups of prospective mathematics teachers for secondary education. Taking into account the available options, we chose to work with the students of groups A and B from the course named "Learning and Teaching Mathematics in Secondary School," included in the Master's Degree in Teaching Secondary Education [14].

In the 2016–2017 academic year at the University of Granada, group A consisted of 33 students, and 41 students formed group B, with regular class attendance. Two of the master courses' university professors collaborated on our research.

In a first class, the prospective teachers of both groups worked on a problem about the filling of a swimming pool. In the first session of the fieldwork, the aforementioned problem was proposed—in the form of a direct problem—and future professors were asked to reformulate it as part of a task enrichment proposal to be used in secondary school courses.

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

The productions of the prospective teachers of both groups underwent a first analysis, and among all the reformulations presented, three of them were highlighted and selected since they had been posed spontaneously in an inverse form. They were particularly interesting, one of them was proposed by a participant from group A, and the other two were proposed by members of group B.

Then, in a second work session, showing these reformulations, they were given a brief explanation about direct problems and inverse problems. At the end, prospective teachers proposed a new direct problem: the sheep problem.

Unlike what happened with the problem of the pool, in this case the participants were asked to reformulate the problem in an inverse manner for task enrichment purposes.

When those prospective teachers worked with the sheep problem, nine different groups of inverse problems were identified—some of them including up to four variants—and in almost all cases the participants added to their proposal the corresponding task analysis. The productions and the most creative reformulations were analyzed in a previous book chapter [14]. Brief descriptions of the nine groups of inverse problems are the following:


2.3 Mathematical modeling

Scheme for inverse causation problems.

Theorizing STEM Education in the 21st Century

Scheme for inverse specification problems.

Figure 2.

Figure 3.

Figure 4.

ous research can be found in papers [31, 34].

Comparison scheme between modeling and application problems.

to be used in secondary school courses.

32

3. Our previous experiences at UGR master courses

In the preliminary discussion document to the International Commission on Mathematical Instruction (ICMI) Study 14 [33], the term "modeling" focuses on the direction that goes from the real world toward mathematics, whereas the term "applications" implies the opposite direction. In addition, the term "modeling" emphasizes the process that is taking place, while the word "applications" stresses the object involved, particularly real-life cases that are susceptible to mathematical manipulation. Taking into account these ideas, we arrive at the following schema (Figure 4).

An extended discussion about modeling and application problems in our previ-

In the University of Granada, the research was designed to work with one or more groups of prospective mathematics teachers for secondary education. Taking into account the available options, we chose to work with the students of groups A and B from the course named "Learning and Teaching Mathematics in Secondary School," included in the Master's Degree in Teaching Secondary Education [14]. In the 2016–2017 academic year at the University of Granada, group A consisted of 33 students, and 41 students formed group B, with regular class attendance. Two

In a first class, the prospective teachers of both groups worked on a problem about the filling of a swimming pool. In the first session of the fieldwork, the aforementioned problem was proposed—in the form of a direct problem—and future professors were asked to reformulate it as part of a task enrichment proposal

of the master courses' university professors collaborated on our research.


After that experience, it was observed that several prospective teachers were particularly creative in both the reformulation itself and in the tasks enrichment; however, the vast majority opted for a standard approach and, in some cases, for a trivialization of the problem.

For these reasons, 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. As an example,

in the year 2017 fieldwork, the participants were not asked to solve their proposed problems, so this was an aspect that needs to be improved in further research.

• When , the sheep can reach the lateral edges, but not the upper one.

• When , the sheep can reach the top edge of the field, but not the

This problem requires modeling and integral calculation, and it can be easily converted into an inverse problem. It is obvious that for every value of r ≥ 0 there exists a unique value of A, but more challenging is to ask the question from another angle. For instance, for any value of A, does a corresponding unique value of r exist or not? To solve this problem, the function f(r) must be studied in terms of conti-

As it was mentioned, the new fieldwork was designed in order to avoid, or at least attenuate, the difficulties observed in the previous experience, carried out during year 2017. In particular, both experimental designs had three main differ-

• Prospective teachers were asked to solve the original direct problem, before

• Prospective teachers were asked to solve their own reformulated problem—or at least write a sketch of the solution—with the aim of reducing the number of

This new design produced different responses that cannot be included in the previous nine groups observed during the year 2017 experience. Some of the most

As already stated, some of the prospective teachers' productions cannot be classified into the nine groups observed in the previous fieldwork. The following

One of the prospective teachers solved the direct problem by integration,

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>R</sup><sup>2</sup> � <sup>L</sup> <sup>2</sup> � <sup>x</sup> � �<sup>2</sup> <sup>q</sup>

<sup>2</sup> � <sup>x</sup> � �<sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup> and

dx. After that, he

<sup>2</sup> � R sin t and several well-

creative and new proposals are analyzed in the following subsection.

4.3 Some of the most creative reformulations for the sheep problem

• Before proposing to them this new task, several examples about inverse problems were discussed. However, none of them were about the sheep problem. The main reason for this decision was to avoid simple imitation or

• When , the sheep can graze all around the field.

Inverse Modeling Problems in Task Enrichment for STEM Courses

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

nuity and growth with r ≥ 0, in order to ensure its invertibility.

proposing their inverse reformulations.

adaptation of a given model.

non-well-posed problems.

examples illustrate this situation.

Example 1: An unusual specification problem.

then the area accessible for the sheep is <sup>A</sup> <sup>¼</sup> <sup>Ð</sup> <sup>L</sup>

observing that the circumference equation can be written as <sup>L</sup>

known trigonometric formulas to obtain the following long formula:

solves the integral by using the change of variables <sup>x</sup> <sup>¼</sup> <sup>L</sup>

whole field.

4.2 The new fieldwork design

ences:

35

The new results showed interesting differences and few similarities which are analyzed in the next sections.
