[go: up one dir, main page]
Next Article in Journal
Experimental Study of Influence of Plant Roots on Dynamic Characteristics of Clay
Previous Article in Journal
Advancing Model Explainability: Visual Concept Knowledge Distillation for Concept Bottleneck Models
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Applied Sciences
Volume 15
Issue 2
10.3390/app15020494
applsci-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

AI-Driven Predictions of Mathematical Problem-Solving Beliefs: Fuzzy Logic, Adaptive Neuro-Fuzzy Inference Systems, and Artificial Neural Networks

by
Seda Göktepe Körpeoğlu
1,*,
Ahsen Filiz
2 and
Sevda Göktepe Yıldız
2
1
Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, 34220 Istanbul, Turkey
2
Department of Mathematics and Science Education, Biruni University, 34015 Istanbul, Turkey
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(2), 494; https://doi.org/10.3390/app15020494
Submission received: 30 November 2024 / Revised: 27 December 2024 / Accepted: 30 December 2024 / Published: 7 January 2025
Browse Figures
Figure 1
<p>A structure of FIS.</p> "> Figure 2
<p>The input–output system’s ANFIS architecture.</p> "> Figure 3
<p>A fuzzy subdivision in two dimensions.</p> "> Figure 4
<p>A full connected neural network’s scheme, showing an example of a single neuron, two layers (L1 with three neurons and L2 with one), one output element, and four input elements [<a href="#B111-applsci-15-00494" class="html-bibr">111</a>].</p> "> Figure 5
<p>The adaptive neural fuzzy rule-based model’s structure.</p> "> Figure 6
<p>The information about testing data of the suggested ANFIS model.</p> "> Figure 7
<p>Neural network diagram (w and b are weight and biases respectively).</p> "> Figure 8
<p>The process of obtaining results that can predict scores with AI methods.</p> ">
Versions Notes

Abstract

:

Featured Application

This study demonstrates the applicability of artificial intelligence methods, such as fuzzy logic, artificial neural network (ANFIS), and adaptive neuro-fuzzy inference system (ANN), in predicting mathematical problem-solving beliefs (MPSBs). The proposed models can be utilized in educational settings to identify and support teachers or students with specific belief patterns, thereby enabling tailored interventions to enhance problem-solving skills. Additionally, the integration of AI techniques into educational research paves the way for innovative approaches to studying cognitive and behavioral traits.

Abstract

Considering that creative thinkers are individuals who can think outside of the box, exhibit original thoughts, and demonstrate problem-solving skills, it is likely that there is a relationship between mathematical problem-solving beliefs (MPSBs) and creative thinking dispositions (CTDs). This study aimed to predict teachers’ MPSBs with their CTDs and some demographic features. Three different artificial intelligence models (fuzzy logic, artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS)) were developed, and artificial data were obtained. The inputs of the research were determined as CTD, gender, age, educational level, school level, and teaching experiences, and the output was determined as MPSBs. Afterward, whether there was a relationship between real and artificial results was examined with statistical analysis. The research results show that there is a statistically significant, positive, and moderate relationship between artificial ANN MPSB scores and real MPSB scores (r = 0.422; p < 0.05), as well as artificial ANFIS MPSB scores and real MPSB scores (r = 0.564; p < 0.05). These results are important sources of evidence indicating that artificial intelligence methods accurately predict teachers’ MPSB scores.

1. Introduction

Among the 21st-century competencies that students need to acquire are creative thinking and problem-solving abilities [1]. To meet the challenges of this century, education must equip students with the problem-solving and creative thinking abilities needed to meet life’s demands [2,3]. The goal of education, according to many experts, should be to help students learn how to reason, think critically, and solve problems [4]. Mathematics educators generally agree that developing students’ problem-solving skills is a key goal of instruction [5]. Generally, researchers focusing on problem solving have traditionally defined the term “problem” as activities or questions that a person or group of people do not clearly have the ability to answer [6]. The ability to think creatively is essential for coming up with and identifying solutions to problems [1]. Solving problems can be achieved through creativity. Generating innovative ideas or approaches can address a lot of problems [7]. By giving students meaningful and appropriate contexts for mathematical learning, problem solving in the real world helps foster creativity in them [8]. Students who possess creative thinking skills can develop a range of ideas and arguments, ask questions, acknowledge the validity of the argument, and even become more open-minded and responsive to other people’s viewpoints [9]. Creativity can be defined as a person’s cognitive ability to solve problems by generating new ideas [10]. Therefore, problem-solving skills and creativity are associated [11].
Skills are characteristics that are necessary to carry out or start an action. Creative abilities and disposition both influence creative actions. These abilities are linked to cognitive attributes such as adaptability, logic, and creativity [12]. Disposition refers to a person’s direction and attitude toward executing an action. Dispositions encourage people to employ their creative cognitive abilities [13]. As a result, it can be concluded that a creative thinking disposition is a motivator for individuals to employ creative thinking skills, as well as a source of knowledge about the individuals’ creativity [14]. In this research, creative thinking disposition is abbreviated as CTD. Creative thinking can help students become more confident and objective. Students can enjoy the beauty of mathematics by developing their self-confidence, as well as objective open-mindedness. These objectives are helpful in preparing for the ever-evolving and expanding future. The attitudes and mindsets of thinking are gradually increasing tendencies towards mathematics [15]. Creativity emerges from students’ tremendous motivation and spirit. A disposition toward creativity will develop cumulatively when the processes of creative thinking are regularly engaged in [16]. To foster creativity in mathematics education, teachers are crucial. They provide a risk-taking, supportive, and safe learning environment where students feel free to take chances, try new things, and learn from their mistakes. They also give students the chance to participate in cooperative learning, open-ended assignments, and real-world problem-solving exercises. Furthermore, they inspire critical thinking, growth mindset development, and reflection in their students. Finally, they model and value creativity in their own teaching methods and interactions with students [17]. In addition, there has been an increase in calls in recent years for schools to encourage and promote creativity from an early age [18]. By identifying and fostering creativity in mathematics, teachers may assist students in gaining the skills and mindsets necessary for success in a world that is rapidly changing [19]. The CTDs of the teachers themselves, who contribute to the creative thinking dispositions of their students in such important dimensions, are also very important because teachers who think creatively and have a high tendency to think creatively will direct their students to think creatively more. However, there are few studies investigating teacher’s CTDs [13,14,20,21,22]. Özgenel and Çetin developed a creative thinking dispositions scale suitable for the psychometric properties of Turkish teachers [13]. The aforementioned studies are the studies in which this scale was applied. In the international literature, there has been no study conducted with teachers using another measurement tool. Studies have been conducted with pre-service teachers or university students [16,23].
Beliefs, like dispositions, have a role in shaping the nature of knowledge and learning. Personal beliefs shape a person’s approach to and response to mathematics, directing their growth in situations involving problem solving [24]. Mathematical beliefs are personal judgments about mathematics based on mathematical experiences [25]. Schoenfeld noted the presence and impact of a belief system that directs students’ actions as they try to solve problems in mathematics [26]. Teachers have a major impact on the growth of students’ learning because they help students become better problem solvers and develop their own understanding of mathematical problem solving [27]. Since teachers’ beliefs about teaching and learning also affect their teaching experiences, teachers’ beliefs can have an impact on how well students solve problems [28,29]. In the meantime, research results show that teachers’ beliefs are effective in forming students’ beliefs [30,31]. Therefore, it is critical to understand the common views held by educators, particularly the factors that shape these beliefs, because many aspects of teachers, such as their demographic characteristics and psychological tendencies, should be evaluated in an integrated manner as part of the same system [32]. In this study, CTDs, gender, age, level of education, school level, and teaching experience, which are thought to affect teachers’ beliefs about mathematical problem solving, were examined and analyzed. In this study, mathematical problem-solving belief is abbreviated as MPSB.
The present study makes noteworthy contributions in three ways. First of all, this research provides evidence as to whether teachers’ CTDs and some demographic characteristics are features that can be used to predict their MPSBs. Thus, it will also be shown whether there is a connection between teachers’ CTDs and MPSBs. Secondly, this research was carried out using artificial intelligence models, which are considered quite new to be applied to educational sciences. In this respect, this research will contribute to the literature as a concrete example of the application of artificial intelligence to education. The fact that it is conducted with teachers can be considered its third important contribution. Because there is limited research examining teachers’ MPSBs [27,29,33,34,35], it is thought that updating these studies and presenting new research results will contribute to the relevant literature.
There are significant gaps in our knowledge of teachers’ beliefs and how they affect classroom practices since, despite the recognized significance of MPSBs, research in this field has been dispersed and frequently restricted to student populations. Previous research has mostly ignored the larger psychological and dispositional components that influence beliefs in favor of concentrating on the cognitive aspects of problem solving. Additionally, although links between instructors’ ideas and their teaching methods have been proposed, there is still a dearth of empirical data, especially when it comes to how these beliefs affect students’ attitudes toward mathematics and academic performance.
By investigating the relationship between teachers’ creative thinking dispositions (CTDs) and their MPSBs, this study aims to close these disparities by taking into account a wide range of demographic and practical factors. The study not only offers novel perspectives on the predictive connections between these constructs but also investigates how AI tools might improve educational research by employing sophisticated AI models like fuzzy logic, artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS).
It is anticipated that the results will have significant effects on educational policy and practice. In order to promote constructive problem-solving attitudes and innovative thinking abilities in both teachers and students, the study provides practical insights for curriculum design and teacher training programs by clarifying the relationship between MPSBs and instructional success.
In this study, we demonstrate how artificial intelligence models can be constructed and interpreted to investigate a research question in education. We have used artificial intelligence to investigate the relationships between teachers’ CTDs, MPSBs, and demographic characteristics in the Turkish context using real data collected for the research. We have trained three distinct models (fuzzy logic, artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS)) to illustrate the various insights that can be obtained from varying data formats. After this stage, we compared the artificial scores obtained with artificial intelligence models and the real scores with statistical methods. Thus, we evaluated the ability of artificial intelligence models to predict real results. In the following sections, we provide a brief overview of mathematical problem-solving beliefs and then describe the creative thinking disposition and each demographic construct that we include in our models. Since the target population for this paper is education researchers who may not be experienced in computer science or artificial intelligence, we offer an overview of fuzzy logic, ANFIS, and ANN models, along with an explanation of how they can offer novel insights into issues like the one we have discussed here. The following research problems are sought to be answered in the study:
  • Is it possible to model teachers’ beliefs about mathematical problem solving with artificial intelligence techniques?
  • Is there a relationship between the real and artificial mathematical problem-solving belief scores of teachers?

2. Literature Review and Theoretical Foundation

2.1. Mathematical Problem-Solving Beliefs

Fitzpatrick identified four key components that have an impact on mathematical problem solving [36]. These components can be conceptualized as mathematical knowledge, beliefs, metacognitive regulation, and metacognitive knowledge. In this study, the concept of belief, which is one of the components, was dealt with. McLeod and McLeod suggest that rather than a sole definition of the word “belief” that is accurate and true, there are several definitions that can be useful depending on the context [37]. Anglophone philosophers of mind often use the term “belief” to refer, roughly, to the attitude we have when we accept or regard something as true [38]. Kloosterman et al. defined beliefs as personal assumptions that learners use when determining what kind of action they will take [39]. They claimed that a belief is the source of the motivation that drives an action. This means that specific behaviors follow specific beliefs. For instance, students are more inclined to put in the effort to study if they believe it will help them in the long run [40]. Beliefs are becoming more and more important in mathematics education, and it is a good idea for students to understand so they can have a positive attitude toward the subject [41]. The process of learning is greatly aided by mathematical beliefs, which are subjective value judgments influenced by one’s prior experiences [42]. Moreover, individuals’ attitudes about mathematics have an effect on how they learn and solve problems [43].
It is now well acknowledged that elements other than a person’s knowledge also influence the procedures involved in solving mathematical problems [44]. Schoenfeld stated that students’ beliefs on the nature of mathematical knowledge and learning have an impact on certain aspects of the problem-solving process [45]. He noted, for instance, that students who struggled with problem solving thought math issues should be answered in ten minutes or less, that mathematicians were naturally talented people, and that students should have confidence in their mathematical understanding. Unsuccessful students may be unaware of their unfavorable attitudes toward mathematics. Additionally, these ideas might be harmful to their development and ability to learn [44]. As such, the development of students’ mathematical beliefs is crucial [41]. An extremely significant portion of one’s beliefs about mathematics are developed in school, and one’s beliefs about mathematics shape how one interprets the mathematical world [46,47]. Therefore, teachers’ beliefs about mathematical problem solving also gain importance.
Ernest outlined three generally accepted and utilized categories of teacher beliefs regarding the nature of mathematics [48]. The first is the instrumentalist perspective, which defines mathematics as a body of knowledge, abilities, and principles to be applied toward some external goal. Mathematics is a static body of preexisting, unified knowledge that is still in need of discovery, according to the Platonist approach, which makes up the second category. The third category identified by Ernest is the problem-solving perspective, which views mathematics as a process as opposed to a result and views it as a continually evolving and innovative human invention [48]. One of the first studies on this subject was Thompson’s pioneering investigation of secondary mathematics teachers’ attitudes, which greatly influenced Ernest’s categories of beliefs regarding the nature of mathematics [48,49]. Although there have been studies since then, it cannot be said that researchers have paid much attention to this issue [50]. Therefore, this study is important in that it will examine teachers’ beliefs about mathematical problem solving in an up-to-date way. In addition, this investigation was carried out with artificial intelligence techniques, which are considered new for educational sciences.

2.2. Creative Thinking Dispositions

After a review of the relevant research, it is not possible to reach a direct definition of the concept of creative thinking dispositions. Generally, researchers have explained creative thinking under this concept. For a more detailed explanation, after explaining the concepts of creative thinking and disposition separately, an explanation was obtained by associating these two concepts. A definition like this might be more useful. From this perspective, we can obtain a definition for creative thinking dispositions by making the following explanations:
Creative thinking is known as going outside of the box to come up with fresh, practical solutions to problems and to develop an idea [51]. Creative thinking processes lead to creativity [52]. For some people, creativity can imply different things. While some define creativity as imagination or inventiveness, others connect it with originality or creating something that has never been done before. Whatever one’s definition of creativity, it is the cornerstone of innovation, which is essential to the growth of any nation, particularly one with a knowledge-based economy. Therefore, the ability of its workforce to be creative is essential for any nation to advance. Fortunately, all people possess the capacity for creativity, which is influenced by their thoughts, feelings, experiences, and personal needs [18]. Plucker et al. identified four dimensions in defining creativity: (1) ability, procedure, and environment in combination, (2) perceivable product; (3) innovative and helpful outcomes in a fresh and valuable product for society; and (4) social context [53].
Creativity and an individual’s attitude are also linked. Individuals can develop an attitude that can help them become creative if they are ready to continue with it and try to do things in their own way [18,54]. There are three basic components of attitude: beliefs, emotions, and behavioral dispositions. Beliefs are our basic knowledge, thoughts, and facts about an object. Emotions express situations such as liking or hating an object. Behavioral tendencies are situations such as being close to that object, being far away, or liking or disliking it [55]. In addition, Gabriel, Signolet, and Westwell considered mathematical disposition as a student’s attitudes and beliefs about learning mathematics. Therefore, creative thinking tendencies are related to individuals’ predispositions towards creative thinking and whether they emotionally like or dislike this type of thinking [32]. Kesici also made a similar definition by explaining and relating these two concepts (creative thinking and disposition) separately [14]. The disposition for creative thinking can be considered a motivating factor for the use of creative thinking abilities and a window into an individual’s creative nature.

Creative Thinking Disposition and Mathematical Problem-Solving Beliefs

Problem solving, which is a skill that people learn and develop throughout their lives, is not innate [56]. It is impossible not to face problems in our lives. It is appropriate to act by knowing how to solve the problems we encounter. The biggest problem in problem solving is that the person is reluctant, does not take steps, and remains passive [57]. It is related to problem-solving beliefs. To achieve the targeted results in the problem-solving process, people should have different perspectives. Thinking creatively and making quick and effective decisions are important factors [58].
Liljedahl and Sriraman stated that mathematical creativity in the classroom can be defined as either the process which yields an original, novel, and/or perceptive solution to a given problem or related problems or the creation of new questions and/or opportunities that enable an old problem to be viewed from a new perspective [59]. This definition makes clear the significance of problem solving and how it fosters mathematical creativity. Furthermore, Tyagi claims that there is a strong, consistent causal link between mathematical creativity and mathematical problem solving [60]. To help students gain a better comprehension of mathematical topics, enhance their problem-solving skills, and become more engaged and motivated, there has been a growing emphasis on promoting creativity in the teaching of mathematics in recent times [61,62]. Research indicates an association between problem-solving instructions and teachers’ beliefs [63,64]. Teachers must also be aware of their own beliefs and how they may affect both their own and their students’ capacity for successful problem solving [65]. Given this information, it was evaluated that teachers’ CTDs and MPSBs were related to each other and CTDs were considered as a predictor variable of teachers’ MPSBs.

2.3. Gender and Mathematical Problem-Solving Beliefs

Studies in the field of mathematics education have consistently discovered that male and female students perform differently in mathematics depending on their gender [66]. Similar research has been carried out on the topic of this study, namely if gender influences one’s beliefs of mathematics or the ability to solve mathematical problems.
Research examining the relationship between gender and beliefs about mathematics reveals that gender can influence one’s beliefs about mathematics [44,67,68,69]. Similarly, some studies conclude that beliefs about mathematics and gender are related [44,69,70,71,72,73,74,75]. For example, Giovanni and Sangcap conducted a study on university students’ perceptions of solving mathematical problems and discovered a significant difference in favor of male students [69]. Soytürk stated that there is a considerable difference in the beliefs of prospective primary school teachers regarding the solution of mathematical problems favoring female students [70]. Prendergast et al., in their study with Irish students, showed that gender significantly influenced three out of the five belief scores related to solving mathematical problems [41].

2.4. Age and Mathematical Problem-Solving Beliefs

Understanding how age and education affect epistemological beliefs can help us understand how education might improve people’s beliefs about the nature of learning and knowledge [75].
Age was found to be a predictor of beliefs about one’s ability to learn by Schommer [75]. When viewed in the context of mathematics, Abubakar and Bada stated that age is an effective variable in predicting mathematics success [76]. Mozahem, Boulad, and Ghanem examined the relationship between mathematics self-efficacy and gender and age [77]. The results show that older girls have lower levels of perceived self-efficacy in mathematics. D’Zurilla, Maydeu-Olivares, and Kant claimed that problem-solving abilities rise in young adulthood (years 17–20) and middle age (ages 40–55), and subsequently fall in old age (ages 60–80) [78]. Thornton and Dumke obtained a similar result in their meta-analytic review of age differences in daily problem-solving activities and suggested that age is an important moderator in terms of the magnitude and direction of the effect [79].
Age’s effect on MPSBs has not been directly measured in any research. However, in light of the research given under this heading, age was taken as an input for this research, considering that age affects beliefs, various factors related to mathematics (such as mathematical success and mathematical proficiency), and problem solving.

2.5. Level of Education and Mathematical Problem-Solving Beliefs

The findings of several regression analyses in Schommer’s study demonstrated that education was a predictor of beliefs regarding the structure and consistency of knowledge [75]. This information indicates that an individual’s age and level of education have distinct effects on their epistemological beliefs. The education levels of the teachers who comprised the participants of this study may also be related to their problem-solving beliefs, because, in a limited number of studies, a certain improvement in problem-solving skills has been observed as the level of education increases. For example, it was found that the problem-solving skills of individuals with master’s and doctoral degrees were higher than those of individuals with bachelor’s degrees [80]. Teachers studying at the postgraduate education level (master’s and doctoral degrees) are expected to be able to synthesize the knowledge they have acquired through the analysis process and use it in their fields [81].

2.6. School Level and Mathematical Problem-Solving Beliefs

The problem-solving approaches and beliefs of educators working at each school level (preschool, primary school, secondary school, high school, or university) may differ because the characteristics and needs of different levels are different from each other. Therefore, their beliefs in mathematics and problem solving will also be different. For example, the experiences underwent by the child at an early age, the knowledge, skills, and habits to be gained, and their problem-solving beliefs will be effective in their later lives. Teachers have an important place among the individuals who have an impact on children’s problem-solving skills. Considering the influence of teachers, it is emphasized that teachers with high problem-solving skills can use them more in education, and this can increase the problem-solving skills of the children they teach [82]. The problem situations encountered in elementary and middle school mathematics are divided into three categories. These are situations that do not have any meaning for the student and that the student encounters for the first time, questions related to four operations that students can usually answer in a short time, and situations that students cannot answer automatically in a short time but can answer with the acquisitions they have acquired [83]. The problem-solving approaches and problem-solving beliefs of teachers who will provide education on the problem situations of students at these levels will also be specific to this level. Similarly, high school mathematical subjects and the characteristics of students at this level are also different. In the face of the learning opportunities provided by age and the technological know-how, the desire to learn, and the energy of the students, to benefit from these opportunities, teachers have to continuously improve themselves [84]. Therefore, working at different levels may be a variable for teachers’ problem-solving beliefs.

2.7. Teaching Experience and Mathematical Problem-Solving Beliefs

The knowledge and experience gained in life is an important element that facilitates problem solving [85]. If people use their experiences properly and correctly, they will have an easier problem-solving process [86]. A person is better able to handle potential problems the more knowledge they possess. This defines the level of experience, which is one of the most important factors in problem solving. Having experience is a key factor in determining success in any field [87]. Research has demonstrated the significance of teacher experience in education, as it is believed that such experience fosters efficacy [87]. The cognitive schemas of beginner teachers were found to be less developed than those of expert teachers, according to Borko and Livingston [88]. They also possessed fewer connected, accessible, and detailed knowledge structures in their cognitive schemas. Borko and Shavelson also showed that inexperienced educators lacked knowledge about how to construct engaging lessons, interact with students, and build complicated learning activities [89]. In a more recent study, Adeyemi suggested that a teacher’s experience has a crucial role in the learning outcomes of their students in secondary schools [90]. He found that schools with more teachers with five years or more of experience in the classroom outperformed those with fewer teachers with less than five years of experience in the classroom. Ewetan and Ewetan stated that students’ academic achievements in mathematics has been greatly impacted by the teaching experiences of their teachers [91]. Schools with a greater proportion of teachers with 10 years or more of experience performed better than those with a lower proportion of teachers with less than ten years. Studies linking problem-solving skills and teaching experience have also obtained similar results. According to the results of Can and Uluçınar Sağır’s research, teachers with 9–12 years of professional experience have high levels of problem-solving skills [92]. Teachers with 29 years or more of experience and teachers with 1–4 years of professional experience have relatively lower problem-solving skills. Therefore, teachers with less experience and teachers with more experience have low problem-solving skills. In another recent study, it was found that the problem-solving skills of primary school teachers showed significant differences according to professional experience [59]. In light of these studies, teachers’ experience over the years was taken as an input variable.

3. Methodology

In this study, teachers’ MPSBs were evaluated and modeled with different artificial intelligence techniques (fuzzy logic, ANN, and ANFIS). In line with this purpose, a cross-sectional descriptive survey design was utilized. Typical to an observational study, a cross-sectional study design is a type of descriptive research that looks at population data at a specific moment in time. Without changing any of the factors or altering the surroundings, researchers look at a group of individuals and show what is already there in the population. Since self-report surveys can be used and no follow-up with individuals is necessary, these studies are rapid, inexpensive, and simple to carry out [93]. Since the current study examines teachers’ perceptions of mathematical problem solving over a certain period of time through some factors, this method was considered appropriate. In addition, in accordance with the method, data were collected through self-reports of individuals through questionnaires.

3.1. Sample Selection

The participants of the study were 133 teachers teaching at different school levels. Research with sample sizes similar to this study is available in the literature [94,95]. The convenience sampling approach was applied in the present study. Convenience sampling, also referred to as accidental or haphazard sampling, is the process of choosing participants from a selected group using practical factors including availability, the desire to participate, and physical closeness [96]. It refers to easily accessible study volunteers within a population [97]. Data were collected online. Data collection tools were created through Google Forms and sent to teachers. Participants completed the forms individually. Before beginning the data-gathering process, the ethics committee approved it and granted the required authorization. Teachers’ participation in research is contingent on their willingness to volunteer.

3.1.1. Demographics

Participants ranged in age from 25 to 73, with an average age of 51 years. Of the 133 participants, 22.6% were male and 77.4% were female. Educational qualifications varied, with 117 holding undergraduate degrees, 15 with master’s degrees, and 1 with a doctoral degree.

3.1.2. Professional Characteristics

Participants represented primary schools (56.4%), middle schools (26.3%), and high schools (17.3%). Teaching experience ranged from 0–5 years (5 teachers) to over 20 years (93 teachers), ensuring a diverse dataset.

3.2. Data Collection and Processing

For the study, three collecting data tools were employed.

3.2.1. Information Form

This form contains sections regarding teachers’ personal information. Teachers were asked about the school level they worked in (primary school, secondary school, or high school), their level of education (bachelor’s, master’s, or PhD), gender, age, and teaching experience.

3.2.2. Marmara Creative Thinking Dispositions Scale

This measurement tool was developed by Özgenel and Çetin to determine teachers’ creative thinking dispositions [13]. The items of the scale are arranged as a 5-point Likert type with values between 1 = never and 5 = always. Confirmatory factor analysis verified the 25-item and 6-component structure from exploratory factor analysis. The factors are innovation-seeking, courage, self-discipline, curiosity, suspicion, and flexibility. The Cronbach’s alpha internal consistency coefficient for the complete scale was found to be 0.87. It was determined that the scale called the Marmara Creative Thinking Dispositions Scale is a valid and reliable tool that measures the overall creative thinking dispositions of individuals [13]. The person possesses the attribute assessed by the related sub-dimension if they receive high scores on each of the scale’s sub-dimensions. The instrument also gives a total creative thinking disposition score. In this study, no evaluation was made according to sub-dimensions. Only the total score was taken into consideration. The lowest score is 25 and the highest score is 125. The range of 25–58 points was considered as a low, 58–92 points as a medium, and 93–125 points as high creative thinking disposition. The Cronbach’s alpha value of the whole scale was calculated as 0.85 for this study.

3.2.3. Mathematical Problem-Solving Beliefs Scale

Hacıömeroğlu created this measurement tool to assess instructors’ attitudes toward mathematical problem solving [98]. The “Belief Scale for Mathematical Problem Solving” was first translated into Turkish from its original language, English, and then the Turkish form was translated back into the original language. The original scale has 5 levels and consists of 36 items. The results of the scale’s validity and reliability examinations demonstrate that 24 of the 36 items in the scale’s original version can also be included in its Turkish version. Five factors were identified in the Turkish version of the scale. These 5 factors were named “Mathematical Skill”, “Place of Mathematics”, “Understanding the Problem”, “Importance of Mathematics”, and “Problem-Solving Skill”. The Cronbach’s alpha internal consistency coefficients of the 5 factors in the scale were calculated as 0.877, 0.775, 0.704, 0.500, and 0.802, respectively. In addition, when the reliability coefficient of the whole scale was examined, Cronbach’s alpha internal consistency coefficient was calculated as 0.768. According to the confirmatory factor analysis results, the fit index values were calculated as c2 =850.41 (sd = 239, p = 0.00), CFI = 0.80, SRMR = 0.079, and RMSEA = 0.10. The ratio of the chi-square value to degrees of freedom (x2 = c2/sd) was calculated as 3.55. It can be said that RMSEA and SRMR values are acceptable in this study. The adapted scale is a valid and reliable instrument that can be used in Turkish culture [98]. The entire scale’s Cronbach’s alpha value for the present study was determined to be 0.77.

3.3. Model Construction

Three artificial intelligence models—fuzzy logic, artificial neural networks (ANNs), and adaptive neuro-fuzzy inference systems (ANFIS)—were developed to predict MPSBs.

3.3.1. Fuzzy Logic Background and Modeling

Despite classical logic, which defines truth values as either true or false, fuzzy logic uses a more complex strategy to simulate logical thinking. Individuals form agreements with one another using language data. By utilizing fuzzy logic in the modeling and computation processes, linguistic ambiguities in everyday spoken language can be taken into consideration [99]. Fuzzy systems are mostly used in the domain of how solutions would be thought of in the event that such knowledge existed. Any difficulties are attempted to be roughly modeled, and FL uses advanced mathematical answers to regulate the problem [100]. By applying language labels stimulated by membership functions, fuzzy set theory may perform numerical computations and offer a systematic calculus associated with such linguistic information.
Designing a fuzzy inference system—a function that converts a set of inputs into outputs using human-understandable rules as opposed to more abstract mathematical formulas—is made possible by fuzzy logic. The real procedure of mapping a given set of input variables to an output that is dependent on a set of fuzzy rules is known as fuzzy inference. Fuzzification, fuzzy rule base, fuzzy inference engine, and defuzzification are the four basic processes of a general fuzzy inference system (FIS) [101]. The steps involved in an inference system are illustrated in Figure 1.
  • The process of fuzzification converts the effort controllers’ feature values into appropriately linguistic fuzzy information. Grouping variables is a difficult operation to complete. Numerous techniques have been established in the literature, including fuzzy clustering [102], genetic algorithms [103], inductive learning [104], neural network-based methods [105], and statistical applications [106]. This paper used statistical applications to partition data spaces into fuzzy sets.
  • The knowledge and rules needed to derive the outputs are stored in a fuzzy rule base. The if–then structure is used to express these rules. Building fuzzy if-then rules can be achieved using the information that is provided and/or expert knowledge. A transition between the input and output fuzzy sets is provided by the rules. While the rules are connected with a logical “or” conjunction, presumed part input fuzzy membership functions (MFs) are combined alternately with the logical “and” or “or” conjunction.
  • A fuzzy inference engine uses relevant fuzzy rules and existing facts to derive a reasonable conclusion while accounting for human emotions, cognitive processes, and logical reasoning. The creation of the consequent MFs based on the membership degrees of the premise (previous) part and finally is known as the implication part of a fuzzy system. A triangle membership function, which can be written as follows, was employed in this research:
    μ ∆ x = 0 ,   x ≤ x 1 x − x 1 x 2 − x 1 , x 1 ≤ x ≤ x 2 x 3 − x x 3 − x 2 , x 2 ≤ x ≤ x 3 0 , x > x 3
    in which x 1 , x 2 , and x 3 are arbitrary-valued numerical parameters arranged according to the order of magnitude x 1 < x 2 < x 3 . A triangle membership function is correlated with each of the six input variables. Inference operators can be broadly classified into two categories: minimization (min) and product (prod).
  • The process of defuzzification is what converts the fuzzy outputs of the fuzzy system into values that are clear. Numerous defuzzification techniques exist, including weighted sum (Wtsum) and weighted average (Wtaver).
There are primarily two methods used in the control and forecasting applications of fuzzy systems: the Mamdani and TS methods [107]. The primary distinction between the Mamdani and TS approaches stems from the process of defuzzification. Each if–then rule in the Mamdani technique results in a fuzzy set for the output variable; hence, the defuzzification step is essential when obtaining the output variable’s values with precision. Instead of a fuzzy set for the output variable, the TS technique produces a scalar as the result of each if–then inference rule. In the TS technique, the defuzzification process is easily finished by calculating the weighted average of the rule result. The selection of the parameters in TS fuzzy logic modeling is the primary source of difficulty. Because of this, Jang (1993) proposed the first proposed ANFIS (Adaptive Network-based Fuzzy Inference System) methodology is used to estimate the membership’s parameters and the subsequent functions [108].

3.3.2. ANFIS Background and Modeling

The adaptive neuro-fuzzy inference system (ANFIS) is an innovative analytical technique that successfully simulates human behavior by combining neural networks and fuzzy logic systems for prediction. Despite their inability to learn on their own, fuzzy inference systems offer robust knowledge representation when expert information is available. Based on the Takagi–Sugeno fuzzy inference system, the ANFIS successfully combines the advantages of neural networks and fuzzy logic to tackle difficult issues. It uses fuzzy techniques to mix linguistic and numerical data and makes use of artificial neural networks’ abilities to recognize patterns and classify data. The ANFIS is a rule-based fuzzy modeling framework that uses a set of data to create fuzzy rules during the training stage. Building a fuzzy inference system (FIS) with membership function parameters derived from training data is what it is used for [109].
Figure 2 shows the general architecture of the ANFIS. Figure 3 displays the two-dimensional input space where x 1 and x 2 are separated into four symmetric triangular fuzzy sets.
Fuzzy sets are only present in the premise portion of Takagi and Sengo’s fuzzy if–then rule. There are fuzzy if–then control rules available as
R r : I F   x 1   i s   S r ( 1 ) ,   x 2   i s   S r ( 2 ) , … ,   x n   i s   S r ( n )
T H E N   y r = f r x 1 , x 2 , … , x n ,
where S r ( i ) is a fuzzy set that corresponds to a partitioned domain of the input variable x j in the r -th if–then rule, y r is the output of the r -th if–then inference rules R r , n is the number of input variables, and f r (.) is a function of the n input variables.
The following is an expression of the TS fuzzy inference system’s general algorithm [104].
  • The function f i in the consequence portion calculates y i for each implication R i :
    y r = f r x 1 , x 2 , … , x n = c r 0 + c r 1 x 1 + … + c r n x n
  • The weights are computed as follows:
    r r = m 1 r ∧ m 2 r ∧ … ∧ m n r P r ,
    where m 1 represents the (membership functions) MFs’ cuts as determined by the r th rule’s input values, P r indicates the likelihood of an event occurring, and L is the minimum or production operation. For simplicity’s sake, P r is assumed to be equal to 1 .
  • Given the weighted average of every y r , the final output y deduced from k implications is given with the weights r r as
    y = ∑ r = 1 n r r y r ∑ r i = 1 n r r

3.3.3. Artificial Neural Network Background and Modeling

It is widely accepted that ANNs are a technology that offer an alternative method for resolving complicated and ill-defined situations. An artificial neural network (ANN) is a type of information-processing model that simulates how the human brain processes data. This information processing system’s architecture is made up of highly interconnected processing units called neurons that operate in parallel to address issues. Neural networks are useful when creating extremely complex algorithms or when identifying patterns in the data that already exist is necessary [110].
By definition, a neural network is a system of basic processing units known as neurons that are linked to a network by a set of weights (Figure 4). In Figure 4, the * sign indicates multiplication. It is affected by the processing element’s mode of operation, the weights’ sizes, and the network’s architecture. Neurons are processing elements that take in many inputs (p), weigh them (w), add a bias (b), add them all up, and then use the result as an argument for a transfer function (f), which is a single-valued function. This procedure results in the neuron’s output (a).
In the most popular networks, the neurons are arranged in layers, with each layer’s neurons only accepting input from the preceding layer’s neurons or from outside sources. A collection of examples showing how the output relates to the inputs is required in order to calculate the weight values. As a result, a set of data is generated that describes the system’s whole working range. Gender, age, level of education, school level, teaching experience, and creative thinking disposition are chosen as the input variables for the neural network that is created. Teachers’ MPSBs are the output variable.
The weight values of the neural network have information contained in them. Training is the process of figuring out the weights from these samples; it is essentially a traditional estimate problem. The most popular approach for this goal is back-propagation, which is employed here. It tends to provide logical responses to inputs that it has never seen before.
In conventional back-propagation, the network weights are changed along the performance function’s negative gradient via a gradient descent method. The process of calculating the gradient for non-linear multiple-layer networks is known as back-propagation. The mean sum of the squares of the network errors between the network outputs and the goal outputs is the common performance function used to train feed-forward neural networks [112]. The training function in this work was the batch gradient descent algorithm [113].
Following network training, the model must be validated using a representative subset of data that were not utilized in the model’s training.
The root mean square error (RMSE) and determination coefficient ( R 2 ) between the modeled output and metrics from the training and validation datasets were used to assess the neural network model’s performance. A model is considered very good when the R 2 is high, ≥0.8, and the RMSE is at its lowest [114]. The neural network model’s design was optimized through the application of varying numbers (1–10) of hidden neurons.

3.4. Data Analysis and Model Validation

The fuzzy system is generated to address the first research problem using the MATLAB R2021b Fuzzy Logic Toolbox. To predict teachers’ MPSB scores, artificial neural network (ANN) and adaptive neuro-fuzzy inference system (ANFIS) models are established.
The implementation of the fuzzy inference systems in this study used triangular membership functions due to their simplicity and computational efficiency. Each input variable (e.g., gender, age, level of education, school level, teaching experience, and creative thinking disposition) was associated with a set of triangular membership functions. These functions were defined based on empirical observations of the input ranges and were adjusted iteratively during the fuzzy system development. Additionally, alternative membership function types such as trapezoidal and Gaussian functions were considered during the initial exploratory phase, but the triangular functions provided the best balance of simplicity and model accuracy for this dataset.
The fuzzy inference system (FIS) was constructed using MATLAB’s Fuzzy Logic Toolbox. Input variables and membership functions were defined within the toolbox, while rules were configured manually and tested iteratively.
A hybrid optimization approach combining back-propagation and least squares estimation was used for training ANFIS models. This approach adjusted membership function parameters effectively to minimize error during training.
ANN models were implemented using a feedforward architecture with one hidden layer. The number of neurons in the hidden layer varied from 1 to 10 to identify the optimal configuration that minimized RMSE and maximized performance. Each neuron used a logarithmic sigmoid activation function to introduce non-linearity, while the output layer employed a linear activation function for regression. Back-propagation with the batch gradient descent method was used to update the weights and biases iteratively. This algorithm minimized the mean squared error between predicted and real scores. Input data were normalized to the closed interval (0,1) using min-max normalization before training to ensure consistency and improve convergence speed. During training, the model’s RMSE and coefficient of determination ( R 2 ) were monitored. The model achieving the lowest validation RMSE while avoiding overfitting was selected as the final architecture. A subset of the data not used during training was utilized for validation, ensuring the network’s generalization ability was evaluated independently. Testing was conducted to confirm predictive accuracy on unseen data.
To evaluate the models, several criteria were employed to ensure robustness and validity:
  • Spearman–Brown Correlation Coefficients: These were used to measure the relationship between real scores (calculated from the Mathematical Problem-Solving Beliefs Scale) and artificial scores predicted by the ANFIS and ANN models. Correlation coefficients were interpreted as follows: 0.00–0.30 (low), 0.30–0.70 (moderate), and 0.70–1.00 (high).
  • Training and Testing Data Partitioning: The dataset was split into training (70%), validation (15%), and testing (15%) subsets to rigorously evaluate the performance of the models. This partitioning allowed us to analyze overfitting and generalization capabilities effectively.
  • Membership Function Validation: For fuzzy models, defuzzification techniques (such as weighted average methods) were employed to ensure precise mapping between inputs and outputs. Predicted results were cross-checked with real values to confirm alignment within acceptable error ranges.
In the second research problem, it was examined separately whether there was a relationship between teachers’ real MPSB scores and artificial MPSB scores created with ANFIS and ANN approaches. Büyüköztürk stated that the Spearman–Brown correlation coefficient is used when the variables are continuous but not normally distributed, while the Pearson correlation coefficient is used when the variables are continuous and normally distributed [115]. Given that the data in this study have a continuous feature, it was first determined whether or not their distribution was normal. Since the number of data was greater than 50, Kolmogorov–Smirnov test results were used for normality analysis [116]. The results of the Kolmogorov–Smirnov normality test for each evaluated variable are shown in Table 1. When interpreting the Spearman–Brown correlation, if the correlation coefficient was between 0.00 and 0.30, it was evaluated as low, between 0.30 and 0.70 was medium, and between 0.70 and 1.00 was high [115]. The significance level of 0.05 was taken into account in the interpretation of the results.
The real and artificial scores, according to the results, did not follow a normal distribution. Spearman–Brown correlation analysis was used to investigate the relationships between teachers’ artificial and real MPSB scores.

4. Results

4.1. The First Research Question Results

This section describes how to create a learning model in the first stage of the research using the ANFIS approach, and in the second stage, an ANN is utilized.
Teachers’ MPSBs are predicted using the ANFIS technique based on replies from multiple learning domains. Using MATLAB R2020b’s Fuzzy Logic Toolbox, the fuzzy system is created. Input variables, gender, age, level of education, school level, teaching experience, and creative thinking disposition are considered along with teachers’ MPSBs as the output variable. The model is created by establishing the membership functions for the input and output values, interval setting, and rule foundation. To find the optimal ANFIS model, a hybrid optimization strategy is used to train the fuzzy inference system (FIS). Figure 5 shows the adaptive neural fuzzy rule-based model’s organizational structure. The input and output variables used in the fuzzy set are shown in Table 2, each of which has a defined membership function. Using a hybrid optimization technique, the FIS training has established predicted outcomes across 100 epochs. Figure 6 shows the information about the testing data of the suggested ANFIS model. The sign # in the Figure 6 means number (For example, “# of input:6” means “number of input:6”).
Table 3 shows 20 examples of teachers’ artificial MPSBs scores obtained through the ANFIS and real MPSBs scores.
First, the data are normalized before being used in the ANN model. To normalize the input data, the arithmetic mean and standard deviation have to be determined.
Table 4 shows 20 examples of teachers’ artificial MPSBs scores in the normalized form obtained through the ANN and real MPSBs scores.
In addition, the result is split by the standard deviation after the arithmetic mean is removed from the input data. A training set (70%), a validation set (15%), and a testing set (15%) are created from the database. The ANN model is trained, validated, and tested in the following step in this study.
The MATLAB toolbox is used to design and create the prediction model. The data file is loaded, the networks are trained and validated, and the model architecture and performance are stored in a file that can be opened in Microsoft Excel using a MATLAB script. The normalization and denormalization of the input and output data are performed both before and after the network’s actual use. Data normalization is necessary since the study’s variables have different units of measurement. As a result, the scale for the data is between 0 and 1. The output layer uses a linear activation function, while the hidden layer uses a logarithmic Sigmoid function. A wide range of flexible parameters for neural network construction are available with the Neural Network Toolbox.
The model with the least amount and highest performance is selected as the best model when adding more hidden neurons does not make it any better. Figure 7 displays the schematic diagram of the generated neural network architecture.
In the second research problem of the study, the relationships between artificial and real scores were statistically analyzed.

4.2. The Second Research Question Results

In the second research problem, it was examined whether there was a statistically significant relationship between the real scores calculated from the teachers’ responses to the Belief Scale for Mathematical Problem Solving and the artificial scores predicted by ANFIS and ANN approaches. Spearman–Brown correlation coefficients and confidence intervals were used to determine the relationship between real and artificial results. Correlation analyses were presented in Table 5.
Based on the results obtained from the Spearman rho correlation coefficient, there is a statistically significant, positive, and moderate relationship between teachers’ real MPSB scores and the artificial MPSB scores obtained with the ANN (r = 0.422; p < 0.05). Also, there is a statistically significant, positive, and moderate relationship between teachers’ real MPSB scores and the artificial MPSB scores obtained with the ANFIS (r = 0.564; p < 0.05).
Confidence intervals were also considered for the correlation coefficient because calculating the lower and upper limits of correlation coefficients can provide useful inferences. According to the value of the upper bound of the coefficient, the size of the relationship can be estimated. Similarly, the lower bound of the coefficient indicates the extent to which the relationship may weaken [117].
Table 5 shows that the real correlation coefficient between teachers’ real MPSB scores and artificial ANFIS MPSB scores has a 95% probability of being within the confidence interval [0.397, 0.718]. That is, there is only a 5% chance that the real correlation coefficient between the real and artificial ANFIS MPSB scores of teachers is less than 0.397 or greater than 0.718. Similarly, the real correlation coefficient between teachers’ real MPSB scores and artificial ANN MPSB scores has a 95% probability of being within the confidence interval [0.254, 0.559]. That is, there is only a 5% chance that the real correlation coefficient between the real and artificial ANFIS MPSB scores of teachers is less than 0.254 or greater than 0.559.

5. Discussion and Conclusions

We have used three artificial intelligence models that have allowed us to investigate how creative thinking dispositions and demographics may act in concert to influence the mathematical problem-solving beliefs of teachers. The aim of this study was actually to provide evidence on whether some teachers’ educational characteristics can be determined without responding to any scale. For this purpose, the following process was implemented and the process is shown in Figure 8. If real data and artificial data are related to each other, it will be interpreted that the scores obtained with artificial intelligence predict the real scores correctly.
Firstly, Turkish teachers’ MPSBs are modeled using three distinct artificial intelligence techniques: fuzzy logic, ANN, and ANFIS. In the study, it was explained with the support of the literature that teachers’ MPSBs and CTDs [60,64,66] and their gender were related [40,44,69,70,71,72,73,74]. No study has explicitly examined the impact of age on MPSBs. On the other hand, age has an impact on problem solving [78,79], beliefs [77], and mathematics success [76]. Therefore, age was also included as an input variable because age was also related to experience and teachers’ experience also affected their beliefs about mathematical problem solving. Individuals can solve problems more easily if they make appropriate and correct use of their experiences [57]. Results show that as individuals’ education levels increase, their problem-solving skills increase [80,81]. In addition, it is thought that working at different levels may be a variable in teachers’ problem-solving beliefs. In conclusion, the inputs of the research were determined as students’ CTD, gender, age, educational level, school level, and teaching experience, and the output was their MPSB. Teachers’ answers to the information form, Mathematical Problem-Solving Beliefs Scale, and Marmara Creative Thinking Dispositions Scale were used to collect real data. This stage, in which the MPSB scores of the teachers are predicted based on their CTD, gender, age, educational level, school level, and teaching experience variables, is related to the first research problem. In particular, any academic program that deals with teachers’ mathematical problem-solving skills can use the artificial intelligence tools designed in this research. This next generation approach can be used to identify qualified teachers to prepare students for twenty-first century skills. The real data used in this study were collected from schools; that is, from the field of education. This means that they are not data that exist in any information system or are readily available in a database. It is believed that this situation significantly adds to the research’s validity. After the operation processes of artificial intelligence models, the participants’ MPSB scores obtained artificially with ANN and ANFIS were obtained.
For the second research problem, the real MPSB scores obtained from the answers given by the teachers to the “Mathematical Problem-Solving Beliefs Scale” and the artificial MPSB scores obtained with the ANFIS and ANN approaches were compared. There is a statistically significant, positive, and moderate relationship between teachers’ real MPSB scores and both the artificially obtained ANN scores and ANFIS scores. Therefore, both the ANFIS and ANN models predict results close to teachers’ real MPSB scores. In a similar study, Göktepe Körpeoğlu and Göktepe Yıldız found a statistically significant but low-level relationship between the artificial scores obtained with the ANFIS and the real scores [118]. They attributed the low relationship to the estimation of artificial scores according to two variables. The inputs of the current study are six. The moderate level of relationship between the artificial scores and real scores may depend on the number of input variables.
The prediction models of teachers’ MPSBs were investigated for future teaching plans and approaches to problem solving. In light of the findings of this investigation, it was concluded that the MPSBs of teachers were predictable by the parameters used in the participant questionnaires. In conclusion, it is observed that fuzzy logic, ANN, and ANFIS methods were appropriate machine learning methods to forecast participants’ skills, such as problem-solving skills, attitudes, and beliefs. On the other hand, this model, as with other predictive models, should not imply that teachers with relatively low mathematical problem-solving beliefs will have negative knowledge, skills, or feelings toward problem solving. As a result, the research suggests that ANN and ANFIS algorithms are significant approaches for accurately and effectively forecasting teachers’ MPSBs.
The results of this study highlight how artificial intelligence is increasingly being used to improve educational research. The study creates novel opportunities for incorporating technology into educational research by proving that ANN and ANFIS models can be used to predict teachers’ beliefs. The results demonstrate how AI techniques can help educators and policymakers determine trends and adjust decisions to promote professional development.
Practically speaking, predicting MPSBs based on demographic and dispositional inputs can assist in identifying teachers who might benefit from specific training or resources to improve their teaching effectiveness and problem-solving beliefs. This predictive capability is in line with the larger objectives of 21st-century education, which emphasize evidence-based decision-making and personalized learning.
The study sheds light on the connection between problem-solving beliefs and creative thinking dispositions, particularly in the context of mathematics education. Understanding these relationships can help build curricula and teacher training programs, creating environments that encourage students’ and teachers’ creativity and critical thinking. The study also opens up the opportunity for using AI models to investigate other important educational concepts including self-efficacy, motivation, and teaching methods.
Results may differ in other educational or cultural contexts. Different countries have different education systems. With differences in education systems, the effects of teachers’ creative thinking dispositions (CTDs) on their mathematical problem-solving beliefs (MPSBs) may vary. For example, the characteristics of teachers in countries like Turkey, which has an education system based on centralized examinations, can be compared with the characteristics of teachers in countries where there is no examination system. These differences can test the generalizability of the model. The results can be evaluated together with the role of cultural norms and gender. Results from Turkey can be compared with other cultures, focusing on gender equality and the perceptions of teacher roles. The influence of school levels (e.g., primary vs. high school) on MPSBs and CTDs in different cultural contexts can be addressed. How the content and teaching style of education at a particular school level affect teachers’ beliefs and creative thinking tendencies can be examined. The issue of how machine-obtained data can work in different cultural contexts is also open to discussion. Studies can be conducted with different cultures to examine whether models, such as the ANFIS and ANN, are suitable for other cultures and educational contexts. When viewed contextually, the issue of how topics such as creativity and problem solving in mathematics education may have different priorities in different cultural value systems can be investigated because the results may differ in a culture that emphasizes individual creativity or a culture that supports teamwork.

6. Limitations and Recommendations

The study’s relatively small sample size of 133 teachers is one of its limitations. Small datasets have also been demonstrated to produce satisfactory results in ANN and ANFIS in recent studies [109,119,120]. In an effort to improve representativeness, stratified or random sampling techniques should be used in future research to include a bigger and more varied sample. Cross-cultural comparisons and insights into the findings’ universality can also be gained by including people from different educational systems and cultural backgrounds. Aside from sample size, the sample comprised solely Turkish teachers; hence, extra caution should be used when extending results to other sociocultural contexts. The dataset is acknowledged as being unlikely to produce the same results, and the authors are willing to provide the data upon request. The participant surveys’ questions can be further modified and varied to reflect different demographic and cultural backgrounds.
In this study, convenience sampling was used and participants were selected according to ease of access. This has led to some limitations. For example, the findings may not be representative as the sample may not reflect the diversity of the population. The results may only be valid for the specific context in which the sample was collected, limiting their validity in different settings. For these limitations of the study, we can suggest the following:
First, we should acknowledge the sampling methodology of the study and its possible biases. As a matter of fact, in cross-sectional descriptive studies, the study is conducted with specific participants in a specific time period. When the characteristics of the participants of this study are analyzed, there is a wide range of participants from 25 years old to 73 years old. The school levels and experience levels of the teachers also vary. We can say that these diversities contribute to the generalizability of the results. The limitation of the sample requires the careful interpretation of the results. Accordingly, the interpretation of the results was as careful as possible. This research is one of the exemplary studies in terms of the application of artificial intelligence methods to mathematics education. The results of the study provide evidence for the applicability of artificial intelligence to mathematics education. This evidence obtained from a small sample will encourage further studies with larger samples. Future research could be conducted using more rigorous sampling methods such as stratified or random sampling to validate the findings in a larger population.
Future studies should include objective measurements such classroom observations, peer assessments, or the examination of instructional artifacts to lessen the biases present in self-reported data. The validity and reliability of the results can be improved by cross-referencing self-reported data with information from other sources.
Further research on prediction algorithms, including machine learning and deep learning algorithms, can be carried out in the future. It may be possible to increase prediction accuracy and capture more complex correlations between variables by incorporating advanced methods for machine learning, like ensemble approaches or deep learning models. Furthermore, including feature selection methods like recursive feature removal or principal component analysis can assist in determining the most significant MPSB predictors.
The explanatory power of the models may be improved by broadening the range of input variables to include systemic and environmental elements like instructional resources, professional development opportunities, and school atmosphere. Longitudinal studies that monitor how teachers’ beliefs evolve over time may also shed light on how dynamic MPSBs are.

Author Contributions

Conceptualization, S.G.K., A.F. and S.G.Y.; methodology, S.G.K. and S.G.Y.; software, S.G.K.; validation S.G.K. and S.G.Y.; resources, A.F. and S.G.Y.; data curation, S.G.K. and S.G.Y.; writing—original draft preparation, S.G.K. and S.G.Y.; writing—review and editing, S.G.K., A.F. and S.G.Y.; visualization, S.G.K., A.F. and S.G.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research work was supported by Biruni University Scientific Research Projects Coordination Unit for the execution and completion of the project within the scope of the research project.

Institutional Review Board Statement

The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of Biruni University (protocol code 2023/78-06 and 21/02/2023).

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

Data are available upon request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Khoiriyah, A.J.; Husamah, H. Problem-based learning: Creative thinking skills, problem-solving skills, and learning outcome of seventh grade students. J. Pendidik. Biol. Indones. 2018, 4, 151–160. [Google Scholar] [CrossRef]
  2. Kurniawan, H. Efektifitas pembelajaran problem solving dan investigasi terhadap keterampilan berpikir kritis matematis berbantuan Google Classroom. J. Pendidik. Surya Edukasi (JPSE) 2016, 2, 56–67. [Google Scholar]
  3. Birgili, B. Creative and critical thinking skills in problem-based learning environments. J. Gift. Educ. Creat. 2015, 2, 71–80. [Google Scholar]
  4. Susiana, E. IDEAL problem solving dalam pembelajaran matematika. Kreano J. Mat. Kreat.-Inov. 2010, 1, 73–82. [Google Scholar]
  5. Haavold, P.Ø.; Sriraman, B. Creativity in problem solving: Integrating two different views of insight. ZDM–Math. Educ. 2022, 54, 83–96. [Google Scholar] [CrossRef]
  6. Lester, F.K. Thoughts about research on mathematical problem-solving instruction. Math. Enthus. 2013, 10, 245–278. [Google Scholar] [CrossRef]
  7. Nuzliah, N. Kontribusi motivasi belajar, kreativitas terhadap problem solving (pemecahan masalah) siswa dalam belajar serta implikasi terhadap bimbingan dan konseling di SMPN 29 Padang. J. Edukasi 2015, 1, 157–174. [Google Scholar] [CrossRef]
  8. Jacinto, H. Engaging students in mathematical problem solving with technology during a pandemic: The case of the Tecn@ Mat Club. Educ. Sci. 2023, 13, 271. [Google Scholar] [CrossRef]
  9. Arifin, Z. Mengembangkan instrumen pengukur critical thinking skills siswa pada pembelajaran matematika abad 21. J. Theorems (Orig. Res. Math.) 2017, 1, 92–100. [Google Scholar]
  10. Maulidia, F.; Johar, R.; Andariah, A. A case study of students’ creativity in solving mathematical problems through problem-based learning. Infin. J. 2019, 8, 1–10. [Google Scholar] [CrossRef]
  11. Isaksen, S.G.; Treffinger, D.J. Celebrating 50 years of reflective practice: Versions of creative problem solving. J. Creat. Behav. 2004, 38, 75–101. [Google Scholar] [CrossRef]
  12. Orhon, G. Yaratıcılık Nörofizyolojik, Felsefi ve Eğitimsel Temelleri, 2nd ed.; Pegem Yayınları: Ankara, Turkey, 2014. [Google Scholar]
  13. Özgenel, Ç.; Çetin, M. Marmara yaratıcı düşünme eğilimleri ölçeğinin geliştirilmesi: Geçerlik ve güvenirlik çalışması [Development of Marmara creative thinking dispositions scale: Validity and reliability study]. Atatürk Eğit. Fak. Eğit. Bilim. Derg. 2017, 46, 113–132. [Google Scholar]
  14. Kesici, A. The effect of digital literacy on creative thinking disposition: The mediating role of lifelong learning disposition. J. Learn. Teach. Digit. Age 2022, 7, 260–273. [Google Scholar] [CrossRef]
  15. Sumarmo, U. Kumpulan Makalah Berpikir dan Disposisi Matematik Serta Pembelajarannya; UPI: Bandung, Indonesia, 2013. [Google Scholar]
  16. Nasution, E.Y.P.; Yulia, P.; Anggraini, R.S.; Putri, R.; Sari, M. Correlation between mathematical creative thinking ability and mathematical creative thinking disposition in geometry. Present. J. Phys. Conf. Ser. 2021, 1778, 012001. [Google Scholar] [CrossRef]
  17. Xu, X.; Zhang, Q.; Sun, J.; Wei, Y. A bibliometric review on latent topics and research trends in the growth mindset literature for mathematics education. Front. Psychol. 2022, 13, 1039761. [Google Scholar] [CrossRef]
  18. Khalid, M.; Saad, S.; Hamid, S.R.A.; Abdullah, M.R.; Ibrahim, H.; Shahrill, M. Enhancing creativity and problem solving skills through creative problem solving in teaching mathematics. Creat. Stud. 2020, 13, 270–291. [Google Scholar] [CrossRef]
  19. Nilimaa, J. New examination approach for real-world creativity and problem-solving skills in mathematics. Trends High. Educ. 2023, 2, 477–495. [Google Scholar] [CrossRef]
  20. Meral, S.E.; Şahin, F.T. Okul öncesi öğretmenlerinin yaratıcı düşünme eğilimleri. OPUS Int. J. Soc. Res. 2019, 13, 311–331. [Google Scholar]
  21. Türkdoğan, M.; Özgenel, M. Öğretmenlerin yaratıcı düşünme eğilimleri ile okul iklimi arasındaki ilişki. E-Uluslar. Eğit. Araşt. Derg. 2021, 12, 190–213. [Google Scholar]
  22. Kesici, A. Ortaokul ve lise öğretmenlerinin yaratıcı düşünme ve yaşam boyu öğrenme eğilimlerinin incelenmesi. Adnan Menderes Univ. Eğit. Fak. Eğit. Bilim. Derg. 2023, 14, 18–28. [Google Scholar]
  23. Kelly, M.O.M. An Examination of the Critical and Creative Thinking Dispositions of Teacher Education Students at the Practicum Point; University of Massachusetts Boston: Boston, MA, USA, 2003. [Google Scholar]
  24. Lerch, C.M. Control decisions and personal beliefs: Their effect on solving mathematical problems. J. Math. Behav. 2004, 23, 21–36. [Google Scholar] [CrossRef]
  25. Wilson, J.W.; Fernandez, M.L.; Hadaway, N. Mathematical Problem Solving. In Research Ideas for the Classroom: High School Mathematics; Wilson, P.S., Ed.; Macmillan: New York, NY, USA, 1993. [Google Scholar]
  26. Schoenfeld, A.H. Beyond the purely cognitive: Belief systems, social cognitions, and metacognitions as driving forces in intellectual performance. Cognit. Sci. 1983, 7, 329–363. [Google Scholar]
  27. Siswono, T.Y.E.; Kohar, A.W.; Kurniasari, I.; Astuti, Y.P. An investigation of secondary teachers’ understanding and belief on mathematical problem solving. J. Phys. Conf. Ser. 2016, 693, 012015. [Google Scholar] [CrossRef]
  28. Barlow, A.T.; Cates, J.M. The impact of problem posing on elementary teachers’ beliefs about mathematics and mathematics teaching. Sch. Sci. Math. 2006, 106, 64–73. [Google Scholar] [CrossRef]
  29. Mkomange, W.C.; Ajagbe, M.A. Prospective secondary teachers’ beliefs about mathematical problem solving. Int. J. Res. Manag. Technol. (IJRMT) 2012, 2, 154–163. [Google Scholar]
  30. Sağlam, Y.; Dost, Ş. Pre-service science and mathematics teachers’ beliefs about mathematical problem solving. Procedia Soc. Behav. Sci. 2014, 116, 303–306. [Google Scholar] [CrossRef]
  31. Wilkins, J.; Brand, B. Change in pre-service teachers’ beliefs: An evaluation of a mathematics methods course. Sch. Sci. Math. 2004, 104, 226–232. [Google Scholar] [CrossRef]
  32. Gabriel, F.; Signolet, J.; Westwell, M. A machine learning approach to investigating the effects of mathematics dispositions on mathematical literacy. Int. J. Res. Method Educ. 2018, 41, 306–327. [Google Scholar] [CrossRef]
  33. Ford, M.I. Teachers’ beliefs about mathematical problem solving in the elementary school. Sch. Sci. Math. 1994, 94, 314–322. [Google Scholar] [CrossRef]
  34. Siswono, T.Y.E.; Kohar, A.W.; Hartono, S. Secondary teachers’ mathematics-related beliefs and knowledge about mathematical problem-solving. Present. J. Phys. Conf. Ser. 2017, 812, 012046. [Google Scholar] [CrossRef]
  35. Harisman, Y.; Kusumah, Y.S.; Kusnandi, K. Beliefs of junior high school teachers on learning process on mathematical problem solving. Present. J. Phys. Conf. Ser. 2019, 1157, 032112. [Google Scholar] [CrossRef]
  36. Fitzpatrick, C. Adolescent mathematical problem solving: The role of metacognition, strategies and beliefs. In Proceedings of the Annual Meeting of the American Educational Research Association, New Orleans, LA, USA, 4–8 April 1994. [Google Scholar]
  37. McLeod, D.B.; McLeod, S.H. Synthesis—beliefs and mathematics education: Implications for learning, teaching, and research. In Beliefs: A Hidden Variable in Mathematics Education? Springer: Dordrecht, The Netherlands, 2002; pp. 115–123. [Google Scholar]
  38. Schwitzgebel, E. Belief. In The Routledge Companion to Epistemology; Routledge: London, UK, 2011. [Google Scholar]
  39. Kloosterman, P.; Raymond, A.M.; Emenaker, C. Students’ beliefs about mathematics: A three-year study. Elem. Sch. J. 1996, 97, 39–56. [Google Scholar] [CrossRef]
  40. Prendergast, M.; Breen, C.; Bray, A.; Faulkner, F.; Carroll, B.; Quinn, D.; Carr, M. Investigating secondary students’ beliefs about mathematical problem-solving. Int. J. Math. Educ. Sci. Technol. 2018, 49, 1203–1218. [Google Scholar] [CrossRef]
  41. Memnun, D.S.; Hart, L.C.; Akkaya, R. A research on the mathematical problem-solving beliefs of mathematics, science and elementary pre-service teachers in Turkey in terms of different variables. Int. J. Humanit. Soc. Sci. 2012, 2, 172–184. [Google Scholar]
  42. Raymond, A.M. Inconsistency between a beginning elementary school teacher’s mathematics beliefs and teaching practices. J. Res. Math. Educ. 1997, 28, 552–575. [Google Scholar] [CrossRef]
  43. Kloosterman, P.; Stage, F.K. Measuring beliefs about mathematical problem-solving. Sch. Sci. Math. 1992, 92, 109–115. [Google Scholar] [CrossRef]
  44. Mason, L. High school students’ beliefs about maths, mathematical problem solving, and their achievement in maths: A cross-sectional study. Educ. Psychol. Int. J. Exp. Educ. Psychol. 2003, 23, 73–84. [Google Scholar] [CrossRef]
  45. Schoenfeld, A.H. Mathematical Problem Solving; Academic Press: New York, NY, USA, 1985. [Google Scholar]
  46. Frank, M.L. Problem solving and mathematical beliefs. Arith. Teach. 1988, 35, 32–34. [Google Scholar] [CrossRef]
  47. Schoenfeld, A.H. Learning to think mathematically: Problem solving, metacognition and sense making in mathematics. In Handbook of Research on Mathematics Teaching and Learning; Groups, D.A., Ed.; Macmillan Publishing Company: New York, NY, USA, 1992. [Google Scholar]
  48. Ernest, P. The impact of beliefs on the teaching of mathematics. In Mathematics Teaching: The State of the Art; Ernest, P., Ed.; Falmer: New York, NY, USA, 1989; pp. 249–253. [Google Scholar]
  49. Thompson, A.G. The relationship between teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educ. Stud. Math. 1984, 15, 105–127. [Google Scholar] [CrossRef]
  50. Beswick, K. Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educ. Stud. Math. 2012, 79, 127–147. [Google Scholar] [CrossRef]
  51. Robinson, K. Yaratıcılık: Aklın Sınırlarını Aşmak [Out of Our Minds: Learning to Be Creative]; Koldaş, N.G., Translator; Kitap Yayınevi: İstanbul, Turkey, 2008. [Google Scholar]
  52. Sukarso, A.; Widodo, A.; Rochintaniawati, D.; Purwianingsih, W. The potential of students’ creative disposition as a perspective to develop creative teaching and learning for senior high school biological science. Present. J. Phys. Conf. Ser. 2019, 1157, 022092. [Google Scholar] [CrossRef]
  53. Plucker, J.A.; Beghetto, R.A.; Dow, G.T. Why isn’t creativity more important to educational psychologists? Potentials, pitfalls, and future directions in creativity research. Educ. Psychol. 2004, 39, 83–96. [Google Scholar] [CrossRef]
  54. Sternberg, R.J. The assessment of creativity: An investment-based approach. Creat. Res. J. 2012, 24, 3–12. [Google Scholar] [CrossRef]
  55. Morris, C.G. Psikolojiyi Anlamak; Ayvaşık, B.; Sayıl, M., Translators; Türk Psikologlar Derneği Yayını: Ankara, Turkey, 2002. [Google Scholar]
  56. Atan, T.; Göçer, S.; Ünver, Ş. Farklı branşlardaki sporcuların atılganlık ile problem çözme becerileri arasındaki ilişki. In Proceedings of the 3rd International Eurasian Conference on Sport Education and Society, Mardin Artuklu University, Mardin, Turkey, 5–18 November 2018. [Google Scholar]
  57. Sağır, M.; Kökocak, S. Sınıf öğretmenlerinin sınıf yönetimi kaygı düzeyleri ile problem çözme becerileri arasındaki ilişki. Ulus. Liderlik Eğit. Derg. 2023, 7, 114–131. [Google Scholar]
  58. Kalaycı, N. Sosyal Bilimlerde Problem Çözme ve Uygulamalar; Gazi Kitabevi: Ankara, Turkey, 2001. [Google Scholar]
  59. Liljedahl, P.; Sriraman, B. Musings on mathematical creativity. For. Learn. Math. 2006, 26, 17–19. [Google Scholar]
  60. Tyagi, T.K. Is there a causal relation between mathematical creativity and mathematical problem-solving performance? Int. J. Math. Educ. Sci. Technol. 2016, 47, 388–394. [Google Scholar] [CrossRef]
  61. Grégoire, J. Understanding creativity in mathematics for improving mathematical education. J. Cogn. Educ. Psychol. 2016, 15, 24–36. [Google Scholar] [CrossRef]
  62. Novita, R.; Putra, M. Using task like PISA’s problem to support student’s creativity in mathematics. J. Math. Educ. 2016, 7, 31–42. [Google Scholar] [CrossRef]
  63. Leikin, R. Problem-solving preferences of mathematics teachers: Focusing on symmetry. J. Math. Teach. Educ. 2003, 6, 297–329. [Google Scholar] [CrossRef]
  64. Andrews, P.; Xenofontos, C. Analysing the relationship between the problem-solving-related beliefs, competence and teaching of three Cypriot primary teachers. J. Math. Teach. Educ. 2015, 18, 299–325. [Google Scholar] [CrossRef]
  65. Chapman, O. Mathematics teachers’ knowledge for teaching problem solving. LUMAT: Int. J. Math. Sci. Technol. Educ. 2015, 3, 19–36. [Google Scholar] [CrossRef]
  66. Mejía-Rodríguez, A.M.; Luyten, H.; Meelissen, M.R. Gender differences in mathematics self-concept across the world: An exploration of student and parent data of TIMSS 2015. Int. J. Sci. Math. Educ. 2021, 19, 1229–1250. [Google Scholar] [CrossRef]
  67. Aksu, M.; Demir, C.E.; Sümer, Z.H. Students’ beliefs about mathematics: A descriptive study. Eğit. Bilim 2002, 27, 72–77. [Google Scholar]
  68. Duatepe Paksu, A. Comparing teachers’ beliefs about mathematics in terms of their branches and gender. Hacet. Univ. J. Educ. 2008, 35, 87–97. [Google Scholar]
  69. Giovanni, P.; Sangcap, A. Mathematics-related beliefs of Filipino college students: Factors affecting mathematics and problem solving performance. Procedia Soc. Behav. Sci. 2010, 8, 465–475. [Google Scholar]
  70. Soytürk, İ. Sınıf öğretmeni adaylarının matematik okuryazarlığı öz-yeterlikleri ve matematiksel problem çözmeye yönelik inançlarının araştırılması. Master’s Thesis, İstanbul University, İstanbul, Turkey. Available online: https://tez.yok.gov.tr/UlusalTezMerkezi (accessed on 25 August 2024).
  71. Abedalaziz, N.; Akmar, S. Epistemology beliefs about mathematical problem solving among Malaysian students. OIDA Int. J. Sustain. Dev. 2012, 5, 59–74. [Google Scholar]
  72. Schommer-Aikins, M.; Duell, O.K.; Hutter, R. Epistemological beliefs, mathematical problem-solving beliefs, and academic performance of middle school students. Elem. Sch. J. 2005, 105, 289–304. [Google Scholar] [CrossRef]
  73. Bal, A.P. Examination of the mathematical problem-solving beliefs and success levels of primary school teacher candidates through the variables of mathematical success and gender. Educ. Sci. Theory Pract. 2015, 15, 1373–1390. [Google Scholar]
  74. Sintema, E.J.; Jita, T. Gender differences in high school students’ beliefs about mathematical problem solving. Int. J. Learn. Teach. Educ. Res. 2022, 21, 395–417. [Google Scholar] [CrossRef]
  75. Schommer, M. The influence of age and education on epistemological beliefs. Br. J. Educ. Psychol. 1998, 68, 551–562. [Google Scholar] [CrossRef]
  76. Abubakar, R.B.; Bada, I.A. Age and gender as determinants of academic achievements in college mathematics. Asian J. Nat. Appl. Sci. 2012, 1, 121–127. [Google Scholar]
  77. Mozahem, N.A.; Boulad, F.M.; Ghanem, C.M. Secondary school students and self-efficacy in mathematics: Gender and age differences. Int. J. Sch. Educ. Psychol. 2021, 9, S142–S152. [Google Scholar] [CrossRef]
  78. D’Zurilla, T.J.; Maydeu-Olivares, A.; Kant, G.L. Age and gender differences in social problem-solving ability. Pers. Individ. Differ. 1998, 25, 241–252. [Google Scholar] [CrossRef]
  79. Thornton, W.J.; Dumke, H.A. Age differences in everyday problem-solving and decision-making effectiveness: A meta-analytic review. Psychol. Aging 2005, 20, 85. [Google Scholar] [CrossRef] [PubMed]
  80. Şekerli, N. Ebeveynlerin Evlilikte Problem çözme Becerileri Ile öz-Anlayış özelliklerinin Bazı sosyo-Demografik Değişkenler Açısından Incelenmesi. Master’s Thesis, İstanbul Gelişim Üniversitesi Lisansüstü Eğitim Enstitüsü, Istanbul, Turkey, 2021. [Google Scholar]
  81. Turhan, M.; Yaraş, Z. Lisansüstü programların öğretmen, yönetici ve denetmenlerin mesleki gelişimine katkısı. Elektron. Sos. Bilim. Derg. 2013, 12, 200–218. [Google Scholar]
  82. Ceylan, R.; Bıçakçı, M.Y.; Aral, N.; Gürsoy, F. Okul öncesi eğitim kurumunda çalışan öğretmenlerin problem çözme becerilerinin incelenmesi. Trak. Univ. Soc. Sci. J. 2012, 14, 85–98. [Google Scholar]
  83. Baykul, Y. İlköğretimde Matematik öğretimi: 6–8 Sınıflar; Pegem Akademi: Ankara, Turkey, 2009. [Google Scholar]
  84. Demirtaş, H.; Dönmez, B. Ortaöğretimde görev yapan öğretmenlerin problem çözme becerilerine ilişkin algıları. İnönü Univ. Eğit. Fak. Derg. 2008, 9, 177–198. [Google Scholar]
  85. Ulupınar, S. Hemşirelik Eğitiminin öğrencilerin Sorun çözme Becerilerine Etkisi, Unpublished. Ph.D. Dissertation, İstanbul Üniversitesi Sağlık Bilimleri Enstitüsü Hemşirelik Anabilim Dalı, İstanbul, Turkey, 1997. [Google Scholar]
  86. Williams, D.J.; Noyes, J.M. Effect of experience and mode of presentation on problem solving. Comput. Human Behav. 2007, 23, 258–274. [Google Scholar] [CrossRef]
  87. Rice, J.K. The Impact of Teacher Experience: Examining the Evidence and Policy Implications; Brief No. 11; National Center for Analysis of Longitudinal Data in Education Research, Duke University: Durham, NC, USA, 2010. [Google Scholar]
  88. Borko, H.; Livingston, C. Cognition and improvisation: Differences in mathematics instruction by expert and novice teachers. Am. Educ. Res. J. 1989, 26, 473–498. [Google Scholar] [CrossRef]
  89. Borko, H.; Shavelson, R.J. Teacher Decision Making 1. In Dimensions of Thinking and Cognitive Instruction; Routledge: London, UK, 2013. [Google Scholar]
  90. Adeyemi, T.O. Teachers’ teaching experience and students’ learning outcomes in secondary schools in Ondo State, Nigeria. Educ. Res. Rev. 2008, 3, 204. [Google Scholar]
  91. Ewetan, T.O.; Ewetan, O.O. Teachers’ teaching experience and academic performance in mathematics and English language in public secondary schools in Ogun State, Nigeria. Int. J. Humanit. Soc. Sci. Educ. 2015, 2, 123–134. [Google Scholar]
  92. Can, K.; Uluçınar Sağır, Ş. Sınıf öğretmenlerinin bilişötesi öğrenme stratejileri, özyeterlik algısı ve problem çözme becerileri arasındaki ilişkinin incelenmesi. Int. J. Soc. Sci. Educ. Res. 2018, 4, 81–95. [Google Scholar]
  93. Simkus, J. Cross-Sectional Study: Definition, Designs & Examples. Simply Psychol. 2023. Available online: https://www.simplypsychology.org/what-is-a-cross-sectional-study.html (accessed on 14 July 2024).
  94. Anelli, D.; Sica, F. The financial feasibility analysis of urban transformation projects: An application of a quick assessment model. In International Symposium: New Metropolitan Perspectives; Springer International Publishing: Cham, Switzerland, 2020; pp. 462–474. [Google Scholar]
  95. Xie, L.; Feng, X.; Zhang, C.; Dong, Y.; Huang, J.; Liu, K. Identification of urban functional areas based on the multimodal deep learning fusion of high-resolution remote sensing images and Social Perception Data. Buildings 2022, 12, 556. [Google Scholar] [CrossRef]
  96. Dörnyei, Z. Research Methods in Applied Linguistics; Oxford University Press: New York, NY, USA, 2007. [Google Scholar]
  97. Saumure, K.; Given, L.M. Convenience sample. In The SAGE Encyclopedia of Qualitative Research Methods; SAGE: Newcastle upon Tyne, UK, 2008; Volume 2, pp. 124–125. [Google Scholar]
  98. Hacıömeroğlu, G. Matematiksel problem çözmeye ilişkin inanç ölçeği’nin Türkçe’ye uyarlama çalışması. Dicle Univ. Ziya Gökalp Eğit. Fak. Derg. 2011, 17, 119–132. [Google Scholar]
  99. Zadeh, L. Fuzzy Sets. Inf. Control 1965, 338, 338–353. [Google Scholar] [CrossRef]
  100. Şen, Z. Fuzzy Logic Modeling Principles in Engineering; Su Vakfı Yayınları: İstanbul, Turkey, 2004. [Google Scholar]
  101. Huang, S.J.; Chiu, N.H. Applying fuzzy neural network to estimate software development effort. Appl. Intell. 2009, 30, 73–83. [Google Scholar] [CrossRef]
  102. Yoshinari, Y.; Pedrycz, W.; Hirota, K. Construction of fuzzy models through clustering techniques. Fuzzy Sets Syst. 1993, 54, 157–165. [Google Scholar] [CrossRef]
  103. Karr, C.L.; Gentry, E.J. Fuzzy control of pH using genetic algorithms. IEEE Trans. Fuzzy Syst. 1993, 1, 46–53. [Google Scholar] [CrossRef]
  104. Ross, J.T. Fuzzy Logic with Engineering Applications; McGraw Hill, Inc.: New York, NY, USA, 1995. [Google Scholar]
  105. Takagi, H.; Hayashi, I. NN-driven fuzzy reasoning. Int. J. Approx. Reason. 1991, 5, 191–212. [Google Scholar] [CrossRef]
  106. Dubois, D.; Prade, H. Fuzzy Sets and Systems: Theory and Applications; Academic: New York, NY, USA, 1980. [Google Scholar]
  107. Takagi, T.; Sugeno, M. Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. 1985, 15, 116–132. [Google Scholar] [CrossRef]
  108. Jang, J.S.R. ANFIS: Adaptive network based fuzzy inference system. IEEE Trans. Syst. Man Cybern. 1993, 23, 665–684. [Google Scholar] [CrossRef]
  109. Parkavi, R.; Karthikeyan, P. Predicting academic performance of learners with the three domains of learning data using neuro-fuzzy model and machine learning algorithms. J. Eng. Res. 2023, 12, 397–411. [Google Scholar]
  110. Mabel, M.C.; Fernandez, E. Analysis of wind power generation and prediction using ANN: A case study. Renew. Energy 2008, 33, 986–992. [Google Scholar] [CrossRef]
  111. Hagan, T.M.; Demuth, H.B.; Beale, M. Neural Network Design; PWS Publishing Company: Boston, MA, USA, 1996. [Google Scholar]
  112. Zupan, J.; Gasteiger, J. Neural Networks in Chemistry and Drug Design; Wiley-VCH: Weinheim, Germany, 1999. [Google Scholar]
  113. Strik, D.P.; Domnanovich, A.M.; Zani, L.; Braun, R.; Holubar, P. Prediction of trace compounds in biogas from anaerobic digestion using the MATLAB Neural Network Toolbox. Environ. Model. Softw. 2005, 20, 803–810. [Google Scholar] [CrossRef]
  114. Kasabov, N.K. Foundations of Neural Networks, Fuzzy Systems and Knowledge Engineering; MIT Press: Cambridge, MA, USA, 1998. [Google Scholar]
  115. Büyüköztürk, Ş. Sosyal Bilimler Için Veri Analizi el Kitabı, 22nd ed.; Pegem Akademi: Ankara, Turkey, 2016. [Google Scholar]
  116. Büyüköztürk, Ş.; Cokluk, O.; Köklü, N. Sosyal Bilimler Için Istatistik; Pegem Akademi: Ankara, Turkey, 2010. [Google Scholar]
  117. URL-1. Available online: https://www.istmer.com/korelasyon-katsayisi-istatistiksel/ (accessed on 27 September 2024).
  118. Göktepe Körpeoğlu, S.; Göktepe Yıldız, S. Using artificial intelligence to predict students’ STEM attitudes: An adaptive neural-network-based fuzzy logic model. Int. J. Sci. Educ. 2024, 46, 1001–1026. [Google Scholar] [CrossRef]
  119. Çetinkaya, A.; Baykan, Ö.K. Prediction of middle school students’ programming talent using artificial neural networks. Eng. Sci. Technol. Int. J. 2020, 23, 1301–1307. [Google Scholar] [CrossRef]
  120. Göktepe Körpeoğlu, S.; Göktepe Yıldız, S. Prediction of metacognition awareness of middle school students: Comparison of ANN, ANFIS and statistical techniques. Avr. Bilim Tek. Derg. 2022, 38, 450–461. [Google Scholar]
Applsci 15 00494 g001
Figure 1. A structure of FIS.
Figure 1. A structure of FIS.
Applsci 15 00494 g001
Applsci 15 00494 g002
Figure 2. The input–output system’s ANFIS architecture.
Figure 2. The input–output system’s ANFIS architecture.
Applsci 15 00494 g002
Applsci 15 00494 g003
Figure 3. A fuzzy subdivision in two dimensions.
Figure 3. A fuzzy subdivision in two dimensions.
Applsci 15 00494 g003
Applsci 15 00494 g004
Figure 4. A full connected neural network’s scheme, showing an example of a single neuron, two layers (L1 with three neurons and L2 with one), one output element, and four input elements [111].
Figure 4. A full connected neural network’s scheme, showing an example of a single neuron, two layers (L1 with three neurons and L2 with one), one output element, and four input elements [111].
Applsci 15 00494 g004
Applsci 15 00494 g005
Figure 5. The adaptive neural fuzzy rule-based model’s structure.
Figure 5. The adaptive neural fuzzy rule-based model’s structure.
Applsci 15 00494 g005
Applsci 15 00494 g006
Figure 6. The information about testing data of the suggested ANFIS model.
Figure 6. The information about testing data of the suggested ANFIS model.
Applsci 15 00494 g006
Applsci 15 00494 g007
Figure 7. Neural network diagram (w and b are weight and biases respectively).
Figure 7. Neural network diagram (w and b are weight and biases respectively).
Applsci 15 00494 g007
Applsci 15 00494 g008
Figure 8. The process of obtaining results that can predict scores with AI methods.
Figure 8. The process of obtaining results that can predict scores with AI methods.
Applsci 15 00494 g008
Table 1. Kolmogorov–Smirnov normality test results.
Table 1. Kolmogorov–Smirnov normality test results.
Kolmogorov–Smirnov
StatisticdfSig.
Real scores0.0891330.003
Artificial ANFIS scores0.0951330.000
Artificial ANN scores0.0721330.036
Table 2. Input and output variables in a fuzzy set.
Table 2. Input and output variables in a fuzzy set.
Input Variables Output Variable
GenderAgeLevel of EducationSchool LevelTeaching ExperienceCreative Thinking DispositionTeachers’ MPSBs
MaleYoungUnder
graduate		Primary school0–5 yearslowlow
FemaleMiddle AgedGraduateMiddle school6–11 yearsmiddlemiddle
AgedDoctorateSecondary school11–15 yearshighhigh
16–19 years
20 and over
Table 3. ANFIS data (20 samples out of 133).
Table 3. ANFIS data (20 samples out of 133).
GenderAgeLevel of EducationSchool LevelTeaching ExperienceCTD ScoresReal MPSB ScoresANFIS Scores
152115108112111.99
24011593103106.50
133223125120119.99
138122100122121.99
23312397114113.99
242123106110110.00
23223186102101.99
153115110106106.00
129122114125124.99
147125100109109.00
23312298113112.99
24112390126125.99
247125117116116.02
15612597109112.86
23911396113113.00
24612511910099.96
24011394110109.99
24411392108108.00
24111591106103.58
243225125114113.99
Table 4. Normalized ANN data (20 samples out of 133).
Table 4. Normalized ANN data (20 samples out of 133).
GenderAgeLevel of EducationSchool LevelTeaching ExperienceCTD ScoresReal
MPSB Scores		ANFIS Scores
00.560010.590.440.44
10.310010.230.320.40
00.160.50.50.510.550.59
00.2700.50.250.400.570.55
10.1600.50.50.330.470.50
10.3500.50.50.540.420.55
10.140.5100.070.310.46
00.580010.640.360.47
00.0800.50.250.730.610.55
00.4500.510.400.400.48
10.1600.50.250.350.460.49
10.3300.50.50.160.630.472
10.4500.510.800.500.49
00.6400.510.330.400.48
10.29000.50.300.460.50
10.4300.510.850.280.49
10.31000.50.260.420.50
10.39000.50.210.390.52
10.330010.190.360.39
10.370.50.5110.470.40
Table 5. Correlation analyses.
Table 5. Correlation analyses.
Correlations
Real ScoresArtificial ANFIS ScoresArtificial ANN Scores
Spearman’s rhoReal scoresCorrelation Coefficient1.0000.564 **0.422 **
Sig. (2-tailed).0.0000.000
N133133133
Bootstrap cBias0.000−0.003−0.004
Std. Error0.0000.0810.078
95% Confidence IntervalLower1.0000.3970.254
Upper1.0000.7180.559
Artificial ANFIS scoresCorrelation Coefficient0.564 **1.0000.357 **
Sig. (2-tailed)0.000.0.000
N133133133
Bootstrap cBias−0.0030.0000.001
Std. Error0.0810.0000.091
95% Confidence IntervalLower0.3971.0000.171
Upper0.7181.0000.523
Artificial ANN scoresCorrelation Coefficient0.422 **0.357 **1.000
Sig. (2-tailed)0.0000.000.
N133133133
Bootstrap cBias−0.0040.0010.000
Std. Error0.0780.0910.000
95% Confidence IntervalLower0.2540.1711.000
Upper0.5590.5231.000
** Correlation is significant at the 0.01 level (2-tailed). c Unless otherwise noted, bootstrap results are based on 1000 bootstrap samples.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Göktepe Körpeoğlu, S.; Filiz, A.; Göktepe Yıldız, S. AI-Driven Predictions of Mathematical Problem-Solving Beliefs: Fuzzy Logic, Adaptive Neuro-Fuzzy Inference Systems, and Artificial Neural Networks. Appl. Sci. 2025, 15, 494. https://doi.org/10.3390/app15020494

AMA Style

Göktepe Körpeoğlu S, Filiz A, Göktepe Yıldız S. AI-Driven Predictions of Mathematical Problem-Solving Beliefs: Fuzzy Logic, Adaptive Neuro-Fuzzy Inference Systems, and Artificial Neural Networks. Applied Sciences. 2025; 15(2):494. https://doi.org/10.3390/app15020494

Chicago/Turabian Style

Göktepe Körpeoğlu, Seda, Ahsen Filiz, and Sevda Göktepe Yıldız. 2025. "AI-Driven Predictions of Mathematical Problem-Solving Beliefs: Fuzzy Logic, Adaptive Neuro-Fuzzy Inference Systems, and Artificial Neural Networks" Applied Sciences 15, no. 2: 494. https://doi.org/10.3390/app15020494

APA Style

Göktepe Körpeoğlu, S., Filiz, A., & Göktepe Yıldız, S. (2025). AI-Driven Predictions of Mathematical Problem-Solving Beliefs: Fuzzy Logic, Adaptive Neuro-Fuzzy Inference Systems, and Artificial Neural Networks. Applied Sciences, 15(2), 494. https://doi.org/10.3390/app15020494

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop