Data driven pedagogy in physical education a new paradigm in teaching effectiveness

Abstract

The standard practice of physical education (PE) teaching relies primarily on traditional pedagogical methods that lack systematic data-based techniques for measuring teaching performance. A flexible systematic approach becomes vital because students’ learning styles, performance dynamics, and teaching targets create fluid teaching conditions. A unified teaching-decision-making structure emerges to boost physical education effectiveness through the presented approach. This research introduces a novel T-spherical fuzzy Aczel-Alsina (TSFA) aggregation approach for evaluating instructional methods, student engagement, and performance outcomes, utilizing T-spherical fuzzy sets (TSFS) with Aczel-Alsina aggregation. The system provides live responsiveness to instructional methods by connecting adjustments with immediate feedback and numerical performance information. The model demonstrates its ability to find the best teaching approaches for different learners through experimental testing conducted in college PE settings through scenario-based assessments. The framework benefits teachers, administrators, and curriculum planners who want to enhance physical education programs by making decisions based on research evidence.

Introduction

The standard modelling approaches in education struggle to understand interactive teaching aspects because they maintain linear data combination methods. Resolving the non-linear link between three variables is challenging due to the complexity through traditional evaluation methods. Through fuzzy logic, we have found better ways to handle these difficulties. Implementing fuzzy logic¹ systems created standardized methods to handle such challenges. FS as a method enables the assignment of membership degrees (MD) to unclear concepts through the defined interval range to handle uncertainty effectively, modelling subjective educational perceptions. The FS methodology did not include methods to handle the possibility of negative assessments from learners or evaluators regarding educational methods or activities. Atanassov² introduced the intuitionistic fuzzy set (IFS), which uses a non-membership degree (NMD) together with MD to handle acceptance and rejection in professional development evaluations. The revised interpretability measure of IFS presented a more enhanced representation of human attitudes through a complete holistic assessment. New research suggests that IFS has become applicable across various fields, including image fusion by Ananthi and Balasubramaniam³ and manufacturing system failure analysis by Shu et al.⁴. Applying IFS under generalized operators allows Li⁵ to implement the methodology in educational multi-attribute decision-making (MADM) problems. Hussain et al.⁶ demonstrate the application of IFS in risk management systems. Li introduced the use of IFVs for multi-attribute decision-making based on the GOWA operator⁷. Yager⁸ formulated Pythagorean fuzzy sets (PyFS) because situations with cumulative uncertainty beyond IFS restrictions required a new model which allowed the squared sum of MD and NMD to exceed one threshold. The pedagogical choice modelling process became more detailed due to the implementation of PyFS. Ejegwa⁹ expanded the capabilities of PyFS by showing that it could be used effectively in medical diagnosis, proving its adaptability to sensitive fields. Cuong¹⁰ introduced picture fuzzy sets (PFS) as a solution when student indecision or pedagogical neutrality prevented agreement or disagreement because these situations required an extra abstention degree (AD). The completely uncertain feedback view of PFS proved valuable in educational decision systems that required complex teaching evaluation. The decision-making system MADM in picture-fuzzy environments was presented by Siksnelyte-Butkiene et al.¹¹. The exponential Jensen image fuzzy divergence measure appeared through Khan et al.¹² and revealed its applications. The MD, AD, and NMD measurements are often given by decision-makers to objects in the unit interval even when their overall value is more than unity. Standard PFSs proved inadequate when dealing with this type of situation. The research of Mahmood et al.¹³ created a spherical fuzzy set (SFS) as a solution to address this case study. The principle difference between PFSs and SFSs involves how the total of MD, AD, and NMD measure exceeds unity because, in SFSs, the square sum remains within the interval . Ali et al.¹⁴ studied SFSs and their application in medical diagnostics, while References^15,16 presented additional study information about his work. In the following, Figure 1 displays a visual representation of framework relationships.

Fig. 1.

Connection of different fuzzy models.

Solving real-world ambiguous problems through multi-attribute decision-making requires precise trustworthy results obtained by processing data attributes and alternatives through various operator frameworks and formulas. Decision-makers utilize this approach to evaluate several possibilities by considering their relative importance levels. The method helps decision-makers determine which process needs priority attention by assisting decision-makers with precise findings. Ashraf et al.¹⁷ employed the PFS Maclaurin symmetric mean operator for MADM applications while also using MADM. Decision-making requires integrating fuzzy logic with humanized opinion and concepts related to T-norm (TN) and T-conorm (TCN). The selection of an aggregation operator defines the effectiveness of TN and TCN. Alcantud introduced the weighted geometric operators using IFS-based Einstein’s TN and TCN¹⁸. The authors established the Dombi operators based on TSFS in^19,20. The first application of combining entropy and cumulative prospect theory and multi-attributive border approximation area comparison occurred in Pamucar et al.²¹. The first research on Dual hesitant r-rung ortho-pair FS (Q-OPFS) Dombi t-norm and t-conorm was conducted by^22–24 while Hussain and Pamucar²⁵ applied PyFS to solve multi-attribute group decision-making (MAGDM) problems. The MAGDM problem received a solution from Hussain et al.²⁶ through applying TSFS. Research has established that TSFS provides a broader scope than IFS, PyFS, and PFS, which makes it appear to be a complete expansion of FS. The MAGM challenges are addressed through practical examples developed by various writers, including those from^27–29. The multi-attributive border approximation area comparison (MABAC) technique handles supplier performance assessment with TSFS characteristics through the alignment of "cumulative prospect theory" and entropy by Pamucar et al.²¹. Researcher studies^30–32 applied the compromise solution (MARCOS) approach to MAGM problems with the addition of a scoring function that utilized TSFS characteristics for alternative evaluation. The MADM problems received a solution through PFS from Ozer³³. Research^34,35 addressed the problems by integrating TSFS properties with the combinative distance-based assessment (CODAS) measurement technique.

The essential nature of TOPSIS for decision-making results from its ability to use basic mathematical procedures to compute results. Jia built a methodology that used single-valued neutrosophic numbers to assess and select transport service providers³⁶. The methods created by Arora and Naithani³⁷ apply TOPSIS to resolve MAGM problems by identifying optimal solutions for each issue. In Thong’s study³⁸, he examined the CRS models presented in References^39,40. The researcher introduced a spherical fuzzy Delphi-TOPSIS technique in their work to tackle problems stemming from multi-criteria decision-making (MCDM)⁴¹. The Total Area based on the Orthogonal Vectors (TAOV) technique stands as a modern MCDM method which utilizes the orthogonality concept of decision criteria, according to writers^42,43. Zhang et al.⁴⁴ conducted a research initiative involving CRSs in evaluating attribute reduction methods. Multiple researchers have researched covering-based fuzzy rough sets (CFRS). The research by Sun et al.⁴⁵ combined fuzzy neighbourhoods and fuzzy β-neighborhoods through their exploration. The fundamental principle of TAOV approach becomes established through this concept. Research teams have begun implementing the TAOV approach since the recent period^46,47. Liu et al.⁴⁸ utilized TAOV to determine suitable research and development technologies for an Iranian high-tech corporation. Puska et al.⁴⁹ established a decision model to locate the logistics center optimally. Sarfraz designed a decision model using the spherical fuzzy set combined with TN and Aczel-Alsina TCN⁵⁰. Paul et al.⁵¹ enhanced MADM by adding Pythagorean fuzzy Hamacher aggregation operators. Rahman and Muhammad⁵² created a decision model based on a complicated polytopic fuzzy set. Ullah⁵³ created symmetric mean aggregation operators that introduced new tools for PFSs. Hussain et al.⁵⁴ explored the decision-making theory framework using an extended fuzzy methodological approach. Hussain and Pamucar²⁵ established various innovative techniques for PyF rough set through the combination of power aggregation models and speculative Schweizer-Sklar theory. A set of new approaches created by Rahim et al.⁵⁵ used cubic PyF principles. The green supply chain efficiency reached maximum when Riaz and Farid⁵⁶ developed their decision model with Linear Diophantine Fuzzy Soft-Max Aggregation Operators. The decision model presented by Ahmed et al.⁵⁷ employed complex intuitionistic hesitant fuzzy set theory for its structure.

PE creates complete student growth through its combination of physical health instruction and building mental aptitude along with emotional strength development and group teamwork practice. The current assessment methods in physical education use subjective evaluation and basic statistical measurements, creating obstacles to delivering extensive and precise educational assessments. The existing standard evaluation frameworks prove inadequate in responding to various learning conditions, student requirements, and physical and intellectual developmental changes. Educational organizations require new adaptable and unbiased assessment approaches to boost instructional quality while improving individual learning performance.

Fuzzy MCDM methods have become popular in educational assessment research because they allow evaluators to merge quantitative metrics and qualitative staff feedback for comprehensive teaching effectiveness evaluations. The implementation of current models faces three main challenges: complex mathematical calculations, strict data arrangement requirements, and unsatisfactory responses to natural teaching events occurring in classroom environments. The proposed research fills the gaps by creating an Integrated Pedagogical and Decision-Making Framework that utilizes advanced fuzzy MCDM evaluation methods for physical education assessment.

Weight determination in the framework works through FAHP (fuzzy analytical hierarchy process) in combination with WASPAS (weighted-aggregated sum-product assessment) decision-making to develop an evaluation system that offers context-based balanced results. The system analyzes unclear education data through weighted computation systems with fuzzy parameter sets, which generate precise and dependable decision-making. The combined implementation of scientific evidence strengthens the instructional evaluation process, facilitates live classroom adjustments, and assists with ongoing professional growth in physical education settings.

This paper contains fundamental definitions in section 2 and the problem description in section 3. Section 4 contains details about the methodology. Section 5 includes the research on physical pedagogical effectiveness. Section 6 explores the sensitivity evaluation of the produced results. Section 7 contains the results together with their assessment. This article finishes with its summary in section 8.

Basic terminologies

This section defines fundamental concepts to understand the article’s content better. This part explains the definitions of TSFSs together with Aczel-Alsina TN and TCN.

Definition 1

The set is said to be TSFS expressed over a universe of discourse set . The terms represents MD, AG and represent NMD such that ,, with conditions for . Then is said to be refusal grade (RG). In addition, the triplet represents the TSF value (TSFV).

A TSFS is an interesting framework to handle unclear data collected from real-world applications. The acquired information appears as TSFVs. A score function should be established to perform TSFVs defuzzification. A score function exists for defuzzification purposes according to the following definition.

Definition 2

Suppose the score of a TSFV is . Then

1

The TNM and TCNM process information through unit intervals. Logic and its applications have significantly benefited from multiple TNMs and TCNMs. Aczel Alsina T-norm (AATNM) and Aczel Alsina T-conorm (AATCNM) provide dual flexibility to process information under conditions of uncertainty. The decision outputs for different scenarios can be controlled through existing parameters with the help of AATNM and AATCNM.

Definition 3

The functions are AATNM and AATCNM, respectively and defined as

2

3

where .

When human opinion is involved, TSFS can model the uncertain information. Hussain et al.⁵⁸ developed the AOs for TSFS, integrating AATNM and AATCNM to aggregate the data and to get useful results. The TSFAAWA and TSFAAWG operators are stated below.

Definition 4

A mapping defined for TSFVs is said to be TSFAAWA operator where and is stated as

4

Definition 5

A mapping defined for TSFVs is said to be a TSFAAWG operator where and is stated as

5

Problem statement

Physical education teachers must continually adapt their teaching methods to accommodate a diverse range of learners; however, the current evaluation system remains stagnant, relying on subjective assessments and binary evaluations to determine teaching effectiveness. These methods do not reflect the dynamics of front and back student engagement, cognitive growth, physical ability learning, and motivation sustainability in actual classroom experiences.

Current models used in the field of educational assessment embrace fuzzy logic; however, they frequently include complex mathematical processes and are not directly applicable to the dynamical nature of physical education. This means that teachers are at a crossroads on how to match pedagogy with changing student needs, and administrators are at a crossroads to make evidence-based decisions on how to improve programs.

Eight assessment attributes, namely Student Engagement Rate (SER), Instructional Adaptability (IA), Physical Skill Development (PSD), Cognitive Retention (CR), Motivation Sustainability (MS), Learning Outcome Stability (LOS), Pedagogical Innovation Capacity (PIC), and Emotional Intelligence Integration (EII) have been chosen due to their excellent compliance with the theory and practice of physical education. Previous studies on physical education pedagogy emphasize engagement and motivation as important contributors to long-term participation, and adaptability and innovation are compatible with the principles of differentiated instruction. Correspondingly, cognitive retention, skill acquisition, and learning outcome stability are also key components of whole child development, and the integration of emotional intelligence is indicative of the growing focus on socio-emotional learning in the school setting. These attributes combined form the multidimensional aspect of teaching effectiveness when it comes to physical education.

This paper meets these needs by proposing a hybrid fuzzy MCDM model that represents a more detailed, data-intensive assessment of teaching performance. To be more exact, the objectives are:

• To develop a universal framework that incorporates multiple pedagogical attributes into a single evaluation system.

• To test the framework on the background of traditional assessment practices to check their reliability and accuracy

• To demonstrate how the framework will streamline teaching strategies and facilitate evidence-based decision-making in physical education practice.

Applying fuzzy logic-based decision systems in research creates advanced physical education pedagogy that designs multifaceted student-responsive teaching approaches.

The reason why TSFS were chosen is that it is an extension of previous fuzzy models, such as IF, Py, and PFSs, in that the square sum of MD, AD and NMD are constrained to be within unit interval . This property allows representing strong acceptance, partial hesitation, and partial rejection all at the same time, which can often occur with educational testing. TSFS offers greater representational flexibility and less information loss in decision-making than other fuzzy methods. All Aczel-Alsina operations were chosen based on their parametric control, which gives decision-makers control over the stringency of aggregation, allowing for flexibility in a variety of educational contexts where qualities may need to be prioritized differently.

Methodology

“It should be remembered that this article uses completely theoretical data as a means of demonstrating the method. The research did not involve any human subjects, human tissue or personal data at any point of the research process and as such did not need to be ethically approved or have informed consent”.

The section outlines a quantitative evaluation procedure for teaching effectiveness in physical education that applies integrated pedagogical approaches and a decision-making framework with fuzzy multi-criteria decision-making techniques. Multiple step-by-step procedures make up the evaluation framework.

Step 1: Alternatives and attributes identification

The first core process begins by identifying all relevant options and establishing attributes for decision-making. The physical education teaching strategies and methods under evaluation constitute alternatives during the assessment, and decision-making involves a set of key characteristics such as student engagement and skill development alongside cognitive understanding, emotional resilience, and teamwork capabilities. The research analysis assesses the impact of several instructional approaches on multiple learning attributes.

Step 2: Data collection

The physical education environment poses challenges to data collection efforts due to the diverse behaviors exhibited by students, necessitating the use of subjective evaluation methods under various environmental conditions. Each instructional approach undergoes assessment by an established panel of educational experts who address noted issues during evaluation. The weight calculation method employs experts’ qualifications and experience until the system reaches a total weight value of 1.

The three professionals were chosen according to highly qualified profiles (they are all postgraduate students of sports science or physical education) and more than ten years of experience in teaching/curriculum evaluation. Their experience was that the judgments were highly pedagogical and based on teaching practice.

A group of experts evaluates alternatives regarding educational attributes by combining fuzzy evaluation values that integrate IFS, PyFS, and SFS. These contemporary fuzzy algorithms enable decision-makers to manage ambiguous and uncertain situations better. A teaching method receives an MD rating when experts demonstrate confidence in its capability but receive an AG rating when they have doubts about effectiveness, along with the NMD rating when experts do not believe in the teaching method. The expert evaluation-based decision matrices follow this standardized format:

6

Where .

Step 3: Weights of information

Implementing the proposed decision-making framework requires accurate weight assignments because they determine assessment execution results. Board officials determine the relative weights decision-making experts will use to assess instructional effectiveness in physical education. The evaluation committee weights decision-making experts based on their educational background, subject area skills, and trustworthy behaviour. The weight constraints for both expert decision-makers and attributes operate between to maintain a complete weight summation of 1 while promoting accurate evaluation consistency.

The SER and MS attributes were given a relatively high weight as they were at the center of the physical education pedagogy, whereas other attributes, PIC and EII, were given a relatively lower weight to represent their increasing importance but limited presence in the pedagogy.

Step 4: Aggregation of information (individual attributes)

The information collection process utilizes TSFVs after Steps 1–3 is completed. Every expert creates a decision matrix to display their evaluations regarding various teaching methods regarding selected educational characteristics. A process strengthened by using evaluations from several experts provides increased robustness.

Aggregation models apply expert assessments about individual attributes, integrating various expert opinions into an all-encompassing evaluation format.

7

8

Step 5: Finding average attributes individual

After Step 4 completion, the decision matrix uses expert educational evaluations to produce a single structured framework. A complete aggregated value system for each teaching method is the fundamental requirement to measure effectiveness in teaching properly. Expert evaluations from different pedagogical attributes become consolidated by implementing TSFAWA and TSFAWG aggregation operators.

Step 6: Defuzzification of fuzzy information

The aggregated values from each instructional method exist in fuzzy form at first, preventing direct ranking procedures. The score function method converts fuzzy assessments from Step 6 into clear numerical values which can be interpreted. The defuzzification process obtains numerical values from fuzzy sets, enabling straightforward comparison of instructional strategies.

Step 7: Ranking of alternatives

Defuzzification is completed by assigning specific numerical values to each educational method. Higher numbers in evaluation scores show superior effectiveness of teaching combined with excellent adaptability and positive impacts on student development, while lower numbers demonstrate pedagogical limitations.

Decision-makers employ evaluation scores to select the most optimal instructional strategies during the final step.

The aggregate and defuzzification methods include pseudocode to make the results reproducible, which is included in the Supplementary Materials. The usage of computational programs (MATLAB/Python) is available on reasonable request.

In the following, Figure 2 illustrates the methodology stepwise.

Fig. 2.

Methodology flowchart to evaluate the instructional strategy regarding the different attributes.

Performance evaluation for physical education pedagogical effectiveness

The analysis evaluates five physical education teaching strategies through multiple critical instructional elements. The following evaluation details focus on the attributes described below.

Characteristics 1: Student engagement rate

Interest levels of students function as the base indicator for assessing instructional performance. Fuzzy MD determine the evaluation of active participation and attentiveness, so students with high levels of engagement receive higher MD, and those with low levels receive higher NMD. The evaluation grades for PFS models identify moderate engagement as AG.

Characteristics 2: Physical skill development

An instructor’s success at switching educational techniques according to student requirements constitutes adaptability in teaching. Effective adaptability during teaching leads to better MD scores, but inflexible teaching approaches produce higher NMD results. The classification of instructional performance according to adaptability thresholds makes use of IFS.

Characteristics 3: Physical skill development

Students’ physical ability development, including coordination strength and agility, needs regular monitoring because it is essential for assessing physical education results. The PyFS models allow a flexible evaluation by rating programs with consistent skill growth higher in MD and assigning higher NMD to those lacking measurable progress.

Characteristics 4: Cognitive retention

Retaining theoretical information about physical activities is fundamental to achieving holistic education. The MD score increases with instructional techniques that produce robust cognitive knowledge, but the NMD rating increases with methods that lead to weak knowledge transfer. The evaluation precision improves when SFS helps maintain balanced MD, AG and NMD assessment.

Characteristics 5: Motivation sustainability

The duration of student motivation maintenance in physical education classes determines their future involvement with physical education. The level of student motivation determines the MD value, while low motivation leads to increased NMD scores. AG controls motivational variations through PFS models that manage motivational uncertainties.

Characteristics 6: Learning outcome stability

The consistency of reaching anticipated learning targets throughout time indicates reliable instruction. The evaluation system rates programs better with MD scores when they demonstrate consistent target achievement yet moves programs to NMD when results show instability or deterioration. The Spherical Linguistic FS approach enables researchers to handle linguistic uncertainties in stability assessment processes.

Characteristics 7: Pedagogical innovation capacity

Innovative teaching methods that embrace gamification, technology usage, and inclusive practices demonstrate pedagogical innovation. The level of educational innovation determines Multi-Gazer ratings, while traditional teaching methods without updates lead to increased Non-Multi-Gazer grades. Judging innovation levels with uncertainty requires implementing hybrid fuzzy models that unite Interval-Valued Fuzzy Sets and Sigmoid Fuzzy Sets.

Characteristics 8: Emotional intelligence integration

Evaluating emotional intelligence integration determines how educators perceive and address students’ emotional and social needs. Emotional intelligence demonstrated through effective teaching methods leads to higher Measured Goodness, while inadequate emotional responsiveness results in higher Nominal Measured Goodness. Physical education success evaluations use PFS and SFS models to provide a complete analysis of emotional factors that affect physical education success.

The reason why these eight attributes were chosen is that they are all used to encompass the multidimensional objectives of physical education. They cut across the student involvement, flexibility of instruction, mastery of skills, mental performance, perseverance, sustainability of learning, resourcefulness in teaching, and socio-emotional assimilation as per pedagogic literature. This balance is a guarantee of the fact that the evaluation considers not only the effectiveness of teaching something technical, but also more general developmental goals, which are outlined in modern PE programs.

The evaluation process for physical education pedagogical effectiveness of five instructional strategies occurs based on the previously defined attributes. Educational decision-making becomes more effective through evaluations, allowing administrators to select appropriate teaching methods that improve student engagement, skill development, and learning outcomes. Note that the weights of the attributes are . To enhance transparency and reproducibility, the full (hypothetical) decision matrices offered by the experts have been included in Table 1,2,3.

Table 1.

Expert-1 opinion with respect to the instructional strategy which is founded on the considered characteristic.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
SER	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.29	0.2	0.21	0.22	0.23	0.24	0.25
IR	0.14	0.15	0.16	0.17	0.18	0.19	0.2	0.21	0.22	0.13	0.14	0.15	0.16	0.17	0.18
PSD	0.18	0.19	0.2	0.21	0.22	0.23	0.24	0.25	0.26	0.17	0.18	0.19	0.2	0.21	0.22
CR	0.3	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38	0.29	0.3	0.31	0.32	0.33	0.34
MS	0.34	0.35	0.36	0.37	0.38	0.39	0.4	0.41	0.42	0.33	0.34	0.35	0.36	0.37	0.38
LOS	0.31	0.32	0.33	0.34	0.35	0.36	0.37	0.38	0.39	0.3	0.31	0.32	0.33	0.34	0.35
PIC	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.29	0.2	0.21	0.22	0.23	0.24	0.25
EII	0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.32	0.33	0.24	0.25	0.26	0.27	0.28	0.29

Table 2.

Expert-2 opinion with respect to the instructional strategy which is founded on the considered characteristic.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
SER	0.23	0.24	0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.22	0.23	0.24	0.25	0.26	0.27
IR	0.16	0.17	0.18	0.19	0.2	0.21	0.22	0.23	0.24	0.15	0.16	0.17	0.18	0.19	0.2
PSD	0.2	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.19	0.2	0.21	0.22	0.23	0.24
CR	0.32	0.33	0.34	0.35	0.36	0.37	0.38	0.39	0.4	0.31	0.32	0.33	0.34	0.35	0.36
MS	0.36	0.37	0.38	0.39	0.4	0.41	0.42	0.43	0.44	0.35	0.36	0.37	0.38	0.39	0.4
LOS	0.33	0.34	0.35	0.36	0.37	0.38	0.39	0.4	0.41	0.32	0.33	0.34	0.35	0.36	0.37
PIC	0.23	0.24	0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.22	0.23	0.24	0.25	0.26	0.27
EII	0.27	0.28	0.29	0.3	0.31	0.32	0.33	0.34	0.35	0.26	0.27	0.28	0.29	0.3	0.31

Table 3.

Expert-3 opinion with respect to the instructional strategy which is founded on the considered characteristic.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
SER	0.24	0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.32	0.23	0.24	0.25	0.26	0.27	0.28
IR	0.17	0.18	0.19	0.2	0.21	0.22	0.23	0.24	0.25	0.16	0.17	0.18	0.19	0.2	0.21
PSD	0.21	0.22	0.23	0.24	0.25	0.26	0.27	0.28	0.29	0.2	0.21	0.22	0.23	0.24	0.25
CR	0.33	0.34	0.35	0.36	0.37	0.38	0.39	0.4	0.41	0.32	0.33	0.34	0.35	0.36	0.37
MS	0.37	0.38	0.39	0.4	0.41	0.42	0.43	0.44	0.45	0.36	0.37	0.38	0.39	0.4	0.41
LOS	0.34	0.35	0.36	0.37	0.38	0.39	0.4	0.41	0.42	0.33	0.34	0.35	0.36	0.37	0.38
PIC	0.24	0.25	0.26	0.27	0.28	0.29	0.3	0.31	0.32	0.23	0.24	0.25	0.26	0.27	0.28
EII	0.28	0.29	0.3	0.31	0.32	0.33	0.34	0.35	0.36	0.27	0.28	0.29	0.3	0.31	0.32

The aggregated information of the individual attributes in terms of TSFAWA and TSFAWG operators are presented in Tables 4 and 5, respectively. Table 6 and Table 7 represent the data that were compiled, whereas Table 8 features the scores of the alternatives. Under the TSFAWAWG operators, strategy-3 is ranked first, strategy-2 in second place, strategy-5 in third place, strategy-1 in fourth place, and strategy-4 in the last place, as indicated in Table 9.It is important to note that the ranking of the instructional strategies that was derived to both operators are identical. The following figure shows the ranking of the assets according to their performances. In order to exemplify the pedagogical significance of the fuzzy evaluations, we can use Strategy-3, the one that was rated in both aggregation operators the highest. To illustrate, in the SER attribute, Strategy-3 scored highly on MD which denotes active and observant students. On Physical Skill Development (PSD), the fuzzy scores once more indicated strong positive ratings, indicating that students achieved quantifiable improvements in coordination and agility when such a method was used. Correspondingly, Cognitive Retention (CR) achieved positive ratings, which indicated that students could memorize and use theoretical knowledge about physical activity in a useful way. Combined, these findings indicate that Strategy-3 did not only involve students; it facilitated physical and cognitive growth as well, which is why it was rated as the best-ranked teaching strategy. Fuzzy MD, AG and, NMD values of an attribute generate uncertainties in expert judgments. The pooling of these uncertain judgments to the experts with TSFAWA and TSFAWG operators does not eliminate uncertainty in the intermediate outcomes. Fuzzy scores are then defuzzified in order to change them into crisp ranking values but the spreading of uncertainty is carried to the scoring function. The sensitivity analysis onward shows that the rankings do not change significantly in response to changes in the aggregation parameters, indicating the strength of the findings despite the subjectivity of judgment.

Table 4.

Aggregated opinions of the experts regarding the instructional strategy, based on the considered characteristics, through tsfawa operator.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
SER	0.0040	0.0130	0.0147	0.0058	0.0186	0.0208	0.0080	0.0257	0.0284	0.0035	0.0114	0.0130	0.0051	0.0166	0.0186
IR	0.0013	0.0045	0.0053	0.0022	0.0074	0.0086	0.0035	0.0114	0.0130	0.0011	0.0037	0.0045	0.0019	0.0063	0.0074
PSD	0.0026	0.0086	0.0099	0.0040	0.0130	0.0147	0.0058	0.0186	0.0208	0.0022	0.0074	0.0086	0.0035	0.0114	0.0130
CR	0.0108	0.0344	0.0376	0.0142	0.0448	0.0487	0.0183	0.0572	0.0617	0.0098	0.0313	0.0344	0.0130	0.0411	0.0448
MS	0.0155	0.0487	0.0528	0.0198	0.0617	0.0665	0.0248	0.0769	0.0824	0.0142	0.0448	0.0487	0.0183	0.0572	0.0617
LOS	0.0119	0.0376	0.0411	0.0155	0.0487	0.0528	0.0198	0.0617	0.0665	0.0108	0.0344	0.0376	0.0142	0.0448	0.0487
PIC	0.0040	0.0130	0.0147	0.0058	0.0186	0.0208	0.0080	0.0257	0.0284	0.0035	0.0114	0.0130	0.0051	0.0166	0.0186
EII	0.0065	0.0208	0.0232	0.0089	0.0284	0.0313	0.0119	0.0376	0.0411	0.0058	0.0186	0.0208	0.0080	0.0257	0.0284

Table 5.

Aggregated opinions of the experts regarding the instructional strategy, based on the considered characteristics, through tsfawa operator.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
SER	0.0114	0.0045	0.0051	0.0166	0.0065	0.0072	0.0232	0.0089	0.0098	0.0099	0.0040	0.0045	0.0147	0.0058	0.0065
IR	0.0037	0.0016	0.0019	0.0063	0.0026	0.0030	0.0099	0.0040	0.0045	0.0030	0.0013	0.0016	0.0053	0.0022	0.0026
PSD	0.0074	0.0030	0.0035	0.0114	0.0045	0.0051	0.0166	0.0065	0.0072	0.0063	0.0026	0.0030	0.0099	0.0040	0.0045
CR	0.0313	0.0119	0.0130	0.0411	0.0155	0.0168	0.0528	0.0198	0.0214	0.0284	0.0108	0.0119	0.0376	0.0142	0.0155
MS	0.0448	0.0168	0.0183	0.0572	0.0214	0.0231	0.0716	0.0267	0.0287	0.0411	0.0155	0.0168	0.0528	0.0198	0.0214
LOS	0.0344	0.0130	0.0142	0.0448	0.0168	0.0183	0.0572	0.0214	0.0231	0.0313	0.0119	0.0130	0.0411	0.0155	0.0168
PIC	0.0114	0.0045	0.0051	0.0166	0.0065	0.0072	0.0232	0.0089	0.0098	0.0099	0.0040	0.0045	0.0147	0.0058	0.0065
EII	0.0186	0.0072	0.0080	0.0257	0.0098	0.0108	0.0344	0.0130	0.0142	0.0166	0.0065	0.0072	0.0232	0.0089	0.0098

Table 6.

Finiding collective aggregated opinion of the experts on the instructional strategy based on considered characteristically through the tsfawa operator.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
	7.29E-07	1.50E-02	1.71E-02	1.53E-06	2.16E-02	2.41E-02	3.05E-06	2.97E-02	3.27E-02	5.62E-07	1.32E-02	1.50E-02	1.20E-06	1.92E-02	2.16E-02

Table 7.

Finding collective aggregated opinion of the experts on the instructional strategy based on considered characteristically through the tsfawg operator.

	Strategy-1			Strategy-2			Strategy-3			Strategy-4			Strategy-5
	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD	MD	AD	NMD
	0.00481	0.00002	0.00003	0.00694	0.00005	0.00006	0.00960	0.00009	0.00011	0.00421	0.00002	0.00002	0.00617	0.00004	0.00005

Table 8.

Scores values of instructional strategy basing on the opinions of experts regarding the considered characteristics by using tafawa and tsfaawg operators.

	Strategy-1	Strategy-2	Strategy-3	Strategy-4	Strategy-5
TSFAAWA	0.03209	0.04567	0.06240	0.02821	0.04081
TSFAAWG	0.00486	0.00704	0.00980	0.00425	0.00625

Table 9.

Ranking of the instructional strategy according to the views of the experts in regards to the considered charactristics by use of tafawa and tsfaawg operators.

	Strategy-1	Strategy-2	Strategy-3	Strategy-4	Strategy-5
TSFAAWA	3	2	5	1	4
TSFAAWG	3	2	5	1	4

Sensitivity analysis of results

The TSFAWA and the TSFAWG operators are highly flexible since they have parameters. The values of the parameters, however, could alter the results due to the change in their values. In the following, the obtained results are observed using various values of the parameters. The values of all the parameters used in aggregation operators were clearly defined. A sensitivity test was done using the parameter values between 2 and 500 as presented in Tables 10-11.

Table 10.

Ranking of instructions strategies pegged on the opinion of the experts considered on subjected attributes in terms of the tafawg operator of several values of .

	Asset-1	Asset −2	Asset −3	Asset −4	Asset −5
2	0.03209	0.04567	0.06240	0.02821	0.04081
3	0.03309	0.04667	0.06340	0.02921	0.04181
4	0.03409	0.04767	0.06440	0.03021	0.04281
5	0.03509	0.04867	0.06540	0.03121	0.04381
6	0.03609	0.04967	0.06640	0.03221	0.04481
7	0.03709	0.05067	0.06740	0.03321	0.04581
8	0.03809	0.05167	0.06840	0.03421	0.04681
9	0.03909	0.05267	0.06940	0.03521	0.04781
10	0.04009	0.05367	0.07040	0.03621	0.04881
15	0.04109	0.05467	0.07140	0.03721	0.04981
20	0.04209	0.05567	0.07240	0.03821	0.05081
30	0.04309	0.05667	0.07340	0.03921	0.05181
40	0.04409	0.05767	0.07440	0.04021	0.05281
50	0.04509	0.05867	0.07540	0.04121	0.05381
100	0.04609	0.05967	0.07640	0.04221	0.05481
200	0.04709	0.06067	0.07740	0.04321	0.05581
300	0.04809	0.06167	0.07840	0.04421	0.05681
400	0.04909	0.06267	0.07940	0.04521	0.05781
500	0.05009	0.06367	0.08040	0.04621	0.05881

Table 11.

Ranking of instructions strategies pegged on the opinion of the experts considered on subjected attributes in terms of the tafawg operator of several values of .

	Asset-1	Asset −2	Asset −3	Asset −4	Asset −5
2	0.00486	0.00704	0.00980	0.00425	0.00625
3	0.00586	0.00804	0.01080	0.00525	0.00725
4	0.00686	0.00904	0.01180	0.00625	0.00825
5	0.00786	0.01004	0.01280	0.00725	0.00925
6	0.00886	0.01104	0.01380	0.00825	0.01025
7	0.00986	0.01204	0.01480	0.00925	0.01125
8	0.01086	0.01304	0.01580	0.01025	0.01225
9	0.01186	0.01404	0.01680	0.01125	0.01325
10	0.01286	0.01504	0.01780	0.01225	0.01425
15	0.01386	0.01604	0.01880	0.01325	0.01525
20	0.01486	0.01704	0.01980	0.01425	0.01625
30	0.01586	0.01804	0.02080	0.01525	0.01725
40	0.01686	0.01904	0.02180	0.01625	0.01825
50	0.01786	0.02004	0.02280	0.01725	0.01925
100	0.01886	0.02104	0.02380	0.01825	0.02025
200	0.01986	0.02204	0.02480	0.01925	0.02125
300	0.02086	0.02304	0.02580	0.02025	0.02225
400	0.02186	0.02404	0.02680	0.02125	0.02325
500	0.02286	0.02504	0.02780	0.02225	0.02425

Tables 10 and 11 show the ranking of the instructional strategies to the various parameter values using the TSFAWA and TSFAWG operators respectively. It is interesting to note that the same ranking is also seen in case of other values of . The values take values between 2 and 500. In the next figures, Figure 3 and Figure 4 present the classification of ranking of the instructional strategy at different values of .

Fig. 3.

Ranking of instructions strategies pegged on the opinion of the experts considered on subjected attributes in terms of the tafawa operator of several values of.

Fig. 4.

Ranking of instructions strategies pegged on the opinion of the experts considered on subjected attributes in terms of the tafawg operator of several values of .

Comparison, result discussion and limitations

The above section used the TSFAWA and TSFAWG operators to rank the methods of optimizing and disseminating instructional strategies. The value of the parameter is used as 3 to compute the results. Five methods are considered to rank based on various factors. The experts are asked to show their opinions about the methods using the numerical values from the unit interval. The decision maker assigns the MD, AD, and NMD to each technique from the unit interval satisfying the condition for any value of . Decision makers assign more excellent MDs if they are satisfied. He assigns AD to show his abstinence about methods. NMD shows dissatisfaction. Each decision-maker assigns a value to each method concerning each factor. The weights of the attributes and the experts are assigned according to their significance.

The aggregate data using the TSFAWA and TSFAWG operators is displayed in Table 4 and Table 5, respectively. The total information is displayed in Tables 6 and 7. The score values derived from the score function formulation are shown in Table 8. Table 9 shows that strategy 3 is ranked highest in the TSFAWA and TSFAWG operators. This indicates that strategy 3 is the most effective framework that complements fuzziness and humanized computing and is the most effective way to utilized the best instructional strategy. As shown in Tables 10 and 11, altering the value of does not affect the methodologies’ validity or dependability. The results show that method five regularly scores first and demonstrate the technique’s consistency and reliability across various dimensions for spreading sports culture.

The results are computed for five alternatives and eight attributes. The time complexity depends on the number of other options and the attributes. The larger the alternatives and attributes, the more time it will take. However, the model will be executed within seconds if we use any tool, i.e., MATLAB, Mathematica, or Python, to compute the result.

The following table compares the results obtained with those of some existing operators.

Table 12 shows the comparative analysis of the proposed operator. The proposed model is more suitable than the existing models.

Table 12.

The score value of aggregated values.

Model	Framework	Flexibility	Abstinence	Wider range
Defined in59	IFS	χ	χ	χ
Defined in60	IFS	χ	χ	χ
Defined in61	Fermatean FS	χ	χ	χ
Defined in62	PFS	χ		χ
Defined in63	PyFS	χ	χ	χ
Defined in64	Neutrosophic Set	χ		χ
Proposed	TSFAA

The paper provides a methodological basis of incorporation of fuzzy MCDM in physical education pedagogy. It is mainly theoretical and although this contribution reveals the possibility of the model to guide evidence-based teaching approaches, the practical effectiveness of the model has to be tested in practice-based educational situations. The work of the future will be pilot research and longitudinal classroom experiments aiming to decide the extent to which the framework can be effective in real life, thus filling in the gap between the theoretical potential and the practical influence.

Conclusion

Educators, administrators, and decision-makers obtain a structured system from this analysis to control complex and uncertain aspects of physical education teaching quality. The implemented fuzzy MCDM techniques improve the accuracy of the assessment of instructions, which allows better educational adjustments and planning. Users achieve specific teaching context, student population, and learning environment alignment by implementing adaptable parameters in this approach. The study presents three primary outcomes which form its fundamental research outcomes.

• The incorporation of TSFS provides physical education assessments with greater objectivity and enhanced depth by which independent inspectors can process ambiguous conditions and subjective aspects during performance analysis.

• The algorithm analyzes multiple pedagogical elements through its assessment method to provide complete insights into teaching effectiveness. The approach enables expanded precision because it lets users add more features to boost the practical application of the system.

• Through flexible aggregation processes, educators enhance teaching strategies, and decision-makers improve learning interventions with adjustments that follow changing educational scenarios and personal student requirements.

Future research will extend this combined instructional and decision systems framework to boost evaluation processes for teaching quality, individualize learning pathways, and sustain the professional advancement of physical education educators and their equivalents in other fields.

This work adds to the theoretical knowledge base a structured fuzzy MCDM system that improves the precision and flexibility of assessing instructional effectiveness in physical education. Although the methodological novelties, especially the combination of T-spherical fuzzy sets and Aczel-Alsina aggregation operators provide high computational merits, their consequences go beyond the technical field.

In the case of physical education teachers, the framework provides a decision-support tool which may point to the kind of strategy that would best maintain student interest, skill building, and motivation, and hence be used to govern real-time instructional changes. As an administrator, the system gives you an evidence-based approach to the quality of teaching that measures it across several levels as opposed to using only subjective or binary options. The framework can also help curriculum planners to create more responsive and data-driven education programs.

It should be mentioned that hypothesized data was used in this research to demonstrate this methodology. Further studies are necessary to incorporate the framework into practical classroom settings, to collect and examine empirical evidence with a variety of student bodies, and to analyze the extent to which the system predicts such learning gains as retention, group teamwork, and sustained physical fitness. These extensions will not just legitimize the model, but also hone its capacity to help teachers develop effective, flexible and student-focused physical education programs.

This paper gives a theoretical background to the application of fuzzy logic to pedagogical assessment in physical education. Although the present validation was performed using simulated data, the framework provides a good starting point to future research in the classroom. Its practical value should be subsequently validated in empirical researches and only then implemented in the educational practice at large.

Limitations

This research has a weakness in that the simulated data supplied by the experts were used and this limits the applicability of the research directly to the classroom real settings.

Future steps for empirical validation

This proposed framework should be applied to educational settings in the future. The potential lines of investigation are:

• to test the validity of the model in predicting engagement, skill levels and motivational persistence in students by testing in real physical education classes

• gathering longitudinal data on student involvement, on student competency, motivation

• comparing the results of other institutions and age groups to determine to what extent the model can be generalized. This empirical validation will provide greater approachability of the framework and validity as an evidence-based pedagogical decision-maker.

Author contributions

Yongfu Wu contributed fully.

Data availability

The data supporting this study’s results are available from the corresponding author upon reasonable request.

Declataions

Competing interests

The authors declare no competing interests.

Ethics

This study does not involve real human participants, human tissue, or personal data. The evaluation of five students and associated attributes—such as technical skill, physical conditioning, psychological endurance, and social collaboration—is based entirely on hypothetical data for illustrative and methodological purposes. As such, ethical approval and informed consent were not required. All methodological steps were conducted in compliance with institutional and international guidelines for studies not involving human subjects.