Some Criteria of the Knowledge Representation Method for an Intelligent Problem Solver in STEM Education

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Introduction
Knowledge representation plays a vital role in designing intelligent systems. Science, technology, engineering, and math (STEM) education emphasizes connections about concepts across different STEM fields to treat STEM education as a whole [1]. STEM education equips knowledge science and their real-world applications for the students. en, the students can develop their ability for discovering and problem-solving. e circle of STEM education is described in Figure 1. "Science" in the STEM circle means the process of scientific innovation from "technology" to "knowledge." In practice, when meeting the technology, scientists always make questions for researching to complete the technology. When finding the solutions for those questions, they will invent new scientific knowledge.
In contrast, "engineering" in the STEM circle uses the scientific knowledge to design new technologies [1,2]. e engineers have to solve problems to apply scientific knowledge to the practice. Science is the scientific process to invent new knowledge, and engineering is also the technical process to create new technologies. Two processes combine to form the scientific and technical innovation cycle, which has a spiral shape. After every turn of this spiral procedure, scientific knowledge improves with the development of the new technologies [2].
Courses of STEM education mentioned in this paper are mathematics, natural sciences (such as physics and chemistry), and basic programming (such as introduction to programming, data structures, and algorithms).
Artificial intelligence applications can be used to support the practice of learning. ree promising applications are intelligent tutoring systems, automated essay scoring, and early warning systems [3]. An intelligent tutoring system (ITS) simulates the instructional experience and interactions between a learner and a human tutor [4]. An intelligent problem solver (IPS) is a part of the ITS that can solve the problems automatically. Learners only declare the hypothesis and goal of problems based on a sufficient specification language [5]. ey can request the program to solve it automatically or to give instructions that help them to solve it themselves. e architecture of an IPS in education is shown in Figure 2.
e primary process of IPS is as follows: rough the user interface, the system recognizes the problems which are specified by the suitable specification language. e hypothesis and goal of the problem are determined by analysing the inference engine, and then they will be recorded into the working memory. e system uses its knowledge base to search objects, facts, and rules. After that, the system uses reasoning rules to solve the problem. When the system finds the solution, it produces a good one. Finally, this right solution is output in a human-readable form for the users via the interface.
For supporting the studying of learners, an IPS in STEM education has to meet requirements [7,8] as follows: RQ1: the program can solve common exercises in the course. Based on the knowledge base, the system can solve the basic and advanced kinds of general problems in the curriculum of the course automatically. RQ2: the input problems are specified by the language similar to the human. e solutions of the program are readable, step-by-step. RQ3: the reasoning of this system uses the knowledge of the learner about the course. Its solutions are similar to the solving method of the student.
For meeting these requirements, the method for knowledge representation in this system has to be built based on specific criteria. e knowledge base of this system is sufficient. It has to be organized fully and exactly. is thing will meet RQ1 of an IPS system. Besides, for satisfying RQ2, the knowledge representation method in this system is also convenient for users. e users can understand the methods for solving exercises.
Moreover, the problem-solving reasoning simulates the way of the learner's thinking. e reasoning steps are suitable for the knowledge level of the learner. erefore, this program satisfies requirements RQ2 and RQ3 of an IPS in education.
In this paper, we propose the criteria of a method to represent the knowledge of an IPS in STEM education. e method for knowledge representation includes a knowledge model, model of problems, and reasoning method to solve problems. For proving the effectiveness of these criteria, a knowledge model has been presented in this paper. is model can represent the knowledge of relations and operators, called the Rela-Ops model. is model satisfies the criteria of a knowledge model for an IPS in education. e criteria in this study develop a representation method about theory and application. Each criterion has also been classified into certain levels. ey have to guarantee the following factors: (i) eory: these criteria tend to ensure the solid foundation of the knowledge representation method. ey make this method developing in-depth.
(ii)Application: these criteria help to build a knowledge model which can be appllied in the real world, especially knowledge domains in STEM education, such as mathematics, physics, and chemistry. is model can be used to design the inference engine for solving practice problems of knowledge domains. e solutions are readable, step-bystep, and reasoning steps are suitable for the knowledge level of learners.
e Rela-Ops model is built based on the ontology approach [9]. It is useful in designing IPS in courses of STEM, such as discrete mathematics in university, solid and plane geometry in high-school, vector algebra in highschool, and direct current electrical circuit in middleschool. is model was presented in [9]. In this paper, we build the method or the process to design the IPS in education which can satisfy requirements of the IPS. Based on the application of this process, we also introduce the Rela-Ops model in general to prove the effectiveness of the proposed criteria. e next section discusses related studies for the criteria of a method for knowledge representation of IPS in education. Section 3 proposes and specifies the criteria of a knowledge representation method for an IPS in education. Section 4 presents a method for designing an IPS in education. e knowledge representation of this method satisfies the proposed criteria. Section 5 presents the Rela-Ops model, which is a knowledge model of relations and operator. is model meets the criteria of a knowledge model for an IPS in education. Section 6 discusses and makes a comparison between knowledge representation methods. e last section concludes the results.

Related Work
ere are many studies for the criteria of a method for knowledge representation. However, the criteria discussed in previous works do not meet the requirements of an IPS in STEM education.
A knowledge model has to represent the practical knowledge domain adequately [10]. It can perform fundamental components of this domain: concepts, relations, and inference rules. is representation method also gives reasoning in the knowledge [11]. Nonetheless, these criteria were still general, and they did not explain how the presentation is adequate; thus, they cannot be applied in practice, especially in building the IPS in education. e result in [12] uses belief rules based on knowledge representation scheme and inference methodology using evidential reasoning rule for representing the uncertainty of knowledge and reasoning.
Besides that, a knowledge model needs to be formal [13]. e components in the model have a solid foundation. e results in [14] study a method for generating a formal ontology by deep learning. e logic-based knowledge also can be represented by linear algebra [15]. Operators and relations in this knowledge are computed based on matrices and tensors. Nonetheless, they are theoretical results and have not yet been applied in STEM knowledge domains.
Solving system of linear equations is a critical problem in linear algebra. e study in [16] presents iterative methods for solving a linear interval system of equations, which is a linear system involving uncertain coefficients appearing as interval numbers. e first method replaces real operations with interval operations based on the conjugate gradient method. e second method solves linear interval systems by using the steepest descent idea. However, those results were not suitable to support the learning of linear algebra in university. ey did not use the knowledge of the course to solve the system of linear equations.
In [17], the study presents the performance of a steam turbine in thermal power plants using an artificial neural network. is method used NARMA to generate data and train network for the controlling model. Although the results of this method are emerging, it cannot show by itself how it works. Hence, it cannot be used to train the user to understand its performance.
In [18], the authors presented some components in intelligent tutoring systems.
ose components have to satisfy educational criteria to tutor the students. However, those criteria still belong to the scenarios, and they cannot be used to develop the system in practice, especially for IPS systems. In [19,20], the authors also proposed the requirements of knowledge representation for ITS. e domain knowledge module can represent structural and relational knowledge. is representation is natural, ease of update. e inference engine is efficiency, and it can reach conclusions from partially known inputs. ese requirements are practical for an ITS. An IPS is a part of an ITS which can solve problems automatically; however, these requirements do not mention to the ability about problemsolving, and thus they have some points which are not appropriate to an IPS.
A set of criteria for software requirements specification had been proposed in [21]. ose criteria are used to evaluate such standards, according to the unique characteristics of organizations and software development projects. However, those criteria are used for industrial software development, and they are not suitable for intelligent systems in education. ey do not mention characteristics in studying, such as naturalness and pedagogy. Criteria of an IPS in education can combine criteria for software requirements specification in [21] to become standards for designing of knowledge representation method for IPS in general. e paper [7] presented criteria for knowledge representation method of an IPS in education. ose criteria are: universality, usability, practicality, and formality. In this paper, some criteria are revised: generality, usability, naturalness, and formality. Each criteria is explained more clearly in each level. ose revisions make the evaluation of a knowledge representation method more easy and more suitable with the practical applications. Applied Computational Intelligence and Soft Computing Table 1 summarizes current criteria of knowledge representation for IPS in education and their novelty in this paper.

Criteria of a Knowledge Representation Method for an Intelligent Problem Solver in Education
For meeting requirement RQ1 of an IPS in STEM education, a knowledge model has to represent the knowledge base sufficiently. is method also has a useful specification language for users and supplies a pedagogical solution of an exercise as the knowledge level of learners [13]. It makes the IPS system satisfying requirement RQ2. Besides that, the reasoning of this method simulates the reasoning of humans, especially the learner. It works based on the specified knowledge as the content of the course. e reasoning meets requirement (RQ3). Moreover, the method for knowledge representation can apply in many knowledge domains, especially in knowledge of courses. It also ensures a solid mathematical foundation [13]. Besides, the IPS in education is also an intelligent software, so it has to satisfy some selection criteria for software requirements specification standard: generality, completeness, precision, practicality, and integration [21]. For these reasons, the criteria of a knowledge representation method for an IPS in STEM education include generality, usability, naturalness, and formality.

Generality.
e generality criterion examines how suitable a knowledge representation method is for different knowledge domains of courses [9]. Most of the current methods are only designed for specific types of knowledge domains [7,19]. is criterion means a method can apply in many knowledge domains of courses.
e generality criterion provides the flexibility of the representation method.
is method can be applied to represent the knowledge of courses, especially the courses about science and technology: mathematics, physics, and chemistry. When representing the practical knowledge, this method can be used directly or only needs some minor improvements for representing. is criterion includes four levels corresponding from very bad to very good. e meaning of each level is shown in Table 2:

Usability.
is criterion is the completeness criterion for intelligent software requirements specification standards. e first aspect of the usability criterion is the completeness of the knowledge model. e knowledge base of the IPS includes the knowledge, the content, and the actual learning content in the curriculum. e second aspect of the usability criterion is the completeness of the reasoning. e reasoning of this knowledge model uses detailed knowledge to solve practical problems, especially joint exercises in courses completely. Moreover, the reasoning steps of solutions are as the solving method of a student.
To achieve these goals, the knowledge model has an adequate structure to represent the practical knowledge domain [20]. e human knowledge domain has numerous components, but it has a foundation, including concepts of the knowledge domain, relations between concepts, and inference rules [5,6]. A model represents these components making a knowledge kernel as ontology. From that, the kernel can integrate with other knowledge, such as operators and functions, to strengthen the ability for representing the practical knowledge. Hence, the representation method needs the ability to represent the knowledge kernel. e inference strategy uses heuristics rules in its processing. Some heuristics rules can be used: arranging the order of rules in priority and using sample problems [25]. It can solve many kinds of exercises in the course.
Levels of the usability criterion are shown in Table 3: Naturalness. e naturalness criterion is the practicality criterion of software. An IPS tends to two main users: knowledge engineers and learners [5,7]. e representation method has to guarantee the specification language and the method of reasoning to accomplish the naturalness criterion.
e system has a knowledge base, which can be updated by the knowledge engineer. e specification language of the knowledge model has a simple structure but can represent the knowledge domain adequately. e representation is naturalness. Users as knowledge engineers can employ it to represent or update the knowledge domain easily.
Besides, the direct users of this system are learners. e method to input a problem into the system is easy to use by learners. e reasoning method of this model also simulates the reasoning for solving problems of the learners. ey can understand the knowledge as solutions for practical exercises. e system can find the pedagogical solutions of the exercises; the reasoning for problem-solving supports learners for studying the corresponding course.
Levels of the naturalness criterion are shown in Table 4: Formality. e formality criterion ensures the correctness of the representation method. is criterion supports theoretical shreds of evidence for the effectiveness of the method [10]. Moreover, by the formality of the method, it can be improved and developed based on the solid foundation.
Firstly, the components of the knowledge model need to be constructed based on a solid theoretical foundation [13].
eir structure and relationships are built formally. e problems of this model can also be modeled. Secondly, the algorithms for solving the problems must be constructed based on the structure of the knowledge model and problems.
ose algorithms must be proven to be finite and active, and their complexity must be evaluated.
Levels of the formality criterion are shown in Table 5:

Method for Designing the Intelligent Problem Solver in Education
For satisfying requirements of the IPS in education, the building knowledge base and inference engine components are essential in designing the system. e knowledge 4 Applied Computational Intelligence and Soft Computing representation method has to meet the criteria in Section 3. e process of analysis and design of the system components consists of seven stages (Figure 3) [5].
Stage 1 collects the real knowledge domain based on the classification of kinds of knowledge. e collection helps to form the model for knowledge representation. Stage 2 builds the knowledge model for the collected knowledge domain. Based on the knowledge model, stage 3 organizes the knowledge base for the IPS. e specification language for the knowledge base, which is studied in stage 4, has to simulate the way of describing the knowledge in practice. Besides the knowledge base, the model of problems on the knowledge domain also has to be studied. ose problems are the foundation for designing of reasoning algorithms. e reasoning algorithms are the demonstration of the problem-solving ability of the system. Stage 5 designs the query language of the system. For the goal supporting of the learning, the query language has to be suitable for the is criterion is the completeness criterion for intelligent software requirements specification standards. It concerns the requirement for building knowledge bases of intelligent systems. e knowledge representation method can represent the components of ITS entirely by using this criterion. However, the current meaning of this criterion does not aim to design the IPS.
In practice, a knowledge domain has many levels, especially the educational knowledge. In this study, the criteria have been classified into levels. Each level has the meaning of being suitable with requirements for designing the knowledge base of each IPS Formality e meaning of the criterion only orients to build formal models, and it did not mention the application in real-world knowledge domain (i) Formal logic methods are proper to meet this criterion [22,23]. However, those methods cannot represent real-world knowledge, especially the knowledge of courses (ii) Algebraic approach is a method based on the mathematical structures. ey are classical algebraic structures [24]. However, this criterion does not mention the ability of reasoning and explaining in the problem-solving process Research the criterion being suitable to apply in practice and ensure the theoretical foundation. is criterion includes: (i) Criteria about theoretical foundation for constructing components of the knowledge model. e structure of those components can be used to design algorithms for reasoning (ii) Criteria can be used to build practical, intelligent systems, especially for IPS in courses Set of criteria for software requirements specification ose criteria are used to evaluate standards according to the unique characteristics of specific combinations of software development projects [21] However, those criteria are not suitable for the characteristics of intelligent educational software: naturalness and pedagogy Build the criteria for software development to adapt to the pedagogical criteria of the intelligent learning system.  e communication of the system is pedagogical and similar to the tutoring of the lecturer. In stage 6 and stage 7, the IPS is completed by designing its interface and testing. Stage 1. Determine the knowledge of courses and scope; then, collect the real knowledge consisting of concepts and objects, relations, operators and functions, facts, and rules. is knowledge collecting can be classified in some ways such as chapters, topics, or subjects; based on this classification, problems and exercises in the course can be collected appropriately and quickly. Problems are also classified according to some methods such as frame-based problems and general forms of problems.
is stage ensures that the knowledge domain will be represented entirely. Stage 2. Build the model for the collected knowledge domain. It is an essential base for designing the knowledge base of the IPS in education. e model has to represent the kernel of the knowledge domain, including concepts, relations, and rules. e kernel can be integrated with other knowledge components to represent the knowledge of the course sufficiently. (i) e specification language is machinery (ii) e representation method cannot solve the common exercises of a course (i) e specification language of the method simulates the human language, but it is not suitable for students (ii) e representation method can solve common exercises of a course (i) e specification language of the method is suitable for students (ii) e system can solve some general problems. e reasoning of its solution is suitable for the level of learners (i) e specification language of the method is similar to the natural language for knowledge representation (ii) Solutions of the system are pedagogy (iii) Besides solving problems automatically, the knowledge base of this system tends to tutor the student on how to solve a problem in the course   Applied Computational Intelligence and Soft Computing e structure of knowledge components in the model has been constructed based on the mathematical foundation. is structure is an integral part for the formality of the model. Stage 3. Establish a knowledge base organization for the system.
is stage makes the representation more natural and suitable for the knowledge level of users. Design the specification language to represent components of the knowledge model. e knowledge engineer uses this language, which is designed to simulate the way of describing the knowledge in practice. Based on this language, a knowledge base can be organized by structured text files [5]. (1) Here, set O is the set of objects, F is the set of facts given on the objects, and G is a list of goals of the problem. ree steps for modeling can develop the design of deductive reasoning algorithms for solving problems and the design of the interface of the system: Step 1: classify problems such as problems as frames, problems of a determination or a proof of a fact, and problems of finding objects or facts.
Step 2: classify facts in the knowledge domain.
Step 3: modeling kinds of problems from classifying in steps 1 and 2. From models of each kind, we can construct a general model for problems, which are given to the system for solving them. e basic technique for designing deductive algorithms is the unification of facts. Based on the kinds of facts and their structures, there will be criteria for unification proposed. en, it produces algorithms to check the unification of two facts. e next important work is researching reasoning strategies to solve problems on the computer. e most challenging thing is modeling for experience, sensible reaction, and intuitional humans to find heuristic rules, which were able to imitate human thinking for solving problems. When designing deduction algorithms, the effectiveness and complexity of those algorithms need to be considered. ose algorithms have to be built based on the way of learners' thinking to solve problems. is stage serves the usefulness of the system to enhance studying. Stage 5. Create a query language for the models. e query language has to be suitable for the knowledge level of students and helps to design the communication between the system and users. Inputting the problem and understanding the solution from the system is more manageable by using the query language. Moreover, the communication of the system is pedagogical and similar to the tutoring of the lecturer. Stage 6. Design the interface of the system and coding to produce the application. Intelligent applications for solving problems in education of mathematics, physics, and chemistry have been implemented by using programming tools and computer algebra systems such as Visual Basic.NET or C#, SQL Server, and Maple [26]. ey are straightforward to use for students to search, query, and solve problems. Stage 7. Testing, maintaining, and developing the application. is stage is similar to what happens in other computer systems.

Rela-Ops Model
Definition 1 (see [9]). A knowledge model of relations and operators, called Rela-Ops model, is a tube: In which: (i) C is a set of concepts. Each concept c is a class of objects, and it has an instance set, called I c . Each concept c is a tube (Attrs, Facts, EqObj, RulObj), which Attrs is a set of attributes, Facts is a set of facts of a concept c, EqObj is a set of equations of a concept c, and RulObj is a set of deductive rules of a concept c. (ii) R is a set of relations between concepts in C. It includes hierarchical relations and binary relations between concepts in C. (iii) Ops is a set of operators between concepts in C. It includes unary and binary operators. (iv) Rules is a set of inference rules of the knowledge domain. In this study, Rules-set is classified into four kinds of rules: deductive rules, rules for generating a new object, equivalent rules, and equation rules.
An inference rule r ∈ Rules is one of the four cases: Rules � Rule deduce ∪ Rule generate ∪Rule equivalent ∪Rule equation . (3) (i) r ∈ Rule deduce : r is a deductive rule, it has the form: e detailed structure of each component in the Rela-Ops model has been presented in [9]. is model is built based on ontology and object-oriented approaches. Each concept in the Rela-Ops model is a class of objects, and each object has the structure and behaviors to solve problems by itself.

Problems on Rela-Ops Model.
In the Rela-Ops model, there are two kinds of problems: problems on an object and general problems on the model. Problems on an object are its behaviors, and they are solved based on the reasoning on their structures. General problems are solved by reasoning method on the rules in Rules-set and solving problems on objects. e solving method combines the knowledge of relations and operators to get new facts in the reasoning.

Problems on an Object
Definition 2 (see [9]). e closure of a set of facts.
Let Obj � (Attrs, Facts, EqObj, RulObj) be an object of a concept in C and F be a set of facts. e closure of set F by Obj, Obj.Closure(F), is a maximum extension of F by using reasoning rules in Obj.EqObj and Obj.RulObj.
ere are three kinds of problems on an object in the Rela-Ops model: (1) determine the closure of a set of attributes, (2) determine the closure of a set of facts, and (3) execute deduction and give solutions for a problem. In this section, we present the algorithm to solve the problem of determining the closure of a set of facts. □ Hence, the complexity of algorithm 1 is O max k 3 , k n 1 ·q 1 , k n 2 ·q 2 � O max k n 1 ·q 1 , k n 2 ·q 2 � O k max n 1 ·q 1 ,n 2 ·q 2 ( ) .
(4) Problems in kind 1 and kind 2 were studied and solved in [6,9,27]. e effectiveness of the algorithms for solving problems in kind 1 has been proven in [6,9] and for solving problems in kind 2 has been proven in [9,27].

General Problems on Rela-Ops
Lemma 1 (see [27]). Let a knowledge domain K as Rela-Ops model and (O, E, F) be the hypothesis of the problem as kind 2 in Definition 3. ere exists a unique maximum set L (O, E, F) such that it contains all facts that can be deduced from (O, E, F). Theorem 2 (see [27]). Let a knowledge domain K as Rela-Ops model and a problem P � (O, E, F) ⟶ G as kind 2 in Definition 3. Suppose S � [s 1 , s 2 , . . ., s k ] is a list of rules. e following statements are equivalent: exists a list of rules S � [s 1 , s 2 , . . ., s k ] such that G.f ∈ S(E, F), with S(E, F) is a set of facts can be deduced from the list S and hypothesis of problem P eorem 2 shows that forward chaining reasoning will deduce the goals of problems. Besides, algorithm 2 is designed based on forward chaining; therefore, eorem 2 guarantees the effectiveness of this algorithm. 8 Applied Computational Intelligence and Soft Computing

Rela-Ops Model and Criteria of a Knowledge Model for an IPS in Education.
Rela-Ops model is a knowledge model, including the knowledge of relations and operators. ese kinds of knowledge are popular in practice, especially in STEM knowledge. is model is flexible and effective in practical applications. As shown in the appendix Let K � (C, R, Ops, Rules) be a knowledge domain as Rela-Ops model, Obj � (Attrs, Facts, EqObj, RulObj) be an object of a concept in C, F is a set of facts. is algorithm deduces the closure of set F by Obj, Obj.Closure(F). Input: Object Obj � (Attrs, Facts, EqObj, RulObj), F is a set of facts. Output: Obj.Closure(F) Step 0: Initialize variables flag:� true; KnownFacts:� F ∪ Obj.Facts; Step 1. Classify kind of facts in KnownFacts Step 2. Determine new facts from facts in KnownFacts by using reasoning rules.
Step 3. Search the closure of facts as an object in KnownFacts.
for fact in KnownFacts do if (fact is an object)  is algorithm will solve problem P through these steps as follows: Input: e problem P � (O, E, F) ⟶ G Output: e solution to problem P. e method for designing this algorithm uses forward chaining reasoning. It combines heuristics rules in the reasoning process. Objects also attend this process as active agents for solving problems on themselves by Algorithm 1. is process is done when it gets the goal.
Step e input and output of exercises in these courses are easy to use and understand. Solutions of them are step-by-step, and their reasoning is like the solving method of students. e detailed structure of the Rela-Ops model and its problems has been presented in [6,27]. e finiteness, the effectiveness, and the complexity of algorithms have also been proved in [6,9,27].

Discussion
Nowadays, knowledge representation methods can be classified into four types: the representation by formal logic, networks, ontology, and algebraic approach.
Formal logic methods are not effective for the complex knowledge domains, especially in education. Besides classical logic methods, description logic has also been studied.
is logic is the formal representation of semantic [22,23]. However, logic methods cannot represent STEM knowledge, mainly structural and relational knowledge. Hence, they cannot be applied to the design of the knowledge base of an IPS in STEM education.
Representation methods by networks are suitable for classifying the concepts. ese methods are not valid for the practical knowledge domain, especially computing knowledge. A semantic network belongs to the language for representation. A knowledge graph is a methodology to perform link prediction between entities. Its nodes represent the item, entity, and user, and its edges represent the linking nodes that interact with each other [29]. e knowledge graph is a useful tool for information searching and giving semantics to textual information. Nonetheless, it is difficult to reason for solving problems, especially the problems of an IPS system in STEM education.
Algebraic approach is a representation method based on the mathematical structures; they are classical algebraic structures, such as groups, rings, ideals, and fields, or they are integrating those structures [24]. e problem of information equivalence of knowledge has been solved in [30] based on the definition of the symmetries of knowledge bases. e knowledge base as logic is also presented by the structure of matrices in linear algebra [31]. e knowledge in these results only has information form; hence, they cannot be applied to solve significant problems that require the ability to reason in the problem-solving process.
In intelligent tutoring systems, ontology is used as a framework to represent the content of a course [32]. ese systems could not yet solve problems automatically. Computational Network Object Knowledge Base (COKB) is an ontology that can be applied to build practical applications in IPS systems [5]. However, the formality of this model has some limitations. e mathematical foundation of COKB's components has not yet been presented clearly.
Rela-Ops model can satisfy the criteria of a knowledge representation method for an IPS in education, especially for technological courses, such as mathematics, physics, and chemistry. It can represent many kinds of knowledge domains in education, such as mathematics, physics, and programming. e IPS systems built based on it are useful for students. ey can solve common exercises in corresponding courses and some hard problems with them. eir solutions are step-by-step. eir reasoning is appropriate to the knowledge level of learners. In practice, some knowledge domains include many subdomains; thus, for representing those knowledge domains, the representation method has to support the integrating of knowledge bases between subdomains. e architecture of the Rela-Ops model can integrate subdomains which have the structure as the Rela-Ops model. e integration between knowledge bases for designing an IPS has been studied in [4]. For example, Appendix C (Supplementary Materials (available here)) presents an integrating model between Rela-Ops model and frames [33]. is model is used to represent the knowledge base of programming and design the intelligent system for learning of courses about algorithms [33,34]. ALGORITHM 2: (see [27]). Solving the problem as kind 2. Table 6 compares the discussed methods for knowledge representation, as far as the satisfaction of criteria of knowledge models for IPS systems are concerned.

Conclusions and Future Work
In this paper, the criteria of a knowledge model for an intelligent problem solver in STEM education have been proposed. ey include generality, usability, naturalness, and formality. Each criterion has certain levels. ese criteria orient to develop a method for knowledge representation about theory and application. e knowledge base, which is built based on those criteria, can meet the requirements of an IPS.
(i) e generality criterion executes the compatibility of a knowledge representation method for different knowledge domains of courses. (ii) is usability criterion is the completeness criterion for intelligent software requirements specification standards. (iii) e naturalness criterion is the practicality criterion of software. It is the nature of the IPS works. e representation method has to guarantee the nature of specification language and the method of reasoning to accomplish the naturalness criterion. (iv) e formality criterion ensures the correctness of the representation method. is criterion supports theoretical evidence for the effectiveness of the method.
For proving the effectiveness of these criteria, the Rela-Ops model is introduced in this paper. It is a model representing the combining knowledge of relations and operators. is model is built based on the object-oriented and ontological approach. Each concept in Rela-model is a class of objects which also have the structure and the ability to solve problems on themselves. Rela-model can be applied to design the knowledge bases of IPS systems for corresponding courses. It also satisfies the criteria of a knowledge model for an IPS in STEM education.
e real-word knowledge domain has many subdomains, so the criteria of knowledge representation method have to mention to the problems about integrating knowledge-based systems. In the future, we will continue to study these criteria of an integrated knowledge model for an IPS. From that, they will be developed to be the criteria of a general knowledge model. ose results will be the foundation for building a supporting tool to design general knowledge-based systems. Besides, the integration method of knowledge bases, which are as Rela-Ops, needs to study for application in IPS.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.