Models and Software for Intelligent Web-Based Testing System in Mathematics Andrey Chukhray and Elena Yashina National Aerospace University «Kharkiv Aviation Institute», Chkalova 17, 61070 Kharkov, Ukraine Abstract In this paper computer models and software for intelligent web-based testing system in mathematics are described. The analysis of computer web-based testing systems in mathematics is carried out, Parameterized mathematical models for different tasks, including algebra and geometry, are synthesized, Architecture and software of intelligent web-based testing system are presented. Knowledges and skills evaluation model on the base of fuzzy logic is developed. Now system is working in two modes: a mode of training and a control mode. Keywords 1 Intelligent tutoring systems, e-learning, educational software, mathematical models 1. Introduction In the past few years e-learning experiences true boom. The driving force of it is continuous computer potential growth: increase of processor power and memory capacity, improvement of I/O facilities, perfection of network technologies, boost of efficiency of modern software development tools. The Covid-19 pandemic gave a powerful impetus to the computer-aided learning growth. Artificial intelligence techniques allow to individualize and personalize the educational process [1, 2] and improve feedback from students [3]. At the same time scientific theory of artificial intelligence for e-learning is developing not so fast as developers would like. Artificial intelligence techniques are successfully used in teaching computer and mathematical sciences [4]. But today it is not possible yet to replace a good teacher with a computer tutor system. There is insuperable complexity of synthesis of models and methods for adaptation to mental work features, repertoire of knowledge and skills of each learner [5]. Therefore efforts of many developers aim at more utilitarian tasks where considerable progress is achieved at interactive intelligent tutoring system, such as evaluation of mathematical expressions [6], models constructing [7], solving algebra [8] and geometry [9] problems, online and web-based tutoring systems [10, 11], web-based test systems [12]. Within the framework of the Ukrainian project of external independent evaluation before authors there was a task of intelligent web-based testing system creation. The purpose of it was increase of effectiveness of school graduates preparation to external independent evaluation in mathematics by means of availability assurance of tests examples through the Internet, introduction of interactive modes of training and control for improvement contents of tasks and technologies of answers filling understanding, economy of expenses for manufacture and distribution of test tasks paper collections, As analysis of world experience of interactive testing systems in mathematics making has shown the most creative way is development of parameterized mathematical models for automatic generation unique tasks, right answers and wrong but plausible answers for closed tests. International Workshop of IT-professionals on Artificial Intelligence (ProfIT AI 2021), September 20-21, 2021, Kharkiv, Ukraine EMAIL: achukhray@gmail.com (A. 1); o.yashina@khai.edu (A. 2); ORCID: 0000-0002-8075-3664 (A. 1); 0000-0003-2459-1151 (A. 2); ©️ 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) 2. Task definition This article is dedicated to solving the problems. 1. To synthesize parameterized mathematical models for automatic generation unique tasks, right answers and wrong but plausible answers for closed tests. 2. On the base of fuzzy logic to develop knowledges and skills evaluation model. 3. To create algorithms and the software for intelligent web-based testing system in mathematics which must work in two modes: a mode of training and a control mode. 3. Task solution 3.1. Parameterized mathematical models We described 26 different mathematical tasks, including algebra and geometry ones, were parametrical. There were 12 tasks of closed type and 14 tasks of open type. Let's consider some examples of parameterization. 3.1.1. Example I To solve equation √𝑥 + 1√𝑥 − 2√𝑥 − 5 = 0. (1) In general we have √𝑥 + 𝑎1 √𝑥 + 𝑎2 … √𝑥 + 𝑎𝑛 = 0. (2) The admitted region is 𝑥 + 𝑎1 ≥ 0 𝑥 + 𝑎2 ≥ 0 (3) { … 𝑥 + 𝑎𝑛 ≥ 0 Hence it follows 𝑥 ≥ − 𝑚𝑖𝑛{𝑎1 , 𝑎2 , . . . , 𝑎𝑛 } (4) and 𝑥 = − 𝑚𝑖𝑛{𝑎1 , 𝑎2 , . . . , 𝑎𝑛 }. (5) The model of the task consists of: 1) n ∈ {1,2, . . . , 10}is randomly chosen; 1) set {𝑎1 , 𝑎2 , . . . ,𝑎𝑛 }, − 100 ≤ 𝑎𝑖 ≤ 100, 𝑖 ≠ 𝑗 → 𝑎𝑖 ≠ 𝑎𝑗 is randomly chosen; 2) variants of answer are generated by the following way: a. equation has no roots; b. 𝑥 = −𝑚𝑖𝑛{𝑎1 , 𝑎2 , . . . , 𝑎𝑛 }; c. 𝑥 = −𝑚𝑎𝑥{𝑎1 , 𝑎2 , . . . , 𝑎𝑛 }; d. 𝑥 = {−𝑎1 , −𝑎2 , . . . , −𝑎𝑛 } 3) variants of answer are arranged randomly; 4) right answer is compared with answer of graduate. 3.1.2. Example II The medial line of trapezium in Ошибка! Источник ссылки не найден. is equal 7. Height is 15√3 . 7 Figure 1: Graphical task statement Angle between diagonal is equal 120°. To find product gets of diagonals of base is equal L and vertex angle is equal 120° BOC and AOD are similar triangles. Hence it follows 𝐴𝐷 𝐵𝐶 (6) = . 𝐹𝑂 𝐸𝑂 Also we know that 𝐴𝐷 + 𝐵𝐶 1 (7) 𝑆= ∗ (𝐸𝑂 + 𝐹𝑂) = ∗ 𝐴𝐶 ∗ 𝐵𝐷 ∗ 𝑠𝑖𝑛(120°). 2 2 Therefore 𝐴𝐷 (8) 𝐴𝐷, 𝐹𝑂 ≤ , 𝐸𝑂 < 𝐹𝑂 2√3 are generated. BC is calculated as 𝐸𝑂 (9) 𝐴𝐷 ∗ . 𝐹𝑂 Then desired value is AD + BC 1 (10) AC ∗ BD = ∗ ∗ EF. 2 √3 3.1.3. Example III To solve equation 2𝑥+2 − 2𝑥 = 96. (11) In general we have (−1)𝑏1 𝑎 𝑥+𝑐1 + (−1)𝑏2 𝑎 𝑥+𝑐2 +. . . +(−1)𝑏𝑛 𝑎 𝑥+𝑐𝑛 = 𝑃. (12) The model of the task consists of following actions: 1) n ∈ {2,3, . . . , 10} is randomly chosen; 2) a,x ∈ {2,3, . . . , 10} are randomly chosen; 3) 𝑐𝑖 ∈ {0,1,3, . . . , 10}, 𝑖 = ̅̅̅̅̅ 1, 𝑛 are randomly chosen; 4) 𝑏𝑖 ∈ {0, 1}, 𝑖 = ̅̅̅̅̅ 1, 𝑛 are randomly chosen; 5) P is calculated: 6) right answer is compared with answer of graduate. 3.1.4. Example IV Areas of bounds of rectangular parallelepiped are equal 𝑆1 , 𝑆2 , 𝑆3 . To find volume of parallelepiped. The model of the task includes: 1) generation random 𝑥, 𝑦, 𝑧 ∈ {2,3, . . . ,10}; 2) calculation 𝑆1 = 𝑥𝑦; 𝑆2 = 𝑧𝑦; 𝑆3 = 𝑥𝑧; 3) check graduate answer against x ⋅y ⋅z. 3.2. Knowledges and skills evaluation model To create evaluation model fuzzy approach was chosen because of its closeness to teacher qualitative reasoning. Let us consider input linguistic variables: 𝑈𝐼 =”Tasks solving speed”, described by the term set 𝑇(𝑈1 )={Slow, Medium, Fast} as in [2]; 𝑈2 =”The number of errors” , 𝑇(𝑈2 )={Zero, Very small, Average, Large, Very large}; and 𝑈3 =”Help reference”, 𝑇(𝑈3 )={Without help, With small help, With help, With great help}. For example, membership functions for 𝑇(𝑈3 ) are presented in Figure 2. Figure 2: Membership functions for input variable Help reference Let 𝑌 = ”𝐾𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒𝑠 𝑎𝑛𝑑 𝑠𝑘𝑖𝑙𝑙𝑠” be an output linguistic variable with term set 𝑇(𝑌)={ Excellent, Good, Satisfactory, Bad } T(Y)={Excellent, Good, Satisfactory, Bad}. Then rules can be expressed in the Mamdani form: If 𝑈𝐼 is Fast and 𝑈2 is Zero and 𝑈3 is Without help then Y is Excellent. If 𝑈𝐼 is Fast and 𝑈2 is Very small and 𝑈3 is With help then Y is Good. … If 𝑈𝐼 is Slow and 𝑈2 is Very large and 𝑈3 is With great help then Y is Bad. 4. Software The main task of this work is intelligent computer system for testing knowledge and skills in mathematics software development. We have collected and analyzed the system requirements. Based on this the architecture of system and main components software was developed. 4.1. System architecture As a result of the analysis of the system requirements, the main components and the connections between them were identified. The system architecture model is shown in Figure 3. The system consists of the following levels: - the database level, which contains the database for accounting information about users (registration, test results); - the business logic level, which includes the APP Server; - the communication level, that is, the level that carries out network interaction with the user; - presentation layer, which includes tools for development and presentation of data, such as ASP .NET, WinForms, as well as directly displaying Web sites, that is, various browsers; - user level. DATABASE Admin LAYER DB (world leadership planner) BUSINESS LOGIC LAYER APP Server Winforms / WPF COMMUNICATI ON LAYER WCF / Remoting / Web services PRESENTA TION LAYER ASP.NET WinForms / WPF IE / FF / opera STPD LAYER Client Figure 3: The architecture of an intelligent computer system for testing knowledge and skills in mathematics Components for generating problem conditions and answer options for them are stored in separate library DLL files. To provide versatility multilevel architecture of system was developed. It consists of four levels: database level, business logic level, communication level and presentation one. Main components of business logic layer are presented in Figure 4. Task Statement TaskPresentation (graphic presentation) < Task Generators > Control *.dll Reflection ITaskGe List Agent nerator AnswerVariant XML WCF Answer / Login Data IPresentation Provider DB Statement XML ASP.NET Pages & Controls (other UI) ASPNetPresenta Equation EquationPres tionProvider Its presentation entation Figure 4: Components of business logic layer After an entrance of the user in system, which happens at ASP layer WCF layer begins work. WCF represents an intermediate link witch connects parts of display and logic. WCF contacts library with tasks and through interface ITaskGenerator by means of reflection operation receives the list of tasks. Each element of the task list, i.e. a copy of class Task, contains a task condition, variants of answers and a right answer as XML-code. Control agent cuts right answer leaving it on a server and directs the remained elements (a condition and answer variants) to layer WCF. Thus, right answers in no way cannot become accessible to the user. Therefore security is provided. The diagram of the main classes of an intelligent computer system for testing knowledge and skills in mathematics is shown in Figure 5. The basic abstract class Task has two inheritors: the EnclosedTask class a of a closed type task; the OpenTask class of an open type task. Both classes override the base class's Check() method and have several constructors. The Task class defines a Statement class that represents a task condition. It is associated with the AnswerVariant class, which generates answer options. The Statement and AnswerVariant classes through the TaskPresentation class form the expression of the condition of the task and the options for responding to it in the form of an XML representation. In addition to the AnswerVariant class, there is an abstract RightAnswer – correct answer class. Its inheritors are the classes of the correct answer to tasks of the private EnclosedRightAnswer and public OpenedRightAnswer types. The Task class, through the TaskPresentation class, generates conditions and response options in the form of XML code containing tags of a specific purpose, for example, the tag contains an expression for building a mathematical formula, the tag is a directive for plotting a function graph. ASPNetPresentationProvider class implements transformation of task XML-code for its presentation on WEB-form. To implement tags such as a formula or graph, the ASPNetPresentationProvider class includes the appropriate libraries to represent them graphically. Figure 5: The diagram of the main classes of an intelligent computer system for testing knowledge and skills in mathematics 4.2. System Interface Development The system must have an advanced user interface for visualizing formulas, graphs and other graphical objects. One of the system components is a module for constructing mathematical formulas designed for visual output of mathematical formulas of any complexity. The formula is specified as a sequence of elementary instructions that determine the location of the formula elements. After the structure of the formula is formed, the elements are arranged, the formula enters to the graphical module, which draws it on the Bimap object and displays it on the web form. The example of formula visualization is given in Figure 6. Figure 6: The example of formula visualization Also, a module for dynamic plotting of various mathematical functions has been implemented. Examples of dynamic output of function graph are shown in the Figure 7. Figure 7: The example of function graph output Screenshots of the system are presented in figures 8 – 10. Figure 8: Screenshot of the system for trigonometric task Figure 9: Screenshot of the system for algebraic task with graphical elements Figure 10: Screenshot of the system for stereometric task Through interface IPresentationProvider the task in the form of a XML-code arrives to object ASPNetPresentationProvider where its analysis and split are executed. The task can consist of the text and pictures with formulas and graphs. In case of formulas object EquationPresentation provides creation of a demanded picture. Similar technology is used for creation plots of functions. Then controls arrive to ASP.NET Pages where they are displayed. Now system is being passed beta testing in site of Kharkov regional centre of education quality evaluation. 5. Summary Parameterized mathematical models, knowledges and skills evaluation model and software for intelligent web-based testing system in mathematics are developed. 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