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.
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 [
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 [
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.
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.
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. (1) (2) (3) (4) (5) 3.1.1. Example I
= are generated. BC is calculated as
To solve equation (7) (8) (9) (10) (11) (12) 2√3 ∗
AC ∗ BD = AD + BC ∗ 1 √3 ∗ EF. 2 +2 − 2 = 96.
(−1) 1 + 1 + (−1) 2 + 2+. . . +(−1) + = . 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) ∈ {0,1,3, . . . , 10}, = ̅1̅̅,̅̅ are randomly chosen; 4) ∈ {0, 1}, = ̅1̅̅,̅̅ are randomly chosen;
Areas of bounds of rectangular parallelepiped are equal 1, 2, 3. To find volume of parallelepiped.
1) generation random , , ∈ {2,3, . . . ,10}; 2) calculation 1 = ; 2 = ; 3 = ;
To create evaluation model fuzzy approach was chosen because of its closeness to teacher qualitative
Let us consider input linguistic variables:
reasoning.
as in [
=”Tasks solving speed”, ( 1)={Slow, Medium, Fast}
2=”The number of errors” , ( 2)={Zero, Very small, Average, Large, Very large};
3=”Help reference”, ( 3)={Without help, With small help, With help, With great help}.
be an output linguistic variable with term set = ” ” ( )={ Excellent, Good, Satisfactory, Bad } T(Y)={Excellent, Good, Satisfactory, Bad}.
If is Fast and 2 is Zero and 3 is Without help then Y is Excellent. If is Fast and 2 is Very small and 3 is With help then Y is Good.
Let …
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.
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.
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. Control Agent WCF
Statement
TaskPresentation
(graphic presentation) AnswerVariant
XML
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
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 <expression /> tag contains an expression for building a mathematical formula, the <plot /> 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.
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.
Also, a module for dynamic plotting of various mathematical functions has been implemented.
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
Parameterized mathematical models, knowledges and skills evaluation model and software for
intelligent web-based testing system in mathematics are developed. The system has multilevel
architecture and flexible interface to provide versatility and security. The next step may be creation of
diagnostic models to adapt system to mental work features, knowledges and skills of each graduate
[
This system joined in single complex meant for learners testing (http://zno-kharkiv.org.ua/cimt/). In the perspective, it is planned to integrate the math web-tests system with other software tutoring products [14]. 6. References