The study of exact sciences includes not only lectures but also the active forms of learning (practical classes, laboratory work and so on). Thus, it is necessary to control not only declarative knowledge but also procedural knowledge of the students (the knowledge of methods for solving problems). This article describes functional requirements, mathematical models, and algorithms for the development of the modules of procedural knowledge testing in the systems of computer mathematics for educational purposes. We present methods of generation of test tasks implemented in the web application “Test on Mathematics”.
There are two approaches to solve the problem of the generation of specific test tasks: saving a series of similar tests in the database and using algorithms for automatic generation of tests implemented as system procedures. Each of these approaches has its advantages and disadvantages. In the first case, it is necessary to spend a lot of time to fill the database, and tests are repeated. In the second case, the large amount of time is spent on the implementation of algorithms for automatic generation, but each test is unique. The weighted approach is to: 1. Develop a single common model for each sufficiently wide class of tests, as well as models and algorithms for generating conditions and answers for this model. 2. Develop common CASE-technologies for description subclasses of test tasks based on a single common model. 3. Develop joint mechanisms for storing and calling of algorithms for generation of specific test tasks.
Test Tasks-Templates The expression ( 1, … , ; 1, … , ) in signature ∑ in subject domain is called expression template.
range of values for metavariables is set ( ∈ ). It follows that in each particular case, each of the variable templates represents one of the variables.
values for parameters is numerical sets ( ∈ ). It follows that in each particular case, each of the coefficient templates is a number.
Example 1 1 1 + 2 2 is expression template.
Specific instances of this template: 2 + 3 , − 2 + 5 and so on. Suppose 3 ( 1, … , ; 1, … , ) is expression template and ∈ , ∈ , then 1,…, ( 1, … , ; 1, … , ) = ( 1, … , ; 1, … , ) 1,…, is called specialization (a special case or instance) of ( 1, … , ; 1, … , ).
Expression templates define the common view of the model and common view of the answer for test task. It follows that the basis of determining the test task is pair < , >, where , are expressions templates (Fig.1), defined on common lists of metavariables ( ) and parameters ( ).Then ( , ) =< ( , ), (
In tests on equivalent expressions transformation means semantic equality in domain . , = ) > , where " = "
In addition to the signature of the domain system function ( ) is used in the definition of a template of test task. It is defined on the set of expressions of the sub
Descriptions for test tasks’ templates are used in a procedure for generation instances of tests. In the simplest case, the procedure has specification: where =<
>. tion “Test on Mathematics”
Suppose the user describes following template for this test task:
, are metavariables, , , , are parameters. Using conditions leads to a reduction of templates’ number and extension of test tasks’ classes. Thus, test task 4 describes all classes of test tasks with templates 1, 2, 3 in conjunction with additional conditions = 1, = 1, = 0 ∨ = 1, = 0 ∨ = 1. Description of the conditions corresponds to the syntax: Each atomic condition divides all set of tests into two classes. First class is a class, where this condition is satisfied, second class is a class, where it fails (in this case corresponding value is randomly generated from the range of values, that are defined by parameter descriptions): ≔ | ≔ (). Independent conditions are listed with separation by commas. All particular cases are defined by all possible combinations of conditions. Conjunction unites independent conditions that describe the same particular case. Disjunction combines dependent conditions, each of which defines different individual cases.
Test generation algorithm returns a single particular test in each of its call. Therefore, the condition is reduced to canonical form Φ where Φ defines one particular case. Let’s use logical variables that are corresponding to atomic conditions: = ( = 1), = ( = 1), 0 = ( = 0), 1 = (
= 1), 0 = ( = 0),
∷= | ∷= ; | & ( = | , )| = ; ∨ ( .
),
fication of tests according to their computational complexity. The complexity of the test is determined by a degree of the monomial (the number of boolean variables). The smaller is the degree of monomial, the more complex is a case of the test. In given example, the simplest tests are tests 0 0, 0 1, 1 0, most difficult test is test with all randomly generated parameters.
Semantics for the
procedure: 1. Randomly substitute one of the variables that are listed in the description of the metavariables into the template instead of each metavariables (Fig.4). 2. Randomly substitute one of the values that are listed in the description of the parameters into the T template instead of each parameters (Fig.5).
in the resulting instance (Fig.6). 4. Return the result of calculations in the form =< interface of web application “Test on Mathematics” In this article, we describe functional requirements, mathematical models, and algorithms for the development of the modules of procedural knowledge testing in the systems of computer mathematics for educational purposes.
application “Test on Mathematics”. The system of computer mathematics for educational purposes “Test on Mathematics” will reduce the burden on teachers via automatic generation of the tests and verification of the answers. Thus, it is easy to control procedural knowledge during the absence of student (during his illness, during quarantine, etc.). 1. Lvov M.S. Proektirovanie logicheskogo vyvoda kak poshagovogo reshenija zadach v matematicheskih sistemah uchebnogo naznachenija / M. S. Lvov // Upravljajushhie sistemy i mashiny. – 2008. –No1. – S.25–32. 2. Lvov M.S. Koncepcija proghramnoji systemy pidtrymky matematychnoji dijaljnosti./ M.S.Lvov // Komp'juterno-orijentovani systemy navchannja: Zb. nauk. praci/ K.:NPU im. M.P.Draghomanova.– Vyp. 7.–2003.– S.36–48. 3. Lvov M.S. Koncepcija, arhitektura i funkcional'nost' gibkoj raspredelennoj programmnoj sredy uchebnogo naznachenija dlja srednej shkoly. Rabochee mestometodista /M. Lvov //Upravljajushhie sistemy i mashiny. –2009. –No 6. –S. 71-78. 4. Lvov M.S. Raspredelennye programmnye sredy uchebnogo naznachenija.
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