Author's methodology of forming modeling skills involves 4 steps: Step 1 - the teacher step by step constructs the curve by means of cloud based service GeoGebra; Step 2 - the teacher ofers a descriptiondefinition of the curve and provides a ready-made algorithm by which students model the curve independently in GeoGebra; Step 3 - the teacher ofers an algorithm for constructing a curve model, and students need to characterize the properties of the curve or give its definition based on the results, Step 4 - students are ofered definitions of curves that they have to model in GeoGebra). An example of realization of the author's methodology is given, the pedagogical experiment on proof of its efectiveness is described.
Modern science operates with various methods, among which modeling is one of the most popular. Method allows you to move away from the object’s ideal representation and use its analogue, which retains the most important characteristics that allow you to talk about the object properties after certain changes or influences on it.
The development of computer technologies has contributed not only to the revival of the
modeling method, but also led to the emergence of specialized environments, where it became
possible to model various objects (processes) based on a constructive approach. At the same
time, the widespread use of smartphones and tablets that have access to the Internet has led
to the emergence of cloud services, which also allow you to model objects of diferent nature
[
The importance of developing constructive skills of youth is emphasized in the findings of
Laksha [
Analysis of the problem shows that with the advent of specialized mathematics software, its
solution has focused not so much on understanding the algorithms of elementary constructions,
but on the ability to use computer tools and the ability to visualize the result. The latter, in
particular, is emphasized in the finding of Bilousova and Zhytyenyova [
Regarding the first (the ability to use tools) it should be noted the emergence of cloud services, including specialized services. Analysis of the latter showed the following.
Diferent aspects of educational using of cloud technologies and services are examined in the
studies of national and foreign researchers. For example, Alkhansa et al. [
Shyshkina and Popel [
Scientific research on the involvement of cloud services for the formation of various skills in
specialists in various fields of knowledge is spreading [
Currently, cloud versions of known environments with mathematical modeling capabilities
are available, including Maple, MATLAB, WebMathematica, Calculation Laboratory, etc. [
Experience in using GeoGebra cloud service (visualization of mathematical objects; organization of not only analytical but also empirical search for answers in determining individual characteristics of mathematical objects; organization of home computer experiment) allowed to consider GeoGebra as means for formation of skills to model based on constructive approach. However, the analysis of scientific findings confirmed the lack of established methods for its use, and therefore the method of formation of skills to model interesting curves by GeoGebra cloud service based on a constructive approach required testing and experimental confirmation of its efectiveness.
The purpose of the paper is to describe the method of forming the skills to model interesting curves through GeoGebra cloud service based on a constructive approach and to confirm its efectiveness.
To test the efectiveness of the methodology of forming modeling skills, a pedagogical experiment, which lasted 3 years (2018–2020) was conducted among pre-service mathematics and computer science teachers.
Makarenko Sumy State Pedagogical University, Borys Grinchenko Kyiv University were the experimental base. The total number of respondents was 86 people (pre-service mathematics and computer science teachers).
The author’s methodology was implemented using the freely distributed GeoGebra cloud service (https://www.geogebra.org) in the special courses. The experimental module “Modeling of interesting curves” with a volume of 10 classroom hours (2 hours of lectures and 8 hours of laboratory classes) was introduced.
Two tests of educational achievements (at the beginning and after studying the module) were organized to confirm the efectiveness of the methodology.
Students were ofered a typical test problem for modeling a conic by its geometric definition (problems difered from each other in the input conditions, for example, the distance between the foci was diferent or one of the foci was in the specific point in a given coordinate system, etc.).
The solution of the problem was evaluated by the following indicators (table 1). Since the results of the tests were dependent and each time provided for the accumulation of marks, the sign test was used. The number of respondents, whose total score decreased (“–”), did not change (“0”) and increased (“+”), was fixed.
In accordance with the purpose of the experiment and the sign test, the null hypothesis was formulated: the author’s methodology does not contribute to the formation of modeling skills by means of GeoGebra cloud service. Then the alternative hypothesis was “the author’s methodology contributes to the formation of such skills”.
The constructed hypotheses define the one-sided sign test for checking the dependent samples.
According to the decision-making rule [
The constructive approach is characterized by the fact that acquaintance with the properties of concepts begins with constructive activities for their “discovery” and assimilation with a gradual transition to definitions and logical proofs. This, in particular, simplifies the perception of the Geometry course, makes it more accessible, while raising the scientific level through the intensification of educatees’ research activities.
The use of the constructive approach contributes to the fact that the activity is manifested
in the gradual transition of actions for the construction of objects from the executive level (is
characterized by external regulation), then to reproductive (is marked by internal regulation of
actions in the construction of known structures), then to the applied level (using the method of
construction), and, finally, the creative level (involves the construction of new objects) (figure 1)
[
We consider constructive tasks to be one of the means of implementing a constructive
approach. Many authors use the term “constructive tasks” in a sense identical to construction
problems. However, the peculiarities of the constructive approach require the use for its
implementation and other tasks, the solution of which requires the implementation of a particular
constructive activity. Therefore, for example, Lisimova [
Constructive tasks can be a base for establishing new properties of figures (concepts
formation), for the consolidation of knowledge, for repetition, control, intensification of research
activities. “The peculiarity of constructive tasks is that they can be solved both logically and
ifguratively or visually efective” [
When solving constructive tasks, there is a productive activity that motivates to think
independently (methods of constructing have to be developed independently), and not reproductive
activity, which is often not an independent mental process, but is a repetition of known steps
[
Mastering the system of knowledge and skills can take place on two levels: constructive and analytical. The main feature of the constructive level of mastery of the material is its visual and constructive awareness. This is manifested: • in the ability to recognize objects that belong and do not belong to the content of this concept, give examples, demonstrate the existence of the studied figures by construction; • in the knowledge of their most essential properties and the ability to apply known properties in solving problems.
The constructive approach provides “points of support” for Geometry study at a higher abstract level, which is characterized by the ability to formulate definitions of concepts, statements and prove already in formal language, rather than the language of geometric images.
Any methodology involves the presence of five elements (figure 2): purpose, content, forms, methods and means. Let us explain them in more detail.
The purpose of the author’s methodology is to form the skill to model interesting curves using cloud services.
Interesting curves of the course of Analytical Geometry are the content on which the author’s methodology is realized. Nowadays, their study in the classical course of Analytical Geometry is possible in three ways: • analytical description and further study of curves – at first analytical (parametric, implicit, explicit) equations, usually of conics, are given and then they are investigated; • study of curves as locus with a given property – at first the geometric definitions of curves are given, on the basis of which their analytical equations are written (as a rule, these are conics, conchoid of Nicomedes, limaçon of Pascal, strophoid, cissoid of Diocles, lemniscate of Bernoulli, Cassini oval); • study of curves generated in the kinematic way (as the trajectory of a point), usually cycloidal curves, folium of Descartes, witch of Agnesi, logarithmic spiral.
Due to lack of time, there is only the first way of the study of interesting curves in the course of Analytical Geometry at universities. This limits the possibility to develop the skills to model mathematical objects.
The form of realization of the author’s methodology is virtual lecture-practical classes or a combination of visualized lectures and virtual laboratory classes.
Among the teaching methods problem method and netnography method are used. GeoGebra cloud service, constructive tasks and instructional materials are learning means. Elementary knowledge in Analytical Geometry is required to use the author’s methodology.
In general, the methodology of forming the skills to model interesting curves is as follows: the teacher on a common online platform informs about constructive approaches to the construction of various curves, which were studied in ancient times (the netnography method). He briefly tells about how such constructions were done (compass and ruler, one compass, two compasses, one or two rulers, etc.), or gives an example of practically oriented problems that are solved using interesting curves. After that, the teacher demonstrates one of the described constructions in GeoGebra cloud service, and then asks students to write an algorithm for the above construction (the problem method). Then the teacher asks students according to the algorithm (provided to each student) to reproduce the construction and demonstrate the result, to analyze errors, and if not, to analyze possible limit cases. After that, students are ofered the definition of curves generated in mechanical way together with the algorithms of their construction in the cloud service; students must model this type of curve according to the existing algorithm. Then together with students, the task of the following type is carried out: algorithms of curves construction are ofered and after their construction, students need to characterize properties of the modeled curves, to give them definitions or kinematic characteristics. After completing this type of task, students are ofered only the definition of the curves or their kinematic characteristics, and they must model the curve themselves.
The methodology should be briefly described as follows.
Step 1 – the teacher ofers an example of a curve model through the definition (the teacher step by step models the curve), and students must independently compile an algorithm for constructing the model.
Step 2 – the teacher ofers a description-definition of the curve and provides a ready-made algorithm according to which students model the curve independently.
Step 3 – the teacher ofers an algorithm for constructing a curve model, and students need to characterize the properties of the curve or give its definition based on the results.
Step 4 – students are ofered definitions of curves that they have to model them.
We will briefly demonstrate the application of the methodology of forming the skiils to model interesting curves based on a constructive approach in GeoGebra cloud service on the example of the hypocycloids study.
The teacher formulates the problem. The calculation of the action of the kneading machine
consists of a description of the trajectory of the point belonging to the working body. The
trajectory of the working body does not always fill the container with the substance, as a result,
the mixing time increases, which leads to unnecessary energy consumption. Determine the
optimal trajectory of the point of the mixer working body [
History of the curve discovery (netnography method). Hypocycloid (from Greek words – “under, below” and – “circle”) – is a curve “under the circle”. The most famous hypocycloids are the astroid (from Greek “star-shaped”) and the deltoid (Steiner curve). The name of the astroid (from “cusp” – exacerbation) was proposed by the Austrian astronomer Carl-Ludwig von Littrow (1811–1877). The shape of the deltoid resembles the capital Greek letter delta, its properties were studied by Leonard Euler in the XVIII century, and then by Jacob Steiner in 1856, after whom it was renamed the Steiner curve.
Definition of the curve. A hypocycloid is a curve generated by the trace of a fixed point on a small circle that rolls within a larger circle.
Next, the teacher demonstrates an example of a ready-made hypocycloid model (figures 3-4), and then ofers to describe the algorithm for its construction to students.
Let the circle 2 roll inside the circle 1. Let’s construct a curve generated by a point on the circle 2.
Algorithm for hypocycloid model constructing 1. Construct a circle 1 with center passing through point . 2. Construct the segment . 3. Construct an arbitrary point on the radius . 4. Construct the circle 3 with center passing through point (dotted line). 5. Construct an arbitrary point on the circle 3. 6. Construct the circle 2 with center , the radius of which is equal to the length of the segment . 7. Construct the ray . 8. Constract the point – the intersection of the circle 2 with the ray . 9. Hide the ray . 10. Let the center of the circle 2 coincides with the point in the initial position. If we move the point around the circle, it will rotate around the point , and remains tangent to the circle 1. Let the circle 2 rotate around it when moving the center to create the illusion that the circle 2 rolls on the circle 1. Only one point of the circle 2 should be investigated. Let this point coincides with the point at the initial moment, and is the point of tangency of the circles 1 and 2. If the circle 2 rolls on the arc , then the center of the circle will return around the point at the angle = . Simple calculations show that all points of the circle 2 will rotate around its center at .
the angle * 11. Construct the point , which is obtained from the point by rotating around the point . by the angle * 12. Hide the point . 13. a) Make the point leave a trace by rolling the point on the circle 2 (Fig. 3). b) Use the Locus tool: the point is the “pencil point”, the point is the “driver point" (figure 4).
Next, the lecturer at the visualized lecture ofers students to explore the analytical equation of the curve, which is described through the parameter.
Let be the radius of a fixed circle, be the radius of a rolling circle, = , then the parametric equation of a hypocycloid is = * ( − 1) * ( − ( − 1) * ) − 1 ), = * ( − 1) * ( − ( − 1) * )
). − 1 The modulus of determines the shape of the hypocycloid: = 2 the hypocycloid is described 2 2 2 by Tusi couple (diameter of a fixed circle), = 4 by the astroid 3 + 3 = 3 .
Study of the curve and establishment of properties (problem method). When the hypocycloid is constructed, we pass to establishment of its properties at the level of practical research.
Note that the curve constructed in this way is open. Let us investigate the conditions under which the constructed hypocycloid will be a closed curve. By changing the radius of the circle 3, thereby changing the ratio = of the radii of fixed and rolling circles, we conclude that the curve will be closed when the ratio = ∈ (figure 5). In addition, in this case, the hypocycloid consists of identical arcs, the ends of which lie on a fixed circle (figure 6).
If the radius of the rolling circle is twice smaller than the radius of the fixed circle, then the hypocycloid degenerates into a segment (figure 7). This statement is well known as Copernicus theorem.
After the lecture, students are first asked to construct curves models according to the given definition and algorithm, for example, to costruct a cycloid, which is a curve generated by a point of a circle rolling on a straight line, reproducing all the steps of the algorithm. Not only instructions but also possible consultations are provided.
The next lesson tasks such as: “Establish the relationship between the number of arcs of the epicycloid, generated by a point of a circle rolling on the outside of another circle, and the ratio of the radii of these circles”, are ofered.
Then, at the next two lessons, tasks such as: “A point and a line, the distance between which is equal to , are given. A line is drawn through an arbitrary point of the line and the point. The points at a distance from point are marked. Find the locus of such points”.
The described methodology difers from the traditional one, as the latter usually involves only step 2 and step 4, which does not allow students to understand the algorithms involved in the constructing the model, to develop the ability to compare step-by-step construction ideas and the algorithm steps. There is also no practice of qualitative analysis of the algorithm (which is provided by step 3 and which is supported by the “step-by-step playback service” in GeoGebra). This is what we consider fundamentally important for efective formation of skills to model and what GeoGebra service makes possible.
Let us note also that in terms of distance learning, it is GeoGebra service that allows the demonstration of constructions and the ability to work with models at anytime from anywhere.
Verification of the efectiveness of the author’s methodology was carried out based on two tests. Students had to solve the problem 1 (for example, to construct a curve, for each point of which the sum of distances to two given points is constant) before studying the topic and the problem 2 (for example, to construct a curve, each point of which is equidistant from the point lying in the third quarter, and a line that forms an acute angle with the abscissa) after studying the topic.
The evaluation of solutions was carried out through the assessment of skills: to take into account the analytical and geometric relationship between the elements; to use the Locus and Trace tools; to visualize the model successfully; to add dynamic text to study numerical characteristics; to write an algorithm for constructing a model; to reproduce the steps of the algorithm to construct the model; to interpret the result of a computer experiment (in more detail in table 1).
The results of the controls are presented in table 2.
These marks were used to determine the number of respondents whose total score decreased (“–”), did not change (“0”) and increased (“+”) (table 3).
According to the decision-making rule [
Since is not included in the acceptance interval of the hypothesis 0, we reject the hypothesis 0 and accept the alternative one with the conclusion that the author’s methodology contributes to the formation of modeling skills. Since the value of is beyond to the segment on the right, it is necessary to conclude there is the positive dynamics of the number of such 5 4 6 6 7 4 3 3 5 6 7 5 4 4 5 6 7 7 7 6 6 7 7 7 7 7 5 4 4 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 students who have developed the skills to model interesting curves using GeoGebra cloud service.
1. Author’s methodology of forming modeling skills involves 4 steps: Step 1 – the teacher ofers an example of a curve model through the definition (the teacher step by step constructs the curve), and students must write the constructing algorithm independently; Step 2 – the teacher ofers a description-definition of the curve and provides a ready-made algorithm by which students model the curve independently; Step 3 – the teacher ofers an algorithm for constructing a curve model, and students need to characterize the properties of the curve or give its definition based on the results, Step 4 – students are ofered to define curves that they have to model). The methodology efectiveness was confirmed by the sign test at the significant level of 0.05. 2. GeoGebra cloud service is efective for being perceived as a means for the formation of students’ (pre-service mathematics and computer science teachers) modeling skills. At the same time, we consider it is also possible to use this methodology in the conditions of traditional training with GeoGebra installed on PC.