Lodz University of Technology Distance Learning and Examining Systems - Theory and Applications edited by Sławomir Wiak Konrad Szumigaj HUMAN CAPITAL - THE BEST INVESTMENT The project is part-financed by the European Social Fund.
Edited by: prof. dr hab. inż. Sławomir Wiak mgr inż. Konrad szumigaj Distance Learning and Examining Systems strategy, logic of the system, architecture, evaluation. (IT platform e matura) Reviewers: prof. dr hab. inż. Maria Dems prof. dr hab. inż. Andrzej Krawczyk The book prepared under the E matura project, part financed by the European Social Fund in the frame of Human Capital Operational Program National cohesion strategy, Prioryty III High quality of the educational system, Action 3.3 Improving the quality of eduaction, Sub action 3.3.4 Modernization of the content and methods of eduaction competition projects.
Book is distributed free of charge Reviewers: prof. dr hab. inż. Maria Dems prof. dr hab. inż. Andrzej Krawczyk Edited by: prof. dr hab. inż. Sławomir Wiak mgr inż. Konrad szumigaj graphic design Niceday The book prepared under the E matura project, part financed by the European Social Fund in the frame of Human Capital Operational Program National cohesion strategy, Prioryty III High quality of the educational system, Action 3.3 Improving the quality of eduaction, Sub action 3.3.4 Modernization of the content and methods of eduaction competition projects.
Joanna Kucner Krzysztof Kisiel Michał Kucner Hanna Drabik - Zalewska Solving stereometry problems using a learning platform 13.1. Introduction In recent years growing interest in online education have been observed. What accounts for this is willingness to gain knowledge, higher qualifications and experience. This situation is also influenced by constant advancements in the field of IT, as those allow to meet students expectations. The Lodz University of Technology conducted projects Mathematical Emergency E-Services and E-secondary school-leaving examination 1 as a learning-support method aimed at pupils of high schools. The latter allowed to examine participants from the distance. Moreover, similarly to Mathematical Emergency E-Services it gave them a chance to benefit from online tutorial sessions. Some students used this assistance when preparing to their current tests. Others, however, took advantage of it on the regular basis to revise in a systematic manner material covering both basic and extended matriculation exam. This thesis is to elaborate on actions the project Mathematical Emergency E-Services consisted of. Organisation of platforms work. Pupils of the chosen high schools were offered the assistance available at learning platform 5 times a week, 5 hours a day. Every day during platform s working hours four tutors were on duty and each of them had only one student to take care of. The length of a session was unlimited and it was determined by each student s knowledge and needs. What should not be omitted is the fact that the number of connections with platform was not restricted and how often it was attended depended on students decisions. On-line help. Tutors were presented with problems of various types and levels of difficulty. This thesis is to address only streometry problems. What is stereometry? Stereometry: the measurement of volumes and other metrical elements of solid figures distinguished from planimetry 2. It is often the case that a student fails to imagine a geometrical object in a threedimensional space. Consequently, they fail to imagine the relations described by the problem. Moreover, the sketches they draw are often not accurate enough to enable noticing those 1 Kisiel K., Galewska E., Kucner J., Szumigaj K.: Wspomaganie nauczania matematyki w szkole ponadgimnazjalnej w oparciu o nowoczesne technologie informacyjne w kontekście e-matury, rozdział w monografii: System informatyczny zdalnego egzaminowania-strategia, logika systemu, architektura, ewaluacja, Politechnika Łódzka, 2013 http://www.merriam-webster.com/dictionary/stereometry 2 http://www.merriam-webster.com/dictionary/stereometry 221
relations. Using mathematical programs such as, for instance, GeoGebry 3. would make solving this type of problems considerably easier due to the fact they supply the user with clear drawing. Although a neat sketch may significantly facilitate noticing some of the relations the given problem is not already solved. What a pupil needs apart from a clear drawing is frequently support offered during the attempt to solve it. It is also feedback which will allow them to comprehend mistakes they have made and somebody who can give a hint that are very helpful for a student. And it was project Mathematical Emergency E-Services that provided them with such a comfort. Exercise 1. Tutor written data of the task and made the appropriate figure. It uses the tools of the educational platforms. During the videoconference student and tutor discussed the exercise. Then the student has used the data contained in the problem and trigonometric functions. In the next step he calculated the length of the proper segment. Tutor wrote a solution to the screen as presented in Figure 1. Figure 1. Source: "Mathematical Emergency E-Services". Solving tasks - calculate the length of the lateral edge of the pyramid. Exercise 2. Tutor drew a rectangle on the panel platform, which is vertical cross section of the cylinder. For the rectangle in this exercise conditions are satisfied. Student applied the Pythagorean theorem and calculated the radius of the base of the cylinder. With these data the student easily calculated as the volume of the cylinder. The student made the same calculation. With a screen just check the correctness of the calculations. The process of solving the problem shown in Figure 2. 3 Kucner J., Galewska E., Kisiel K.: Stosowanie darmowego oprogramowania matematycznego do rozwiązywania zadań maturalnych, rozdział w monografii: System informatyczny zdalnego egzaminowania-strategia, logika systemu, architektura, ewaluacja, Politechnika Łódzka, 2013 222
Figure 2.Source: "Mathematical Emergency E-Services". Solving exercises - calculating the volume of a cylinder. Exercise 3. In this exercise the drawing is also necessary. Using the platform tools tutor drew the regular quadrangular pyramid and using a pen charted data from the task. After an exchange of questions and answers student applied the Pythagorean theorem and calculated the length of the diagonal of the base. Not all calculations are given Figure 3. The student did some calculations on a piece of paper. 223
Figure 3. Source: "Mathematical Emergency E-Services". Solving exercise - calculate the length of the base of the regular quadrangular pyramid - teacher Hanna Drabik-Zalewska. Exercise 4. In the next exercise is considered quadrangular prism and appropriate rectangle. Figure 4 helps to solve exercises. Figure 4. Source: "Mathematical Emergency E-Services". Solving exercises - calculate the length of the base quadrangular prism - teacher Hanna Drabik-Zalewska. 224
Exercise 5. We started of course from the application of the data of the exercise. The teacher drew of the cone and appropriate angle. The student noted that the vertical cross section of the cone is an isosceles triangle. Using the information contained in the problem the student has calculated the vertical cross section area of a cone. Fragment of solving the problem is presented in Figure 5. Figure 5. Source: "Mathematical Emergency E-Services". Solving exercises - the calculation of the vertical cross section area of a the cone - teacher Hanna Drabik-Zalewska. Exercise 6. Students log on to the platform. The student had a problem with the calculation of the volume of a regular quadrangular pyramid. Solution exercises started by drawing the pyramid satisfied the conditions of exercise. After discussion and exchange comments and questions the student realizes the different steps leading to the solution of the problem. Figure 6 shows a portion of solution of the problem.ia. 225
Figure 6. Source: "Mathematical Emergency E-Services". Solving the exercise - calculating the volume of the pyramid - the teacher Hanna Drabik-Zalewska. Exercise 7. After preparing Figure 7 and the discussion the student quickly noticed what to do to solve an exercise. 226 Figure 7. Source: "Mathematical Emergency E-Services". Solving exercises - calculating the volume of the prism - teacher Hanna Drabik-Zalewska.
13.2. Conclusions The platform gives users the opportunities to see the exercise in space. It allows also to fast check if students ideas, way of thinking and methods of solving are correct. When it comes to the sterometry that fact is crucial. Student is not always able to imagine the object in space, moreover is not able to see connections between the element. Those possibilities of platform and their orginal way of solving exercises help students to understand the topic of sterometry. The tutor asked, marked leght and angles on the pictures, wrote the formulas, tried to help students in finding properly way of solving exercises. In the case of platform cooperation between tutor and student, that second received almost immediately information about hers/his progress in learning. The knowlegde about self achevements is very possitive for them. It guided into having higher self-esteem and better learning progress. 13.3. References 1. Kisiel K., Galewska E., Kucner J., Szumigaj K.: Wspomaganie nauczania matematyki w szkole ponadgimnazjalnej w oparciu o nowoczesne technologie informacyjne w kontekście e-matury, rozdział w monografii: System informatyczny zdalnego egzaminowania-strategia, logika systemu, architektura, ewaluacja, Politechnika Łódzka, 2013 2. http://www.merriam-webster.com/dictionary/stereometry 3. Kucner J., Galewska E., Kisiel K.: Stosowanie darmowego oprogramowania matematycznego do rozwiązywania zadań maturalnych, rozdział w monografii: System informatyczny zdalnego egzaminowania - strategia, logika systemu, architektura, ewaluacja, Politechnika Łódzka, 2013 227