1. How to teach function by using of DERIVE Ingrida Kraslanová Mária Slavíčková Faculty of mathematics, physics and informatics Comenius University, Bratislava http://www.ddm.fmph.uniba.sk

2. Introduction  Using Derive in Austria 1993-2005: CAS I - V projects, CAS VI in preparation CAS educated students better results than non-CAS students (always) Textbooks – CAS in appendix  Using Derive in Slovakia Every secondary school – at least 1 PC room with DERIVE Lack of teacher interested in CAS

3. Secondary school 1 st phase (pilot study): 64 participants PC1- 33 NPC1 – 31 2 nd phase: 63 participants PC2 – 33 NPC2 - 30

4. Aims of the experiment  to teach in both groups the subject matter Graphs of goniometric functions and  to prepare a test for all respondents of experiment,  to evaluate the test and to compare the results of two types of groups (experimental and control group, 1st and 2nd phase),  to find out whether the computer aided teaching is more effective than the traditional.

5. Investigating the family of functions

6. Example In the figure below is sketched graph of function Find the coordinates of signed point A

7. Example of the test question In the figure below is sketched graph of function. Choose one of 7 lines (k, l, m, n, o, p, q), which represents the x-axis and one line which represents the y-axis?

8. Evaluation 2 nd phasePre-testTest PC 272,84%47,40% NPC 273,85%50,00% 1 st phaseTestPost-test PC 156,49%25,32% NPC 138,25%17,51%

9. Worksheets 1/4

10. Worksheets 2/4

11. Worksheets3/4 f´(x) > 0 f´(x) < 0 Exercise 3 Investigate the function f: y = x 4 -8x 2 -12

12. Worksheets4/4

13. University level Find the bounded extreme of the function on border

14. Finding a global extremes Find the lowest and highest value of the functionon the close area

15. Developing conceptual and procedural knowledge 1 st step: graph of real function of the two real variables + modeling using some plastic mass 2 nd step: teaching formula and procedures + using CAS for calculating derivates 3 rd step: counting on the paper, CAS = control function

16. Structure 1 st level: CAS as a tool for teaching/learning 2 nd level: CAS as a tool for helping to making calculation 3 rd level: CAS as tool for controlling our own calculation Developing a conceptual knowledge – students must understand, what is going on Using conceptual and procedural knowledge for solving similar tasks, or tasks where the students can use these knowledge for solving tasks from specific topics Developing a procedural knowledge – students must learn a procedure for solving some type of the tasks

17. Some ideas of solving the problems with mentioned topic  Printed graphs  Using notebook and projector + CAS  CAS + some plastic mass  Worksheets and homework

18. Worksheets Lets have a function f(x,y)=px 2 +qxy+ry 2 and point A[0, 0]. Find the values for p, q, r to fulfill: a)The function has in A saddle point b)The function has in A maximum c)The function has in A minimum

19. Results  Almost excellent procedural knowledge of the students Problems with basic operation Problems with understanding basic terms or sketching basic graphs  Better imagination of 3D  Generalization of the knowledge about the function from 2D to 3D and 4D and so on  Better understanding of the procedure

20. Thank you for your attention! Web: http://www.ddm.fmph.uniba.sk E-mail addresses: ingrida.kraslanova@gmail.com maria.slavickova@centrum.sk