1. Róbert Vajda Computer-assisted Assessment of Mathematical Knowledge

2. Introduction Assessment: An important part of the learning-teaching process, its purposes are - informing learners about their own learning - informing teachers of their learning strategies - characterizing quantitatively the grade of achievement - informing colleagues and researchers of the status of the given sucessfully absolved course in the eduacation system [Houston 2001]

3. Integration of new tools Question: should we also enable the use of new technological tools such as graphic calculator, computer algebra systems, etc. during exams? If the assessment is an integral part of the whole teaching-learning process then our answer: YES!

4. Tasks - Reevaluation of the traditional assessment exercises (CAS – trivial, CAS – required, CAS - excluded, …) - Construction of new types of exercises and problems that that fit to the new assessment environment [Macogain 2000], [Brown 2003]

5. Two(three)-tiered exams [Kutzler 2000] 1.Assessing mental fitness – no technology is allowed 2.Assessing student’s problem solving skills – every technology is allowed

6. Two(three)-tiered exams [Schirmer 2000] Extension of the two-tiered system with a third component  3.self-elaborated “focus-papers” and project works

7. Assessment of a course: Aspects - Type of a course - The place of the course in the curriculum; how it is connected to other parts of the curriculum  Two different types of mathemtical curriculum regarding the use of technology

8. First Model - Introduction of CAS (syntactical knowledge and programming skills) at the beginning, usually parallel with the basic theoretical mathematical courses - After the introduction comes the full integration of the new technology to mathematical courses

9. Second Model - At the beginning, basic theoretical mathematical courses are instructed without the ‘interior’ knowledge of CAS - At a higher level, a module - “Computer Aided Mathematical Modeling” – makes students understand how these systems can help to elaborate mathematical problems

10. Second Model at University of Szeged Reasons: -large number of students, no sufficent lab and instructional capacity -at a freshmen level the explicit use of CAS opens new possibilities but also new difficulties  [Gélis – Lenne 2002]

11. Difficulties -CAS are command oriented with rigorous syntax, they were originally designed for research purposes  -their didactic capabilities must be improved  An innner or outer new learning environment should be created based on the services of CAS

12. Web-based learning environment [Vajda - Kovács 2003 ] - A new outer layer as an interactive mathematical webportal WMI offers an intelligent leaning environment for students who have no prior programming and syntactical knowledge as well

13. http://wmi.math.u-szeged.hu WebMathematics Interactive is a software that makes it possible for a wide range of students of universities, colleges, and high schools to practice exercises of mathematical nature, to check the solutions of exercises, and to test their knowledge. The user only needs internet connection and a web browser.

14. Conclusion It makes sense to consider the computer-assisted assessment at the beginning of the second curriculum model as well, but the two cases have to be discussed separetly : 1.Assessing the basic mathematical skills of students who have no prior programming knowledge (2 nd Model, Freshmen level) 2.Assessing the mathematical problem solving skills of students who have programming and previous mathematical knowledge with (2 nd Model, second stage, 1 st Model)

15. First case Examples from the Thematic Modules “Integrals”: - open-ended tests Definite Integrals (Double Integrals) - mutistep tests Antiderivatives 10 (Rational Functions) Definite Integrals (Double Integrals) Antiderivatives 10 (Rational Functions) - (more) complex exercises with allowed tools Local Extrema of Functions 1 Local Extrema of Functions 1

16. Second case Example of a problem created and to be solved with Mathematica Let us consider the picture below which represents a planar region constructed by the aid of a cubic polynomial. All the following questions are related to this plot.

17. Problem in Mathematica a)GIVE a cubic polynomial p that fits to the points P 1 (-1,-1), P 2 (0,0), P 3 (1/2,7/8), P 4 (1,1). b)INVESTIGATE the parity of the function p. c)GIVE the local extrema of the function p (exact values). d)Using symmetry CONSTRUCT AND DRAW the closed curve shown in the diagram by the aid of CAS without coloring the region H and without the points P 1 -P 4 How can the symmetrical properties of the closed curve be exploited during the construction? e)CALCULATE the (exact) area of the filled region H.

18. More information, contact addresses:http://wmi.math.u-szeged.huhttp://www.sf.net/projects/wmihttp://www.math.u-szeged.hu Márta K. Németh kalmarne@math.u-szeged.hu Róbert Vajda vajdar@math.u-szeged.hu SZTE Bolyai Intézet, 6720 Szeged, Aradi vértanúk tere 1. tel.: 62/544-647