Rbert VajdaComputer-assisted Assessment of Mathematical Knowledge

IntroductionAssessment: 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]

Integration of new toolsQuestion: 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!

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]

Two(three)-tiered exams[Kutzler 2000] Assessing mental fitness no technology is allowedAssessing students problem solving skills every technology is allowed

Two(three)-tiered exams[Schirmer 2000] Extension of the two-tiered system with a third component

self-elaborated focus-papers and project works

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

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

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

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 [Glis Lenne 2002]

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

Web-based learning environment [Vajda - Kovcs 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

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.

ConclusionIt 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 :

Assessing the basic mathematical skills of students who have no prior programming knowledge (2nd Model, Freshmen level)Assessing the mathematical problem solving skills of students who have programming and previous mathematical knowledge with (2nd Model, second stage, 1st Model)

First case

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

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.

Problem in MathematicaGIVE a cubic polynomial p that fits to the points P1 (-1,-1), P2 (0,0), P3 (1/2,7/8), P4 (1,1). INVESTIGATE the parity of the function p.GIVE the local extrema of the function p (exact values).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 P1-P4 How can the symmetrical properties of the closed curve be exploited during the construction?CALCULATE the (exact) area of the filled region H.

More information, contact addresses:http://wmi.math.u-szeged.huhttp://www.sf.net/projects/wmihttp://www.math.u-szeged.hu

Mrta K. Nmethkalmarne@math.u-szeged.huRbert Vajdavajdar@math.u-szeged.huSZTE Bolyai Intzet, 6720 Szeged, Aradi vrtank tere 1.tel.: 62/544-647