Graphical calculators in pre- and in- service teacher training

1. Graphical calculators in pre- and in- service teacher training Tünde Berta Eötvös Loránd University, Budapest

2. Content • Use of graphical calculators in Hungary- present situation • Didactical and psychological background of use of graph. calculators • About a didactical course at Eötvös Loránd University • Some concrete examples

3. Use of graphical calculators in Hungary - present situation • In Hungary it is not admitted to use any graphical calculators at the secondary school final exam or at any mathematics exam – this is one of the reasons why the use of graph. calc. is not widespread. • Several years ago Casio and TI tried to enter this market unsuccessfully. • As an experiment we started a CASIO project at the university last September.

4. Didactical and psychological background • The use of different representations at the same time and experimentation with different models, computer animations and tools and observation has an extra value added from the point of view of Psychology of learning and development of Problem-solving skills. • All of 3 different levels of representations ( in sense of Brunner enactive, iconic and symbolic) are present in any phase of the teaching process, but they play different roles. • These levels of mathematical representations cannot be separated during a teaching process in a classroom. • The learning process of the children can be made more effective by a conscious change of these modes. • Graphical calculators are able to connect these 3 levels of mathematical representations, mainly the iconic and symbolic level, because these kind of calculators are use it parallel. • From the researches of human brain we know, that brain memorize a symbolic and iconic information in a two different part. The graphical calculator can play role of the “bridge” between this two part and information

5. The importance of traditional goals, methods and conceptual elements of mathematics teaching have decreased. We cannot allow essential computing skills, solution of equations and in equations cannot fall victims to these tendencies. Moreover the use of graphical and numeric possibilities we can highlight the background. • We can anticipate a shift in mathematics education. We cannot avoid students using their calculators more frequently, even if we try to eliminate it by various head counting exercises. • Until the appearance of WHY button on the keyboards we will not be able to avoid thinking! • It is on a teachers how they use calculators on their lessons, if they use just for a routine calculation or if they starting to use for more effective (from learning and teaching process too) visualizations.

6. Didactical course with graphical calculators at Eötvös Loránd University • We started a seminar in September, 2006 with 20 students as a part of our didactical education in pre-service teacher training. • During the seminar we put emphasis on the optimal combination of the development of technical knowledge, didactical assumptions and the development of mathematical problem-solving skills. • We try to find such didactical methods that do not interfere with pure mathematical solutions. • We do not intend to modify the mathematical content and curriculum but we search for connection points where graphical calculator has a value added to better understanding. • The central aim of these lessons is to give personally experiences in addition to the theoretical discussions.

7. Curriculum • From 2002 Hungary has a new national curriculum, and with it a new mathematical part statistics. • Statistics is a very good example for using computer- spreadsheets, but also graphical calculators in which is integrated spreadsheet editor. And this is a part in which our teachers needs more help from mathematical, methodological and didactical side too. • Beside the national curriculum in the ministry of education prepared a list of teaching tools. For example statistics there is written that minimum is use of calculators but additionally use of computers concretely Excel and for algebra and function is written the same for geometry additionally is recommended use of computers and dynamic geometry programs.

8. Tasks • Some concrete task form secondary schoolbooks and exercises books. • 1. Functions: Function f(t)=50*0,82 shows how the coffee goes cold in the thermos. Time is given in hours and temperature in Celsius. a) Fill the table with t = 0, 1, 2, 3, 4, 5 hours, b) Calculate the f(-1), f(-2), f(-3), c) Calculate the domain of f(t) if we cast the 95 Celsius degree coffee at 10 Celsius temperature of air. d) What temperature should have a coffee after 29,30 and 45 minut and 1,5 hour later? e) Draw all pair of points in this task to the coordinate system. Connect them and explain it if it is significant proposition?

9. Geometry 2. Lets given a circle and a point B on it. Find the locus of midpoints of chords given from point B to the circle. Proof your conjecture! 3. Find the angles of the triangle, if you know the coordinates of its edges. A(2,0), B(5,4) and C(-1, 3)

10. Statistics 4. We are asked 50 people how many books do their red in last year. Answers are given in a next table. Calculate the average and standard deviation. Prepare a diagram!