2. COMPUTERS SIMULATION IN MATHEMATICS EDUCATION Dr. Ronit Hoffmann Kibbutzim College of Education, Israel

3. PME 25 - Dr. Ronit Hoffmann2 Many educators agree that a more extensive use of computers in school in general, and in mathematical education in particular, is desirable. The NCTM Standards (1989, 2000) and the Harari Committee Report (1992) already recommended this over ten years ago.

4. PME 25 - Dr. Ronit Hoffmann3 "Computer-Oriented Numerical Math" In our efforts to integrate the use of the computer in teaching mathematics in teachers training colleges, we have developed a "Computer-Oriented Numerical Math" course, in which the computer is used as a tool for solving mathematical problems and as a tool for exploring, discovering and illustrating abstract terms

5. PME 25 - Dr. Ronit Hoffmann4 In this presentation, I will present a teaching module, meant for teachers intending to teach in the higher grades of elementary school and/or in middle-schools.

6. PME 25 - Dr. Ronit Hoffmann5 To demonstrate the teaching module I will describe The Monte Carlo simulation for area approximation

7. PME 25 - Dr. Ronit Hoffmann6 Focusing on the computation of the area of a unit circle (a topic taught in the 6 th grade in Israel) which leads to the approximation of the number 

8. PME 25 - Dr. Ronit Hoffmann7 According to many mathematicians the number  is one of the five most important and interesting numbers in mathematics. Leonhard Euler (1707-1783) even linked these five numbers in the equation e i  +1=0.

9. PME 25 - Dr. Ronit Hoffmann8 The historical origin of the number is in the investigation of the ratio between the circumference of a circle to its diameter, but in time it was found that its importance went beyond the field of geometry. Today plays a central part in various applications of mathematics, physics, engineering, statistics etc’.

10. PME 25 - Dr. Ronit Hoffmann9 “Monte Carlo method” The term “Monte Carlo method” is a general concept. A Monte Carlo simulation uses random numbers to model some sort of a process. This technique derives its name from the casinos in Monte Carlo.

11. PME 25 - Dr. Ronit Hoffmann10 It refers to numerical methods based on probabilistic or randomized algorithms Which use elementary statistical methods Allowing rapid approximate solutions for problems for which computational solutions are either not known or are inefficient.

12. PME 25 - Dr. Ronit Hoffmann11 Monte Carlo computational science Monte Carlo methods are used in various fields of computational science: economics statistics nuclear physics chemistry biology mathematics, and the like.

13. PME 25 - Dr. Ronit Hoffmann12 We will show how modern day technology enables embedding this topic within the school curriculum.

14. PME 25 - Dr. Ronit Hoffmann13 Area Approximation Based on the use of random numbers S 2 s 1 x y Y=f(x) X=aX=b In order to approximate the area s 1 under the graph of a nonnegative function f(x), we built a rectangle whose area is s 2.

15. PME 25 - Dr. Ronit Hoffmann14 S 2 s 1 x y Y=f(x) X=aX=b Imagine throwing darts or arrows towards s 2. (all of them hitting s 2 ) What is the probability to hit s 1 ? P( s 1 )= s 1 / s 2 =Lim(n/m) m-arrows hitting the rectangle n- arrows hitting the area s 1

16. PME 25 - Dr. Ronit Hoffmann15 The Monte Carlo Method to approximate  In a similar way, we shall now find the area of a quarter of the unit (r=1) circle. P(s 1 ) =Lim(n/m) = s 1 / s 2 = (  /4)/1=  /4 y x s1s1 s2s2

17. PME 25 - Dr. Ronit Hoffmann16 (n/m)*4 The intuitive notion of probability implies that for large m the quantity (n/m)*4 should be a fair approximation to .  ~ (n/m)*4 We also expect the quality of this approximation to improve with increasing m (though not monotonically).

18. PME 25 - Dr. Ronit Hoffmann17 The coordinates (x,y) of the points m, in the unit square satisfies 0  x,y < 1. In order to find n, we have to count the number of cases in which the point (x,y) is situated inside the quarter circle, i.e – to count those points for which x 2 +y 2 <1.

19. PME 25 - Dr. Ronit Hoffmann18 Now we can harness the computer to simulate the experiment of throwing darts or arrows towards the unit square.

20. PME 25 - Dr. Ronit Hoffmann19 We suppose that our computer has a built-in Random-number generator Random-number generator RAND(x), which produces a random number x, 0  x< 1, all numbers in this range having an equal chance of being picked. Now the computer will generate pairs of random numbers x,y such that 0  x,y< 1. We`ll think of each pair (x,y) as coordinates of m.

21. PME 25 - Dr. Ronit Hoffmann20 algorithmthe program  The Monte Carlo Method Now the students are ready to write the algorithm and then the program to obtain an approximation for  using The Monte Carlo Method : 1.Input m 2.a  0, n  0 3.While a  m do a  a+1, x  rand(), y  rand() if x 2 +y 2 <1, then n  n +1 4. Print pi = (4*n)/m

22. PME 25 - Dr. Ronit Hoffmann21 We believe that there are many advantages to students building the mathematical computer programs themselves, after exploring and discussing the various existing methods. During the next stage, the students 'run' the programs they have written, check the results obtained, discuss them and draw conclusions.

23. PME 25 - Dr. Ronit Hoffmann22 The students in all the classes engaged in this module with great enthusiasm and interest. mathematical concepts The topic created opportunities to discuss important mathematical concepts such as: Area the use of simulation Probability Approximations, etc`

24. PME 25 - Dr. Ronit Hoffmann23 In addition, the students were introduced to yet another facet of computer use in mathematics Were lead to experience mathematical studies from a new perspective.