Probability and Statistics in Primary and Secondary School Education in Republic of Macedonia

1. Probability and Statistics in Primary and Secondary School Education in Republic of Macedonia V.Bakeva, M.Georgieva Institute of informatics Faculty of Natural Sciences and Mathematics P.O.Box 162, Skopje Republic of Macedonia verica@ii.edu.mk

2. System of education in Macedonia • Starting from 2007, the primary and secondary education of Republic of Macedonia is constituted as 9+4 system. • It is a fairly new system of education. • To 2007, there are 1+8+4 years of primary and secondary education: • 1 year is spent for preparation, • 8 years are spent in primary school and • 4 years in high school.

3. Some history… • Some topics of probability and statistics are taught as parts of mathematical curriculum for many years during the primary and secondary education. • About 25 years ago, they were taught in the fourth year of the secondary schools. • From 1982 until 1992, there existed special high schools that were oriented toward natural sciences and mathematics. • In these schools “Probability and Statistics” was taught as a separate subject.

4. Some history… • After that it became part of the subject “Applied Mathematics”. • Now, students start studying some topics of probability and statistics when they are 9 years old, and then continue to explore these subjects during their secondary and higher education.

5. Probability and Statistics Education on the Primary School Level • Some rudiments of probability and statistical education are studied in the primary education mathematics curriculum of Macedonian schools. • The students start to study these rudiments as part of the subject Mathematics in their fourth grade (when they are 9 years old) in the sections titled “Working with data” (or “Basic description of a set of data”). • There are not organized just in one chapter of the book, but there are few separate sections in order some applications of mathematical operations defined there (as summation, multiplication, e.t.c.) to be given.

6. Probability and Statistics Education on the Primary School Level : IV grade • obtaining frequency tables for a given set of data; • show frequencies in bar graphs; • obtaining min and max of a set of data; • reading information of a bar graphs;

7. Probability and Statistics Education on the Primary School Level : V grade • using questionnaire to make a table; • calculate the mean of data; • sort the data in ascending or descending order; • draw bar graphs; Example: Given the lengths of 6 jumps of each of 10 jumpers, find the average jump length of each jumper, sort obtained data in ascending order and present them in bar graph.

8. Probability and Statistics Education on the Primary School Level : VI and VII grades • VI grade: Revision of concepts learned in previous two years • VII grade: • percentages; • pie graphs; • comparing data (absolute and relative) • mean • median and mode • range

9. Probability and Statistics Education on the Primary School Level : VII and VIII grades • VIII grade • revision of concepts given before • population • sample, representative sample • random events • probability • certain and impossible events • IX grade • revision of concepts given before • classical definition of probability (elementary examples)

10. Expected knowledge • At the end of this component of the curriculum, students are expected to be able to: • draw bar graphs of information collected locally (using questionnaires); • prepare tables for graph work; • calculate the mean, median and range of data; • introduce population and sample; • introduce elementary probability concept.

11. Probability and Statistics Education on the Secondary School Level • In Macedonia, the students start their secondary education when they are 14 years old. • Duration of the secondary education is 4 years. • There are two types of secondary schools in Republic of Macedonia: high schools (gimnazii, in Macedonian) and professional high schools (struchni skoli, in Macedonian).

12. Probability and Statistics Education on High Schools (gimnazii) • According to the last reform, parts of probability and statistics have been taught each year in high schools (gimnazii). • During the first two years, some statistical concepts learned in primary school are reviewed and then studied in more details. • In the last two years, some advanced concepts of probability and statistics are introduced and studied.

13. Syllabus of Probability and Statistics on High Schools (gimnazii) • : I grade (6 hours) • Population and sample. • Using data to make a table. • Presenting data in a graph. • Mean. Median and mode. • II grade (6 hours) • Population and sample. Mean. Median and mode (revision). • Range, quartiles. • Variance, standard deviation. • Standardize data.

14. Syllabus of Probability and Statistics on High Schools (gimnazii): III grade (15 hours) • Combinatorics • Variations, permutations, Combinations. • Binomial formula. • Probability • Experiments. Event. Random event. • Classical definition of probability. • Probability properties. • Statistical probability.

15. Syllabus of Probability and Statistics on High Schools (gimnazii): IV grade (26 hours) • Probability • Random events and probability. • Set of elementary events (space). Set operations. (only discrete space are consider) • Definition of probability and properties. • Conditional probability. Independent events. • Discrete random variables. • Mean. Variance.

16. Syllabus of Probability and Statistics on High Schools (gimnazii): IV grade (26 hours) • Statistics • Population. Sample. Statistics. • Hypothesis and errors; • Tables of normal, t- and 2-distributions. • Test of value of mean when the variance is known • Test of value of mean when the variance is unknown • Test of value of mean of variance • Contingency tables

17. Syllabus of Probability and Statistics on Professional High Schools: Economic and Technical • The syllabus of probability and statistics in economic schools (in II year) and technical schools consists of: • Statistical and classical definition of probability. • Conditional probability. Independent events. • Population. Sample. • Presenting data in tables and graphs. • Mean. Median. Mode. • Range. Variance. Standard deviation. • In medical schools, the syllabus contains only statistical parts.

18. Advantages and Disadvantages Advantages in primary school: • The probability and statistical education starts during the early stage of the education; • The study of probability and statistics in primary school is not forced. • The students are introduced into the secrets of probability and statistics as application of other mathematical operations in a very relax way. • Exercises are well chosen and completely support the concepts which are introduced. They are well measured and interesting.

19. Advantages and Disadvantages • Advantages in high school: • The syllabus of probability and statistics gives good foundations in order to continue education to universities. • many revisions during each year which is good for student to become familiar with the subjects. • Disadvantages in high school: • low number of classroom hours during the first two years.

20. Some conclusions • The concept of teaching of probability and statistics in primary schools is well chosen • In our opinion, it would be better if all concepts of probability and statistics in high schools are incorporated in one-year mathematical curriculum (one or two semesters depends of hours per week) or returning of subject “Probability and Statistics” as year long class.