Discrete Mathematics, A New Course for Iranian National Curriculum

Ali Rejali

Isfahan University of Technology
IRAN

Some information about the Iranian system of education, the motives for developing this new course, and the text book for the course plus some preliminary results from our study during the in-service programs and the first semester teaching of the course in Isfahan province will be presented

Introduction

Our system of education consists of an optional kindergarten for 6 years old students, and a compulsory 5-years elementary school, 3-years intermediate school, 3-years high school and a one-year newly established pre-university program. The students have their choice for selecting their branch of study when they pass the second year of high-school. Some of them, say about thirty percent do not enter the pre-university program.

In the intermediate school, the students see concepts such as divisibility, greatest common divisors, and the least common multiples, the idea of being a prime in elementary number theory and some elementary descriptive statistics.

In the first years of high school, they get to know the idea of the integer part of a real numbers as a real function, some more about statistics, combinatorics and very elementary chapter on probability. The students have also a chance to work with matrices. In the last year of high school, the mathematics-physics students take a course in Algebra and Probability in which among the ideas for mathematics reasoning, they see mathematics induction, recursive methods and pigeon hole principle.

Working with logical construction of sets especially finite sets starts from intermediate school and the students work more with operations on power sets, in this course. In studying the equivalence relations, concepts such as congruences in elementary number theory is studied in Algebra and Probability Course. Probability, as a mathematics tool for studying random phenomena, is introduced in this course, and about 5 pages is devoted to elementary graph theory.

Motivations

As I mentioned in the introduction, there are some background information among pre-university students. On the other hand, we have a long history of interest and a good and deep literature on number theory. The students and teachers are interested in solving problems in number theory. In an study on the international and national olympiads, the author [1] shows a good achievement among Iranian students for number theory problems.

The positive effect of solving problems in Number Theory for strengthening students in reasoning and mathematical thinking, the interest among students for studying this subject due to a long and beautiful history of it and wide range of applications for the subject, were some motivations to devote a major part of a course to Number Theory.

Although we do not have any literature on Graph Theory in Persian language, and even most of the high-school teachers do not take any course on this subject and Combinatorics during their studies, but some aspects of graph theory like number theory made us believe that this is a good and useful subject for high school students. Meanwhile most of the students used to study graph theory and combinatorics to prepare themselves for mathematics olympiads.

Discrete probability has a wide range of applications and like graph theory is not only useful for mathematics majoring students, but other students can get benefit from the course, also solving problems in elementary probability improve the ability of mathematical reasoning among students.

Everybody knows about the NCTM Curriculum Standards on number theory, probability, statistics and Discrete Mathematics in general, for example one can see the strand 12 about Discrete Mathematics, as follows:

In grades nine through twelve, the mathematics curriculum should include topics from discrete mathematics so that all students can:
Represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations.
Represent and analyse finite graphs, using matrices.
Solve enumeration and finite probability problems.

And so that, in addition, students planning to attend college can:
Represent and solve problems, using linear programming and difference equations.
Investigate problem situations that arise in computer validation and the application of algorithms, [2].

Looking at mathematics frameworks in different states and countries, one can see some discrete mathematics in all curricula, either optional or compulsory.

I looked at 4 different place that I had chance to visit:

California Public Schools [3].
Australian Schools [4].
Hungarian curriculum for high schools [5].
Italian curriculum [5].

History of the Book

Since the text books in Iran is produced nationally, I and three of my colleagues were invited to write a book on Discrete Mathematics for pre-university students with mathematics major. I think the reasons for choosing me for this job are:

My field of study is probabilistic number theory.
I thought many courses in number theory, probability and discrete mathematics at university and also at in-service programs for high school teachers.
I have a long history of study in curriculum standards and the problem of mathematics education in my country.
I have a good and deep relations with mathematics teachers and students and also with other mathematics educators and university professors, so I could use their advices and share their ideas in my writings.

So we decided to study backgrounds and the needs of students and many text-books from our old curricula and other countries.

After writing the book, we passed it to about 100 teachers, professors and mathematics educators and we taught our drafts in an in-service program in which many teachers from all our the country participated.

Then I presented our drafts in our weekly continuous-in service program in Isfahan.

The outcome is being examined by a 10 percent students who entered the new system in last semester.

Content:

Graph and Its Applications (Graphs, special graphs, tree and matrix)

Number Theory (Principles and divisibility, prime numbers, congruences)

Combinatorics (Intuitive modeling in combinatorics, recurrence relation

Probability (Discrete probability Space, conditional probability, discrete random variables)

In the Preface of the book one can read:

In recent years, courses and books with titles `Discrete Mathematics', `Combinatorics', and `Finite Mathematics' are popular. Most of them deal with finite sets and sometimes with infinitely countable sets. Graphs, Enumeration Methods, Combinatorial Structures, parts of Number Theory, Probability and some of their applications such as Coding and Cryptography are included in these courses.

Their common structure of most of them is discreteness, and although concepts such as limits and continuity are not often used, but they are well equipped with other useful and extraordinary beautiful tools.

Due to dynamic nature, having many easy imposed, but unsolved problems, wide range of applications, beauty, and the power of strenghtening mathematical thinking of Discrete Mathematics, the study of its foundation is popular among many societies, and for Iranian curriculum is a necessary.

In parts of the book, there are some mathematics magazine for introducing either the history or new findings on the subjects, such as Four Color Problem, Fermat's last Theorem, etc. ...

Some References for the Book

Graham, R.L., Knuth, D.E., & Patashnik, O., Concrete Mathematics, Addison-Wesley, 1992.
Behzad, M., Chartrand, G- & Losniak-Foster, L., Graphs and Digraphs, Wadsworth Internationa, 1979.
Ore, O. & Wilson, R.J., Graphs and their Uses, MAA, 1990.
Burton, D.M., Elementary Number Theory, Allyn and Bacon, 1976.
Rosen, K.H., Elementary Number Theory and its Applications, 4th. ed., Addison-Wesley 1992.
Ghorbani A., Saffari H., Number Theory 1968, (in Farsi).
Mosahab, G.H., Introductory Number Theory, Dehkhoda 1974, (in Farsi)
Grimaldi R.P., Discrete and Combinatorial Mathematics: An Applied Introduction, 2nd ed., Addison- Wesly 1989.
Ross, S., A First Course in Probability, Macmillan 1976.
Amidi, A., Probability and its Applications, Peyameh Noor, 1995, (in Farsi).

Some Preliminary Results:

Although most of the teachers opposed the new course, especially graph theory and combinatorics parts, but after one in-service program they forgot their resistance.
The students like the book, and enjoy the material.
The teachers who are used to teach number theory, wants to put more emphasis on number theory. The study is still continuing and I welcome suggestions from the readers and the audience.

References

Rejali, A.(1988). Statistical Eraluation of the Past Mathematical Competitions of Iran, Presented in TA3, ICME-6, Budapest.

Curriculum and Evaluation Standards for School Mathematics, NCTM, 1989.

Mathematics Framework for California public Schools, Kindergarten Through Grade Twelve, California Department of Education, 1992.

A National statement on Mathematics For Australian Schools, Australian Educational Council, 1991. Personal Communications.

ICME8 - WG13 - 02 JUL 96

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