Why is it important for schools to spend more time studying discrete mathematics?
The author of the material argues in favor of studying discrete mathematics at the school stage.
Most mathematics programs [in the USA] for middle and high schools follow a clearly defined pattern:
Predalgebraic Problems → Algebra 1 → Geometry → Algebra 2 → Trigonometry / Mathematical Analysis Begin → Mathematical Analysis
In some other schools, preference is given to a more integrated approach, within which elements of algebra, geometry and trigonometry are served in a mixed manner over a 3 or 4 year course. However, both methods lack a significant emphasis on discrete mathematics and such sections as combinatorics, probability theory, number theory, set theory, logic, algorithms, and graph theory. Discrete mathematics appears very little in most of the "critical" intermediate exams of middle and high schools. The situation is similar with the entrance examinations for universities and colleges, such as the SAT. Because of this, discrete mathematics is often given little attention.
Nevertheless, this area of knowledge in recent years has become an increasingly important area. And there are a number of reasons:
Discrete mathematics along with numerical methods and general algebra is included in the list of fundamental components of high-level mathematics. Pupils who have received a substantial amount of knowledge in discrete mathematics before entering college receive a significant advantage during their further studies.
All calculations of modern computer science are almost entirely based on discrete mathematics, and in particular, combinatorics and graph theory. This means that in order to study the fundamental algorithms used by computer programmers, students need to have solid knowledge in these areas. Indeed, to obtain a diploma in computer science, most universities have a corresponding mandatory course in discrete mathematics.
Many students often ask questions about where, in real life, traditional higher mathematics, that is, algebra, geometry, trigonometry, and other directions may be useful to them. Often, looking at the abstract nature of these objects, they lose interest in them. Discrete mathematics, and in particular, combinatorics and probability theory, allow pupils even at the secondary school level to very quickly come to the study of interesting and non-trivial problems that are directly related to the problems of the real world.
Prominent mathematical olympiads like MATHCOUNTS (high school) and American Mathematics Competitions (high school) include a significant number of tasks in discrete mathematics. In more difficult competitions for high school students such as AIME, the number of tasks increases even more. Students who do not have an appropriate knowledge base will have much less chance of success in such competitions. One well-known teacher involved in preparing students for MATHCOUNTS even devotes half the time to preparing for tasks in combinatorics and probability theory. So he considers them important.
Algebra is often taught as a set of formulas and algorithms that students should memorize. For example, the formula of the roots of a quadratic equation, or the solution of a system of linear equations by replacement. Geometry is often taught as a series of exercises proving theorems and explaining their essence, which are often suggested to be learned by heart. Despite the undoubted importance of studying such material, in general, it does not contribute very well to the development of creative mathematical thinking of students. In contrast, students of discrete mathematics learn to think flexibly and creatively from the very beginning. The number of formulas you need to know by heart is relatively small. In this area of knowledge, the emphasis is rather on the need to study a number of fundamental concepts, which can then be applied in completely different ways.
Many students, especially the gifted and motivated, find algebra, geometry, and even the methods of mathematical analysis boring, not causing lively interest. As for discrete mathematics, such topics are rare in it. When we are interested in students in their favorite topics, the majority calls combinatorics or number theory. The most unpopular topic is geometry. In other words, most students find discrete mathematics more interesting than algebra or geometry.
Based on all these arguments, we strongly recommend building a program so that after studying geometry, schools will devote some time to familiarizing students with the elementary ideas of discrete mathematics, and in particular, combinatorics, probability theory and number theory.
Most mathematics programs [in the USA] for middle and high schools follow a clearly defined pattern:
Predalgebraic Problems → Algebra 1 → Geometry → Algebra 2 → Trigonometry / Mathematical Analysis Begin → Mathematical Analysis
In some other schools, preference is given to a more integrated approach, within which elements of algebra, geometry and trigonometry are served in a mixed manner over a 3 or 4 year course. However, both methods lack a significant emphasis on discrete mathematics and such sections as combinatorics, probability theory, number theory, set theory, logic, algorithms, and graph theory. Discrete mathematics appears very little in most of the "critical" intermediate exams of middle and high schools. The situation is similar with the entrance examinations for universities and colleges, such as the SAT. Because of this, discrete mathematics is often given little attention.
Nevertheless, this area of knowledge in recent years has become an increasingly important area. And there are a number of reasons:
Discrete mathematics plays a significant role in the study of mathematics in colleges, universities and higher levels.
Discrete mathematics along with numerical methods and general algebra is included in the list of fundamental components of high-level mathematics. Pupils who have received a substantial amount of knowledge in discrete mathematics before entering college receive a significant advantage during their further studies.
Discrete mathematics is the mathematics of computational processes.
All calculations of modern computer science are almost entirely based on discrete mathematics, and in particular, combinatorics and graph theory. This means that in order to study the fundamental algorithms used by computer programmers, students need to have solid knowledge in these areas. Indeed, to obtain a diploma in computer science, most universities have a corresponding mandatory course in discrete mathematics.
Discrete mathematics is closest to the problems of the real world.
Many students often ask questions about where, in real life, traditional higher mathematics, that is, algebra, geometry, trigonometry, and other directions may be useful to them. Often, looking at the abstract nature of these objects, they lose interest in them. Discrete mathematics, and in particular, combinatorics and probability theory, allow pupils even at the secondary school level to very quickly come to the study of interesting and non-trivial problems that are directly related to the problems of the real world.
Discrete mathematics is a popular area of most middle and high school math competitions.
Prominent mathematical olympiads like MATHCOUNTS (high school) and American Mathematics Competitions (high school) include a significant number of tasks in discrete mathematics. In more difficult competitions for high school students such as AIME, the number of tasks increases even more. Students who do not have an appropriate knowledge base will have much less chance of success in such competitions. One well-known teacher involved in preparing students for MATHCOUNTS even devotes half the time to preparing for tasks in combinatorics and probability theory. So he considers them important.
Discrete mathematics develops logical thinking and teaches proof techniques.
Algebra is often taught as a set of formulas and algorithms that students should memorize. For example, the formula of the roots of a quadratic equation, or the solution of a system of linear equations by replacement. Geometry is often taught as a series of exercises proving theorems and explaining their essence, which are often suggested to be learned by heart. Despite the undoubted importance of studying such material, in general, it does not contribute very well to the development of creative mathematical thinking of students. In contrast, students of discrete mathematics learn to think flexibly and creatively from the very beginning. The number of formulas you need to know by heart is relatively small. In this area of knowledge, the emphasis is rather on the need to study a number of fundamental concepts, which can then be applied in completely different ways.
Discrete mathematics is fun.
Many students, especially the gifted and motivated, find algebra, geometry, and even the methods of mathematical analysis boring, not causing lively interest. As for discrete mathematics, such topics are rare in it. When we are interested in students in their favorite topics, the majority calls combinatorics or number theory. The most unpopular topic is geometry. In other words, most students find discrete mathematics more interesting than algebra or geometry.
Based on all these arguments, we strongly recommend building a program so that after studying geometry, schools will devote some time to familiarizing students with the elementary ideas of discrete mathematics, and in particular, combinatorics, probability theory and number theory.
Source: https://habr.com/ru/post/409943/