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The Importance of Teachers Understanding and Teaching Basic Number Theory
Mathematics is one of the most important academic disciplines because it is woven into the
very fabric of life, and affects almost everything, if not everything, that we do. Mathematics is
not only important to cognitive development, but is essential for the development of certain life
skills. Important to these developments is the area of elementary number theory. This paper
looks at a brief history of number theory, and its relevance to mathematics education. It also
examines the importance for a teacher to understand, and to be able to teach basic or elementary
number theory, because it is foundational to the teacher’s competence to teach mathematics, and
is the key to students having a strong base in mathematics.
Basic or elementary number theory involves the areas of number theory which are generally
taught in elementary and high school math programs (Wikipedia). “Number theory or in older
usage, arithmetic, is the branch of pure mathematics devoted primarily to the study of integers,”
(Wikipedia). The history of elementary number theory dates back to several thousand years. For
example, writing which dates back to the Babylonian era shows that the Babylonians knew how
to factor natural numbers into prime factors, ( Encyclopedia of Mathematics). “In the 5th century
B.C. the Pythagoreans established the so-called doctrine of even and odd numbers and justified
the proposition that the product of two natural numbers is even if and only if one of its factors is
even,” (Encyclopedia of Mathematics). In many ways, the history and development of number
theory has led to an increased understanding of number relations and the development of several
principles and theorems. Euclid is claimed to have developed a general theory of divisibility
around 300 B.C. He also brought about the creation of an algorithm “for finding the greatest
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common divisor of two integers and on this basis he justifies the main theory of the arithmetic of
integers,” (Encyclopedia of Mathematics). These divisibility rules and theorems are
foundational to many areas of mathematics at all levels. The four operations: addition,
subtraction, division and multiplication are primarily rooted in number theory, and are integral to
all areas of mathematics.
So why should a teacher understand basic number theory? Since understanding this area of
mathematics is crucial to the development of a strong mathematical base, it is important for a
teacher to understand number theory so that he or she can communicate it effectively. The
teacher can only effectively communicate what he or she understands. In other words, teachers
“need to know the mathematics they teach as well as the horizons of that mathematics,” (The
National Academics of Sciences, Engineering, Medicine). This aspect of mathematics is
fundamental to the way students understand and appreciate mathematics, but as Selden, A, and
Selden, J, pointed out, “number theory appears to be a rather neglected area of mathematics
education research literature.” There is therefore, generally speaking, a lack of understanding of
the subject and how it relates to other areas of mathematics. If teachers develop a thorough
understanding of number theory, they will be able to connect the dots between it and the other
areas of mathematics, and students’ appreciation for this fundamental area of mathematics will
grow. One very good reason why a teacher should understand basic number theory is that it is an
integral aspect of the mathematical content and practice of the Common Core Standards. For
example, Common Core Standards practice require students to be taught to construct viable
arguments and critique the reasoning of others, model with mathematics, look for and make use
of structure. Regarding content at varying levels, student should be taught counting and
cardinality, operations and algebraic thinking, number and operations in base ten, (Wikipedia). In
The Importance of Teachers Understanding and Teaching Basic Number Theory
Mathematics is one of the most important academic disciplines because it is woven into the
very fabric of life, and affects almost everything, if not everything, that we do. Mathematics is
not only important to cognitive development, but is essential for the development of certain life
skills. Important to these developments is the area of elementary number theory. This paper
looks at a brief history of number theory, and its relevance to mathematics education. It also
examines the importance for a teacher to understand, and to be able to teach basic or elementary
number theory, because it is foundational to the teacher’s competence to teach mathematics, and
is the key to students having a strong base in mathematics.
Basic or elementary number theory involves the areas of number theory which are generally
taught in elementary and high school math programs (Wikipedia). “Number theory or in older
usage, arithmetic, is the branch of pure mathematics devoted primarily to the study of integers,”
(Wikipedia). The history of elementary number theory dates back to several thousand years. For
example, writing which dates back to the Babylonian era shows that the Babylonians knew how
to factor natural numbers into prime factors, ( Encyclopedia of Mathematics). “In the 5th century
B.C. the Pythagoreans established the so-called doctrine of even and odd numbers and justified
the proposition that the product of two natural numbers is even if and only if one of its factors is
even,” (Encyclopedia of Mathematics). In many ways, the history and development of number
theory has led to an increased understanding of number relations and the development of several
principles and theorems. Euclid is claimed to have developed a general theory of divisibility
around 300 B.C. He also brought about the creation of an algorithm “for finding the greatest
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common divisor of two integers and on this basis he justifies the main theory of the arithmetic of
integers,” (Encyclopedia of Mathematics). These divisibility rules and theorems are
foundational to many areas of mathematics at all levels. The four operations: addition,
subtraction, division and multiplication are primarily rooted in number theory, and are integral to
all areas of mathematics.
So why should a teacher understand basic number theory? Since understanding this area of
mathematics is crucial to the development of a strong mathematical base, it is important for a
teacher to understand number theory so that he or she can communicate it effectively. The
teacher can only effectively communicate what he or she understands. In other words, teachers
“need to know the mathematics they teach as well as the horizons of that mathematics,” (The
National Academics of Sciences, Engineering, Medicine). This aspect of mathematics is
fundamental to the way students understand and appreciate mathematics, but as Selden, A, and
Selden, J, pointed out, “number theory appears to be a rather neglected area of mathematics
education research literature.” There is therefore, generally speaking, a lack of understanding of
the subject and how it relates to other areas of mathematics. If teachers develop a thorough
understanding of number theory, they will be able to connect the dots between it and the other
areas of mathematics, and students’ appreciation for this fundamental area of mathematics will
grow. One very good reason why a teacher should understand basic number theory is that it is an
integral aspect of the mathematical content and practice of the Common Core Standards. For
example, Common Core Standards practice require students to be taught to construct viable
arguments and critique the reasoning of others, model with mathematics, look for and make use
of structure. Regarding content at varying levels, student should be taught counting and
cardinality, operations and algebraic thinking, number and operations in base ten, (Wikipedia). In
