Equality in Proper Context

Share Cognitive Science Implications for Teaching Mathematics
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For the last 10 years of my teaching career I always pointed out the following important considerations for viewing mathematics.  I think every student will benefit if their view of mathematics includes the following considerations.  Teachers should help students to establish this habit by regularly identifying mathematical objects, unary operations, binary operations, and relations.  Provide students with low-stakes questions for self-study about these topics.  Ask high-stakes questions about these topics on tests.

One of the things we do in mathematics is use deductive reasoning to study relations and operations on mathematical objects. It will help your understanding if you identify each “thing” as either:

  • or a mathematical object.
  • an operation,
  • a relation,

It will help you understand mathematics, especially new mathematics, if you make that identification as soon as you are introduced to a new mathematical “thing”.  The following examples will help in that identification process.

Mathematical objects: Some familiar mathematical objects are numbers, variables, algebraic expressions, equations, formulas, and geometric figures.

Operations: In mathematics, an operation is a calculation involving some number of mathematical objects. The objects involved in a particular operation are called operands of that operation.  The number of operands involved in an operation leads, quite naturally, to a classification of operations.  For elementary algebra courses, two classes of operations are important; unary and binary.

Unary Operations: Unary operations are those operations involving one operand.  Calculating the reciprocal of a number, calculating the opposite of a number, calculating the square of a number, calculating the multiplicative inverse of a matrix, and calculating the conjugate of a complex number are each examples of a unary operation.  Other unary operations which will be introduced and used as needed.

Binary Operations: Binary operations are those operations involving two operands.  Calculating the sum of two numbers, calculating the product of two numbers, calculating the sum or product of complex numbers, and calculating the sum or product of matrices are examples of binary operations.  The operations of addition, subtraction, multiplication, and division are the most common binary operations.  Other binary operations which will be introduced and used as needed.

Relations: Relations in mathematics are comparisons. There are only three important relations in basic algebra courses.  These are the comparisons of “less than”, “equal”, and “greater than.” It is very important to realize these are the named concepts as opposed to the symbols (<, =, >) used to represent them.  The meanings of these concepts are not “a priori” knowledge.  In every situation where we use any of these concepts we must provide a stipulative definition.  For example, two complex numbers are equal if and only if their real components are equal and their complex components are equal.  (Certainly not a priori knowledge).  Most, as this definition of equality of complex numbers, stipulative definitions of equality rely on the definition of equality of two real numbers. Therefore, we must establish a firm understanding of equality of real numbers.  A firm understanding does not require the most elaborate and set theoretic definition.  A firm understanding is obtained by defining these concepts in terms of their relative position on the real number line.  The following definitions also establish the real number line as a very important basis for much of mathematics.

Definition:  Two real numbers are equal if and only if they are represented by the same point on the real number line.

Definition:  A first real number is less than a second real number if and only if the real number line representation of the first number is to the left of the real number line representation of the second number.

Definition:  A first real number is greater than a second real number if and only if the real number line representation of the first number is to the right of the real number line representation of the second number.

I will make another plea to teachers.  Please refer to the real number line as often as appropriate.  Please explicitly use the real number line to explain concepts.  Please use the real number line to instill a sense and appreciation for the one-to-one correspondence between the real numbers and points on the real number line.  Please use the number line to help your students understand that irrational numbers exist and are real. Help your students to establish a basic understanding of the fact that the real numbers can be ordered.

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