I learned, and internalized, the different uses and meanings of the equals (=) sign. I accomp-lished this through inference as I can not recall ever being made aware that there were different uses.

Even now when I see it used in different ways I look at the expression – equation – relation and figure out what is meant.

I am (and was) was able to do this because I knew enough already to use inference to determine the use.

Most student do not have this ability. They lask the background and with modern electronics, it is unlikely they will ever gain the background. Below is an except from an article describing the equals sign:

To illustrate the psychological complexity that mathematics education researchers see and the difficulties students encounter in school algebra, Usiskin (1988) presents five equations (in the sense of having an equal sign in them) involving literal symbols. Each has a different feel.

A = LW
40 = 5x
sin x = cos x * tan x
1 = n * 1/n
y = kx

As Usiskin observes, “We usually call
(1) a formula,
(2) an equation (or open sentence) to solve,
(3) an identity,
(4) a property, and
(5) an equation of a function of
direct variation (not to be solved)” (p. 9).

Each of these symbol strings reads differently. Furthermore, Usiskin (1988) argues that each of these equations has a different feel because in each case the concept/idea of variable is put to a different use:

In (1), A, L, and W stand for the quantities area, length, and width and have the feel of knowns.

In (2), we tend to think of x as unknown.

In (3), x is an argument of a function.

Equation (4), unlike others, generalizes an arithmetic pattern.

In (5), x is again an argument of a function, 􀝕 the value, and k a constant (or parameter, depending on how it is used). Only with (5) is there a feel of “variability” from which the term variable arose.

(Usiskin, 1988, p. 9)

In (1), A, L, and W stand for the quantities area, length, and width
and have the feel of knowns. In (2), we tend to think of 􀝔 as unknown.
In (3), 􀝔 is an argument of a function. Equation (4), unlike others,
generalizes an arithmetic pattern. In (5), 􀝔 is again an argument of a
function, 􀝕 the value, and k a constant (or parameter, depending on
how it is used). Only with (5) is there a feel of “variability” from
which the term variable arose.
(Usiskin, 1988, p. 9)

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I am (and was) was able to do this because I knew enough already to use inference to determine the use.

Most student do not have this ability. They lask the background and with modern electronics, it is unlikely they will ever gain the background. Below is an excerpt from an article describing the equals sign:

To illustrate the psychological complexity that mathematics education researchers see and the difficulties students encounter in school algebra, Usiskin (1988) presents five equations (in the sense of having an equal sign in them) involving literal symbols. Each has a different feel.

A = LW
40 = 5x
sin x = cos x * tan x
1 = n* 1/n
y = kx

As Usiskin observes, “We usually call
(1) a formula,
(2) an equation (or open sentence) to solve,
(3) an identity,
(4) a property, and
(5) an equation of a function of
direct variation (not to be solved)” (p. 9).

Each of these symbol strings reads differently. Furthermore, Usiskin (1988) argues that each of these equations has a different feel because in each case the concept/idea of variable is put to a different use:

In (1), A, L, and W stand for the quantities area, length, and width and have the feel of knowns. In (2), we tend to think of x as unknown. In (3), x is an argument of a function. Equation (4), unlike others, generalizes an arithmetic pattern. In (5), x is again an argument of a function, x the value, and k a constant (or parameter, depending on how it is used). Only with (5) is there a feel of “variability” from which the term variable arose.
(Usiskin, 1988, p. 9)