Look at these five lines:

$$x+3=7,\qquad A=b\cdot h,\qquad a+b=b+a,\qquad y=kx,\qquad \sin^2x+\cos^2x=1.$$

All of them use letters, and all of them look like algebra. But in none of them do the letters do the same thing.

  • In $x+3=7$, the $x$ is a fixed, unknown number. Only $4$ will do. It is an unknown.
  • In $A=b\cdot h$, the letters hide nothing: they say how three quantities are related. It is a formula, and the letters are quantities that vary together.
  • In $a+b=b+a$, the letters are arbitrary numbers. The sentence holds for every pair. The letter generalises.
  • In $y=kx$, $x$ and $y$ move, and $k$ stays still. The $k$ is a parameter: fixed in each problem, but arbitrary.
  • In $\sin^2x+\cos^2x=1$, the $x$ is also arbitrary, but the sentence is not an equation to solve: it is an identity, true for every $x$.

Usiskin’s conceptions

In 1988 the American educator Zalman Usiskin, in an article that became a classic, proposed that school algebra mixes at least four conceptions, each giving the letter a different role:

Conception of algebra What the letter is Example
Generalised arithmetic an arbitrary number that generalises a pattern $a+b=b+a$
Problem solving an unknown, to be discovered $3x+5=20$
Study of relationships a variable, which changes along with others $y=2x+1$
Study of structures an arbitrary symbol, handled by rules $x^2-y^2=(x-y)(x+y)$

In the last row nobody thinks of numbers: $x$ and $y$ are marks moved by rules, like chess pieces. To these four we add a fifth here, which appears throughout the book: the letter as a name, of a function ($f$), of a constant ($\pi$, $e$, $c$) or of an object ($A$, the point; $r$, the line).

The text does not warn you

The semiotic problem is that the writing is the same for every role. Nothing in the signifier distinguishes the unknown from the variable: both are an italic $x$. What decides is the context, what Morris called pragmatics (see chapter 2). The clues are usually small words, which pupils learn to ignore and should not:

  • “Find $x$ such that…” announces an unknown.
  • “For every $x$…”, or “let $a$ and $b$ be real numbers”, announces an arbitrary number.
  • “When $x$ increases, $y$…” announces variables.
  • “Given $a>0$…” announces a parameter.

Professional mathematicians read these clues without noticing. Pupils often do not even know they exist, and the result shows in the errors we will see in text 8.

Exercise: what is the role of each letter?

(a) $2n$ is always even. (b) $2x=10$. (c) $s=s_0+vt$. (d) $(a+1)^2=a^2+2a+1$.

(a) $n$ is an arbitrary integer: the letter generalises. (b) $x$ is an unknown, and is only $5$. (c) $s$ and $t$ vary together; $s_0$ and $v$ are parameters, fixed for each motion. (d) $a$ is arbitrary: it is an identity, and the letter can be treated as an arbitrary symbol in the manipulation.

Exercise: the same writing, two roles

The expression $x^2=4x$ can be read in two ways. As an equation, it asks for the values of $x$ that make it true: $x^2-4x=0$, $x(x-4)=0$, so $x=0$ or $x=4$. As a comparison of functions, it compares the curves $y=x^2$ and $y=4x$, and the solutions become the points where they cross. The writing is identical; what changes is what the reader does with it.

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