Here's what I tell my high-school student’s:

Derived from the Arabic word al-jabru, which was the title of a
book written by Arabic mathematician al-Khwarizmi (pronounced
suspiciously like algorithm); algebra is the branch of math
that describes problems--often real world—by first assigning
them to a category such as 'quadratic' or 'ring.' Once the
category is known, the solution may be derived
"algorithmically"—by following a series of prescribed
operations—often in a prescribed order. This last statement
seems obvious—even trite—until we realize that what we are
actually saying, is that the ability to solve ONLY ONE of them
is equivalent to being able to solve EVERY problem of this sort.

The thrill of algebra—my favorite branch of math--is that the
solution to any particular problem is WHOLLY INDEPENDENT of the
values that define that very problem.

If I may take a moment to pontificate: The diminished
readiness that we are seeing in our students is the result of
current--wrong--beliefs that algebra should first be understood
conceptually, then mastered. In truth, algorithmic processes
should be mastered first. Understanding follows naturally as
the result of the human brain's instinct to recognize and
qualify patterns.

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