I like this explanation. High school students are surprisingly smart and a little perspective -- that "algebra" is just manipulations in a symbolic system (and not even a special one, just very widely used) -- is sometimes all it takes to help them "get it" a little more.
It's kind of funny, because students are taught how to use these things like numbers and operations and variables, but it's never explained in an abstract way what "algebra" is or why it's even called that. Hell, I didn't know until I took linear algebra in college.
When my sister (who is currently taking algebra) asked me what "algebra" meant, I told her:
An algebra is a combination of a set of objects and a set of operations. The algebra you're learning has real numbers as the set of objects. You know what the operations are: they're things like multiplication, addition, sqrt, whatever. There are other algebras too that use things other than the reals.
As an aside, this is a great way to teach fractions as well (or at least, I like it a lot). People try to give intuitive explanations with pieces of pie or whatever but students tend to find them very confusing. You would be surprised how fucking confused students can get about fractions, I mean some just can't wrap their heads around it ever. Even as adults. They're not dumb, they try to learn fractions by "intuition" and, frankly, fractions aren't intuitive. The simplest route is to just lay it out: these are objects with one expression and another expression and a line between them, and you can multiply them like this, and add them like this, and "move" expressions from one side of the line to the other by taking the reciprocal like this, and that's it.
It's kind of funny, because students are taught how to use these things like numbers and operations and variables, but it's never explained in an abstract way what "algebra" is or why it's even called that. Hell, I didn't know until I took linear algebra in college.
When my sister (who is currently taking algebra) asked me what "algebra" meant, I told her:
An algebra is a combination of a set of objects and a set of operations. The algebra you're learning has real numbers as the set of objects. You know what the operations are: they're things like multiplication, addition, sqrt, whatever. There are other algebras too that use things other than the reals.
As an aside, this is a great way to teach fractions as well (or at least, I like it a lot). People try to give intuitive explanations with pieces of pie or whatever but students tend to find them very confusing. You would be surprised how fucking confused students can get about fractions, I mean some just can't wrap their heads around it ever. Even as adults. They're not dumb, they try to learn fractions by "intuition" and, frankly, fractions aren't intuitive. The simplest route is to just lay it out: these are objects with one expression and another expression and a line between them, and you can multiply them like this, and add them like this, and "move" expressions from one side of the line to the other by taking the reciprocal like this, and that's it.