Wednesday, July 20, 2011

Thoughts on Algebra 1

After a five week mad sprint to teach my students everything they'll really need to know about Algebra 1 going forward, I have a few thoughts on the subject.

First, as far as I can tell, there are two "Big Ideas" of Algebra 1:
  • Everything students already know about arithmetical expressions (adding, subtracting, multiplying, exponents, radicals, fractions, simplifying, word problems) can be extended in a consistent way to situations involving variables.
  • Equations with variables give you the power to describe situations more generally than a single set of numbers allows.
If you've taught Algebra 1, do you agree with those? Am I missing some? If I ever find myself teaching this class again, I'd like to try to present those up front and highlight them repeatedly as they come up, rather than gradually discovering them for myself and trying to make them clear to my students over the course of the class.

I think one of the most problematic concepts of Algebra 1 is "Simplifying". It's so very easy to slip into "here are the rules for what constitutes simplified in this case, and this case, and this case..." (polynomials, exponents, rationals, radicals, ...) and so very difficult to access a big idea like, "We're trying to make our lives easier when we move on to the next step of solving or using a complicated equation that describes some situation in real life like particle physics or the stock market or the operation of an engine or the forces on a bridge." I can say that to students, but I have a hard time showing them or getting them to experience it...

Second, I find that I have some startlingly clear memories of my own Algebra 1 class in eighth grade in 1990, probably because some of the subject matter I learned never got used again. It seems like there's a lot of manipulation and variations on a theme that we teach students but aren't actually critical to their ongoing life as mathematical practicioners.

For example:
  • I've used a lot of math in my life and graphed a lot of linear functions and data. All you really need is a deep and thorough understanding of y = mx + b. You do not need the point-slope form of a line. You might need the standard form when you get around to doing linear functions at a deeper level in Linear Algebra (and all of its applications in physics and economics and ...) in college, but could it wait until then?
  • I've used a lot of math in my life, and while being able to factor out the greatest common factor of a polynomial is a deeply practical skill for all the math it can make easier, and factoring x^2 + bx + c offers a sort of aesthetic pleasure and can be used to ensure that students understand FOIL at a deeper, backwards operation level, no one ever tries to factor ax^2 + bx + c . You pull out the quadratic formula, because it's so much more general and so much less guess and check. Why do we then waste time teaching an essentially guess-and-check approach to solving it?
What's your perception of which parts of the standard Algebra 1 curriculum are actually essential and useful to practicing users of math in this modern, computational age, and which parts are historical cruft?

1 comment:

  1. First, a quote:
    "'When are we ever going to use this in real life?' has been the cry of bored math students since time immemorial. Though the vast majority mightn't a clue what 'real life' entails, it's still a fair question that deserves an answer. Coming up with practical examples of math in real life is a reasonable approach.

    But I would go further. I would deny the very implicit premises that the question is based upon: first, that the only really legitimate knowledge worth having is practical 'real life' knowledge, second that anything which lacks an immediate, direct application is by definition impractical."


    I understand that this doesn't necessary apply to summer school, though.

    I find it useful to be able to factor ax^2 + bx + c in the trivial cases: a^2 + b^2, a^2 + 2ab + b^2, etc. But, I doubt I'd recognize these as trivial without the more extensive exposure I got in school.

    I have less to say on the point-slope form, except that being exposed to different views of the same information seems like a useful way to bring up that there are often multiple ways of approaching a problem. We like to chose the easiest, but we don't know which way will be the easiest unless we know which ways there are, and that requires a passing familiarity with the ways in the first place. Reinventing the wheel is fun once in a while, but giants have shoulders for a reason. :)

    (Real life example: If you're writing a computer program to solve a Sudoku puzzle, there are many ways to approach it. You aren't going to realize it is trivially convertible to a graph coloring problem unless you're familiar with graph theory, though.