Analytical, Numerical, or Graphical?

Despite using technology all of the time, I am a late-adopter when it comes to teaching upper school math.  I was convinced that an analytical approach to problem-solving was best, since it requires critical reasoning skills.  What’s an analytical approach?  It’s the kind you probably learned in school as well: for the equation x^2 = 3x+4, you move everything to the left-hand side, factor, and solve.

And this is great for strong math students, but some people think differently.  For limits on rational expressions of x as x approaches infinity, the best way to convey what happens may be to just use really, really big values of x and plug the expression into a calculator.  This type of approach is called numerical.  For the example above, you can show how the x^2 value gets smaller and then larger again, and how 3x+4 just gets larger.

A graphical solution would involve graphing the left-hand and right-hand expressions and seeing where they cross.  There are downsides, of course; you need to know that your graphing window includes all of the solutions you’re looking for, and while that point looks like (4,16), is it exactly (4,16)?

But while I used to look at the graphing calculators as a crutch to critical thinking skills, I now see it as a useful tool in allowing the student to use numerical and graphical methods to solve problems.  And I feel this is beneficial because each method can strengthen the comprehension of the other methods.  For a student who thinks visually, the use of the graphing calculator may aid in his understanding of why the analytical method works, and so on.  So while I’m a little late in incorporating the graphing calculator into the precalculus curriculum, we’re doing it this year.

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2 Responses to Analytical, Numerical, or Graphical?

  1. lmhenry9 says:

    Interesting thoughts. I have taught for 20 years and I have used a graphing calculator most of the time in my Algebra 2 classes. My emphasis has always been on the analytical and knowing how to graph/do the process by hand, However, you bring up some very valid points – ones I had not thought about before. This year I will have both calculators in my class. I will have to think how to use the graphing calculator in the approaches you mention. Thanks for sharing!
    –Lisa

  2. David Wees says:

    I personally think that students should learn all of the approaches, regardless of their strength in math. Being able to see that different approaches are equivalent is a useful abstraction in math.

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