Easy or Worthwhile?

I was walking by our copy machine this week, and saw a handout for the same material that I was about to work on in a class.  I took a look, and reacted a bit strongly to what I saw.

The basic idea on the handout was this:

The easy way to solve these equations is to enter one side as Y1, the other side as Y2, and have the calculator find the intersection.

I have to admit that using the calculator can be easy … not as easy as just looking up the answer, but sometimes easier than a human being solving the problem.  The question is:  Is it just easy, but not worthwhile?  Do students gain anything from using a built in program to solve a problem?

I face this issue in our Applications for Living class.  A bit later in the semester, we will talk about medians and then about quartiles.  Students discover that the calculator will find all of that for them.  Should students start to use the calculator to find the quartiles and median right away, to avoid the tedious work of ordering sets of 12 to 20 numbers?

In this statistics example, the material is worthwhile if the student can answer this question easily:

A set of 100 numbers has a median of 40, a lower quartile of 25 and an upper quartile of 70.  How many of those numbers are between 25 and 70?

A basic understanding of quartiles gives a good approximation (50); I’d be thrilled if a student said ‘about 50 but we don’t know for sure’.  In the practice of statistics, technology is always used to find the calculated parameters … and we need to know how to interpret those values.

The content for the handout I saw was ‘solving absolute value equations’, one of my least favorite topics because it tends to be hard to understand while there are a relatively small number of places where this needs to be applied.  However, the understanding of absolute value statements contributes to some common themes in mathematics — multiple representations in general, symmetry in particular.  Technology (as used for an ‘easy way’) avoids all of this stuff that makes it worthwhile.

A focus on the ‘easy way’ presumes that the only purpose for a topic is to get the corresponding correct answers.  To me, a student that uses the calculator to solve |x-5|=7 is just as dependent as a student who uses a calculator for 8 + 5.  The solution is simple enough that it can be done mentally; even writing out all steps gets it done quicker than a calculator process.  If all we do is show students how to obtain correct answers, what is the value that we have added to their education?  If we need to solve |25.8x + 4/3|=8.52, I will certainly tell students … ‘well, we understand how to solve this problem ourselves, so let’s set it up that way — and here is how to check that on a calculator’.  Of course, I know of no place, outside of an algebra textbook, where such a problem would be needed.

Easy is not the primary goal.  Worthwhile learning, and education, are the main things.  Every time we avoid learning we detract from our students’ education.  Technology has a role to play; ‘easy’ does not.  Understanding is a lot more valuable than a hundred correct ‘answers’.

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