Wednesday, February 27, 2013

Are you smarter than a calculator?

Lately, I have been noticing how dumb our calculators are. It's become kind of a running theme in my classes, where I've been teaching how to use different graphing tools, and I say several times a week "and remember, you have to be smarter than your calculator" or "you have to help your calculator..."

The calculator doesn't think like humans. God created us--not calculators--in his image, and I believe one aspect of that is the ability we have to reason, to notice patterns, to create, to organize, to see. These are all things that calculators, and in general computers or machines, are all pretty bad at. They are improving, because our minds are helping to generate better and better machinery, but they still don't work like humans.

Perhaps the best example of this is the Captcha messages at the bottom of so many websites. Computers and bots are still horrible at "seeing" things. Most humans can interpret those pictures and type letters or numbers properly, but that relatively simple operation is difficult for most computers. That's because we think about the problems entirely differently.

Likewise, our calculators think about calculations entirely differently than us. We can think algebraically and manipulate symbols, variables, and even numbers in symbolic ways that allows us to simplify problems, or calculate values exactly. Most calculators don't think in that way at all, but are programmed with different algorithms that work with really precise approximations and quick calculations. Even the slowest earliest calculators can do this sort of thing faster than all but the freakest of humans -- but I haven't seen any calculators that are good at playing What's the Word.

Here's a bullet list of items that I've noticed lately:
  • In PreCalc we've been studying complex numbers. One assignment the other day was to calculate i17 which is easy to calculate by recognizing a pattern.  i, i5, i9, i13, and i17 are all equal to the purely imaginary number i, but the calculator spit out -1E-13 + i.  Yes, the -1E-13 is a ridiculously small number, close to zero, but it shouldn't be there AT ALL! What strange algorithm does the calculator use to calculate that instead of just recognizing the pattern like humans?
  • Similarly, some versions of the calculator were not able to convert some of our operations involving complex numbers into exact fraction form -- where as we could. Some calculated approximations (admittedly better approximations than we could find in anything short of five/ten minutes) but several others gave an ERR: data type message instead
  • In Algebra 2, we gave been using the calculator to calculate summations, and the notation for summations is sum(seq(function,VAR,start,end)) and we have been laughing at the fact that even though our functions only have one letter in them, we still need to write that variable again. I understand you could certainly have many variables in a function and then you'd have to specify which one is the index -- but when there's only one, you'd think the calculator would be able to figure that out.
  • In Algebra 1 we've been graphing systems of equations, and numerous stupid calculator quirks have popped up. Though we set the word problems up with sensible variables like N for the number of nickels and D for the number of dimes, when we went to graph things, we had to use the letters X and Y. Again, you could maybe give your calculator the benefit of the doubt because maybe those letters are going to be used for constants (like I do in physics storing 6.67E-11 in for G) but...
  • Then we try to calculate the intersection of two lines and we have to tell it which lines we're interested in and help guide it towards the solution. Seriously?! There's only two lines on the screen! And they're lines! Not curves!  
  • If the intersection isn't on the visible window screen, the calculator won't be able to find it for you -- you need to realize that those lines will intersect above, left, right, etc. of the screen and adjust the window yourselves.
  • And what's with providing the answer as 1.999946 when it's clearly and exactly 2?  The algorithm that calculates the intersection necessarily has limits to its precision, and sometimes those fall short. My students better not ever report an answer of x=1.999946 to me.
  • To be fair, let's pick on non-TI84 calculators -- one of my newfound favorites is the app MyScript Calculator which interprets my handwriting and calculates things for me.  I'll admit, I played with it for a good 30 minutes after downloading it -- only true math nerds play with their calculators right?  But I noticed pretty quickly that it's trig values didn't always calculate properly, which was a bug that their updated version supposedly has fixed. I knew that because I knew the limits of sine and cosine values, and even had several memorized -- and also can estimate relatively well and had ideas of what the answers should be ahead of time.

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