Walking the Mine Field: Misadventures in Mathematics

Doing a complex calculation is not the simple matter than many people perceive it to be. “It’s simple,” I’ve heard repeatedly, “Just memorize the equation. Then it’s just ‘plug-and-chug’.”

In truth, performing calculations requires a great many steps, any one of which can be mistaken (or miss-taken), leading to disaster. It doesn’t matter if we’re doing calculus or statistics; either way, there’s a formula to choose and data to crunch through.

Identify the data from the problem. Not as straight-forward as you’d think; often there is extraneous data running around in there that’s not needed. We also run into charming instances of professors and books using different terms to mean the same thing, or handwriting that makes the same term look like a completely different character. “Is that a sigma or a delta?” Pro mea lingua Graeca est. For added entertainment value, let’s have a hand-written test.

Be able to correctly transform the data into the necessary forms, pre-flight. Are the numbers in the correct units? Do we first need to find means and standard deviations of the 30 values listed in the table? Do we need to flip through 15 pages of lecture notes and a chapter of textbook pages to find or verify the transformational technique? Yes, I have a page of formulae that I’m building. As we slog through the class, this is how I know what to put on the page. Let us hope that I don’t have a transcription error on my formula page!

Be able to correctly transcribe the data, without transpositions, morphing of numerals, or loss of data. This is often where I get into trouble – there’s a sense of spatial meandering as the numbers seem to wander around like ants. Pages with several problems on them make this worse, cluttering the search image. Sometimes I cover over parts of the page with index cards to reduce the visual clutter, but then I have to turn the pages… Double-check the numbers you’ve entered into the calculator before punching Enter. Oops, dropped a digit; re-enter all the numbers again.

Identify which procedure is being called for. Often indirectly stated, especially on tests. Once you know what you need to do, then you need to select which of that mass of massive, messy equations does that trick. It sure would be nice if the professors would spend less time explaining how someone derived the formula – for all I care, it could have arrived fully-formed, like Athena from the forehead of Zeus. Copying down all the steps someone used to transform one formula into another nifty new formula is not helpful to me – it just gives me pages of notes of half-cooked formulae that I need to puzzle through while trying to track down the one I really need. I’d rather they spent more time taking us through a flow-chart of how we determine which formula we need.

Transcribe the formula onto the homework page, without error. Then be able to correctly transcribe the data into the formula format, without transpositions, morphing of numerals, or loss of data. Stop and compare the numbers here with those in the problem. All systems “GO”? Clearance from the control tower?

Be able to remember where in the sequence of functions you are in the procedure. What was a bit awkward in “borrowing” during subtraction, became confusing in long division, and is downright maddening in regression analyses where each problem is a series of computational subsets. (I sure hope this problem doesn’t take more than one side of the page.) Sometimes I put labels alongside the subsets so I know what the pieces are, but sometimes writing ∑xy or s2 on the page only adds more ants. The page is already messy looking from several erasures. Flick rubber crumbs off the table.

Double-check the numbers you’ve entered into the calculator before punching Enter. Got “decimated” – transposed a zero and the decimal point. Re-enter all the numbers again.

Be able to correctly transcribe the correct data, in its proper transformation, without transpositions, morphing of numerals, or loss of data. Yes, we’re stuck in a loop of trying to keep track of a swarm of answers, some of which are raw, some of which are cooked, and it’s not impossible for one to roll off the counter and end up forgotten on the floor.

Next step of the procedure: double-check the numbers you’ve entered into the calculator before punching Enter. So now what do we plug this answer into?

Be able to interpret the significance of the numeric result. So what does “17.2” mean? (Do I care?) Re-read the problem again. Did I use the right formula? Oh, yeah. Write out the answer verbally, because by tomorrow in class this home-work page will have reverted to an unintelligible ant-farm of digits. I really do NOT recall what I did on a math problem from one day to the next.

Congratulations. You have finished the first homework problem. Only fifteen more to go. Um, are we doing problem 56 or 65? Did I get the right answer, or am I practicing doing the problem incorrectly?

In the Final analysis. Of course, in a homework assignment, you know what formula(e) you’re supposed to be using; it’s the one related to that section of the book. Now let’s go to a test, where we’re doing several different kinds of problems.

The test questions written by the professor state the problems differently than the book did, and require using the formulae in different ways than in the homework, to asses our understanding of the concepts. Naturally, this means that the problems on the tests don’t look at all familiar, because they aren’t set up the same way that the problems were on the homework. Before tackling the brute calculations, we have to decipher just what is in front of us. (Where are we going, and what am I doing in this hand-basket full of eraser crumbs and ants?)

“It’s simple,” they tell me, “Just memorize the equation. Then it’s just ‘plug-and-chug’.”

::sigh::

1 Comment

  1. Jacqueline said,

    9 November 2006 at 17:43

    This is why my favorite prof would practically rap our knuckles if we tried any sort of “plug-and-chug” approach to problem solving. Digits be damned, if you what was going on and why, you had succeeded. Of course this meant that most of the rest of the class hated him (expecting fundamental understanding seemed rather harsh to them when they’d been getting by on the old formula memorization methods), but I really appreciated it. I’m not sure how this would work for statistics though, I was studying electrical engineering and never really got friendly with the world of statistics.


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