Tuesday, October 04, 2005

Why Was Butthead Angry at Numbers?

Welcome to the inaugural math rant...
Butthead: "I'm, like, angry at numbers."
Beavis: "Yeah, there's like, too many of 'em and stuff. Heh heh."
Tangent (to return to the main rant, simply integrate):
In the UK, mathematics is abbreviated as maths; in the US, the preferred term is math. Of course, those stateside who are concerned with "The Three R's" should be made aware of the following:
  • Neither writing nor arithmetic begin with the letter "R"
  • Although arithmetic is an important foundation, the discipline does not even adequately encompass the level of mathematics education that is required for high school graduation.
The Problem:
I have a theory regarding why so many people profess a dislike of math. This theory is based in part on the educational methods employed, and in part on a great deal of social/societal baggage that tends to travel with the subject. You see, in my opinion we've been teaching math incorrectly for several generations, which has created a self-perpetuating cultural and societal bias.
Take, for example an arbitrary parent of a junior high / middle school aged child. In America, the vast majority of such parents will have found math to be "a difficult subject." The parent's recollections of their unpleasant math-related experiences then inadvertently socialize their child to believe that math must be difficult. Said socialization is then reinforced when the parent ceases to have the necessary skills to answer the child's questions regarding the subject. Test scores decline, and the media takes up arms proclaiming that math must be too hard, otherwise our children wouldn't perform so poorly. And the cycle goes on...
Aside, Mark 1:
Do I have any data to support this? No, it is simply a hair-brained theory that I'm posting on a practically unread blog that I write for the fun of it. Did I happen to mention that my robots.txt file doesn't allow any search engines to index anything on my site. Do the math...
The Root Cause?:
We teach math in a manner that places the emphasis on arriving at a solution and then provides a rote, mechanical, formulaic method for transmogrifying the problem into an answer. The vocabulary that we use to describe the process -- working a math problem -- reinforces this fundamentally flawed methodology. The focus on mechanics and procedures removes intuition from the equation. However, intuition is the key to shifting the focus toward understanding the problem. Only by focusing on the actual problem itself and devising a means of approching a solution can we instill understanding, therefore removing the fear and baggage that are often the only remainder of the current system.
Aside, Mark 2:
It's really easy to insert bad math-related puns into a math rant. I'm sure there will be more. Stay tuned. Oh, and no, I don't know exactly how to "grade" a rational thought process that doesn't necessarily converge onto the expected solution. Perhaps that whole concept needs to be explored more carefully...
That's Nice, but So What?, Mark 1:
I haven't quite figured out the answer yet, so I will try to distract you with an example. For this example, you will require the skills expected at the culmination of somewhere in the 3rd-to-5th grade range of arithmetic. However, you will also require an understanding of what numbers, and particularly digits, represent. This example focuses on adding a set of four four-digit numbers. Please play along by putting away your calculator (even the one on your computer), putting down your pencil (or other writing implement), and solving the following in your head: 1234 + 5678 + 9012 + 3456
The Way You Were Taught:
Consider the problem, and apply the method that has been demonstrated to arrive at an answer to the problem. Be sure to mentally visualize everything in nice little columns, so that you'll be able to "carry" any digits as required. Then, start turning the "add like this because it works" crank. The process works something like this (starting from the right-hand side):

  1. Four plus eight is twelve, plus two is fourteen, plus six is twenty. Write down zero and "carry" the two.
  2. The carried two plus three is five, plus seven is twelve, plus one is thirteen, plus five is eighteen. Write the eight to the left of the zero, and "carry" the one.
  3. The carried one plus two is three, plus six is nine, plus zero is still nine, plus four is thirteen. Write the three to the left of the eight and zero, and "carry" the one.
  4. The carried one plus one is two, plus five is seven, plus nine is sixteen, plus three is nineteen. Since we've run out of places to "carry," simply write nineteen to the left of the three, eight, and zero.
  5. Congratulations, you have magically arrived at an answer of 19380. Based soley on this mechanical solution, I defy you to explain why you arrived at that answer or to justify whether it makes sense.
The Way That Makes More Sense:
What are you being asked to do? You are computing a sum of four four-digit numbers. What does a four-digit number represent? Well, any arbitrary sequence of four decimal digits WXYZ represents W thousand, X hundred, etc... So, let's solve the problem intuitively, applying something akin to "iteratively refined estimation, until exhaustion of data." With regard to acquiring the sum of numbers, I call this solving the problem "forwards instead of backwards," and it works something like this (starting from the thousands, and working "bitwise" down to the ones):
  1. One thousand plus five thousand is six thousand, plus nine thousand is fifteen thousand, plus three thousand is eighteen thousand. [Estimate one]
  2. Two hundred plus six hundred is eight hundred, plus nil plus four hundred is twelve hundred. Eighteen thousand plus twelve hundred is actually nineteen thousand two hundred. [Estimate two]
  3. Thirty plus seventy is one hundred, plus ten is one hundred ten, plus fifty is one hundred sixty. Nineteen thousand two hundred plus one hundred sixty is nineteen thousand three hundred sixty. [Estimate three]
  4. Four plus eight is twelve, plus two is fourteen, plus six is twenty. Nineteen thousand three hundred sixty plus twenty is nineteen thousand three hundred eighty, or 19380. [Yes, that is my final answer]
That's Nice, but So What?, Mark 2:
That little exercise is meant to demonstrate that often, the best methodology for solving math problems is exactly the same methodology employed for solving arbitrary problems. First, start with the data that has the highest significance and formulate a first-order estimated solution. Then, apply the additional data, in order of significance, to refine the estimate. Finally, when all of the data have been exhausted, the solution becomes apparent. Does this sound like troubleshooting to you? It should.
Consider another problem, such as "My car won't start." Applying my as-yet-really-poorly-named (I'm open to ideas) "iteratively refined estimation, until exhaustion of data" principle, a mechanic would be faced with a vast plethora of possible solutions to the stated problem. By performing investigations and asking questions, the mechanic gains data that provides additional granularity to the problem. This data presents other avenues for investigation or raises other questions, which in turn provide further granularity. Eventually, when the available data have been exhausted, our competent mechanic will have arrived at a solution to the problem (or at least a small discrete set of potential solutions that can be tested and verified).
Ah, but how does our mechanic become a better troubleshooter? Although the question sounds rhetorical, it is actually easy to answer. The key to successful troubleshooting is rooted in repeated observation and experience. The same is true for any subject of study...
Aside, Mark 3:
Granted, the vocabulary that I have used is heinously raw, the one type of example I provided is the most trivial, and even the concept of working a summation bitwise in order of decreasing significance isn't something that we can immediately go shoving down the throats of our children today. However, what if the whole system was grabbed by its ear and yanked down to the principal's office? What if we built a foundation based on number theory as well as arithmetic? If we're willing to trust that little Johnny and sweet Sally are capable of understanding what numbers signify, perhaps Butthead can learn to forgive them...