Math on the Brain

During the holiday break in December, I was in the company of a teenager who was eager to get outside and enjoy a snow day but for some geometry problems I had committed to review with her.  It was standard school stuff as vacation break homework goes, so we both expected to breeze through it.  Mindful of the axiom that we learn/digest information most thoroughly when we explain it to others, I asked her how she came to each answer.  She sighed at the obvious delay my inquisitive method would cause but settled in to do what must be done. 
I am not one to shy away from math, but this selection of problems were in my least favorite area.  (Pun intended? Not really.)  We proceeded rather quickly through the first few problems, the softballs, where the rule to follow and the algebra required were simple.  But the curve balls came only a few pitches later.  While the questions didn't become more difficult, most of the nearly 100 items were encumbered by multiple layers of work one had to peel through to arrive at the solution. These questions required knowledge of terminology including chord, tangents, secants (the last of which I do not recall ever knowing though I successfully completed calculus in my school days); an understanding of geometric shapes; and problem-solving skills.
As it turned out, this doughty teenager had all the requisite skills but had not applied them to each question and did not always recall her method for solving the problems, so we traversed the bumpy land of geometry together, with her class notes and regents prep website as our guides.
During one of several backtracks to figure out what we had done wrong when our solution was not one of the available choices, I realized that this endeavor extended beyond math skills, this was math, this was literacy, and this was practical application of multiple skills.  We found, as is usually the case when one errs, that one of several of our assumptions and decisions could have veered us off course:

• faulty multiplication - calculators should not be allowed for all casual classwork use; her ultra-reliance on them and my disallowance of them for all simple computations slowed our progress and increased her contempt for manual labor;

• difference in our meanings during our discussions - when we spoke about a shape without drawing it, we often visualized it differently which led to differing approaches; our verbal communication was less effective alone while we traversed mathematical mountains;

• difference in usage of theorums - (Yes, theorums, ugh!) while it was not clear to me whether she had spent much time proving them, she was apt to rely on a theorum so long as it applied to the items involved, even when it did not address the problem we were to solve.  (A chord and a secant mentioned? This must be the right theorum.) She had arrived at many of her incorrect answers using this impratical application method.  I spent some time directing her towards thinking about what matter the theorum addressed so she would choose based on analysis of the information rather than based on keywords or visual cues alone. 

 
Overall, it was an interesting, multi-hour journey through the mathematical dregs of high school that renewed my already ample appreciation for educators who do this everyday with partially attentive students who often misappropriate the shortcuts and keywords given to guide them.  It especially renewed my appreciation for teachers who go to great lengths to help students fully digest mathematical concepts.  I met one middle school teacher who uses dance, song, rhyme, written assignments, and other suggestions and ideas he could implement to facilitate the development in his students of more than rote mathematical skills but complex problem-solving ability using all available resources.  When and wherever you encounter such an educator, be grateful and demonstrate your gratitude.
 

Stoudty is a TEN!!!