Saturday, March 12, 2011
From my child's point of view, every time that I asked for help, it yielded extra work for me. There was no point in asking for help if it meant extra work.
Once I was in the second grade, early in the year, and I was learning how to add two-digit numbers. My worksheet had a page of problems, but there was no symbol to indicate what I was to do with them. Even though the instructions tersely stated, "Add," I was confused why there was no "+" symbol. It didn't make any sense. We had never seen problems like this before. In addition, I was not clear on the concept of carrying units from the ones to the tens, and I often miscounted in the first place, counting one of the numbers itself, as in "11+12" to mean I would count on my fingers or number line: "11, 12, 13, 14, 15...22"...or is is "12, 13, 14...23?" I never did get the idea right in Kindergarten, so I kept repeating the same problem with everything I added.
In any case, the combination of confusing factors led me to ask for help. There was one too many items that had thrown off my orientation, and I needed assurance. Mistakenly, i went to a parent: my dad. My dad is an engineer by training, and he is very strong in math such that he estimates large and irregular numbers whenever we are roughly calculating numbers, percentages, or rates. It's almost like a magic trick to me. Poof! And he can come up with something akin to a calculator response. My daughter and oldest nephew love to test him versus the calculator to see how close he can get to accuracy with multi-step problems or very large numbers. Especially in today's computer age when computations can be quick and accurate, the human brain is truly spectacular at its ability to challenge something that appears automatic and mechanical.
But being his daughter, I did not really know that asking for math help should be out of the ordinary. Maybe other dads did this sort of mathemagical thing. Maybe other kids needed math help. Maybe he was the right person to ask, as all adults appeared to be. I was wrong. First, my dad explained that it was always assumed to be addition when a mathematical sign was absent because that was the default. Other symbols were always written in. This baffled me because it seemed illogical. Why wouldn't the workbook authors simply take the time to include the symbol and stop confusing me? Textbooks always included the symbol. It was just a law of nature. I was still looking quizzical when he went on to address my second issue: carrying over to the tens' column. I was not strong on the concept, so his quick explanation escaped me as I was probably still mulling over the issue of including or not including the math symbol for the function. His tone turned serious when he realized I was lost, and he buckled down to see if I was needing an explanation of the concept. This is where things turned very hairy.
My dad likes concepts. And he liked to make sure I understood them--conceptually. I was not a conceptual math thinker. I was a reproducer. I could reproduce models over and over, but they had to be the same way. By doing this many times, I finally figured out patterns and, at some point much further down the pipeline--probably much further along than was truly acceptable by math teacher standards--I "got" the concept. It was a backwards approach for me, but I was willing to take math on faith. No proof was needed. And if proof were eventually given, I was even more pleased, for I could now apply this idea in many ways and in new contexts unlike my rote approach prior to my gestalt. However, this would take years for me to realize, and I was happy at the time to simply accept math wisdom as it was handed down through my workbooks.
My father, however, was not. He realized how inflexible my thinking was and that this would not serve when thinking about the nature of a problem. Problems needed to be understood on a conceptual level or the answers could not be scrutinized as appropriate or out-of-the-ballpark. The test of reason always had to be applied to a solution. This was starting to look like a very involved math tutoring session. I was remorseful for ever having asked. When I struggled through the problems I was given, trying to use this "conceptual" approach, he harumphed in his dissatisfied way and proceeded to write out a page of math problems for me of his own. He used a thin, almost college-rule pad of paper with a greenish tint on which he wrote his "business" work. He was just rolling up his sleeves. After I finished the page to the lowest level of acceptable quality, I hoped I was out of the woods though I knew in my heart of hearts, there must be more. But what?
He then asked about my subtraction abilities. All I knew was that I was even worse in subtraction. He wrote out a page of those, and he included a number of two-digit subtraction problems. I had not seen those yet, but he was not concerned. "Conceptually," they were the reverse of the addition. The concept should apply regardless. I was lost. I had no idea what he was talking about. I had no idea how to solve them. My teacher must have believed I could not for she had not even taught me this yet. I was sunk. I knew I could not get out of these exercises until I could prove I "got" them, but that enlightenment was so far off as to be statistically impossible. I do not remember how it ended or that it ended at all, but I do remember that I only asked for math help twice after that: once in middle school and once in high school. They were moments of clear desperation, and I vowed to avoid my dad as a possible source of help for anything in the future, especially if it were quantitative.
The final time that I asked for help, I remember I was stuck one of two challenge problems that my teacher had given me. I don't know if my teacher thought that students would actually solve each of the problems assigned. Maybe an attempt was decent. For me, homework was homework, and it must be completed on time. So I worked and worked, but I could not solve this final problem of the night. Realizing that I might never solve it, and realizing that my only source of help might be retiring to bed soon, I went to him and asked. Having my few prior experiences with him over math homework, I was trying my best to make clear what I needed: this was the Book's way of solving these types of problems; these were all of the Book's examples. I did not want any deviations from the Book. I liked the Book. But this problem was a variation that was not handled in the examples, so I needed some guidance about how I was misapplying these approaches so that I could then apply the Tried-and-True Approaches to find a solution.
My dad must not have heard a single thing I said. Immediately, he began tackling the problem using an approach that was completely dissimilar to anything I had learned to this point. I had no idea what he was doing. His first attempt failed. He was puzzled. His second attempt came to nothing. He was most intrigued by this math problem. I was thinking, "Holy crap! What kind of math is this book covering that my dad can't even solve it?...and that I'm supposed to!?!?" Finally he arrived at a solution. He had to explain it to me about three times. The first time, I simply lamented that nothing he said matched up with the Book's explanations. There was nothing I could hang onto. The second time, I realized he had to rely on other mathematical truths about arcs, diameters, radii, lines, and angles to put together a few steps to solve for the unknown. The third time, I was able to understand what he had done so that I could explain how I had cheated on this final problem the next day in class.
When I got to class, we went over the problem set and finally, we arrived at the two challenge problems. For the last one, the teacher asked if anyone had gotten the solution. No one raised a hand. After a few seconds, one of the smartest math students in class raised her hand. She explained her answer (which was also different from the Book's), and the teacher was impressed. We all continued to bow down on the ground she walked on. After no one else came forward, I admitted that my dad, not I, had solved the problem, but that he had taken a different approach than either the Book's or the student's. I explained how he had calculated for different pieces to set up a problem that he could solve for. I was embarrassed and felt cheap.Everyone in class oohed and aahhed over the thinking behind it. My teacher was amazed. She was stunned by the cleverness of the approach. In the end, it was quite elegant and simple.
Elegant? Simple? Clever? Amazing? Was this how other people viewed my father? I was shocked to think that these words might apply to My Dad. My Dad?! Wow, this was a different view than I had ever taken to his way of teaching math. For the rest of the day, I pondered this awe about my dad. I couldn't wait until he came home for dinner late that night, and I told him how his approach had been received by my classmates and teacher. He smiled happily with his hands folded across his tummy, saying nothing. Though my resolve continued to be that I not ask for help unless critically needed, I was able to enjoy the moment, that I felt proud that he felt appreciated.