For those of us who are old enough to remember classrooms with walls, the methods we used to learn math were teacher-centered and method-based. Those who came of age before cooperative learning became prevalent in schools probably remember learning one method of solving problems and some of us may have felt the sting when we could not understand or even use that method at all.
My own encounter with The Mathematical Wall came in the second grade, when my class was learning to borrow in subtraction. I could not get the hang of it, much less make sense of the concept of borrowing anything from a number. I had only borrowed things from my older brother and it simply made no sense to me to cross off numbers in what seemed like a game of arithmetic Three-Card Monte. Panic set in and I thought I would never get past the second grade into the third. Although I was short, I envisioned a life of sitting cramped at a too-small desk until such time as those odd chicken-scratches made some sense.
Fortunately, before permanent depression set in and just in time to save me from imagining a lifetime consigned to the most elementary of work, my mother showed me an alternative method using dots instead of cross-offs and additions instead of depriving those poor numbers of their values. We both cautiously approached the math teacher and asked her if we could use this method, which made infinitely more sense to me than her game of arithmetic monopoly, She said - with no hesitation - yes. To this day I use that method, which has proven to be neater, faster and more accurate for me than the one I had been expected to learn.
Math tutors encounter children who have experienced both an abundance of traditional uniform instructional techniques and the more free-wheeling individual-based experiential constructivism. Those who learn algorithms for multiplication and division by visual means only and without a feel for the movement of the quantities throughout the multiplication or division process are likely to be the ones who forget how to proceed over summer vacations.
Without an experientially based understanding of the process and without having the process concretized and made "solid," they are learning with only a visual memory if at all. Visual memory alone just does not do it for most of us. And yet, when math students are left without substantive guidance to "discover" truths and methods on their own, confusion, uncertainty and inaccuracy prevail. When it becomes more time-consuming to discover, prove and assess truths than it does to reason the logic of truths, we are engaged in a counterproductive exercise.
The general principle of constructivism, a stream of learning theory used in many classroom environments, is that we all learn differently and we all construct our knowledge bases differently. While it may have been fine for many in my class to learn to "borrow" in subtraction as a mechanical process, I needed to understand why that process worked in order to be comfortable with it. The idea that "all learning is experiential" is key to constructivism and it manifests itself in the math classroom in the inquiry-directed activity.
Does your child's teacher use a base of constructivism for classroom learning activities? Ask your child if s/he works in groups more than in teacher-based classroom instruction. Ask your child if s/he is asked to write procedures and definitions in math topics. Ask your child if s/he knows how to perform basic operations with whole numbers, fractions and decimals. If your child's answers indicate that groups prevail and procedures and definitions take a back seat to project-based cooperative inquiry, your child may be experiencing the constructivist method.
Under the best of circumstances, our teachers are versed in more than one method and they are blessed with the judgment to discriminate as to when to use which method. However, we are creatures of habit and large bureaucracies may dictate the method used. Parents owe it to themselves and their children to determine which method is the foundation the teacher uses for instruction.
One might suppose that the world lost a highly competent "jill-of-all-trades" by the flexibility shown by my teacher, and yet I still do flip burgers and mop floors occasionally. The only difference between what might have been and what my reality is, is that now I can flip burgers and mop floors with a mathematical understanding of the notion of how gravity affects those burgers in their earthward descent and how my dirty floors re-affected by the application of water. I never could have done it without my second-grade teacher's wisdom.
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