Metaphor for quelling the growth of mathematical misunderstandings:

Last summer, I had more time for yard work than the previous years and decided that I would finally tackle the burdensome weeds propagating my landscape. The first morning out, with a small shovel and trowel in hand, I began in the farthest southwest corner of my lot. I worked steadily across the area to rid my yard of the multiple Creeping Charlies, Purple dead nettles, and Shepherds purse dandelions that had made my backyard as popular as cholesterol at a hot dog stand. After a long morning and part of the afternoon, I gave up in exhaustion and went inside to get some water.

Upon my return to the outdoor sun, I paused to check my progress. Surveying my results from my back deck, nearly half of the backyard was untouched and still weed infested. But a greater surprise, much to my chagrin, were the weeds I now noticed that were still swaying in the breeze, and mocking me within the areas I had so painstakingly worked. My callused hands, sweat-filled shirt, and aching back confirmed that I had been working hard at the task. It seemed implausible that there was still so much left undone, and still more to do within the places I focused my energies?

I ended up spending several more Saturday mornings in similar fashion last summer, with improvement along the way, but always more to be done. During my sweat-filled drudgery out  in the hot sun, it struck me of the hopeful, yet supercilious expectations we often have to eradicate all mathematical errors from our classrooms. Why do we expect a simple lesson or explanation will quell the misunderstanding that propagate our mathematics classrooms? Are we so foolish, as was I, to believe that one diligent pass of rooting out the collective misunderstandings will be sufficient to eradicate the confusion with our learners?

This connection became clearer in my mind when I listened to the weekend garden show on NPR. This particular show interviewed an environmental studies professor, Dr. Nancy Gift,  who is a renowned weed expert. She stated that even among the experts, there is disagreement and ambiguity as to what actually constitutes a weed. Because often, weeds are simply "botanical growths out of place", and that trying to eradicate them was a futile as trying to eradicate the bacteria in the world. Rather we need to spend our energies mitigating the worst of the bunch, and trying to find the right place for the others.

 Often, students' mathematical misconceptions run the gamut from malignant to benign, just as the weeds do. And while we as teachers do need to be diligent in our effort to root out the nastiest ones, and replace them with what we know to be better growth, we should not expect that our daily efforts this summer (or next fall) will preclude any mathematical weeds from returning again next year. They will be back, and they'll try to find their place. It is our job  as educated gardeners to get back on our hands and knees, and manage them appropriately. For in doing so, we never should expect to eliminate them entirely, but we can create the space for the seedlings and flowering plants that we hope to nurture and grow.