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