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A TOOL FOR INQUIRY:
Most individuals, no matter their age, see the world through
experienced eyes and offer explanations for the quantitative relationships
they observe. These interpretations and models make sense; they are the
result of years of observing and sense making beyond the context of formal
schooling. Our students bring mathematical "private universes",
to echo the elegant image from the Annenberg/Harvard-Smithsonian film,
into our classes. For most students arithmetic has been adequate to
describe quantity to date. Why change to an algebraic view that may appear
very confusing. "What is that 'X'?" The shift from arithmetical
understanding to algebraic understanding is difficult for many. The common
sense pedagogical strategy of building models or classroom experiences that
challenge naive views should serve as a starting point. Base Ten Blocks
modeled the number system. Other physical objects can adequately model
algebraic relationships for understanding symbols and functions. Using
elegantly designed microworlds we begin to learn and teach effectively the
elements of algebraic thinking. Students need to break conceptual barriers
as in the example from the Renaissance drawing at the left, Empedocles
Breaks through the Crystal Spheres. The National Council of Teachers
of Mathematics challenges teachers to see that all students attain an
understanding of variable, symbol, and functional relationships. How can
we meet that challenge?
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In a world that does not use the language of algebra outside a math
classroom, how does one build understanding of algebraic ideas? It is
frequently the
case that natural language impedes the development of understanding in
mathematics. The power of ordinary language is strong: our everyday words and
descriptions adequately describe the world around us as depicted in the
"ordinary" view in a segment of the drawing to the right. "Sun", "stars",
"air", "sunlight", "earth", "plant" are all full of meanings and associations,
many of which are not at all "mathematical", other than how many are there?
Where is the algebra in this image? How can we support learning a language
that seemingly has no physical correlate? |
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The answer lies in engagement in activities and laboratory
experiences that challenge us to pose problems, design ways to develop
answers to problems, challenge old meanings, develop new ones, and persuade
our peers of the validity and utility of solutions. Algebraic ideas and
language need to be represented in multiple ways, through symbols, physical
phenomena, graphs, and dynamic situations. Another essential component if
students are to develop conceptual understanding of algebra is extended,
quality reflection, both individually and communally, on their
experiences. When and where do variables make sense? Do parts of equations
have different meanings? In situations where educators must teach a set
curriculum, extended hands-on investigation is very difficult to sustain or
justify. This module presents a set of activities designed to bridge
arithmetic understanding and an understanding of quantity based on algebra.
The tools are used and abandoned once a rationale and facility with
algebraic process is attained.
The Algebra module uses the LAB GEAR tools developed by Henry Picciotto and Anita Wah. One activity uses Pattern Blocks and a video from Ricky Carter and Fadia Harik of BBN Learning Systems and Technologies. These activities have been tested in hundreds of classrooms. They are designed to facilitate collaborative investigations; to integrate experience from hands on studies; and to provide extended opportunities for inquiry, reflection and discussion that lead students to conceptual understanding. The materials may also serve as the basis of district workshops on transitional algebra for non-INTEC participants.
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