"Let's view the process of factoring of polynomials with a new representational toolãalgebra as seen through quantity," I suggested. "These methods treat algebra as a language of symbols."

In recent research an understanding of quantity has been shown to be key in conceptualizing algebraic relationships (Early Algebra, James Kaput, ed.). These authors hold that multiple representation of a process such as factoring can be a bridge to building conceptual understanding. The language of "quantity" and "multiple representation" carried little meaning for my audience. Something concrete was needed.

"Consider the polynomial X³-1. Who can factor it?" I said.

I asked, "Do your students believe this, or do they just do it?"

Not much challenge there. An algebra teacher confidently took some chalk and wrote X³-1=(X-1)(X² + X + 1).

"How do you know that these expressions are the factors?"

She responded, "You just multiply them. I teach them that multiplying polynomials is like a dance. Each term must partner up with every other. We call this the distributive law also." Her voice was assured, her method well practiced. "Each term matches every other." She continued the "dance" of the matching terms. Negatives and positives canceled to reveal X³-1.

The dance metaphor was popular. I had encountered it on many other occasions within the same context. The dance was a bridge to procedural competence, not conceptual understanding. The teachers realized this.

Everyone nodded knowingly, and offered a resigned assent: the students know the procedure but not why it is true.

"What other ways can you show that these terms are the factors of X³-1?"

Someone responded that she sometimes used manipulatives like Algeblocks but they did not seem to help here.

I asked the group to explore the meaning of the expression X³-1=(X-1)(X² + X + 1) using the Multi-link cubes. We explored the expression for X=3. Can you build a representation of X³-1 when X=3? We stared at the blocks. There was a clatter of plastic, and pairs began their constructions. X³-1 could be viewed as a cube with one block removed.

"What does a 'factor' mean?" I asked.

There were many responses. "Factors are parts of numbers." "They are related to addition, or rather addition a multiple number of times." In this case one of the factors is X² + X + 1.

"In what sense is X² + X + 1 part of X³-1?" I asked.

"You should be able to take X² + X + 1 out of X³-1," one teacher responded.

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