by F. Joseph Merlino
January 2001
Metaphors can evoke powerful imagery that helps us interpret and organize information. They help us decide what is important, what to attend to, and what to ignore. They establish boundaries for how we think. But boundaries are doubleedged. Metaphors can also limit how we think, can prevent us from "thinking outside the box," can constrain our imaginations, and can veil our eyes from seeing things as they really are.
The mathematics metaphors that dominated my education in the 1950s and 1960s were from manufacturing and masonry. Mathematics was about "laying a good foundation." Memorization was the mortar that held the bricks of mathematics facts in place. Practice and drill was the mortar mix. If the walls had too many gaps, the structure would be weak. Going back to "fill in the gaps" was necessary. The goal was to build up one's "storehouse" of mathematics facts and computational competences.
This kind of imagery about mathematics made perfect sense to me. After all, that seemed to be exactly how my mathematics textbooks were laid out. Each succeeding chapter involved problems with slightly more algorithmic complexity, for example, division of fractions by another fraction, then by a whole number, then by a mixed number, then division of mixed numbers by another mixed number.
Much has changed since I took high school mathematics 35 years ago. The world has changed. So too has thinking about mathematics education, as evidenced by such publications of the National Council of Teachers of Mathematics as Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: NCTM, 1989) and Principles and Standards for School Mathematics (Reston, Va.: NCTM, 2000). The newer integrated mathematics curricula and pedagogy developed during the 1990s with funding from the National Science Foundation are based on learning theory that views human cognition as essentially a biological process rather than a manufacturing one. These integrated curricula can be better understood if we view the act of thinking as a living system, like a growing tree, rather than like a structure made of bricks and mortar. Using "living" metaphors has enormous implications for how we think about mathematics education.
For example, one hallmark of living systems is their hierarchical integration of parts into wholes. In turn, lowerlevel wholes become parts to more complex systems. Thus, "integrated" textbooks strive to integrate mathematical topics conceptually through using such large organizing ideas as function and variable. These concept "arcs" organize daily lessons into large unit ideas. These large organizing ideas are then repeated across the grade levels, serving to bundle units together into "strands," thus providing conceptual coherence for students to otherwise disjointed and juxtaposed mathematics topics and procedures. Think of cells organized into tissues, tissues into organs, organs into subsystems, and subsystems into a body. Brick walls do not have such hierarchical integration. The brick on top looks just like the brick at the base.
Properly teaching an integrated curriculum requires sustained, intensive, multiyear professional development. In our project, we train teachers for 60 hours per year for three to four years. We also provide 50 hours of inclassroom coaching per teacher. The result: higher test scores for all types of students, and more students alive to mathematics, not dead to it.
F. Joseph Merlino is the project director of the Greater Philadelphia Secondary Mathematics Project at La Salle University. He and his staff are training more than 1000 teachers in various integrated mathematics programs in the middle and high schools. 
