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 double-edged.
Metaphors can also limit how we think. Keep us from “thinking outside the box.” Constrain our
imagination. Veil our eyes from seeing things are they really are.
The math metaphors that dominated my education in the 1950s and 60s were from manufacturing and masonry. Math was about “laying a good foundation.” Memorization was the mortar that held the bricks of math facts in place. Practice and drill was the mortar mix. Too many “gaps’ in the walls and the structure was weak. It was necessary go back and “fill them in.” The goal was to build up one’s “storehouse” math facts and computational competencies.
This kind of imagery about math made perfect sense to me. After all, that seemed exactly how my math textbooks were laid out. Each succeeding chapter involved problems with slightly more algorithmic complexity, e.g., division of fractions by another fraction, then by a whole numbers, then by a mixed number, then division of mixed numbers by another mixed number.
Much has changed since I took high school math 35 years ago. The world has changed. So too has thinking about math education as the 1989 and 2000 NCTM Standards illustrate. The newer “integrated” math curriculum 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 bricks and mortar structure. Using “living” metaphors has enormous implications for how we think about math education.
For example, one of the hallmarks of living systems is their hierarchical integration of parts into wholes. In turn, lower level wholes become parts to more complex systems. Thus, “integrated” texts strive to conceptually integrate mathematical topics through the use of large organizing ideas, such 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 math topics and procedures. Think of cells organized into tissues, tissues into organs, organs into sub-systems, sub-systems 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, multi-year professional development. We train teachers for 60 hours per year for 3 to 4 years. We also provide 50 hours of in-classroom coaching per teacher. The result: higher test scores for all types of students, and more students alive to math, not dead to it.