Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
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Consider two representations for a moment: "2 + 2 = 4" and "a + b = c." Now imagine an x-axis and y-axis grid with no scale indications.
What are the distinctions? They differ in level of abstraction: Neither the symbolic expressions nor the graph has explicit meaning in the sense that "2 + 2 = 4" does. The graph and the equation are based on conventions that can be learned, and their interpretation depends upon other information and aspects of the problem situation.
In a more general sense, however, "2 + 2 = 4" is also a representation with a level of abstraction that must be mastered. Just as a 6-year-old learns this representation -- how to write and interpret this operation on these values -- a high school student learns how to use the formal representations necessary for more advanced work. Initially, more abstract representations, such as those in algebra classes, may be as elusive to the high school student as "2 + 2 = 4" is to the 6-year-old, but in both cases, understanding the meaning and purpose of the representation will help build understanding.
At any level, one goal for teaching is sufficient understanding of representations so that they can be "read" by a student and connections can be made. For example, a linear equation, such as y = -(2/3)x - 4, can be written in slope-intercept form to easily give the slope of the related graph, while the slope can also be found from the graph or from two points, or by examining the triangles.
Similarly, students with a strong understanding of quadratic equations can look at an equation and determine the orientation of the related graph, the coordinates of its vertex, and its axis of symmetry. And seemingly disconnected representations such as tree diagrams and matrices can be used to find the same solution to the same probability problem. The important point is that each set of representations for a particular problem has a common underlying mathematical structure; for example, as a linear or quadratic relationship.
In general, graphs give a geometric interpretation of a problem and give an indication of the type of growth involved, if any, and a sense of possible asymptotic or tangent behavior. Examining the shapes of graphs also makes it relatively simple to venture an educated guess about the type of function. For example, two seemingly different inverse variation situations both produce hyperbolae.
On the other hand, equations offer the possibility of finding specific values of f(x) such as maximum or minimum values, and, when the user is experienced, give information through various parameters. With graphing technology, specific solutions can also be determined from a graph.
As students become more familiar with relationships between representations, they will develop the habit of looking for underlying structure. They will be less likely to be misled by surface details, as when a graph of time vs. speed is misinterpreted as showing the steepness of the hill during a bike ride.
"We want our students to be able to recognize the geometry of an algebraic situation, or to recognize the algebra of a geometric situation. And when they are able to see the relationships between and among the various representations of a mathematical idea, they're more empowered to become problem solvers." (Henry Kepner and Martha Brown, Connections Standard video)
Watch the video segment (duration 0:44) in the viewer box on the upper left to hear reflections from educators Henry Kepner and Martha Brown.
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