The first installment in an ongoing series of articles to help prepare for the new prospective course, Making Math Real: Geometry
LOOKING FORWARD TO MAKING MATH REAL: GEOMETRY
Geometry is Different from Algebra and Elementary Mathematics
Geometry is a vastly different developmental experience than either elementary school mathematics or algebra. From a developmental perspective, the major difference distinguishing geometry from other math is its perceptual demand: synthesizing a complete big picture from given partial information. This is much like a visual or auditory closure task in which part of a picture is provided and the student must visualize the complete whole of the actual picture or listen to a word with some of its phonemes left out, and the student must synthesize what the actual whole word is.
Geometry is therefore a multisensory closure task requiring visual, auditory, and kinesthetic closure to synthesize the complete whole from partially given information. The developmental/perceptual demand of algebra is entirely different. The developmental/perceptual demand of algebra requires students to generate the working memory/comprehension picture of the math from the interrelationship of the symbolic details. In algebra, how do we know in looking at equations for parabolas whether it opens upward or downward? How do we know in looking at polynomial products whether to factor them using x² + Bx + C, Difference of Squares, Perfect Squares, Greatest Common Factor, or Ax² + Bx + C? It is the interrelationship of the symbolic details that activates the appropriate comprehension picture: in the case of equations for parabolas, the symbolic detail that determines how the parabola opens is whether the lead coefficient is positive or negative; and for factoring polynomials, there are many different detail relationships that determine the appropriate factoring style such as the number of terms, the values of lead coefficients, the values of first and last coefficients, etc. Geometry is also different than elementary mathematics. Elementary math provides the fundamentals of all future math development and as such, its developmental/perceptual demands are basic. In elementary math the developmental/perceptual focus is for students to generate the working memory/comprehension picture entirely from the symbols. The symbols in elementary math are symbolic codes that express real and concrete relationships. Students learn to decode and encode these symbols to connect that the concrete, real math and the symbols are identical: they are both making the same picture and telling the same story.
For elementary mathematics, students get their entire working memory/comprehension picture directly from the symbols. This is why symbol imaging is the primary sensory-cognitive development supporting the successful decoding and encoding of the elementary math codes. In algebra, in which students get their working memory/comprehension picture from the interrelationship of the symbolic details, the major sensory-cognitive development for successful algebraic processing is detail analysis. Geometry is about synthesis, and therefore requires the synthesis of both symbol imaging and detail analysis in support of students’ ability to generate the complete whole picture from partially given information.
The Different Developmental/Perceptual Demands of Geometry and Algebra are Related to Different Processing Styles
The different developmental/perceptual demands of geometry and algebra can be related to different processing styles: synthesizing the complete whole picture from partial information in geometry is more closely aligned with a big-picture processing style, while the developmental perceptual demand of algebra in which the comprehension picture is determined by the interrelationship of the symbolic details is more closely aligned with a linear processing style. As the cognitive demands of higher math increase in algebra II/trigonometry, pre-calculus, and calculus, students need equal access to both big-picture and linear processing capabilities to be successful. In this manner, the K-12 wavelength of cognitive development necessary for successful processing in mathematics is a clearly sequential progression of symbol imaging in the elementary grades, detail analysis in the pre-algebra/algebra grades and an ongoing synthesis and deepening of the two starting with geometry and continuing through calculus.
The next installment is this series, LOOKING FORWARD TO MAKING MATH REAL: GEOMETRY PART II: Success with Deductive Proof: Synthesizing the Perceptual Big Picture
First Published 2/23/16