LOOKING FORWARD TO Making Math Real: Geometry Part I
Part III
Developing the Perceptual Tools for Geometry:
You Can’t See the Geometry Through “Algebra Eyes”
Parts I and II of this series referred to how the developmental/perceptual demands of geometry rely on students’ ability to synthesize a complete whole picture from partially given information. This presents an entirely new perceptual experience for students and requires specific, developmental and incremental structures first helping them to integrate the new perceptual abilities necessary for geometry. We can never assume or expect students to immediately perceive the geometry without support, especially since they have been using an entirely distinct perceptual experience throughout algebra; and without receiving explicit structures from their teachers to help them shift to the perception needed for geometry, students will likely continue attempting to perceive the geometry in the same manner as they perceived algebra.
As referenced in Part I of this series, algebra requires students to generate their working memory/comprehension picture of the math from the interrelationship of the symbolic details, which is mostly a linear, detail-to-detail processing linkage: “detail analysis”. The developmental/perceptual demands of geometry are not linear, and are mostly about synthesis of the big picture. Analysis and synthesis have different brain activations and processing requirements, and therefore, different perceptual demands. If our students start high school geometry without the educator first helping them to integrate the new perceptual abilities necessary for geometry, they may experience significant and unnecessary challenges in their geometry classes, because you can’t see (perceive) the geometry through algebra eyes (perception).
The Perceptual Shift Needed for Geometry May Take Certain Processing Styles Longer to Achieve
Also stated in Part I, the different developmental/perceptual demands of geometry and algebra can be related to different processing styles: synthesizing the 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 perceptual demands of geometry are more related to a big-picture processing style, students who are big-picture processors, may connect to the perceptual demands of geometry more readily than those with linear processing styles, as some linear processing styles may have underdevelopment in synthesizing partially given information into a big picture.
I have received hundreds of comments over the past 40 years that reference this distinction between algebraic and geometric processing. The following are some representative examples: “I just loved geometry because I could see it and it held still on the page, but I could never understand algebra. In fact, I’ve taken algebra four times and I still don’t get it.” As opposed to: “Algebra made so much sense to me, but I never could get what I was supposed to do with geometry, and I just hated doing proofs.” Subsequent investigations into the respective processing styles of the individuals who expressed a strong ability for one, but not the other, generally indicated big-picture processing styles had experienced better success in geometry, while linear processing styles had been more successful in algebra. Possibly, big-picture processing styles may not need as much structured guidance in transitioning from algebra to geometry as linear processors with underdeveloped synthesis abilities. In my experience, students with underdeveloped synthesis abilities may require two to three months (or longer) of structured incremental interventions to help them achieve the perceptual shift necessary for the successful processing of geometry.
I have encountered a few students with extremely rigid linear processing styles on the autism spectrum who have not been developmentally able to achieve the perceptual shift from their linear, analytic processing styles to the perceptual ability to expand certain visual-spatial relationships to their intended big picture. For example, these students may not perceive a three-dimensional drawing of a rectangular prism. To them, the 3-d drawing looks flat, more like a distorted, flat hexagon, and the lines intended to show depth of the prism are instead confusing and perceptually unrelated to length, depth, or height, especially the dotted lines intended to show depth behind the front of the prism. These types of visual-perceptual underdevelopments may also manifest in challenges with reading, making, and interpreting graphs, charts, diagrams, the x-y coordinate plane, the distinction between vertical angles and a linear pair, and numerous other diagrammatic, visual-spatial relationships.
The Perceptual Shift Needed for Geometry is Necessary at Any and All Times, Not Just for Transitioning to the High School Geometry Class
Whether in second grade or in middle school, students need structured support from their teachers to help them become perceptually ready for success with any geometry unit or lesson. Therefore, just because it is slated to be the next lesson, it is not educationally appropriate to jump into a new geometry unit without first preparing students for the necessary perceptual shifts. In the upcoming Making Math Real: Geometry, participants will be provided with the specific, simultaneous multisensory structured incrementation to help students perceptually prepare for the unique cognitive demands of geometry, whether for teaching metric units, how to read the ruler, or how to understand finding the area of a trapezoid, and much, much more.
Success in High School Geometry is Critical for Success in Higher Math
Students’ successful ability to synthesize the big picture from partially given information in high school geometry, in combination with the development of detail analysis from algebra I, provide a critical developmental foundation for their continuing success in higher-level math classes. The increased cognitive demands of algebra II, pre-calculus, and calculus require the developmental ability to simultaneously access analysis and synthesis abilities, frequently within the same problem solving tasks. For these higher-level math classes, students’ preference for algebra (linear processing/analysis) or geometry (big-picture processing/synthesis) is not indicated. Students need equal access to both processing abilities.