# 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.

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# Success with Deductive Proof: Synthesizing the Perceptual Big Picture

Although the ability to perceptually synthesize the complete whole picture from partially given information is needed in all geometry, it is most particularly required for success with deductive proof. With deductive proof, some partial information is provided from which the student must prove a specified relationship such as congruence or similarity. To successfully structure a deductive proof, students must be able to generate the complete whole picture of the proof from the partially given information prior to picking up their pencil. In the correct incremental teaching of geometry, it is necessary to help the student synthesize the complete big picture from the partially given information, then mentally hold that synthesized perceptual picture as the basis from which to structure each step of the deductive proof. This means the student has already “proven” the relationship prior to writing anything down. It is in this manner students can successfully manage the challenges of structuring and organizing deductive proofs. The students’ ability to synthesize the big picture from a partial picture requires significant developmental practice and experience, and is therefore not indicated until well into the high school year, and not immediately at the beginning of the year as it is typically presented in textbooks. Presenting deductive proof prior to students’ development of synthesizing the perceptual big picture can make it unnecessarily difficult for students to be successful with deductive proof, which is most unfortunate since the developmental outcome of successful proof is directly related to significant increases in executive processes such as organization, structure, sustaining focus, sequential processing, cause and effect, inhibiting distractors, and mental flexibility.

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# Looking Forward to Making Math Real: Geometry Part I

The first installment in an ongoing series of articles to help prepare for the new prospective course, Making Math Real: Geometry Part I

# LOOKING FORWARD TO MAKING MATH REAL: GEOMETRY PART I

## PART I

### 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

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# IF YOU BUILD IT, WE WILL COME

Over the years so many of you have fervently requested a particular MMR course only to hear that particular course will not be offered for another year or two. We understand how difficult it is to wait, but we can only offer a course once we know we have sufficient enrollment.

Double Bonus Incentive

Therefore, the Making Math Real Institute is going to try an experiment: “If You Build It, We Will Come.” This means if we can get sufficient enrollment several months prior to the course offering, we will put that course on the calendar far in advance of when it would normally be scheduled. As a double bonus incentive, you can have the course you want much sooner AND receive a 15% discount on the price.

Pre-Algebra & Algebra

For example, so many of you have requested Pre-Algebra this summer, which would be great because it is not scheduled again until Spring 2018. However, to make that happen we need you to “Build It” by taking advantage of the 15% discount and enrolling in July 2016 Pre-Algebra course by May 20, 2016. If this experiment works, then we will do it again to offer Algebra in July 2017.

Triple Bonus for Out of Town Participants

If we can offer Pre-Algebra and Algebra during the summers, this will be the first time we have offered these courses during the summer, making it much more accessible for participants from out of town, state, or the country to attend. Please help us to help you. Fulfill your requests for the MMR courses you want and need. Register now for Making Math Real: Pre-Algebra.

First Published 2/9/16

# Introducing a Unique Opportunity: Making Math Real: Geometry

INTRODUCING A UNIQUE OPPORTUNITY:

## Making Math Real: Geometry

March 11-12, 25-26, April 8-9, 22-23, May 6-7, 2017; Berkeley, CA

I am extremely excited to announce the prospect of offering Making Math Real: Geometry in Spring of 2017. This would mark the first time in the 20 year history of the Making Math Real Institute that Geometry would be offered publicly as a part of the Making Math Real series of courses.

Making it Happen

I am excited about the possibility of including Making Math Real: Geometry as a part of the MMR series of seminars to show my commitment to those of you who have been so committed to Making Math Real. I have never been able to offer the Geometry course because I have not been able to get sufficient enrollment to make the course a viable addition to the MMR series – until now – maybe. A number of highly committed, long-term MMR veterans asked what it would take for me to offer the Geometry class. I told them I would need a minimum of 20 participants who were so committed to attending this course, they would be willing to pre-pay for the course one year in advance. This same group has taken it upon themselves to organize and network to bring together so far a group of 17 people who are good to go. Therefore, if three more MMR veterans also want Making Math Real: Geometry as much as these 17 by May 1, 2016, I will officially schedule the course for Spring of 2017.

What it Takes

The following are the requirements for Making Math Real: Geometry:

1. Successful completion of Making Math Real: Fractions, Decimals, and Advanced Place Value; Making Math Real: Pre-Algebra; and Making Math Real: Algebra.*
2. 50% non-refundable deposit (\$874.50) towards the \$1,749** cost of this 10-day seminar (all deposits will be returned if the course is either not offered and/or cancelled) received by May 1, 2016; and remaining 50% of enrollment received by January 2017.

* For this first offering of Making Math Real: Geometry only. Future offerings will require Making Math Real: The 4 Operations and the 400 Math Facts, Making Math Real: The 9 Lines Intensive, and Making Math Real: Kindergarten.
** There are no discounts offered with this course.

What’s in Store?

Making Math Real: Geometry is a synthesis course and will cover content spanning second grade through high school geometry including standard and metric units for length, weight, capacity, measurement for standard and metric rulers, conversions for standard and metric units, polygons and polyhedra, area, perimeter, surface area, and volume; unit analysis, coordinate geometry, using the protractor, fundamentals of geometry: points, lines, and planes; transformations: reflections, rotations, translations, and dilations, congruence and similarity, right triangles and the Pythagorean Theorem, trigonometry, properties of circles, arcs and sectors; area of regular polygons, and deductive proof.

Algebra in Spring 2016

Offering Making Math Real: Geometry as soon as Spring 2017 is only possible because I am scheduling Making Math Real: Algebra in Spring 2016. If you need Making Math Real: Algebra to complete the prerequisite requirements for Geometry, this upcoming Algebra class will be the last opportunity as the next Algebra class will be scheduled after the 2017 Geometry class. This is a most unique opportunity for you. Don’t miss out. Stay tuned for more information about the structure and design of the Making Math Real: Geometry course and updates in the count of reaching our goal of at least 20 participants by May 1, 2016 to make the course official.

First Published 1/28/16