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The 4 Operations and The 400 Math Facts: The Essential Building Blocks for All Future Mathematics

The 4 Operations and The 400 Math Facts:
The Essential Building Blocks for All Future Mathematics

The Four Operations Through Fractions And Decimals
Directly Connect Elementary Math To Algebra And Beyond

The principal foundation for students to be ready for pre-algebra is the full concept-procedure integration of the four operations through fractions and decimals and automaticity with the 400 math facts. Why do students need the four operations through fractions and decimals to be ready for pre-algebra and beyond? Because the major fundamental for elementary grades mathematics is addition, subtraction, multiplication, and division through fractions and decimals when we know how much we have; and the major fundamental for algebra is also addition, subtraction, multiplication, and division through fractions and decimals, but in algebra, we may not know how much we have, because in algebra, there are variable expressions and variable equations.

All students need the complete concept-procedure integration of the four operations through fractions and decimals because much of the mathematical structure of algebra IS based on the four operations through fractions and decimals: the order of operations to simplify variable expressions and the inverse order of operations to solve variable equations.

To cover this voluminous and essential foundation in elementary grades mathematics, I have created two seminars: Making Math Real: The 4 Operations and the 400 Math Facts (4 Ops) and Making Math Real: Fractions, Decimals, and Advanced Place Value (FDAP). At 12 days each, these are the two longest seminars the Making Math Real Institute offers, and since they provide the critical foundation for all future mathematics, they are the only two large-scale (seminars of 10 or more days) institute courses offered once per year, every year.

In the appropriate developmental sequence, I intend for Making Math Real: The 4 Operations and the 400 Math Facts to precede Making Math Real: Fractions, Decimals, and Advanced Place Value because much of the content in FDAP is a direct extension and application of the 4 Ops content:

MMRI Recommended Course Sequence Guide for Elementary and Middle School
To see the complete MMRI RCSG, click here.

  • 4 Ops provides the concepts of place and place value for ones, tens, and hundreds; FDAP continues with advanced concepts and applications of place and place value for the thousands through the billions
  • 4 Ops provides complete concept-procedure integration for addition, subtraction, multiplication, and division through multi-digit operands with renaming in addition and subtraction, and through double-digit operators for multiplication and long division; FDAP applies all concept-procedure integration of the four operations with whole numbers to addition, subtraction, multiplication, and division for fractions and decimals
  • 4 Ops provides the structures to develop automaticity for the 400 math facts*; FDAP applies all math facts to develop concept-procedure integration for the four operations with fractions and decimals. In particular, automaticity with the multiplication and division facts directly supports the abilities to factor products, prime factor products, mentally find the greatest common factor of two products, mentally find the least common multiple of two products, generate equivalent fractions, simplify equivalent fractions, rename two or more fractions as equivalent fractions with least common denominators, transform mixed fractions to improper fractions/improper fractions to mixed fractions, and cross simplify fractions in multiplication – all of which are presented in FDAP.

* The 100 multiplication facts are only taught in the 3-day Making Math Real: The 9 Lines Intensive, and NOT the 4 Ops course. Consequently, it is strongly recommended to take the 9 Lines course prior to 4 Ops and FDAP

In Addition to all the Content, Math Comprehension is a Focus of the 4 Operations and 400 Math Facts Course

Making Math Real: The 4 Operations and the 400 Math Facts course presents the most essential foundational 1st through 5th grade content and cognitive developments necessary to prepare students for all future mathematics. Embedded within the teaching of the content, this course emphasizes the development of math comprehension as the basis for the learning of all mathematics, because the definition of success in math is the confidence that comes from the deep understanding of “I know that I know”, rather than “I remember what to do”.

Concept-Procedure Integration Develops Comprehension

All of the operations content in the 4 Ops course is based on concept-procedure integration, which means the concept of the mathematics and its respective procedure(s) are integrated as one. The teaching of concept-procedure integration will be fully covered throughout the 4 Ops course.

Concept-procedure integration is vastly different from the way math has been traditionally and currently taught: as a series of disconnected procedural commands intended to program students to remember what to do rather than develop the comprehension of what they are doing. Often, the phrase “Just do…” is the sum total of the teaching. Some representative examples of disconnected “Just do…” procedural commands:

  • For renaming (regrouping) in addition: “Just carry the 1
  • For subtracting integers: “Just do the opposite”
  • For increasing or decreasing decimals by powers of 10: “Just move the decimal to the left or right”
  • For division with fractions: “Just keep, change, flip” or “Just invert and multiply”.

All of these procedural commands are an educational “hope and a prayer” that students can remember what to do and are completely disassociated from the actual mathematics, and even worse, are frequently mathematically incorrect. For example, “Just carry the 1”: it’s not a “1”, it’s a ten (or it could be a hundred, a thousand, etc.); “Just move the decimal”: the decimal is a symbolic wall that separates whole units from parts of a unit, and therefore, can never be moved, rather, it’s the numbers in their respective places that increase or decrease in relation to the decimal point (the “wall”).

Active Working Memory = Math Comprehension

There is no math comprehension with procedural commands, and consequently, students must rely on memorizing procedural steps rather than building knowledge of correct mathematics. The major failure with using procedural commands is even deeper and more critical. Teaching math procedures disassociated from comprehension means students’ working memory for math is never activated. Working memory is the perceptual understanding of what we are currently doing, and we need and use working memory for every active experience (as opposed to receptive experiences, such as watching TV), spanning from basic and/or habituated experiences to extremely complex and challenging ones. Activating and sustaining working memory throughout an activity is how we maintain conscious awareness of what we are doing. Sustaining working memory enables us to complete a task without making mistakes, or if we make mistakes, noticing and self-correcting any error(s) if and when they occur.

Active working memory in math is the perceptual experience of comprehension and knowing what we are doing throughout problem solving. Math is like performing surgery. Just as the surgeon cannot afford to make a mistake during a procedure, so, too, must a person doing math sustain a clear and robust working memory picture throughout problem solving. In math, even though all of the other details within the problem have been completed correctly, just one detail mistake within that problem will make for an incorrect solution. In math, we have to bat 1,000, or be 100% accurate from the free-throw line if we are to complete a problem correctly. The better we understand and know what we are doing, the stronger our working memory picture, and the better we can manage all the details in our problem solving.

Losing our working memory picture during an activity is extremely risky. The moment we lose our working memory picture we become perceptually blind because we no longer can perceive what we are doing, and are now eminently capable of making mistakes without noticing them. Frequently, these mistakes can be silly ones about which we “should have known better”, and may ask ourselves, “How could I have done that?”

Activating and sustaining working memory is an executive function and requires significant cognitive effort, therefore, there are numerous factors that can contribute to making us lose our working memory picture. Some of these factors can be multitasking, becoming distracted, going on autopilot, getting confused, overloading working memory, or following procedural commands disassociated from comprehension.

Working Memory: One of the Great Developmental Gifts of Math

One of the major developmental purposes of the successful math experience across the grades is the ever-increasing ability to activate and sustain working memory for longer and more challenging periods of problem solving. This powerful gift of the successful math experience means, if we can activate and sustain working memory in math, then we can transfer that ability to activate and sustain working memory for almost any task. The ability to activate and sustain working memory provides a key exit ticket  from school for becoming successful independent adults. We need a strong working memory to navigate the challenges of being an adult as this helps us manage the details of all that we do including our jobs, our responsibilities, and our households. Developing working memory is one of the most significant contributions we can give to our students to foster their future success as independent adults, and having a sustained, successful math experience is one of the best pathways to do so.

It’s the TEACHING, Not the Program

All students need and deserve a well-trained teacher, because it’s the teaching that ensures success, not the program. This truth is the heart and soul that drives the vision and mission of the Making Math Real Institute. All of the time, effort, and integrity in creating, expanding, and sustaining the growth of the MMR simultaneous multisensory structured clinical methods exemplifies the focus, design and purpose of MMR: empower educators to foster success for all students.

A powerful example of the Making Math Real Institute’s vision and mission successfully applied comes from an article written by Karen Cavallaro who operates a learning center in Reno, Nevada, and has thus far completed 300 hours of MMR courses. The title of her article is “Living with Dyslexia: Are Math Programs the Answer?” published in the Summer 2017 edition of the International Dyslexia Association’s Dyslexia Connection.

Please read Ms. Cavallaro’s entire article here: dyslexia-are-math-programs- the-answer

Ms. Cavallaro’s article provides a wonderful and authentic experience of the successful outcomes when teaching is the focus, not the program. Using Ms. Cavallaro’s article as a starting point, I will publish an ongoing 10-part series of articles entitled: “It’s the Teaching that Ensures Success, Not the Program” in which I will explain what I intend by this statement and how we can achieve it.

“It’s the Teaching that Ensures Success, Not the Program”
Part 1: The Introduction

A Barrage of Goodness: MMR Changing Lives | An Invitation to Share

In reflecting back over the 20 years since I decided to go public with the Making Math Real methods I had created, the most outstanding, consistent, and singular result has been the unsolicited responses from all of you sharing your excitement, passion, and deep reward in having changed the lives of your students/children, their families, and yours, too.

Effecting positive change has always been at the forefront of the Making Math Real Institute’s mission and vision and is the entire reason I am so unrelentingly vigilant in protecting the integrity that is MMR. During these difficult times, I feel the most we can contribute is the goodness we do each day; and in the name of MMR, I would like to contribute a barrage of goodness. Therefore, in celebration of the 20 years of service in providing positive change, I am inviting you to share with us at the MMRI any and all stories you would like to contribute to the Barrage of Goodness Campaign.

Your stories need not include any identifying information of your students or yourselves, (unless you want to) just expressions of positive change through your provision of MMR that you have observed in your students/children, their families and in yourselves as well. No story is too big or too small. These can be in any format and any length, and it is my intent to publish every one of these that come in now and into the future in its own section on the website. In addition, I will select some of these to include in the May 13, 2017 Free Seminar: Staying the Course: Changing Lives for Students, Families, and Teachers.

Hopefully, over time, we will have collected a great volume of goodness to share with the rest of the world, comprised entirely of your authentic stories of changing lives. I look forward to hearing from all of you.

David Berg, ET
Founder & Director, Making Math Real Institute

Special Feature: David Berg, Artist

In addition to being Founder, Director, and Creator of Making Math Real, David Berg is also an artist, musician, and composer.

As an artist, David has exhibited internationally and his work is in public and private collections around the world including the Los Angeles County Museum of ArtSan Francisco Museum of Modern ArtThe Fine Arts Museums of San FranciscoThe Contemporary Art Museum of Hawaii and more. Recently, David was included in his fifth exhibition at the Los Angeles County Museum (LACMA), “Lens Work: Celebrating LACMA’s Experimental Photography at 50”, and LACMA has just acquired two more pieces of David’s work for their permanent collection.

Click for full image


Click here to view David Berg’s art résumé.


Click here to view the full artist list for Lens Work.

page1_articleAs a musician and composer, David has performed and recorded his music internationally, including on television and radio on PBS stations and at the Banff Centre for the Arts.

Click here to read the full article.





listen Click the play button below to listen to “What’s On Your Belly?” (9:06) from the CD “Portraits of My Grandmother’s Daughter” Spoken Word/Music Collaboration, recorded 1994 at the Banff Centre for the Arts, Alberta, Canada. Darryl Keyes: Poet and spoken word; David Berg: composer, arranger, musical director, and plays all musical parts.

mvc-013sAs a founder and director, David is frequently asked, “Where did Making Math Real come from?” Part of the answer to this question is directly related to the significant development resulting from a long-term immersion in solving creative problems in the arts: finding solutions for expressing creative ideas in physical form. From a developmental perspective (not content), MMR’s seminal period coincided with David’s simultaneous immersion in the fine arts. David’s significant bodies of work in painting and music occurred between 1983 and 2002, as did the major initial developments of MMR.  In 2002, David stopped painting, exhibiting, and performing to focus entirely on the continuous development of MMR, channeling the creative drive from art and music into the expansion of math education.

The creation of MMR, like painting and composing, has been, and continues to be, an analysis/synthesis activity, requiring full lateralization and front-to-back connections (critical thinking) in the brain. The artist’s model of problem solving is entirely about this ongoing analysis/synthesis experience. The artist’s technical facility and detail analysis provide the support of solving the expression of creative ideas, questions, and problems. The artist engages limitless possibilities as potential means of solving creative problems and is best served by remaining completely open to encourage access to maximum creative resources. The artist’s model of problem solving, in combination with 40 years of intensive clinical observation of thousands of students of all ages, cultures, and diverse processing styles, has been the ongoing genesis of Making Math Real.

Making Math Real: Kindergarten: The Genesis of All the Making Math Real Methods

The Genesis of All the Making Math Real Methods

I have been pleasantly surprised of late to hear from a number of people who have taken all of the Making Math Real classes that the Kindergarten class was their favorite, especially from those who took the Kindergarten class last in their sequence of courses. Extremely curious as to why Kindergarten was their favorite, I asked each of them to please explain. The overarching response was that the Kindergarten class was their favorite because it felt like the most important of the Making Math Real courses since it was the source from where all of the Making Math Real methods started.

I have often expressed how math is a perfect architecture in which all of its content and codes interconnect from the earliest developments of number sense through calculus. Consequently, the content taught in a current grade has profound implications and connections for the content yet to come across the higher grades; and that math educators, therefore, must know all the content, if not k-12, then at least four years prior to, and four years subsequent to, their current grade levels. In this manner educators can teach their respective grades with precision because they know what content developments precede their grades, and they know exactly how to teach their current content to prepare students for the connections of that content to the future grades. The following is a quote from a 20-year veteran fifth grade teacher who also maintained a private practice using MMR methods:

“I have been teaching for 20 years, about half in first and the other half in fifth grade. Having taught first grade for so long, I was familiar with the math that came before fifth grade, but it was not until I thoroughly taught algebra that I knew how to teach my fifth graders exactly what they needed to learn to be prepared for algebra.”

Making Math Real: Kindergarten is an extremely valuable course because, as the earliest fundamental for the interconnected architecture yet to come, the Kindergarten class provides the foundational connections for all of the subsequent Making Math Real methods.


To be ready to receive formal instruction in math, kindergarten students must have integrated the earliest developments of number sense:

  • one-to-one correspondence for accurately counting objects and/or images
  • conservation of number to maintain stable quantities despite the interference of numerous distractors such as changes in location and configurations of quantities
  • number-symbol relationship to decode and encode numerical symbols, to connect numerals to quantities and quantities to numerals, and comparing quantities

Teaching students to independently solve word problems is at least a 10-year development spanning kindergarten through sophomore year in high school. Each grade year contributes specific developmental increments toward this overarching goal, and kindergarten gets it all started with the earliest developments of solving word (story) problems.

In addition, the earliest developments of symbol imaging take place throughout the kindergarten year in helping students to image the numerals from 1-10, the names of the shapes, the names of the coins, etc. These kindergarten symbol imaging activities are the origins of the 9 Lines symbol imaging methodology and prepare first graders to use the 9 Lines to learn the doubles addition facts.

By the end of kindergarten students need to be mentally able to increase and decrease single and double-digit numbers by one, especially in crossing 10s counting either forward or backward. The ability to mentally increase and decrease numbers by one provides the necessary neural structure for first graders to be developmentally ready to begin mentally integrating and applying the addition and subtraction facts.

All of the above mentioned developments: counting, sequencing, decoding and encoding, comparing, mentally increasing and decreasing by one, and symbol imaging provide the necessary developmental supports for the working memory demands of first grade math.


As kindergarten provides the earliest fundamental developments of working memory for math, it also provides helpful assessment insight for those students of any age who are not yet able to integrate these fundamentals. Recently, I have worked with numerous students from ages seven to thirteen, who at their initial assessments were unable to sustain one-to-one correspondence. In all these cases, the students’ inability to count (or any of the other significant developmental indications) was not observed and/or assessed by their respective schools, and all of these students were nonetheless subjected to ongoing attempts to teach them math. Until a student has achieved the earliest developments of number sense, formal instruction in math is not indicated because students are not able to establish effective working memory from which to learn. Consequently, everyone of these children was still at a pre-kindergarten level of math development, despite having attended school for years, while all involved educators were confounded by the “students’ inability to learn”.


Understanding the fundamental kindergarten developments provides an effective baseline for determining possible indicators for why students beyond kindergarten age and with average intelligence are still not able to accurately count objects and/or images one for one. In the cases mentioned above there were strong correlations to significant under developments in sensory-motor integration, bilateral coordination, crossing the midline, proprioception, manual dexterity, muscle tone, sequencing, and in some of the cases, receptive and expressive language. All of these issues may indicate under developments in finger gnosis.

As explained in the Making Math Real: Overview, finger gnosis, or finger knowledge, directly activates the left angular gyrus (AG) in the parietal lobe. Finger knowledge refers to deriving proprioceptive meaningfulness by using the fingers to count and helps all of us experience what various quantities feel like. This is how quantity enters the body, directly from finger touch and movement to the parietal lobes. The parietals deal with quantity and are known as the “math brain”. The left angular gyrus is also activated to encode the quantity experience in the parietals into language and is located near Wernicke’s Area in the temporal lobe, the area known for language input at the mouth of the phonological language pathway, and is itself the direct activation of the comprehension language pathway, both in the temporal lobe.


The temporal lobes are known as the “language brain”, and as also stated in the Overview course, the activation and development of exact math is a parietal-temporal integration.


Therefore, from experiencing the concept of the math to expressing the math either in language or with pencil and paper, the angular gyrus is used for both, but development of the AG is required. However, if finger gnosis is underdeveloped, so too, might the angular gyrus be underdeveloped, which may contribute to a student’s ongoing inability to sustain one-to-one correspondence, and since the angular gyrus is also crucial to both the phonological and comprehension language pathways, receptive and/or expressive language may also be underdeveloped.

If finger gnosis is underdeveloped and the student cannot sustain one-to-one correspondence, it is premature to begin any formal instruction in mathematics until a prescriptive, developmental intervention develops the finger gnosis-left angular gyrus activation connection that supports the ability to count accurately. Once one-to-one correspondence is established, then conservation of number and number-symbol relationship can be developed, thereby supporting the activation of sufficient working memory for the student to receive formal instruction in mathematics.

* click here for references list for finger gnosis and the left angular gyrus


I consider the k-2 sequence to be the most essential period for the development of mathematics as it relates to the development of number sense, sequencing, and especially, working memory. Underdeveloped working memory during the primary grade years can be a strong indicator of future struggles with math, and I have encountered innumerable students of all ages whose struggles with math could be traced back to under developments during the k-2 period. Therefore, we need our best and our brightest educators to be working with children at this invaluable developmental age, because in my experience, I have found it much more clinically challenging to appropriately and effectively foster these critical math developments in five to eight-year olds than it is to teach higher math to older students. Furthermore, I strongly believe, if we can successfully provide kindergarten through second graders with these necessary developments, we would see a profoundly positive change in the overall mathematics achievements of these children as they continue across the grades.

Looking Forward to Making Math Real: Geometry Part IV

LOOKING FORWARD TO Making Math Real: Geometry Part I

Part IV


“Just the Facts, Ma’am”: In Geometry, Assumptions Are Not Welcome: If It’s Not a Fact, We Say Nothing; We Write Nothing

Imagine a classroom in which the teacher asks a question and a student is jumping out of her seat, frantically waving her raised hand in the unmistakable gesture of “call on me, please!” This affective behavior indicates a student who irrevocably knows that she knows the correct solution. The experience of certainty, “knowing that you know”, is the definition of successful mathematics integration. Guiding students to the awareness of knowing that they know prior to expressing the solutions is the goal and objective of all math education. This “knowing” is deeply integrated in the body – not merely remembering what to do. This depth of body integration is another way to feel and express mastery. When we have learned to mastery, that learning represents the highest level of proficiency, and as such, the learning is fluently and automatically activated when needed, and our ability to sustain that learning is maintained at a high degree of accuracy and consistency.

Math is Surgery

Do you want your surgeon to possess mastery or to guess what to do? Just as in the midst of an operation you would not want your surgeon guessing what to do, so too, in the midst of mathematical operations you do not want to be guessing what to do. Math provides us with specific tools for problem solving applications, and our knowledge of how to use these tools enables us to be precise, direct, and effective with our problem solving. In terms of surgery, I would want my surgeon to be precise, direct, and effective in knowing where to aim those sharp tools… Like surgery, math is never guessing.

Knowing That You Know

Frequently in algebra, the solution to given problems may be a simplified or transformed version of an equation or expression. The solution may not be a conveniently cogent numerical value from which one could easily the check the solution for accuracy. For example, how do we expediently check the following solution for accuracy?


If solutions turn out to be a transformed version of the original equation rather than easily recognizable numerical values, then how do we know with confidence our solution is correct without having to do cumbersome checking of the solutions? Answer: we know our solution is correct because throughout the algebraic problem solving of transforming to equal equations we can sustain the certainty of “knowing that we know” each new transformation is accurate and correct. Knowing that we know before expressing the solution requires the development of a strong executive function directly supporting the ability to sustain a robust working memory picture throughout problem solving. The development of a strong executive function that monitors and regulates a robust working memory picture is how we learn to manage the details in our procedural work (detail analysis), and is also the necessary development all of us need to be successful as independent adults: manage the details – it is how we keep getting a paycheck.

Geometry Deepens What Algebra 1 Starts

As mentioned in the previous installments of this series, from a developmental perspective, the principal cognitive demand of algebra is detail analysis: activating the perceptual working memory picture from the interrelationship of the symbolic details while managing the details throughout the procedural work. In algebra, well-developed detail analysis helps us to sustain knowing that we know so we can effectively manage all of the numerous details comprising extensive algebraic transformation-based problem solving. The developmental outcome is students’ success with simplifying expressions and solving equations because they are able to sustain managing the details and therefore, maintain knowing that they know throughout problem solving.

Geometry introduces a perceptual, structural code that provides factual certainty. The provision of this code of factual certainty deepens the integration of students knowing that they know, because geometry, within all of its problem solving, requires certainty before expressing any statement, solution, or conclusion. Using language and/or physical markings on figures, the codes of geometry tell us explicitly that lines are perpendicular, planes are parallel or two angles are congruent. Applications of geometry also help to string together two or more facts to find new facts, such as two angles are congruent because they are vertical angles; and two lines are perpendicular because the angle they form at their intersection is 90˚. It is never correct to assume lines are perpendicular just because they appear to be in the figure provided; or that angles are right angles, because to the eyes, they look like 90˚ angles. Geometry is not the mathematics of appearances and assumptions, only facts.

For this purpose, it is essential that teachers require students to consistently extend given information, e.g., given language, relationships, markings, etc., to fully “mark up” figures to indicate all known factual information. If there are two intersecting lines forming vertical angles, students need to mark the congruent vertical angles on the figure, not merely hold that congruence in mind. If students are given that the two lines cut by a transversal are parallel, students need to mark the congruent alternate interior angles; and if students are given that on triangles ABC and DEF, sides AB and DE are congruent, students need to mark respective corresponding sides to show that congruence. By fully marking up figures, students are now equipped with multiple facts from which they can begin to string together individual facts to generate new facts. Furthermore, all of the facts marked on the figures hold still on the page, which reduces cognitive load significantly, because the increased cognitive demand of mentally holding multiple visual-spatial facts simultaneously can lead to cognitive overload, which in turn may cause students to lose or weaken their working memory picture resulting in a higher frequency of errors and missed relationship connections.

Developing Deductive Proof

The ability to string together two or more facts is the basis of developing deductive proof. For example, students can prove two triangles are congruent because they can connect and justify a series of individual facts that lead to a specific, certain, and factual conclusion: two triangles are congruent because each of the two triangles has two corresponding sides of equal length and each has an included angle of the same degree measure. When two triangles share congruent corresponding sides and an included angle they make a specific, factual relationship called side-angle-side; and side-angle-side provides sufficient factual basis to be certain that the two triangles are congruent. The progression of connecting and justifying a connected series of individual facts to a culminating factual conclusion is the basis and developmental purpose of geometry. By requiring students to exclusively express, justify, and connect each fact until culminating in a factual conclusion, geometry provides the maximum development of “knowing that you know prior to expressing the solution.”

Geometry Prepares Students for “Court”

Imagine a trial attorney who does not manage the details, assumes facts not in evidence, and asks questions for which the answers are a surprise. Not only does this lead to disasters in the courtroom, but this same lack of preparation also leads to breakdowns for students in math classrooms across the grades, nationwide. Math students are not preparing for a case in court, but they may be preparing for something equally as challenging – the standardized tests. The well-known high school and college entrance exams are more based on measuring executive function development than math content knowledge, which may explain why the test questions mostly do not resemble the content students encounter in their math classrooms, and why these tests contain so many “trick and trap” questions. The ability to successfully navigate the minefield of the math sections on these standardized tests requires a highly developed executive function to sustain a strong math-based working memory picture rather than succumb to the seductive tricks and traps set by the test makers.

One of the major ways the tests attempt to trick the test-taker is to present a perceptual conflict between the mathematical facts and how the eyes perceive the problem. For example, a test problem may present a misshapen quadrilateral in which clearly none of the sides are parallel or of equal length and all of the angles are clearly a combination of acute and obtuse angles – no angles remotely resemble right angles. However, this misshapen quadrilateral has two opposite angles in which the mark indicating right angles has been placed. In the remaining two angles, there are indicators of angle bisectors in each in which each bisected angle has “x˚” and “y˚”, respectively. The question for students to solve is: x˚ + y˚ = ? Lastly, the problem has the ominous statement: “figure not drawn to scale” as an appropriate and necessary inclusion for solving the problem.

The answer to this problem is x˚ + y˚ = 90˚, because, despite the visual impossibility that this misshapen quadrilateral contains right angles, it is nonetheless a rectangle because of the factually marked indicators that two opposite angles are right angles, and half of each right angle is 45˚, and 45˚ + 45˚ = 90˚. The test makers set up a nefarious conflict to determine which processing tool will override the other – the executive processing tool that uses math-based factual knowledge (I know that I know) the figure is in fact a rectangle, despite the powerful visual distractor of the misshapen quadrilateral; or the visual system overrides the executive because there is no way that misshapen quadrilateral could ever be a rectangle, despite the inclusion of “figure not drawn to scale”.

Why do these tests attempt to trick students, because as confusing as math classes have become in our nation’s public and private schools, it is not the specific intent of the curricula or the teachers to present problems to trick students in this way? Perhaps the purpose of these trick questions is to measure students’ executive function development, because admissions teams understand, the higher the executive development, the more likely students will be successful, not only at their schools, but across the board in all of their endeavors for now and on in the future, because a well-developed executive is highly correlated to success in life as an independent adult.

Geometry, the Math of Certainty, Continues the Ongoing K-12 Development of Executive Function

The ongoing developmental outcome of having students engage the executive to “prepare for court” to exclusively express, justify, and connect each fact until culminating in a factual conclusion provides profound development for the integration of “I know that I know prior to expressing the solution”. Therefore, it is imperative in teaching geometry to students across the grades, we structure all geometry lessons to require factual basis, not assumption; and that deductive reasoning requires mathematical knowledge to override the visual distractor of what the eyes may assume to be facts. Any visual information unsupported by facts cannot be determined until factual information is given:


Developing “Knowing That You Know Prior to Expressing the Solution” is Necessary for Success in the Higher-Level Math Classes

As stated in the third installment of this series, “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.” The simultaneous blending and shifting of analysis and synthesis abilities is a high-level executive function that benefits significantly from the ongoing experiences and integration of knowing that you know. The students’ developmental abilities to hold their working memory picture with certainty makes it possible for them to simultaneously shift back and forth between analysis and synthesis processing without experiencing cognitive overload – an essential developmental foundation for success in higher mathematics.

I look forward to seeing all of you in the first ever Making Math Real: Geometry Part I.

∞  REGISTER for Making Math Real: Geometry

LOOKING FORWARD TO Making Math Real: Geometry Part III


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.

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LOOKING FORWARD TO Making Math Real: Geometry Part II


LOOKING FORWARD TO Making Math Real: Geometry Part I

Part II

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



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

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First Published 2/23/16


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