All posts by mmradmin

NEW Algebra II Course

Announcing NEW Algebra II Course

This first time offering of Making Math Real: Algebra II is the result of a successful “If You Build It, We Will Come” campaign. If You Build It, We Will Come is a grassroots “by popular demand” campaign in which people who want and need a particular MMR course either to be created or to be scheduled sooner than its regularly scheduled date can “build” a new course or a new date for the course. In this case, Making Math Real: Algebra II, was not a part of the Making Math Real series of courses until a successful IYBI campaign made it happen! – Read more

Making Math Real: Algebra II
March 7-8, 21-22, April 4-5, 18-19, May 2-3, 2020  |  10-Day Course  |  $1,999
Mandatory prerequisite: Algebra I

Algebra II Registration Now Open — Click Here to Register

In the developmentally appropriate scope and sequence starting with pre-algebra, algebra II follows algebra I and high school geometry. The developmental focus of pre-algebra and algebra I is analysis and the developmental focus of geometry is synthesis, thereby making algebra II the first significant mathematical experience of applying analysis-synthesis to activate and sustain working memory for math.

The development of analysis-synthesis has been activated across the grades starting in kindergarten and occurs any time students are connecting the symbolic details (numerical or otherwise) to the big picture of comprehension; and from the big picture of comprehension back to the component symbols that comprise an expression or equation. For example in the elementary grades, analysis-synthesis occurs every time we prompt students to explain how they see the concrete model and the symbols are telling the same story. In addition, analysis-synthesis is the principal developmental foundation for solving word problems. Although analysis-synthesis has been developing along the way, algebra II is the first time analysis-synthesis is the main developmental emphasis.

The development of analysis from the successful completion of pre-algebra and algebra I refers to the activations and applications of detail analysis. Detail analysis is an executive function-based sensory-cognitive development that directly supports the ability to activate and sustain working memory from the interrelationship of the symbolic details, which is the basis for the successful processing of algebraic expressions and equations. For example, detail analysis makes it possible to perceive which of the 7 factoring styles is indicated from either product form or factored form, if the slope of the line is positive, negative, 0, or undefined from slope-intercept form, to identify which form of the linear equation is given: standard form, slope-intercept form, or point-slope form, etc.

The development of synthesis from the successful completion of geometry refers to the activations and applications of synthesizing the big picture from partially given information. In supporting the development of deductive reasoning, specifically for solving deductive proofs, the application practice of geometry provides partial information about certain relationships, within or in conjunction with, lines, angles, polygons and/or circles, from which the student must systematically deduce all pertinent information about those relationships to reach specific conclusions (incontrovertible proof) about those relationships. For example, synthesis makes it possible for students to deduce two triangles are congruent based on the partially provided information that two corresponding sides and a corresponding included angle of the two triangles are congruent, therefore guaranteeing that all six corresponding relationships of sides and angles are congruent.

The ability to synthesize partially given information and expand it to comprehensive proof means that all component details comprising the big picture have been accounted for, and before writing or expressing any solution or conclusion, it must be a fact, not an assumption. The ability to be certain about a solution prior to expressing it defines the successful experience of math: “I know that I know”.

Analysis-synthesis is necessary for successfully activating and sustaining working memory for algebra II content. The principal content focus of algebra II is functions, their graphs, and their applications, and the overall function of algebra II is to prepare students for pre-calculus. This presents a significant extension of both algebra I and geometry content, and analysis-synthesis is activated to support the perceptual connections necessary to decode and encode the equations and their graphs, the graphs and their equations, and especially for all the applications of word problems. Within the first three units, algebra II content will require full activation of all the algebra I content: every component detail of simplifying expressions, solving equations in one variable through level XIV, solving linear equations graphically and algebraically for single equations and two equations using systems of equations as well as systems of inequalities, and more than 12 categories of algebra I word problems.

All of the algebra II content will be structured through functions, their graphs, transformations of the parent functions, and their applications. Topics include comprehensive units on:

  • Functions and relations, linear and absolute value functions including linear programming and solving systems for three equations and three variables
  • Quadratic functions and parabolas including the factoring of prime products: completing the square and the quadratic formula, imaginary numbers, complex numbers and their graphs, and real and imaginary roots
  • Polynomial functions of higher degree, the shape of their graphs and end behavior, factoring polynomials of higher degree including real and complex solutions, polynomial long division and synthetic division
  • Rational functions including their transformations to polynomial rational form; graph behavior: asymptotes and holes, and direct and inverse variation
  • Applications of functions including piecewise functions, the four operations with functions, composition of functions, inverse functions, and even and odd functions
  • Square root functions, radical expressions and equations, exponential expressions and equations, rational exponents, and transformations between radical and exponential forms
  • Exponential and logarithmic functions, expressions, and equations including the natural exponent and the natural logarithm: problem solving with continuous growth and decay, simple and compound interest, half-life and doubling
  • Arithmetic and geometric sequences and series, sigma notation, and infinite sums
  • Synthesis applications of all the algebra II functions

And if time permits:

  • Probability, using the counting principle for permutations and combinations.

For everyone who has completed Algebra I, I hope you will join us in the Spring for this vital new course in the Making Math Real series.

Sincerely,

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


SPECIAL NOTE: Since this course will use all of the algebra I content immediately, you are strongly encouraged to have fully integrated all of the math content in the Making Math Real: Algebra I course and have it fully refreshed and activated to receive full developmental benefit from the Making Math Real: Algebra II.

Mandatory pre-requisites: All course participants must have successfully completed the Making Math Real: OverviewFractions, Decimals and Advanced Place ValuePre-Algebra, and Algebra I

Strongly recommended: The 9 Lines IntensiveThe 4 Operations & the 400 Math FactsGeometry Part 1Geometry Part 2

Stories from the Front Lines

Searching for SOLUTIONS & Sharing EXPERIENCES

Stories from the Front Lines

On a good day, Alison, a middle school student, was indifferent about math. On a bad day, she could not stand it. After receiving prescriptive math instruction utilizing Making Math Real methods, she sees math in a whole new way. Check out this short animated film where she expresses her feelings about the difference a good picture can make:

Several years ago I introduced you to the “Changing Lives” section of Making Math Real’s website. We launched this special page of the website to provide relief and support for those who have been searching for solutions and a forum for sharing experiences to help those who feel isolated and alone to know how common the struggle is and how difficult it can be to find the solutions they need.

I am very pleased to announce that the number of stories on the Changing Lives webpage has quadrupled since it was initially launched! I want to thank both the dedicated teachers and parents who have written in to MMR to share their wonderful stories of success and those who have reported back letting us know of their ongoing successes in the face of their many obstacles and challenges. I also want to celebrate the many students of all ages who have written to me personally and agreed to openly share their letters in the hope of helping other kids like themselves.

Please visit Changing Lives to find inspiration, to learn from others who are paving the way, and to know that you are not alone in your valiant efforts to advocate, teach, and learn how to make math real for your students and your children.

I encourage everyone, those who have yet to send us their story as well as those who have already sent one or more, to SUBMIT YOUR STORY to MMR to join and bolster our community of dedicated parents, educators and students who strive for authentic success. I want all of you to know your stories of success provide the best evidence basis for those just starting on the path toward finding a solution, and have given, and continue to give, significant hope for those feeling overwhelmed, discouraged, and demoralized by circumstances and problems that can feel impossible to solve. One of the best ways we can contribute to effecting positive change is by letting the world know a solution exists.

Thank you for sharing,

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


Published on: November 22nd, 2017

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… Read more from the Blog…

Welcome to the Making Math Real: Overview

WELCOME TO THE MAKING MATH REAL: OVERVIEW

Transformative as an Introduction, Invaluable as a Repeat

Recently I have been extremely pleased to note the number of experienced Making Math Real practitioners who have elected to take the Making Math Real: Overview course again. The unanimous motivation: “There are so many layers of learning and understanding in the Overview, and now that I really understand how MMR works, I am ready to make all those deeper connections”.

Making Math Real: Overview, K-12
September 7-8
$399 | 1 optional academic unit

What Makes Making Math Real Different?

This is one of the most frequently asked questions we receive, and its answer is the basis of the Making Math Real: Overview. The Overview course is extremely voluminous and provides the necessary introduction to the structure and methods of Making Math Real to prepare educators for the up to 680 hours of content courses that follow (see the full list of courses here). The introduction is necessary because the Making Math Real simultaneous multisensory structured methods are historically unprecedented. Making Math Real is the first and only comprehensive pre-K through calculus prescription for teaching that emphasizes integrating the development of executive function and working memory within every math lesson. The focus on the development of the essential self-regulatory executive processes that directly support students’ abilities to initiate, activate and sustain working memory distinguishes Making Math Real from every other method, program, textbook, or software.

Functional Math Education: Addressing the Root Cause Rather than Treating the Symptom

Working with thousands of students and teachers nationwide over the last 43 years has consistently indicated to me that any degree of math struggle is not related to students’ lack of math ability, intelligence, or motivation. Rather, the root cause(s) for challenges in learning math is more related to teaching practices that do not activate students’ working memory and/or relative underdevelopments in students’ executive processes that support working memory, because without sufficient ability to activate and sustain working memory, students cannot access their native intelligence, and therefore, are unable to express what they know.

Therefore, a major emphasis of the Making Math Real: Overview is the introduction to the connection that we can successfully teach all students once we understand this developmental basis for successful teaching and learning, because the profound limitation affecting math education, has been and continues to be, focusing entirely on math content skills disconnected from the development of the self-regulatory executive processes that directly support students’ abilities to initiate, activate and sustain working memory. Without working memory activated, students are “perceptually blind” and consequently must rely on procedural memory rather than understanding and knowing what they are doing.

I have seen innumerable math programs come and go, and each new version is a repackaging of math content that has yet to address the developmental basis for successful teaching and learning. This is why there has been no effective positive change in math education outcomes across the decades (according to research data), in particular for the gap in achievement and special needs populations.

From the “2018 Brown Center Report on American Education: How Well are American Students Learning?”:

“…Since NCLB’s (No Child Left Behind) early years, scores have largely plateaued at levels of nationwide performance that many Americans find underwhelming, leaving still-large gaps between historically advantaged and disadvantaged groups.”

And from “NAEP (National Assessment of Educational Progress), the Nation’s Report Card: 2015: Mathematics and Reading at Grade 12”:

“In comparison to 2013, the national average mathematics score in 2015 for twelfth-grade students was lower” and “In comparison to the first year of the current trendline, 2005, the average mathematics score in 2015 did not significantly differ.”

Research has identified the critical value of executive function and working memory in student success in math, even to the extent that by kindergarten, a student’s relative development of several key self-regulatory executive functions and working memory predicts future success or struggles in math. Research has identified the source of the problem, but unfortunately has yet to propose any specific, direct, and practical way to incorporate it into teaching.

Making Math Real is the first and only to successfully integrate research within its methodologies, and this is another major emphasis of the Overview: present the research, define working memory, present which self-regulatory executive processes directly support working memory, and how to structure these developments while teaching math.

The Main Takeaways from the Making Math Real: Overview

In support of the multifaceted focus of the Overview, the course is organized into three main sections: 1) Pedagogy 2) Structure 3) Sensory-Cognitive Development. The following are some of the main takeaways from each of the three sections.

Pedagogy: The Research Connections
from Cognitive Science and Neurobiology

  • Making Math Real is for all students, not just for those who struggle or who have special needs
  • Math should never hurt
  • How Making Math Real is different from every other method, program, textbook, or software
  • Direct, explicit, sequential, incremental, systematic, connected, and simultaneously multisensory structured
  • Math is a perfectly interconnected architecture in which any concept/application taught in any grade has direct connections to the math coming in later grades
  • “It’s the teaching that ensures success, not the program”: the art and science of teaching math
  • The research connections from cognitive science and neurobiology that define how the brain does math and how Making Math Real puts this research directly into teaching practice
  • The connection of finger gnosis and the left angular gyrus and their relationship to number processing
  • Define executive function, working memory, and processing as well as their function in supporting math learning and how to integrate their development within every math lesson
  • Top-down and bottom-up: the two-way relationship between executive function development and processing development
  • The development of executive processes takes time, many of which needed for success in math develop beyond the k-12 years
  • Working memory is comprehension and comprehension means “You know that you know”, not the memorization of procedural commands disconnected from comprehension
  • Diagnostic teaching: clinical observations of students’ affective and cognitive behaviors provide basis for adapting curriculum delivery
  • Define simultaneous multisensory structured methods
  • Making Math Real is a clinical methodology that empowers educators to prescriptively reach all students and is not a program (curriculum)
  • Making Math Real simultaneously serves Response To Intervention (RTI) tiers 1 through 3 students
  • The two strands that develop numeracy, what numeracy means, and the two distinct brain activations that support the development of numeracy: exact math and approximate math (mental math)
  • Define “Codes of Math”: a system of interconnected codes from pre-k through calculus that provide user-friendly symbolic codes to express what is real
  • Decoding and encoding in math, equally as important in math as they are in reading

Structure: The Hands-On Section

  • Nothing succeeds like success. Frustration, anxiety, and confusion are not helpful in fostering student grit and persistence
  • The structure, what it means and its role in successful teaching: guaranteeing successful processing
  • Two different structures spanning k-12: Concrete to Abstract for k-5, Integrity of Incrementation for pre-algebra through calculus
  • The developmental focus for activating working memory: symbol imaging for k-5, detail analysis for pre-algebra through algebra 2
  • Simultaneous multisensory structured methods: in-class demonstrations of concrete to abstract and integrity of incrementation
  • Incrementaton: each current learning activity builds the tools for the next learning activity, and each next learning activity adds only one new element
  • The successful transfer of the concrete, hands-on experience to the abstract symbols
  • A true concrete experience is far more than using manipulatives
  • Using the mathematically correct manipulatives: not just any manipulatives will do
  • Developing the language of math
  • The essential use of gestures and prompts: centralizing students’ perceptual focus
  • Providing effective scaffolding to support students’ working memory and executive function
  • Developing students’ independent processing ability
  • Differentiated structures of teaching
  • Creating comprehensive and prescriptive problem sets

Sensory-Cognitive Development:
Building the Tools of Working Memory

  • The role of symbol imaging for learning and retaining the math facts
  • The activation and development of symbol imaging while teaching the multiplication facts
  • The critical distinction between authentic processing and rote processing: rote processing is anathema and must be avoided at all times
  • “The Antidote to Rote”: automaticity with the math facts: fluent retrieval and storage does not mean rote memorization
  • Automaticity with the math facts significantly reduces cognitive load enabling increased access to working memory to support math reasoning and problem solving
  • Focus on central processing: perception and association: imaging and organized storage
  • Developing processing speed, cognitive efficiency, and cognitive endurance
  • Math facts speed tests are strongly contraindicated

Epiphanies for Some, Synthesis for Others

The Making Math Real: Overview course is an intensive, exciting, and rewarding learning experience, and whether you are new to Making Math Real or a returning Making Math Real practitioner seeking synthesis, I look forward to seeing all of you in a future Making Math Real: Overview course.

Please CLICK HERE to access a selected bibliography of research articles that provide the research basis for the Making Math Real: Overview.

Geometry Part I and Pre-Algebra in the Same Semester

Unprecedented, Unique Opportunity: Geometry Part I and Pre-Algebra in the Same Semester

Since we have never done this before in our 23 years of service, we are extremely excited to announce the 2019 Summer Institute is offering the unique opportunity to continue your progression along THE TWO major content domains, algebra and geometry, within the same semester. This summer, scheduled back-to-back in June and July are:

Geometry Part I
June 24-25, 27-28, July 1-2

Pre-Algebra
July 8-9, 11-12, 15-16, 18-19, 22-23

The Pre-Algebra course is part of the algebra series*: Fractions Decimals and Advanced Place ValuePre-AlgebraAlgebra I, and Algebra II (offered for the first time in spring 2020); and Geometry Part 1 is part of the geometry series*: The 4 Operations & the 400 Math FactsFractions, Decimals and Advanced Place ValueGeometry Part 1Pre-AlgebraAlgebra I, and Geometry Part 2.

Analysis-Synthesis:
The Developmental Basis for Pre-Calculus and Calculus

The domains of algebra and geometry together provide the essential basis for the higher mathematics of pre-calculus and calculus and are completely different from one another. Not only is the content of pre-calculus and calculus based on a seamless blend of algebra and geometry content, more significantly, its developmental imperative, analysis-synthesis, is a blend of the developmental structures of algebra (analysis) and geometry (synthesis).
For a full explanation of the analysis-synthesis of algebra and geometry, please read the following articles:

Pre-Algebra: The Foundation of the Algebra Strand Through Calculus
and
Geometry is Different from Algebra and Elementary Mathematics

Analysis-synthesis refers to the cognitive abilities, which are executive function-based, to activate the working memory, comprehension picture from the interrelationship of the symbolic details (detail analysis: algebra: analysis) as well as synthesizing the big perceptual picture of working memory from partially given information (synthesizing the big picture: geometry: synthesis). Students’ abilities to access analysis-synthesis simultaneously supports the activation of working memory for the concept comprehension and applications of the following representative samples of pre-calculus and calculus content: all trigonometry relationships and applications for all triangles, trig functions, inverse functions and identities, their graphs and their applications including the unit circle and sinusoidal graphs, conic sections, limits, derivatives and integrals.

Further Incentive:
After This Summer, Pre-Algebra and Geometry Part I
Will Not Be Offered Again Soon

Further incentive for taking advantage of this unique opportunity is Pre-Algebra is not scheduled again until spring 2021, and Geometry Part 1 is not currently scheduled again, so will likely not be offered until some time in either 2021 or 2022 at the earliest.

Of special note, the Pre-Algebra class this summer is the result of an “If You Build It, We Will Come” campaign. If You Build It, We Will Come is a grassroots “by popular demand” campaign in which people who want and need a particular MMR course to be scheduled sooner than its regularly scheduled date can “build” a new date for the course. In this case, Pre-Algebra, originally scheduled for spring 2021, is now being specially offered in summer 2019, thanks to a successful IYBI campaign that completed on 1/31/19! Please note that course discounts do NOT apply to IYBI course offerings. To successfully “build” a new date for a course requires a minimum of 20 people to fully commit by registering in advance.

CLICK HERE FOR MORE IYBI INFO

We look forward to seeing all of you in Pre-Algebra and Geometry Part 1 this summer.

* Please see List of Prerequisites required for the courses in these series.

If You Build It, We Will Come: Pre-Algebra

NEW CAMPAIGN PROPOSED FOR SUMMER 2019

IMPORTANT ANNOUNCEMENT AS OF 2/1/19:  Now that the IYBI campaign has been successful and Making Math Real: Pre-Algebra has been officially scheduled for summer 2019 rather than spring 2021, as part of the IYBI campaign, the price of tuition is fixed at $1,999, so discounts do not apply for this specially scheduled course.

If You Build It, We Will Come: Pre-Algebra

If You Build It, We Will Come is a grassroots “by popular demand” campaign in which people who want and need a particular MMR course to be scheduled sooner than its regularly scheduled date can “build” a new date for the course. In this case, Making Math Real: Pre-Algebra, originally scheduled for spring 2021, is now being offered in summer 2019!

SUCCESSFULLY “BUILT” PRE-ALGEBRA DATES:
July 8-9, 11-12, 15-16, 18-19, 22-23, 2019

To successfully “build” a new date for a course requires a minimum of 20 people to fully commit by registering in advance. This commitment requires payment in full from each person by an advance-registration deadline date, in the case of Making Math Real: Pre-Algebra summer 2019, the deadline for advance-registration was January 31, 2019.

If you want and need this course, help make it a reality and join the campaign!

If You Build It, We Will Come Campaign Information & Policies:

MMR accepts credit cards, checks, money orders and purchase orders from schools and districts. [No course discounts are provided for “If You Build It” campaigns.]  If MMR cancels this course between now and the 1st day of class, all registrations will be fully refunded. Any participant canceling registration at any time or for any reason, after payment has been made, will not receive a refund.

 

Pre-Algebra: The Foundation of the Algebra Strand Through Calculus

Pre-Algebra: The Foundation of the Algebra Strand Through Calculus

Within the K-12 scope and sequence, pre-algebra marks the beginning of the second major developmental milestone spanning kindergarten through calculus. The first major development, typically within the domain of grades K- 5 math instruction, is the concept-procedure integration of the four operations through fractions and decimals and the 400 math facts.

The algebra strand relies heavily on the developments achieved in the elementary strand 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. Additionally, the combination of learning fractions and decimals across the four operations and developing automaticity with the 400 math facts during the elementary grades directly supports the development of automaticity with the integer facts and rational numbers applications across the four operations necessary for simplifying variable expressions and solving variable equations.

Getting Ready for Pre-Algebra

Completing fifth grade does not necessarily indicate students’ readiness for pre-algebra – only the full concept-procedure integration of the four operations through fractions and decimals and automaticity with the 400 math facts does. Therefore, readiness for pre-algebra can occur at any time the student(s) has achieved this developmental integration, because if students are required to begin pre-algebra without having achieved the necessary first major fundamental development, the prognosis for success in algebra is questionable.

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 the entire algebra strand, they are the only two large-scale (seminars of 10 or more days) institute courses offered once per year, every year.

The Working Memory Demands for Algebra are Different Than Those for Elementary Mathematics

Elementary math provides the initial fundamentals of all future math, therefore its developmental/perceptual demands are basic. In elementary math the developmental/perceptual focus for students is to activate their working memory/comprehension picture entirely from the symbols, which is achieved by guiding the students from the concrete to the abstract. This means students’ original working memory/comprehension picture of all the math content is derived entirely from a direct concrete experience of the math, and once established, teachers structure the systematic transfer of the concrete working memory picture to the identical working memory picture expressed by the symbols: math symbols are a short hand code to express what is real.

Symbol imaging for numbers is therefore one of the principle sensory-cognitive developments that supports the working memory demands of elementary mathematics. Symbol imaging is the combination of the following four self-regulatory executive processes: perceive, hold, store, and retrieve sequences of numerical symbols. Symbol imaging is directly engaged both for learning the math facts and for concept-procedure integration of the four operations through fractions and decimals. In learning the math facts, symbol imaging allows us to fluently perceive, hold store and retrieve the sequences of symbols comprising each of the math facts; and in learning the four operations through fractions and decimals, symbol imaging is also required for successfully activating the concrete working memory picture from the symbols.

Once the algebra strand begins with pre-algebra, the working memory demands of the math change and become more sophisticated. In algebra, students will no longer activate working memory as a concrete experience expressed by the symbols, and this is why concrete models of algebra are strongly contraindicated. Instead of activating a concrete picture of the math from the symbols, in algebra, the perception of the interrelationship of the symbolic details activates the working memory/comprehension picture. This symbol-based detail relationship determines our comprehension of the algebra. For example, the detail interrelationship of the symbols indicates:

  • The slope of a line and its y-intercept from an equation in slope-intercept form
  • If an expression is in simplest form
  • The difference between a perfect square trinomial from x(2) + bx + c when both are presented in product form

The sensory-cognitive development necessary for activating working memory from the interrelationship of the symbolic details is called detail analysis. Detail analysis is a higher order executive process than symbol imaging and depends on the development of symbol imaging as its basis, because to be able to analyze the interrelationships of symbolic details, students must be able to perceive, hold, store, and retrieve sequences of numerical symbols to support analyzing them mentally. Detail analysis refers to the ability to select out irrelevant details, focus in on key details and link key details back to the big picture of comprehension. Detail analysis also includes task monitoring as part of its function as a cognitive editing tool, enabling us to check and monitor our work as we proceed through problem solving.

All math has a developmental focus and a content focus. They are entirely distinct, yet the two interconnect constantly in support of one another. The developmental focus of pre-algebra is to engage and strengthen students’ applications of detail analysis in support of their activating and sustaining working memory by perceiving the interrelationship of the symbolic details. The mathematical content focus of pre-algebra is to introduce students to a new kind of problem solving: simplifying variable expressions and solving variable equations.

The Two Posters on Permanent Display In My Algebra Classrooms

In support of the new sensory-cognitive demands for activating working memory in algebra, I prominently place two posters in my classroom to which I refer consistently:

  • Poster #1: Manage the Details, Your Grade Depends on it.
  • Poster #2: Get Your Picture Before Expressing Any Transformation or Solution

The purpose of these posters is to help students become increasingly aware of their own working memory/comprehension pictures through the activation of detail analysis. The way to manage the details is by getting the picture (activating working memory) before expressing any next transformation or solution. The high incidence of detail errors in students’ algebraic problem solving (incorrectly referred to as careless mistakes) indicates detail analysis is not (fully) engaged, meaning working memory for algebra is not established, and therefore, students are proceeding through problem solving while not being able to successfully perceive the details. This is why I require all students to show all detail transformations and do not allow for mental math to replace it.

“I can do this math in my head”, boasts a student, hoping I will be impressed with the mental acumen on display. The expression and tone of my response do not indicate that I am impressed as I point to poster #1: “Doing the math in your head is for elementary school. We are in a new math universe in which the main purpose is to manage the details, so you are required to show you are managing the details by expressing every detail transformation.” It is not about the arithmetic of algebra,” I continue, “It is about your ability to manage all of the details throughout problem solving, no matter how easy you feel any particular detail is. This is helping us to learn to keep our picture for longer and longer periods without making detail mistakes. Managing the details is how we will continue to get a paycheck as adults.”

“But negative seven is close to positive seven,” laments another student who received no credit for a solution by not including the negative signage, feeling unfairly treated because a negative sign is such a small, little mark, rather than comprehending the details were not sufficiently well managed, and the difference in solutions is significant.

To What Does the “Pre” in Pre-Algebra Refer?

Over the last ten years I have observed a lack of consensus on what math constitutes PRE-algebra. The range of content presented in textbooks and in public/private school classrooms around the country is so varied, scattered, and disconnected, I cannot determine any common basis for what is intended by pre-algebra. The frequency with which content from algebra 1, algebra 2, and high school geometry is randomly dropped in pre-algebra students’ laps can create inappropriate scopes and sequences that may seriously impair their ability to learn the pre-algebra content. Some representative examples in which pre-algebra students are expected to:

  • Solve problems with parabolas, but have yet to be introduced to the x-y Cartesian plane or the graphs of lines
  • Solve exponential functions and they have yet to be introduced to constants, variables, expressions, or equations
  • Solve surface area and volume of polyhedrons, 3-d space figures, and composite figures and have yet to be introduced to what prisms, pyramids, cylinders, cones and spheres are as well as finding the area of polygons.

I believe we can all agree on the meaning of the prefix, “pre-”. In the case of “pre-” algebra, there are distinct parameters that distinguish pre-algebra from algebra 1. The function of pre-algebra is to provide the initial foundation and problem solving tools from which all future algebraic processing will emanate. This is especially critical because the developmental demands of activating working memory for algebra are different from and more challenging than those for elementary mathematics, and students’ introduction to algebra needs to be successful and strong to support the higher level expansions coming in algebra 1, algebra 2, pre-calculus, and calculus.

Some Parameters That Distinguish Pre-Algebra From Algebra 1

The “pre-” in pre-algebra is about preparing students for most of what comprises algebraic problem solving: simplifying variable expressions, solving variable equations and graphing. Therefore, it is within the domain of pre-algebra that students learn the integer and rational numbers facts across the four operations, as these number facts will be applied throughout simplifying and solving for all algebra. The extent of simplifying expressions in pre-algebra is basic only, mostly to learn what constants, variables, expressions and equations are and the Order of Operations, GEMDAS, and not for combining like terms which, is an early algebra 1 unit in which students learn what a polynomial is, the degree of the polynomial, and expressing polynomials in standard form.

For solving equations, pre-algebra covers only the first four levels of solving equations in one variable and the first four levels of solving inequalities in one variable. Levels five through fourteen of solving equations in one variable are all covered in algebra 1. The parameter that distinguishes the pre-algebra equation solving levels from the algebra 1 levels is all pre-algebra level equations are ready to solve and all the algebra 1 levels require extra transformations and/or simplifying of expressions before the equations are ready to solve. The pre-algebra equation solving levels provide students with a solid foundation for all future equation solving: an equation is ready to solve when one variable expression in simplest form equals one constant expression in simplest form; and to be appropriate equations for pre-algebra, all variable expressions and all constant expressions are already in simplest form. Any variable or constant expressions requiring combining like terms, use of the distributive property, or having variable terms in both expressions are all exclusively parts of the algebra 1 equation solving levels.

In the ratio, proportion, percent unit, pre-algebra covers the four levels of fraction, decimal, and percent equivalence, the introduction of ratios, rates and proportions, the three forms of percent problem solving, and solving basic proportions with a single variable and three constants. Solving percent change and proportions with binomial numerators and/or denominators is covered in algebra 1.

The pre-algebra probability unit covers all basic probability including theoretical and experimental probability, simple, multiple, and compound events, dependent and independent events, dependent and independent events with and without replacement, and the counting principle. Permutations (nPr) and combinations (nCr) are typically presented in second semester algebra 2.

The pre-algebra linear graphing unit starts with graphing on the x-axis and solving levels one through four of inequalities. Pre-algebra follows with graphing on the x-y Cartesian plane, by first introducing solving equations in two variables and connecting the infinite solutions generated by 2 variables in one equation to the graphical expression of the infinite solutions as infinite points comprising a line. From this experience students learn slope-intercept and standard forms of linear equations. The extent of the linear graphing unit for pre-algebra is students’ ability to graph any line with positive, negative, and special slopes from any given equation, and to generate the equation of any graphed line with positive, negative, and special slopes. Algebra 1 continues this development by teaching students to generate the equations of lines algebraically without seeing the line on a graph.

Prescribed Sequences to Maximize Learning and Integration for Each MMR Course

Prescribed sequences to maximize learning and integration for each MMR course

For years educators have asked me to help guide them through the MMR course series, according to the grade levels of the students they work with. Now that the MMR professional development course series has grown to 12 seminars, I would like to introduce the Recommended Course Sequence Guide to help you navigate the course schedules and maximize your learning and integration of the course content.

All of the Making Math Real courses are designed with a highly structured and sequentially developmental architecture. All of the content from each course emanates directly from the content of the course(s) that precede it. The intended sequence for the current 12 seminars is:

  • The Overview
  • Kindergarten
  • The 9 Lines Intensive
  • 4 Operations & The 400 Math Facts
  • Time & Money
  • Fractions, Decimals & Advanced Place Value
  • Games!
  • Geometry Part 1
  • Pre-Algebra
  • Algebra I
  • Geometry Part 2
  • Algebra II

Since it would be overwhelming to require participants to follow the intended order or to schedule courses exclusively by the intended order, I have created several recommended course sequences that may facilitate participants getting what they most need more quickly and more directly. The Making Math Real Recommended Course Sequences Guide provides the prescribed sequences to maximize learning and integration for each course.

Since it is not always possible for course participants to schedule enrollment in Making Math Real courses according to these recommended sequences, there are numerous options available, once the Overview has been completed. If you have any questions about the courses and sequences that interest you, please do not hesitate to contact the Institute at info@makingmathreal.org. We look forward to hearing from you!

Choose from 3 Sequences

An Invitation to Share

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.

READ STORIES WE HAVE RECEIVED

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.

SHARE YOUR STORY WITH US

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:
dyslexiaida.org/living-with- 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.

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