 
# Early Immersion Math

Tapping into the Linguistic Genius  
of Young Children

by Evelyn Raiken Lewis

Copyright 2016 Evelyn Raiken Lewis

Smashwords Edition

License Notes

Thank you for downloading this ebook. This book, previously called High-level Math for Little Tykes, remains the copyrighted property of the author and may not be redistributed to others for commercial or non-commercial purposes. If you enjoyed this book, please encourage your friends to download their own copy from their favorite authorized retailer. Thank you.

Also visit my blog: <https://gamechangerforkids.wordpress.com/>

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Table of Contents

This discussion is organized in three sections: The first section, Chapters 0-3, is a discussion of the hypothesis and backup research. The second section, Chapters 4-5, is an evaluation of the failures of the current education model. The third section, Chapter 6-7, is a discussion of how to tweak the education model to be in alignment with the biology of learning language.

Chapter 0: Preface

Chapter 1: The Hypothesis—The Punchline

Chapter 2: Young Children are Natural Mathematicians

Chapter 3: The Systematic Decline

Chapter 4: Descent into Math Hell

Chapter 5: Irreparable Harm

Chapter 6: A Model for Learning Math

Chapter 7: New Approaches to Teaching Mathematics

Recommended Links, Reading and Resources

Acknowledgements

About the Author and Contact Information

# Chapter 0: Preface

Only babies and young children can learn language by immersion. They learn language by interactive conversations and by listening to conversations of others. No lists to memorize, no workbooks, no tests. Using language as a comparison basis, I will ask us to consider if babies and young children can just as easily learn the language of math with no lists to memorize, no workbooks, no tests.

This isn't to say that no effort will ever be required, but the rewards of that effort will be greater. Again, using language as a comparison basis, at five years, children competently speak in their native language. Effort expended will be in discovering new thoughts and ideas. Compare this to a teen or adult learning a new language. There will be lists of words to memorize, conjugations of verbs to practice, grammar, and sentence structure to get right. There is no time for humor, irony or the profound. Learning a new language requires hard work and perseverance.

Similarly, learning math is thought to require hard work and perseverance, but perhaps this is because children have missed learning the language of math in the critical years for learning language when the human brain is automatically doing the statistical analyses that learning a new language requires.

### Tapping into the llinguistic genius of children

Consider the computational analysis required to learn the algorithms of the structure of a language by analyzing sound. Then consider if kindergarteners who have sorted out their native language through statistical analysis of sound are capable of more than adding with pebbles.

I will ask us to reexamine the conventional wisdom which asserts that children must work progressively step by step from simplistic arithmetic before moving on to more complex algebraic and geometric concepts.

I will explore the idea that children can learn mathematical vocabulary and concepts as easily as their native language if they are given the opportunity at the critical time to learn language. A solid understanding of vocabulary and concepts so easily learned in early childhood would provide the foundation for enhanced proficiency, more sophisticated calculations, and profound discoveries.

### Challenging our educational model

The failures of the current education models challenge us to look at reasons for the weaknesses in these models. The proliferation of dual-language immersion preschools and early elementary programs speaks to our understanding of the advantages of learning language in early childhood. One critical component of these programs is having proficient teachers fluent in each language.

Similarly, with math, for a program to be effective teachers must be proficient and fluent in mathematical thinking. Current math programs for young children might be constrained by the level of fluency and expertise of preschool and early elementary school teachers.

The idea to teach math as a language in the earliest years of life is not yet validated, but one argument in its favor is that if we think of math as a language, then it follows that math and mathematical thinking can be easily learned and become integrated with one's native language.

### The madness behind the method

The idea of bringing high-level math vocabulary and concepts to young children was born of my experience as a math tutor. My son had done well in math and I thought I could help remedial math students catch up. After all, I thought at the time, these students had learned the lyrics to every popular song on the charts, they could certainly keep a few math algorithms straight.

As I worked with students, I could see that the students did not have a picture in their mind's eye of the problem they were trying to solve. Students had memorized formulas but often could not read clues in a problem to determine which formula to apply.

### Meet Mary

One high-school student I tutored, I'll call her Mary, was struggling with a simple math problem: Find the area of a tennis court given its length and width. Mary scrolled through several formulas she had memorized looking at me for clues on which to use. Mary was trying to solve the problem without a picture in her mind's eye of the terminology. I made a labeled paper cut-out. Mary was surprised, "That's a square foot?"

I thought about how other students would come to understand concept of a square foot. They might have seen a parent measuring a room to figure out what size sofa to buy or they might have measured a plot in a garden.

I asked Mary if any of her teachers had demonstrated examples of this problem, such as measuring the length and width of their classroom or a classroom blackboard to calculate square footage. Mary said that they had not. Mary never asked her teachers for help, nor had she seen this problem brought to life with a demonstration. So, like many mathematical problems, finding the area of a tennis court was an abstract idea that she thought she would never understand. And the problem was irrelevant in her mind, something she would never have to do in "real life."

Once Mary understood the term 'square foot,' the problem was easy for her. However, her increased understanding of concepts did not inspire her to skip her TV shows in the evening and spend time working on additional problems.

### Priorities change in teen years

Even though students caught on quickly I could not motivate them to practice even in the face of an important test like the SAT. That would require belief in themselves which was long gone.

I gained clarity on my failure to motivate students when I read current research on learning and brain science as part of my job at the University of Washington (UW) Human Subjects Division. In particular, the publications of  Doctor Patricia Kuhl.

Kuhl discusses how quickly and effortlessly babies learn language and describes this "as nothing less than rocket science." Kuhl identifies the critical time to learn language and maps the systematic decline in the ability to learn a new language.

We can all see that children learn language easily but perhaps don't consider the temporal nature of this capability. I considered that the research mapping the systematic decline in the ability to learn a new language might apply to math. I compared the students I was tutoring in math with the memories of students in my high-school French class back in the day.

### My foray into French

The idea that young children learn language easily is not a new one. However, until I read Dr. Kuhl's research, I took for granted that I easily learned English. I didn't consider it a "breathtaking feat" as Kuhl describes the early acquisition of language. Whereas Kuhl asserts "Babies are geniuses until they turn seven." I never considered myself a genius for learning English. It was the most common of accomplishments.

Allowing that I might have been a genius learning English, I was no genius in my high-school French class. I don't remember much French, considering two years of classes, but I do remember my French teacher who would become so rattled by our responses she would stop class and take a moment to shuffle through her purse for one of her little white pills. She always kept a glass of water on her desk for these emergencies. I can look back and understand her frustration. Afterall, as a native of France, she learned French easily. Why couldn't we get it?

Unlike my French teacher who needed pharmaceuticals to deal with our incompetence, my expectations were low. Some students were able to get good grades by studying more, but there was not one student who was inspired to study beyond class assignments and test prep. We didn't care. The same refrain heard in math classes, "what do we need this for?" was palpable, though unspoken. High school was not the optimal time to be learning a new language, and we seemed to know this. None of us felt dumb for not getting it. We were much more engaged with the wisecracker in the back of the class than the French language. Our attitudes were strikingly like the remedial math students I was tutoring for their easy distractibility.

Moreover, if our prospects were judged by our ability in the French language, expectations would have been low. Fortunately, incompetence and lack of interest in foreign-language classes are not judged in the same way that incompetence and lack of interest in math classes are judged even though the obstacles are perhaps the same.

### The missed opportunity

Kuhl's research led me to consider that the language of math might have the same learning trajectory as spoken languages. If true, then it follows that math could be as effortless to learn as a child's native language.

Researchers have not studied the optimal age for learning the language of math; however, using language as a basis of comparison, I am asking us to consider the critical time for learning the language of math is also the first seven years of life.

Back to Table of Contents

# Chapter 1  
The Hypothesis—The Punchline

"Is China genetically engineering super-smart babies? How else can you explain that by the time they are three they can all speak Chinese?" - Stephen Colbert

### Genetically engineered babies

Babies might not appear to understand mathematics but behind the scenes they are brilliant analysts genetically engineered to decode any language they hear. If babies hear French, they will learn French. If they hear Swahili, they will learn Swahili. If babies hear two languages, they will learn two languages and keep them straight. (Keep this thought.)

This inquiry will consider the possibility that if babies and young children hear the language of math as part of their speech community, they will learn the language of math.

### The Hypothesis

Math vocabulary and concepts are most easily learned in the first years of life.

It is the delay in building an early foundation in math that makes math difficult for students.

Applying this hypothesis:

Is it a good idea to introduce the vocabulary and concepts of economics and finance to a toddler who can't get a spoonful of food in her mouth without getting it all over her face, and clothes and the floor?

Yes, it would seem.

Is it a good idea to introduce vocabulary and concepts of algebra and geometry to a first-grader who has a meltdown if he doesn't get as many fries as his sister?

Yes, again.

### A once-in-a-lifetime opportunity

Patricia Kuhl, PhD of the Center for Learning and Brain Science (ILABS) at the UW studies how babies and young children learn language. She has mapped the critical time for learning language and the systematic decline in the ability to learn language after this period. The graph below and other graphs in this publication are based on Kuhl's publications and TED talk  "The Linguistic Genius of Babies."

Kuhl explains:

What we see here is that language has a critical period for learning. The way to read this slide is to look at your age on the horizontal axis. And you'll see on the vertical your skill at acquiring a second language. The babies and children are geniuses until they turn seven, and then there's a systematic decline. After puberty, we fall off the map. No scientists dispute this curve.

### A unique biological ability to learn language by immersion

Babies, it seems, will put anything in their brains (for lack of a more precise term.) They are genetically engineered to take in random sounds and organize them into linguistic thoughts. They analyze, categorize, and make correlations with any conversation they hear. Automatically.

Babies and young children love interaction, but they will also listen in on adult conversations whether on serious subjects, profound ideas, or mathematics. By the age of five they understand most of the conversations they hear. They have learned their native language by immersion. They not only parrot what they have heard, they manipulate language to express their thoughts and ideas.

A family speech community is not necessary. Young children who immigrate to a new country can learn a new language at school by listening to teachers in the classroom and children in the school yard. Within a few years are translating for their parents who immigrated at the same time have taken classes and memorized lists of words.

### Cracking the speech code

Young children are able to crack the speech code of the dominant language(s) they hear. Like super sleuths they are always on the job. They don't miss a clue. They never give up. They are proficient detectives uncovering the elements of the language common to their immediate culture. They absorb all vocabulary within earshot (unfortunately, in some cases). They uncover the rules of grammar, however complex. They study the pitch, the tone, and the rhythm of language. They have a once-in-a-lifetime ability to organize the speech sounds they hear into language.

### They're still throwing temper tantrums

The idea of introducing high-level math vocabulary and concepts to babies and young children is counterintuitive. Their underdeveloped motor function and tenuous emotional stability suggest that they would not be good candidates for learning anything beyond basic arithmetic. And this is exactly how most children spend their pre- and elementary-school years—doing basic arithmetic.

There are many things that have to

### The speech community

The lack of a speech community with a proficiency in mathematics is often the limiting factor in early education programs. Even teachers with proficiency in mathematics are held to a curriculum of basic arithmetic. Many students do not experience a speech community proficient in mathematical thinking until they are past the age when they can learn language by immersion.

To understand the disadvantage this presents for children, suppose pre- and elementary teachers in the U.S. were mandated to teach Swedish, having never learned to speak Swedish themselves. Teachers would make a presentation copied from the teachers' guide and assign a couple of pages from a workbook. Then they would use the teachers' guide to grade tests. Of course, any students whose parents speak Swedish at home will do well.

The other students would recognize that Swedish is not part of the teachers' vocabulary, or relevant other than a required task, and will put minimal energy into it. Any Swedish they learn for a test will go into short-term memory, which purges at the end of the school year, or even at the end of the weekly test.

### Aging out of biological ability to learn language by immersion

We take it for granted that children easily learn their cultural language(s) and we also know that learning a new language later in life is challenging. This ability in babies and young children to learn language by immersion and the systematic decline in the ability to learn a new language by immersion in later years is the basis for the hypothesis that the language of math, its vocabulary and concepts, like spoken languages, are most easily learned in the first years of life.

This discussion will consider whether children given early exposure to mathematical thinking have better potential to progressively develop proficiency over time. If we consider that it takes time and experience to understand the mathematical properties of the physical world, we might consider facilitating this knowledge with an early start.

Back to Table of Contents

# Chapter 2  
Children are Natural Mathematicians

### Babies are natural statisticians

Math comes naturally to babies. Kuhl's discusses how babies absorb statistics in the acquisition of language in her TED talk "The Linguistic Genius of Babies" saying,

During the production of speech, when babies listen, what they're doing is taking statistics on the language that they hear.

They take statistics to determine what mathematicians call the order of operation. In the English language the sentence structure order of operation is subject, verb, object. By the time children are two, they use this algorithm. My son would say, "Danny do it." By five years old they are well on their way to speaking with precise verb tenses to convey how action relates to time.

While everyone takes for granted that children easily learn the rules, or algorithms, of complex languages, a true appreciation for the brain's ability to master the complexities of language would make it difficult to negatively regard a child's capacity to learn math.

###  What is math?

While libraries categorize math as a science, math has more recently been thought of as a language, a philosophy, or a system of logic. It has widely differing definitions, some of which are quite amusing such as Vi Hart's definition "Mathematics is about making up rules and seeing what happens." But more relevant to my inquiry is a R. L. E. Schwarzenberger's definition:

Mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the fundamental theorem of algebra using the English language.

Math is a language that can help us more precisely describe things. Without math, we might be limited to adjectives and vague descriptions, such as "a long way," "as big as an elephant," or "it's curvy at the top and gradually flattens out."

Math can describe what we cannot see and forecast what is yet to come. Math can be used to model scenarios based on current and past data to make decisions such as "Can I really afford this car?"

If we think of math as a language that is easy to learn in early childhood but difficult to learn in teen years, teachers and administrators would understand that they cannot put teens in remedial math classes and expect them to catch up with students who have incorporated mathematical thinking into their native language from early childhood.

###  Naming brings things into existence

Young children, as they are learning to name things, are also starting the analysis of more abstract concepts. It does not enter their minds that mathematical concepts are too difficult to learn, nor will they decide that mathematical concepts are not worth the trouble to analyze. They want to be part of the conversation. They do not discriminate as they interpret sound patterns into ideas and concepts. They sort out whatever dialect they hear. They do not lock in value judgments such as, "I'm not smart enough." They never question, "Do I really need to know this?"

Most young children are given the vocabulary to express hundreds of conceptual ideas such as sharing and compassion. In the case of black communities, children have a complex rubric to follow to keep safe by not offending those who might have racial biases. They are given instructions regarding clothing options, word choices and body language.

I would argue that these concepts are more complex than properties of triangles and circles.

### Children love to calculate

By the time children can crawl, they intuitively understand that there is an interrelationship between distance, rate, and time. They crawl faster when they want to get somewhere sooner. As toddlers, they run to the end of the block to get there before their mothers.

Since children intuitively know the relationship between distance rate and time, we should consider introducing them to the terminology distance, rate and time and the formula of calculating one variable given the other two. With an early start and relevancy based in their experience, they might be less likely to zone out years later in algebra class when they see a problem that begins, "Two trains leave the station 20 minutes apart."

Toddlers are dependably interested in the physics of the displacement of water. They try to get the biggest splash when they jump in a puddle. Yet, in our current educational model, toddlers, who intuitively understand displacement of water are not thought ready to learn the terminology for this fascinating phenomenon. Schools wait until the tween years and then assign problems to calculate the displacement of water in the context of a container ship. Who can blame students for finding their phones more interesting?

Children analyze outcomes. If their intentions and outcomes are not in line, they get immediate feedback—through a disappointing splash, a throw that falls short, or a jump that results in a fall. They make adjustments and try again. This dynamic exploration of their bodies to the world might the optimal time to add mathematical terminology to their calculations.

We should ask ourselves if it would be more effective to teach math to young children in the context of their interests than wait until they are in algebra class and expect them to do calculations which seem to have no relevance. From my experience, it could not be less effective.

### Safe to daydream

"Learning will not occur unless children are comfortable and safe to daydream, to indulge in 'the sweetness of a moment's detachment." - Ruthy Alon, _Mindful Spontaneity_

When children learn about themselves and their world, they feel more empowered to be themselves in the moment and express themselves appropriately in the moment. They develop situational awareness. Children's books and PBS children's shows help children unpack emotional thoughts and challenges. Music, art, and sports activities allow children to feel empowered by developing the interconnection of their brains and bodies.

Perhaps we can also develop children's sense of empowerment and situational awareness by offering children an understanding of how the world works in a mathematical context.

### The speech community

Many children learn mathematical reasoning from the earliest years of life by listening in on family discussions. This allows them to automatically integrate the language of math into their native language.

Children might hear parents making household decisions involving measurements and materials such as, "We need to put up a six-foot fence to keep the dog in the yard," or, "Should we buy a 19 or 26 cubic-foot fridge?" They might hear terminology such as price-per-square-foot or engineering concepts such as load-bearing walls. They might hear parents making purchase decisions based on bandwidth or download speed. They might also hear these terms used out of their original context such as "He doesn't have the bandwidth to solve this problem."

Children might listen in while parents make investment decisions. Terms like compound interest or rate of return are not as easily comprehended as the word 'dog' but hearing discussions that include financial terminology establishes a foundation for financial literacy. These discussions, which seem unremarkable as they happen, can help normalize the language of math for children and become another way of connecting mathematical concepts to lived experience.

Adding definitions in math class will add insight as if a search engine is on the lookout. When relevancy has not been established, the concept of compound interest might seem overly complicated and irrelevant.

Children who have been introduced to these terms as part of everyday conversations have the advantage when they come up in math class. Children who have incorporated mathematical thinking in this way are likely to become increasingly proficient. Thus, mathematical reasoning is passed generation to generation and becomes part of a family's native language.

### "Unless you're an infant or young child"

"You won't be picking up any complex grammar structures through osmosis. Unless you're an infant or young child, advanced proficiency or even fluency will take more than having foreign words yelled at you." GoAbroad.com

In the promotional text of the travel website GoAbroad.com, is a warning to prospective travelers or their parents who might have "romantic notions" that they will become proficient in a new language in a language immersion program. This warning might be the result of complaints of disappointed travelers who returned home without picking up more than a few words and phrases in a new language. Travelers, who in all measurable ways seem more competent than babies, might have thought they could do what babies do so easily.

Those of us who speak intuitively often don't appreciate the incredible amount of analysis required to master a language. Take the English language for an example. English has at least a dozen verb forms depending on relative time. There are many irregular verbs due to the many languages from which English is drawn. English uses split infinitives, most typically depending on whether the negative form of the verb is used. English has a different sentence structure for statements, questions and negative statements. Most sentences are made of several clauses which often have different verb tenses and sentence structure. This is just a start describing the complexity of the English language.

Yet, babies figure it out.

### Students who "get" math

Educators often view math as a difficult subject better left to later years when students have mastered arithmetic. Algebraic and geometric terms and concepts are not introduced until middle school. By then, children's natural ability to learn a new language is dropping off the charts. Children's genius for learning language is sacrificed. Children start failing.

Research suggests that early learning provides a foundation for expanding concepts in later years. Kuhl writes in "Early Language Acquisition: Cracking the Speech Code:

[E]arly learning promotes future learning that conforms to and builds on the patterns already learned but limits future learning of patterns that do not conform to those already learned

If we think of math as a language, it follows that an early introduction to the vocabulary and concepts of math would promote future learning. For example, if children have exposure to the mathematical term average, they can easily grow this knowledge in algebra class to include the terms median, mean, and mode.

Children that hear conversations that include terminology such as percentage, derivative, and exponential growth will be able to build on these concepts. If we apply Kuhl's research that "early learning promotes future learning that conforms to and builds on the patterns already learned" it follows that children who have this early foundation will become progressively proficient. Their understanding will be at a deeper level than top-down instruction ever can be.

When children hear mathematical terms in context, they also have the advantage of understanding the relevance of the terminology, a relevance which is often missing in school math curricula.

These children became the "smart ones" because they have been evaluating math vocabulary and concepts since infancy, even in utero, as research suggests. They were introduced to terminology and concepts in the context of real-world problems through family conversations. These children were listening in, and analyzing math terminology, even if they are seemingly in their own world, eating, daydreaming, or playing with their toys.

It takes time for children to fully understand conceptual ideas, and longer to verbalize what they understand, and even longer to compute with variables. This may be why students who have listened to conversations from early childhood that include mathematical terminology "get" math.

Recognizing this, and recognizing the ease that young children learn language, there is the potential to provide students with a foundation that gives them an intuitive knowledge of variables so they might be less confused when asked to solve problems in algebra class and beyond.

Our current educational system does not provide this foundation. Rather, it is geared toward procedural computations that don't require teachers to have mathematical proficiency. Elementary curricula are increasingly designed for worksheets and bubble tests rather than a young child's curiosity and propensity to learn.

Specific ideas on providing young children with a speech community proficient in the language of math are discussed in Chapter 7.

### Meet Rodrigo

Rodrigo is a young mathematician from Argentina who recently graduated from the University of Washington. Since Hispanic mathematicians graduating from the UW are something of an anomaly, I asked about his childhood experiences and how they led him to be a mathematician. Rodrigo grew up in an economically and culturally advantaged family. His parents and their friends would all chat around the dinner table about things that didn't interest Rodrigo at the time. But the high-level conversations that Rodrigo and his two brothers listened in on may have contributed to their choosing math-based disciplines as careers, Rodrigo studying math, one brother is a computer scientist and another an engineer.

Back to Table of Contents

# Chapter 3  
The Systematic Decline

We casually assume that children will easily learn their native language. But anyone who has tried to learn a new language later in life knows that there is nothing easy or automatic about it. Kuhl has mapped this decline in the ability to learn a new language.

If we apply this map to learning math, we can see the disadvantage students experience in middle and high school. Many students do not experience a math teacher until they are well beyond the critical time to learn language as shown in the graph below.

One reason that students fall behind might be that they do not get exposure to mathematical thinking at an age when it would be most beneficial, when their brains are in high gear for learning language, including the terminology and concepts of math.

In a standard curriculum, children learn to count to 10 in kindergarten and learn a few basic geometric shapes, in first grade they learn to count to 30, add and subtract with single-digit numbers. In second grade, children learn double-digit addition and subtraction. Second graders are seven years old, the age when they are losing the ability to learn a new language.

In the standard curriculum second graders have learned fewer than 100 math terms and that includes numbers. Even teachers with mathematical expertise who could teach higher-level math will likely follow this standard curriculum.

Contrast this with children's comprehension of their native language. Five-year-old children comprehend about 10,000 words, 20,000 at age six, and about 30,000 at age seven but only about 100 math terms if their exposure is limited to preschool and elementary school programs.

Patricia Kuhl writes in "Early Language Acquisition: Cracking the Speech Code:"

[L]earning produces a neural commitment to the patterns of an individual's native language, and this constrains future success at acquiring a new language.

Many seven-year-old children have not been introduced to the rich vocabulary of math. They are aging out of the critical time to learn language when learning this vocabulary would have been easy for them. If they follow the trajectory of the decline in the ability to learn a new language, they will likely have more difficulty acquiring a sizeable mathematical vocabulary. This difficulty is often interpreted as laziness or a bad attitude.

### Learning a new language gets harder as years go by

Unlike immigrants coming to a country who expect struggles with an unfamiliar language, students who have not become acquainted with mathematical concepts in early childhood question their abilities when they don't understand what others do.

Math teachers know too well the frustration of teaching math to t(w)eens who do not have a foundation on which to build. These teachers are often heard in the teachers' lounge or even the hallways lamenting their wasted lives.

Math teacher John Bennett even went so far as to give a TED talk arguing against math requirements for middle and high school students. In his talk, "Why Math Instruction Is Unnecessary" Bennett argues that most people don't need algebra in "real life" anyway. In his talk, Bennett details how he got to this dispirited place. He says that he didn't always believe that math education was a waste of time. He did, after all, become a math teacher.

Rather than debate the necessity of algebra in "real life" it might be more useful to consider math a language and note that his students' brains are doing what they are designed to do; that is; blocking out new language patterns. Bennett and others have found that there is nothing they can do about this reality which is usually not attributed to biology.

Going out on a limb (where I often like to hang out,) teachers who are now teaching remedial math in high school might be more effective teaching toddlers and early elementary students as specialists in schools. They would be teaching math concepts to children at the age when they are most interested in learning about the algebraic and geometric properties of the world.

Even motivated university students taking an elective language course face an uphill battle to learn the basics of a new language as James Romm, Professor of Ancient Greek, described in an article "Beginning Greek again and again:"

Sisyphus would sympathize with my condition. Every year I begin rolling my stone up a four-month-long hill, my hopes high. Every year I end up far closer to the bottom than the top. Some of my students still, after 120 hours of instruction, take the first noun in a sentence as its subject, no matter what form it's in. Their habits of 15 years of reading English will not give way to the methodology that an ancient language demands.

Presumably, children of ancient Greece had no difficulty learning the order of operation of the Ancient Greek language.

### "The early catastrophe"

Over a half century of research into academic achievement ties academic achievement to a rich early language exposure but our school system does not consider that a rich language exposure in early childhood applies to math. They do not consider the systematic decline in the ability to learn a new language might apply to math.

This research correlating academic achievement and a rich early language exposure began in the 60s as part of former President Johnson's War on Poverty. Researchers Betty Hart and Todd R. Risley studied early language development and found that "the sheer number of words heard varied greatly along socio-economic lines."

They wrote in "The Early Catastrophe: The 30 Million Word Gap by Age 3" that a child from a high-income family will hear 30 million more words within the first three years of life than a child from a low-income family.

[C]hildren from families on welfare heard about 616 words per hour, while those from working class families heard around 1,251 words per hour, and those from professional families heard roughly 2,153 words per hour. Thus, children being raised in middle to high income class homes had far more language exposure to draw from.

This research indicated an obligation to compensate for inequality in economic conditions. The Head Start program was created to address this inequality and better prepare children of low-income families for school. It has been successful in helping these children start on par with their middle-class schoolmates, and there are long-term positive outcomes for children enrolled in the Head Start programs. Continued academic proficiency is not one of the positive outcomes.

While Head Start Children begin kindergarten on an equal footing with their middle-class peers, academic gains regress as children move through elementary school. In the third grade children start experiencing a "fade-out effect."

Head start does not provide a linguistic foundation for higher-level math which might in part explain this regression. The Head Start program does not introduce algebraic and geometric mathematical ideas but sticks to repetitive simplistic arithmetic.

A policy analysis from the Southern Regional Education Board addresses this issue which applies to Head Start as well as kindergarten:

The researchers found that kindergarten teachers spent more days a month on basic, repetitive content than on advanced reading and math material.

We can ask ourselves if young children are capable of so much more. Are precious years of easy language acquisition wasted pushing pebbles around? Are we squandering the mathematical potential of our young children?

The language exposure in middle-class and professional families likely included math terminology so that mathematical thinking would become part of a child's native language. Children, who have been listening to a rich vocabulary at home, including the vocabulary of math, seem to have the true head start.

### The correlation between wealth, education and SAT scores

It seems that the Johnson-era recognition that family wealth and education play a role in their children's academic achievement continues to hold true. In 2014, the Washington Post published an article correlating parent education and wealth with SAT scores.  "These four charts show how the SAT favors rich, educated families" by

Zachary A. Goldfarb writes:

A student with a parent with a graduate degree...on average scores 300 points higher on their SATs compared to a student with a parent with only a high school degree...[I]t also dispels the notion that students in America have good opportunities to advance regardless of the family they're born to.

The reasons for the linear correlation between students' SAT scores and parents' education are complex and varied; but higher SAT math scores of children of college educated parents suggest a possibility of a correlation between hearing math terminology and concepts during early childhood, and mathematical problem solving in later years. The exposure to math concepts at the dinner table could be a factor in students' math competency.

There is also a linear correlation between family wealth and SAT scores. The reasons for this correlation are similarly complex and varied, but one simple advantage that applies to my hypothesis is that parents, with the means to do so, tend to make long-term investments and buy more stuff, and these investments and purchases often include discussions that include math-related terms. Children of wealthy parents might hear strategies for making financial investments with vocabulary such as capital gains, price/earnings ratios, or rate of return.

Family discussions of purchases might include features including dimensions and properties of materials. Discussions might include percentage off the retail price. Children who hear their parents calculating the price of an item at 25% off and adding 10% tax to determine the cost will convey the importance of working with percentages.

Children might hear discussions of car features when their parents are buying a car such as fuel efficiency expressed in miles per gallon, traction control, or seconds from zero to 60. They might hear deliberations on the benefits of short-term verses long-term financing, or they might hear discussions about the total cost of owning a car including insurance, gas, licensing, and depreciation. All these discussions would inform children about mathematical concepts.

Limited purchasing power can limit conversations including math-related considerations. Low-income parents might go to used-car dealerships wherever they can get financing and buy from a limited stock. In these cases, there might not be a dinner-table conversation in which children would hear the mathematical considerations of car features and financing.

Children of single parents and/or children whose parents work long hours away from home are less likely to hear math-related conversations at the dinner table.

### Justifications for the low math competency of our students

When students don't understand math, the biological decline in the ability to learn language is not considered. Rather, children's confusion leads to erroneous perceptions about the students' character and intellect. Insults start flying suggesting attitude problems and substandard family structure.

My child/this student is creative, not academic. (Suggesting that creative children cannot learn math)

My child/this student has other interests. (Putting a positive spin on low math competency)

No one in our family is good with numbers. (Genetics)

Students who understand math are naturally gifted. (Genetics)

Children these days are too lazy to do their homework. (Substandard students)

Parents of children in some racial, ethnic groups don't value education. (Substandard families)

Children are from broken, dysfunctional families. (Substandard families)

Inner-city children have too many other problems to concentrate on math. (Substandard communities)

It's difficult to teach math with a culturally diverse student population. (Even though math transcends cultural barriers.)

Calculators and computers have replaced the need to learn math. (Math is irrelevant.)

I was never good in math and look how well I did. (Math is irrelevant.)

Class sizes are too big. (Schools need more money.)

The school day/year is not long enough. (Schools need more money.)

School funding has been cut. (Schools need more money.)

Some races of people are smarter. (Systemic Racism)

These "reasons" impede progress to address underlying shortcomings in our educational programs.

Back to Table of Contents

#

Chapter 4  
Descent into Math Hell

###  Hooked on instant gratification

There is a persistent idea in our education system that learning math is a series of sequential steps starting with the most simplistic arithmetic. Young children are generally successful with this. Their success at these simplistic problems provides instant gratification for parents and educators. Children's prospects look bright and educators appear to be setting children up for future success. No one seems to notice the discrepancy between the child's incredible ability to analyze algorithms of language and the simplistic math curriculum they are offered.

No one notices deficiencies in the program until about fourth grade when the confusion of finding the lowest common denominator leads many students to decide they are not smart enough for math. A few years later, algebra seems incomprehensible.

This confusion creates the perception that math is a drudgery. Filling out worksheet after incomprehensible worksheet often becomes an exercise in frustration more than a learning tool. The number of wrong answers becomes the focus. Math becomes a chore, and worse, something to fail at.

Unlike the social, collaborative way children learn language, children fill out math worksheets in isolation. They are not allowed to "cheat" by copying. Talking is not allowed.

Year after soul-crushing year, these math worksheets are collected and graded, checkmarks are tallied up, and the papers are returned with incorrect answers marked in red ink. Students are aware of every mistake they make. They don't see the red checkmarks as an opportunity for advancing their knowledge. More often, students sit alone with their worksheets and are sometimes required to take their embarrassing papers home for a parent's signature. They internalize the red checkmarks as their own failure.

Alternately, in some math curricula, students work in groups. These collaborative groups are equally ineffective because they are arbitrary groups of similarly confused students. Even if there are students in the group who know how to solve the problems, they may not be able to communicate their intuitive way of solving problems to others. Working in a group of similarly confused students will not have the same benefit of working with a person who understands math is able to explain the concepts.

To illustrate, imagine putting children in arbitrary groups to learn the Chinese language without a proficient speaker. I have used this example using Swedish, but I think the idea is worth repeating. Hard as the students might try to learn Chinese from a non-speaker of the language, they will have limited success. Of course, there might be students who hear Chinese spoken at home. They will be considered the smart ones. To add insult to injury, students in such a language group might hear, if that curriculum were anything like the usual math curriculum, "Why can't you get it? Some of your classmates get it. A billion people on this earth have learned to speak Chinese without any problem at all."

The outcome of both the isolationist and collaborative curricula is that many students experience a downward spiral as they go through school. They cannot even imagine successfully learning math, just as I, myself, cannot even imagine successfully learning Chinese.

Math workbooks are as incomprehensible to many students as if they were written in Chinese (for those who do not speak Chinese.) The result of this worksheet-dependent curricula is that after a decade of math classes many students need remedial help in basic arithmetic and elementary algebra.

###  The sorry state of American education

There are several cross-national assessments that rank students in math competency. A 2015 international math assessment by the PEW Research Center ranked the U.S. 38th out of 71 countries tested. Among those with higher rankings included Estonia, Viet Nam and Latvia. The U.S. was not even in the same league with most industrialized countries.  https://www.pewresearch.org/fact-tank/2017/02/15/u-s-students-internationally-math-science/

Yet, the U.S. spends more on education than any of these countries? The average student in Singapore is 3.5 years ahead of her U.S. counterpart.  https://www.theguardian.com/us-news/2018/sep/07/us-education-spending-finland-south-korea

Acknowledging that the global economy favors math-dependent skills, low scores of U.S. students on standardized tests are causing a great deal of anxiety in parents, teachers, school administrators, and policy makers. This anxiety, as is so often the case, is leading to misguided reasoning in curriculum development and irreparable harm to students, discussed in chapter 5.

### Extrapolating conceptual math from task-oriented math

One explanation for low U.S. achievement scores could be attributed to the years spent filling out rote worksheets in basic arithmetic. This is due in part to our perception that arithmetic must progress step by step from the simple to the more complex. For this reason, it is not thought necessary to provide young children with math specialists. While this is true for math calculations, it is not necessarily true for understanding concepts.

Students who have been limited to the elementary school curriculum of arithmetic often have difficulty when they start middle school and are asked to solve conceptual problems which they have no means to understand. They cannot extrapolate higher-level math concepts from their experience with rote worksheets, so they end up accepting failure as inevitable. They will have to memorize math vocabulary and formulas, and we all know how well that works!

###  No need to be a math whiz

Because math education in early preschool and elementary years is limited to basic arithmetic, teachers' proficiency in higher-level math it is not thought necessary. There are no state educational requirements for most daycare or preschool teaching positions. Requirements for teaching in public elementary schools are graduation from college and a teaching certificate. Generally, the only math course required to get an elementary teaching credential is a review of basic arithmetic and an introduction to latest "reform" math curricula. Many are drawn to early education because it is a field without math requirements.

The Institute of Medicine and the National Research Council published a report "Transforming the Workforce for Children Birth Through Age 8." They describe early education as problematic. The study points to stress and lack of proficiency undermining children's development and learning. It states:

"Adults who are underinformed, under-prepared, or subject to chronic stress themselves may contribute to children's experiences of adversity and stress and undermine their development and learning."

For many preschool and early elementary teachers, confusion is their last remembrance of math. Children learn from their teachers to be anxious about math in the same way they learn to be anxious about people of dissimilar demographic groups when their teachers are of that mind.

Teachers who do not have a math background, however nurturing and dedicated, cannot be depended on to provide a foundation in mathematical vocabulary and concepts. External pressure for students to pass standardized tests is no substitute for math literacy. Meanwhile children are expected to understand what their parents and teachers do not understand themselves. Because of the struggles inherent in this situation, failure can feel inevitable.

Many educational initiatives have been developed and implemented. In addition to these educational initiatives, discussed in part in Chapter 5, longer school days and year-round school have also been promoted. However, many young children spend their day with math-averse teachers, and then go home to math-averse parents. In this scenario, math proficiency is not more likely in full-day kindergarten than half-day. And this is just when children's brains are working on overdrive to analyze any terminology and concepts they hear.

### The analogy of the "lazy" eye

Using the analogy of the "lazy eye," a term describing a baby's loss of eyesight if an eye is covered too long due to injury; children whose exposure to language has been withheld too long seem to experience a comparable loss.

The brain does not readily acknowledge images seen by the "lazy eye." Similarly, the brain does not seem to acknowledge math vocabulary and concepts of math without exposure in early childhood. Both are biological phenomenon. The mechanisms of the "lazy eye" are understood so there is an effort to avoid the loss of eyesight.

### Auto-delete in the teen brain

"Infants learn not only because they are computational wizards, but also because they are social beings, with a strong drive to communicate with other social beings." - Patricia Kuhl, "Decoding How Babies Learn Language"

Using the analogy of a storage warehouse, babies start building organizational storage units for expanded knowledge based on their experiences. Storage units are a framework of what is relevant. Once there is a storage unit in place, it is easy to stock related concepts and expanded vocabulary. Information has a place to hang its hat, so to speak.

These storage units must be built in early years. In later years concepts that cannot find a storage unit go into short-term memory with its famous auto-delete feature. No amount of drills and tests can change that. This is what happens to new content in math class, which "goes in one ear and out the other."

### Failure in math does not build character

There are romantic notions of the nobility of failure and the idea that failure that builds character. This noble failure paradigm is not often applicable when children fail at math. Instead, failure to learn math locks in children's perceptions of the extent of their abilities in other areas. The humiliation of not understanding what others do causes a baseline of stress that they can't distinguish, inhibiting their ability to learn.

Moshe Feldenkrais writes in _Awareness t_ _hrough Movement_

The feeling that something is 'too difficult' will spread and engulf other activities. It is difficult to estimate the importance to the individual of the qualities he lacks and of the things he therefore never tries, and thus the losses he incurs without knowing it are incalculable.

When children are faced with difficulty learning math, they don't think, "This education program sucks." They think, "I suck at math." They don't think, nor are they encouraged to think, "I didn't have any trouble learning the English language which has two roots (Germanic and Romance), at least a dozen verb forms, irregular verbs and words and idioms from every corner of the globe; I must be pretty smart."

Children accept the low expectations set by parents and teachers who themselves are math averse, low expectations which, due to ineffective educational opportunities, are often realistic predictions.

### A blow to dignity and status

"When the individual repeatedly experiences a certain difficulty, he usually abandons the activity that he has found too hard to master, at which he has not succeeded or that has proved disagreeable in some way. He establishes a rule for himself, saying, for instance ... "I shall never understand mathematics." The limits that he thus sets for himself will stop his development not only in the fields that he has decided to abandon, but I other areas; they may even influence his entire personality." - Moshe Feldenkrais, _Awareness through_ _Movement_

The human brain has parts, or sectors, which are often in competition with each other, as anyone who has tried to diet or kick a habit will know. These sectors work something like a bureaucratic committee. In the first seven years of life, the committee is open to all new ideas. It has a whole team of analysts and statisticians that are at the peak of their power and efficiency and dedicated to understanding every linguistic term and concept they hear.

After about seven years, new terms and concepts are increasingly evaluated for the possibility of causing discord with established norms or causing a blow to dignity and status. The analysts and statisticians of the committee are quieted to protect against these perceived dangers. For children who don't "get" math, the potential of failure is a definite status risk. Once children see too many red checkmarks, a math worksheet becomes a potential source of humiliation to be avoided at all cost. Children become afraid of making mistakes, afraid to fall, as it were.

As children age out of the critical time to learn language, the sectors of the brain that protect children from the loss of dignity, unwisely ignore the possibility of long-term gain. The analysts and creative members of the committee, so powerful before the age of seven are either silenced or become party hacks. Over time, the brain becomes a lazy trickster, justifying ill-advised behavior. It relies on the quick fix. "I hate math." Period. The brain doesn't know how to do things differently.

### The scam-artist brain

"The hardest thing to change in people is their belief they cannot change." - Moshe Feldenkrais, _Awareness t_ _hrough Movement_

Our brain, in sheltering us from potential humiliation, doesn't always have good long-term judgment. Thus, the brain becomes something of a scam artist in avoiding new mathematical concepts. It will set up a shortcut such as, "Math sucks," or "I'm a girl." and be done with it. That's the way the brain is wired.

As young children grow increasingly self-consciousness and more vulnerable to external validation, they stop asking questions and stop asking for help. New math concepts blow right over them. As they grow into teens, they run into additional roadblocks and landmines due to fluctuations in hormone levels. If relevance has not already been established, it's often too late. Cumulative frustrations make learning impossible.

Popular music reflects what is relevant to teens, (and it's not math). Here's a sampling: "Take me into your loving arms. Kiss me under the light of a thousand stars" (Scheeran), "I want to pull you over, pull you under. Make your body surrender to mine" (Mohombi), "Fuck tha police" (Ice Cube)

### Feedback loop hell

We can imagine, based on our own experiences. that when children feel confused and unprotected, they create a feedback loop where trauma feeds back into itself. It might sound something like, "I must be dumb. Why can't I get it? I hate math. I must be dumb." Boing, boing, boing. Every iteration of the loop degrades self-confidence and often leads to rebellions of the usual kind.

Children may try to stop this loop, but even feedback loops children have been taught to use, such as, "I think I can, I think I can" reduce awareness of what they are doing in the moment. Gabriele Oettingen, in her book _Rethinking Positive Thinking,_ writes that the advice "think positively" is detrimental to awareness and saps energy for achieving goals. She writes that positive thinking results in reduced effort and creativity.

### Labels become internalized

After a couple of years of failure, and sometimes even less than that, children are negatively labeled. If there is no one to contradict the labels, the labels are internalized. Adults who have avoided math themselves can be very judgmental. At try to believe that learning math is possible, but this flies against their own brain processes, and everything they have been told.

Dr. Bernard Lake, in his introduction to Ruthy Alon's _Mindful Spontaneity_ writes:

Part of adolescent turmoil springs from an intensely focused realization of the insecurity which drives the world and their lack of resource to deal with it.

Failure to learn math encroaches on other endeavors, and limits perceived ability. By adolescence, it is nearly impossible to regain a sense of wonderment. The teenage brain sets limits on how much confusion, shame, and failure it will tolerate, which is not much. When it gets to its limit it develops a response such as, "I want to be finished with school. Now." In the U.S. over 1.2 million students drop out of high school every year About 25% of high schoolers fail to graduate from high school on time.

As often happens, failure impacts students' identities. The part of our consciousness that protects us from perceived humiliation will self-medicate with the usual destructive activities and substances.

Back to Table of Contents

#

Chapter 5  
Irreparable Harm

###  Inspiring-sounding strategies, less than inspiring results

Accountability has been the guiding buzzword of twenty years of educational initiatives including No Child Left Behind, Race to the Top, and Gates Foundation Common Core Standards Initiatives. These initiatives, however inspiring their titles, have not improved math competency of American students. Rather they are associated with negative school experiences and outcomes.

These initiatives, colloquially known as "Drill and Kill," have, as indicated by their nickname, resulted in more drills, more homework, more tests, more disciplinary action, more racial segregation and disparities and less money for school programs which might provide a safety valve for these brutal curricula. They are all programs whose function is to transfer school district money to corporate investors.

### Fuzzy math

In response to the embarrassingly low math scores of U.S. students on cross-national tests, and an understanding of the importance of math in career development, new curricula were developed to teach math. These curricula, which include "The Inquiry Model," "The Discovery Approach," and "Constructivist Math," are collectively known as "reform math."

In response to these curricula a consortium of mathematicians, scientists, community members and parents ask:

Where's the math?

Below are a few critiques of these programs that highlight some serious flaws in the curricula. There are dozens more critiques across all media.

\- Reform math curricula require students to "work problems to death." Students are expected to explain how they got an answer, provide alternate ways to get to an answer, or write verbose explanations to simple problems when they have not been introduced to the vocabulary. They are asked to think of two ways to solve a problem and even draw pictures when a single calculation could solve the problem.

\- Reform math books are filled with projects that don't relate to math. The _Everyday Math_ book for fourth graders includes a 48-page world atlas to plan a world tour. The fifth-grade math book includes a 60-page atlas to plan an American tour.

\- _The Connected Math Program_ ( _CMP2)_ discovery approach expects fifth graders without a comprehensive foundation in mathematics to develop their own algorithms. For example, the assignment, "Writing a Division Algorithm" asks fifth graders to discover "One big algorithm" for dividing fractions. (There isn't one, unless the possibility of zero as a number on the number line is eliminated.)

\- Reform math curricula teach complicated and error-prone techniques to solve simple problems like addition. Students are expected to forgo any efficient method they have learned like the stack method to add a column of numbers. The _TURK Investigations_ math curriculum teaches students to add a column of numbers by first converting the numbers to cubes, squares, lines, and dots. It works something like converting numbers to Roman numerals. After the conversion students combine every 10 dots to make a line, every ten lines to make a square, every ten squares to make a cube. This requires a whole page of these symbols and conversions to add three four-digit numbers.

Presumably, students get experience with the decimal system by adding in this cumbersome error-prone way. Not only is it an inefficient way to add, but it is also an inefficient way to give students an understanding of the decimal system. There is a reason that Arabic numerals have been universally adopted and much better methods for teaching the decimal system.

\- There is no evidence in many curricula including _Connected Mathematics 2 (CMP2)_ that numbers are mathematically relevant such as in quantity or measurement. The first middle school math assignment states:

Many people have a number that they think is interesting. Choose a whole number between 10 and 100 that you especially like ... Explain why you chose that number.

Students are also asked to keep a journal of this number. They are asked to "Show off your number," "compose a poem," "make a poster" or "find some other way to highlight your number." I, myself, can think of a couple of numbers that teenagers find interesting, but I don't think this is what they are asking for.

### Math and racial profiling

Math competency is a big factor in systemic racism and racial injustice. James Milgram, Professor of Mathematics at Stanford University noted on the Cliff Mass Weather Blog:

[The Connected Math Project (CMP2) is] "worse than weak mathematically." "it is patronizing to minorities, if not on the margin of being racist. The book is full of children of color that are having trouble with math. For example, Luis does not know how to use decimals. If you talk to advocates of books like this, they often talk about how minorities are not ready for the 'regular' math. This is simply insulting: all parents want their kids to learn the mathematics required for success in the real world."

Since "regular math" is not defined, it seems a dog whistle for the racist ideology that minorities do not have the aptitude for learning math. However, I think that one could argue that minority domination in sports suggests that minorities could rock mathematics should they have access. Sports, particularly team sports, require athletes to have analytical skills, focused concentration, precise communication skills and the ability to take decisive action. By these standards, math is a piece of cake.

### "Mr. Calculator" to the rescue

Reform math curricula assure students that they don't really have to know math because "Mr. Calculator" will solve their math problems. The _Everyday Mathematics_ teachers' manual states:

The authors of _Everyday Mathematics_ do not believe it is worth students' time and effort to fully develop highly efficient paper and pencil algorithms for all possible whole-number, fraction and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor; one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. Mathematical mastery is not worth the cost, particularly because quotients can be found quickly and accurately using a calculator."

Most critics of the "reform" curricula disagree. It is only when students can create a picture in their mind's eye of the type of problem they are trying to solve, is a calculator of any use. Students must be able to recognize the clues in the problem to know what function to use, and the order of operation. Spending years on repetitive rote problems does not provide this insight.

That is not to say "Mr. Calculator" is not a useful tool. It is currently not used in elementary curricula with its heavy focus on memorization. Students are expected to use calculators in later years on more complex problems. More on using calculators in early elementary programs in Chapter 7.

### STEM schools

Many high schools have responded to the current employment marketplace by converting to STEM schools. They have changed their focus to teaching science, technology, engineering and math. These programs often have disappointing results, possibly because students are first introduced to vocabulary of these important fields well after they have established cultural norms.

It could also be problematic if liberal arts teachers are reassigned to classes outside their realm of expertise.

It is possible that making a commitment STEM education in early childhood might have more profound results. STEM education offered to children the first years of life might provide them with a better foundation for more in depth inquiry.

### More fallout from focus on repetitive arithmetic

Mathematics challenges unconscious biases. Repetitious arithmetic problems do not teach children the important skill of challenging mathematical assumptions. For example, many people think if a coin flip lands on heads three times, it is more likely to land on tails the next flip. But the next flip is still random. The odds are still 50-50. Gamblers are vulnerable to this thinking. When playing the slot machines, not only are they vulnerable to applying this assumption, they may not know that their odds are controlled in real time by the casino.

Another example is a couple with three sons, sure that the next pregnancy will result in a daughter. They don't consider that the odds are still 50-50, or maybe less, since there might be some biological reason that would favor another son.

Basic arithmetic doesn't address decision-making under uncertainty, where one decision is dependent on an uncertain other. Uncertainty is an important concept that children are always exploring except in the context of math. "What will happen if I (insert experimental idea here)." Taking uncertainty into consideration is an important skill in financial and other decisions.

### The No Child Left Behind Act (NCLB) of 2001

"In the systems of teaching generally accepted today emphasis is placed on achieving a certain aim at any price, without regard for the amount of disorganized and diffused effort that has gone into it." –Moshe Feldenkrais, _Awareness through Movement_

The failures of our school system create an open season for politicians. The most successful among the political campaigns to exploit our failing education system was "No Child Left Behind," the campaign slogan of former President George W. Bush. This slogan resonated with voters in a country where many children, particularly those from minority and impoverished communities, had been left behind by the education system.

Once Bush was elected, Congress passed the "No Child Left Behind Act" (NCLB). It was based on the premise that the U.S. could improve on its dismal record by establishing high standards and measurable goals.

While the slogan was admirable, in practice it did not take inequities of school funding or the crushing effects of poverty and racism into consideration. Students who could not meet rigid standards would not advance to the next grade. Teachers who failed to bring their students up to grade level would be dismissed. Under NCLB, even school funding depended on test results and many schools in the poorest neighborhoods were 0closed.

With a mandate to produce good test scores, teachers started "teaching to the test." School instruction was on a need-to-know basis, resulting in bulimic gorge and purge educational practices. Activities that were considered "unnecessary" to achieving good test scores were dropped, including music, art, and drama programs to pay for standardized curricula, testing and infrastructure for scoring tests.

Researchers in brain science have found that enrichment programs develop different parts of the brain, parts which cultivate a positive relationship with the world. They can also act as a relief valve. Eliminating enrichment programs to have more time to work on math drills has been shown to be an ineffective way to improve math competency. In addition to the loss of brain development, children have no respite from the anxiety of test preparation and testing.

NCLB ultimately placed the responsibility for school funding on the shoulders of our most vulnerable children who have the fewest resources to deal with this burden. Teachers and principals were given mandates over which they had no control. Although NCLB has been replaced, it was replaced with similar programs with equally inspirational titles. Taken together, they will have a lasting negative impact on many communities.

### Race to the bottom

"The federal government, states and school districts have spent billions of dollars to phase in the standards, to prepare students to take the tests and to buy the technology needed to administer them online. There is nothing to show for it. The Race to the Top demoralized teachers, caused teacher shortages and led to the defunding of the arts and other subjects that were not tested." –Diane Ravitch, "The Common Core Costs Billions and Hurts Students"

The "Race to the Top Fund" was part of President Obama's "American Recovery and Reinvestment Act of 2009" (ARRA). It recognized that the current testing system was deeply flawed and spoke to our moral obligation to give students the best education possible. There were beneficial Race to the Top programs; yet, the top two mandates were adapting standards and assessments and building data systems to measure student growth and success. It established Common Core Standards using the same assumption as NCLB, that by establishing standards we could test children to the top.

Four of five options for receiving federal money from Race to the Top required replacing the principal of the school if students didn't meet pre-determined standards. While there is a range of competencies in principals (see next sections), many schools would lose good leadership because their students did not have the foundational preconditions to succeed.

### Common Core: Crushing good sense at every turn

"[I]n the end, nothing seems more natural than that to which he is accustomed, even if it is opposed to all reason or necessity." –Moshe Feldenkrais, _Awareness through Movement_

The Common Core Standards Initiative high-stakes testing program was continued due to an effort of The Gates Foundation under the pretext of providing every child with a good teacher. School districts continue to invest billions in the unfounded claims that collecting data would result in greater accountability.

Although the Common Core concept of aligning the curricula with tests did improve test results, U.S. students have not improved their standing on international standardized tests. After $80 billion of public-school expenditures the Gates Foundation reports "The percentage of high school graduates who have enrolled in postsecondary institutions has remained flat." The Gates Foundation does not acknowledge that spending $80 billion for flat results would be considered a loss in any other context.

Instead of dropping the idea of high-stakes testing. many school districts have reacted to continued low test scores by increasingly repressive policies that limit free time and enforce rigid behavior expectations. So the limited focus, test-heavy curricula continues to prevail in many school districts while money flows from education budgets to corporate providers of tests, curricula materials, and testing infrastructure. This transfer of funds from school budgets to corporate providers leaves a shortfall in school funding which leaves schools in low-income districts unable to provide students with the softening effect of arts and other motivating programs.

Below is the Gates Foundation K-12 vision of education as featured on their website homepage. Students sit shoulder to shoulder in a segregated classroom. Only two of the students are engaged/in focus. In this top-down model, focused attention on the teacher is required. The rule is "Sit still, be quiet, and listen to me."

In recognition of the impulse to be engaged, the Gates Foundation vision for education has a solution. Backpacks the are hung in the back of the classroom. This seems to indicate a lack of trust in students which I assume would be insulting. No middle-class or upper middle class school would demonstrate this lack of respect for students.

Compare this "vision" to the images on the homepage of Lakeside School, the school the Bill Gates attended. Students are engaged in small groups doing many projects and activities. Their approach to education as stated on their homepage, "This is a place for curious minds." Lakeside, irrespective of a huge budget does not waste money on the Common Core Standards materials, testing or testing infrastructure.

### Teaching to the test

Common Core Standards create considerable anxiety around meeting standards. In districts of low-income families many factors converge to make it challenging for students to pass tests. Factors include underfunded schools, the burden of racism and food and housing insecure students. To meet standards, teachers teach to the test with little time for any topics that students might find interesting. It's the education equivalent of what accountants call "cooking the books." School children are merely collateral damage because teaching to the test to the exclusion of conceptual ideas is the worst possible path to long term gain.

The limited focus, test-heavy curricula continues to prevail in many schools with money flowing from education budgets to corporate providers of tests, test-aligned curriculum materials, and testing infrastructure. This transfer of funds from school budgets to corporate providers leaves a shortfall in school funding so that schools cannot provide students with the softening effect of arts, and other motivating programs.

In the teaching to the test environment, most students learn the survival technique of looking like they are paying attention. They spend their energy trying to control the impulse to look at something interesting, like their phones.

###  The school-to-prison pipeline

"[P]rison is where society sends its failures." (Ice Cube)

In response to the high stakes standardized testing where teachers and schools are evaluated based on their students test scores, many schools have reacted by increasingly authoritarian policies such as limiting free time and attempting to maintain rigid behavior expectations. The anxiety of school districts, teachers, and parents, and students to meet Common Core standards brings out the worst in everyone, liberating mean-spirited racist tendencies. This is most apparent in methods that schools employ to handle minor infractions, such as disturbing class.

In wealthy suburbs, students are sent to the principal's office for minor infractions. The principal assigns detention, which means that the students stay an hour after school in a room full of others who were similarly disruptive. It was my experience that detention was a party.

Schools with low-income and minority students often implement "zero-tolerance" policies that criminalize minor infractions of school rules. "Zero-tolerance schools" operate on the idea that discipline is necessary for learning, and top-down authoritarianism will help insure discipline. Schools apply rigid discipline, without evaluating the disconnect between intention and outcome.

Some students cannot cope with soul-crushing militarism in the schools. Those who don't have enough emotional calm to look like they are paying attention when they don't understand the material or don't see the relevance to their lives are accused of disrupting class. All too often they are handled by police and funneled into the criminal justice system.

### Criminalizing failure

"The task of the educator lies in seeing that the child does not confound _good_ with _immobility_ , and _evil_ with _activity_ , as often happens in old-time discipline." - Maria Montessori, _The Montessori Method_

A high-school student in South Carolina recorded a cell phone  video of how her school handles lapses in paying attention. In this video, a white male police officer comes into her math class and approaches her friend, an African American student who is looking at her phone. The officer picks up the student desk and all, flips her backwards, and body-slams her against the floor. Then he drags her out of her desk and throws her. This did not seem to be surprising to other students seen in the video.

The teacher did not intervene on her student's behalf. He was never questioned about what he was teaching that was so important to justify this aggression. It seems no one in the administration had questioned this tactic, or its relevance to educating youth.

The black vertical shape to the left of the officer is the student's leg and foot. Her head is jammed where the floor meets the wall.

Once the video went viral, "School district superintendent Debbie Hamm made a statement that the district "is deeply concerned" about the confrontation. "Student safety is and always will be the District's top priority." "The District will not tolerate any actions that jeopardize the safety of our students." (She's quite a Hamm!)

To show the District's concern for "any actions that jeopardize the safety of our students," the student who recorded the event was arrested and charged with disturbing school. The police officer, who did what he was expected to do, was terminated after the video went viral. There were no consequences for either the Superintendent Debbie Hamm, the principal of the school, or the math teacher. None of the federal programs designed to improve student outcomes by replacing ineffective teachers and administrators impacted them.

### "Asses in classes"

"[F]or-profit college students, graduates and dropouts combined, earned less after leaving school than they did before they enrolled." David Halperin, "Trump University: A Scam, But a Familiar One"

Many young people including single mothers and those back from military service are lured by television commercials urging them to inquire about "university" programs that can help them "work toward the future you want" or "Earn a degree in Tech". When perspective students call to inquire, they are given similar assurances by commission-dependent recruiters.

David Halperin, of the _Republic Report_ writes that recruiters have a quota of "asses in classes."

"Many of these colleges have been caught using deceptive advertising and misleading prospective students about matters like program costs, accreditation, transferability of credits, job placement rates, and likely starting salaries. Although for-profit colleges often promise that their programs are affordable, the real cost can be nearly double that of Harvard or Stanford. But the quality and reputation of the programs are often weak, so even students who manage to graduate often struggle to find jobs beyond the Office Depot shifts they previously held."

Even when students are not keeping up with the program, the recruiters, who are called counselors, keep students strung along as long as they can secure federal loans. Students hang on to the promise that by working hard, they can make up for the twelve years of schooling which they were not able to follow. Students and their parents end up with debt that they will never be able to repay.

The fraud is exacerbated by the system where recruiters get commissions on enrollment, not graduation or employment rates. Those with more sales are rewarded with more leads. (Leads are people who have called the number on their TV screen.)

Gillian White in "The Empty Promises of For-Profit Colleges" published in _The Atlantic_

The amount of debt owed by those attending for-profit colleges was $229 billion in 2014. Many of the students who are carrying this debt are not more qualified for a job then before they enrolled.

In 2016 ITT Technical Institute was shut down after the Department of Education cut off new federal financial aid funding for ITT students.

"ITT, which has been getting as much as $1.1 billion annually in taxpayer funds, and got $664 million last year, has for years engaged in predatory and reckless practices, coercive and deceptive recruiting, unconscionable student loan practices, financial aid abuses, poor quality programs, but sky-high tuition." David Halperin, "Friends in High Places: Who Endorses America's Troubled For-Profit Colleges?" _Republic Report_

Several other for-profit colleges have abruptly closed, some mid-year leaving students scrambling to find a way to complete a degree.

### Foreclosed homes and blighted communities

"The housing crisis was also a major migration event, although we seldom think of it that way. As many as 10 million families lost their homes to foreclosure. As a result, nearly all of them had to move." Emily Badger, _The Washington Post_

People who lack math literacy are also easy prey for mortgage scams. Prior to the 2006 financial meltdown, banks targeted low-income homeowners urging them to borrow more money than they could afford to repay. Many families lost their homes when they could not make monthly payments. Foreclosed homes fell into disrepair and entire neighborhoods became blighted by these homes. Cities have been tearing down thousands of these houses. Failure to improve our school math programs has led to incalculable tragedy.

Back to Table of Contents

# Chapter 6  
A Model for Learning Math

There is a great disparity between the way children learn their native language(s) and the way math is taught in our current educational model. Since children learn their native language so reliably and so many children have difficulty learning math, it is worth analyzing the way children learn language and use this as a model for helping children learn math.

Stepping back for an overview, it is well established that children benefit by an early start in many endeavors. Learning language is just one example. Throwing a ball is one example. This is a skill which must be learned at two years of age as I learned at a mommy and me class. This explained why I could never throw a ball more than ten feet even though at one time as a teenager I worked with a friend to improve my range. Armed with the knowledge of why I could not throw, I took my then two-year-old son to throw rocks in the lake. He later became a pitcher on his high-school baseball team. Similarly, gymnasts, ice skaters and dancers begin serious programs in early childhood. Great musicians often grow up in families with a musical tradition and get an early start learning music from their families.

Although it defies logical reasoning that toddlers are more competent at some things than teens or adults, mathematical reasoning may follow the same trajectory as learning language and throwing a ball.

### The gentle way babies learn language

We talk to babies in an infantile way but there are no artificial barriers where children are kept from exposure to the full range of linguistic expression. It might be dinner-table conversations or conversations they hear at the grocery store.

We start reading to babies and singing songs before they have the vocabulary to understand the stories and songs. We don't tell babies to sit still and listen for the purpose of learning language. There is trust in the process. Babies listen to anyone who puts words to ideas and ideas to words even though they don't yet know the meaning of the words. Babies seem equally attentive in listening to readings from daily newspapers and academic textbooks as story books.

As babies begin to understand language, they find it just as interesting to hear about the phenomenon of the natural world as hearing stories about bunnies that run away from home and trains with goals.

### As babies grow into toddlers

Toddlers speak in short sentences but have already figured out the order of operation, as it is referred to in mathematics, of their native language. In English it is subject, verb, object. As they grow, they fine tune their communication skills to match their cultural language more closely. They start with limited vocabulary and make grammatical mistakes, but no one tallies up mistakes or compares children's language skills to others. Uncle Wob doesn't correct his nephew's pronunciation. He uses unclewob as his user name.

Children who have exposure to a rich vocabulary become more precise in their communication. They learn to detect humor, find meaning in what is not said and even differentiate bureaucratic boilerplate.

Betty Hart and Todd R. Risley study language disparity comparing children of different socio-economic groups. They write in _Everyday Differences:_

The professors' children simply seemed to know more about everything.

The gentle way babies learn language results in nearly universal success. By the age of five, without worksheets, grades or tests, children can navigate the meaning of most conversations they hear. They will even speak metaphorically. The timeline is the same in every corner of the world.

### Reading and writing

At about five years, children start learning the phonetic sounds of the alphabet. They transition to reading with someone by their side to help interpret the symbolic representations of words. It doesn't take long before children can not only read, but "read between the lines."

Soon after, they learn to write by practicing one letter at a time. Then they write words, and finally sentences. When children move on to writing stories, they are not judged on content or grammar.

As an example, when elementary school teachers present a unit on Columbus and then ask their students to write a paragraph on what they have learned; teachers expect young children will get their facts mixed up and/or focus on the most inconsequential details. The discrepancy of what was presented and what their students write is quite amusing. Teachers would not think of putting a big red X on papers to indicate incorrect facts, as they do on math papers. Teachers understand that facts are not as important as the process of learning to write. There is an assumption that children will sort everything out and grow their knowledge in later years.

The gentle way babies learn language contrasts with the difficulty teens and adults experience when they try to learn a new language. Teens and adults will not become proficient without putting in a great deal of effort. They will have to memorizing lists of nouns, practicing verb conjugations, and study sentence structure. Because most teens have not learned the language of math they will similarly have to put in a great deal of effort. They will learn in an inefficient tedious way having to memorize and practice and study.

If we use language as a comparative basis, children can only language from fluent speakers. Teachers can only teach the language(s) they know. With this in mind, we should evaluate our current educational model of expecting young students to learn math from teachers who are uninterested, unknowledgeable, or even math-averse.

### Differentiation

"The most interesting thing about babies is that they are so enormously interested; the most interesting thing about them is their infinite capacity for wonder." - Alison Gopnik, Andrew N. Meltzoff, and Patricia Kuhl, _The Scientist in the Crib_

Children try to figure out the how and why of things long before they are asking "How?" and "Why?" The game of peek-a-boo illustrates a child's process of learning about the world.

Mothers play peekaboo with their babies by covering their faces with their hands. Babies are momentarily distressed when they don't see their mothers, but all is well when mothers uncover their faces and say, "Peekaboo." Babies come to understand that their mothers do not disappear when they hide their faces. Then, babies play the game by covering their own eyes which is the mother's clue to say, "Where's baby?" Babies uncover their faces and consider this great fun.

Peek-a-boo evolves to the game hide and seek. Toddlers will hide behind a curtain where the shape of the curtain and their feet poking out gives them away. Mothers ask, "Where did John/Jill go?" Toddlers are very amused when they reveal themselves.

A year or two later children are more sophisticated. They understand that they can be seen, even if they cannot see others. They will find hiding places that are out of the visual range of the seeker. They have refined the abstract concepts that things exist that they cannot see. They understand object permanence. I would like us to consider that many complex mathematical ideas introduced early in a child's life would help them develop comprehension over time.

### Math must have relevance to stick

If young children hear, "The ratio for making rice is two cups water to one cup rice," they might not immediately understand what ratio means, but they will store the sound pattern and will keep searching for meaning. Later, they come to understand the concept of ratio and will be able generalize and apply their understanding to ratio problems that come up in math class. Relevance is established early so math problems using ratios build competency, not frustration.

Just as in all experience, a broad base of ideas from which to express themselves is empowering. With continuous exposure, children gain an intuitive sense of how to express mathematical ideas, in the same way they come to have an intuitive sense of expressing other ideas in their native language.

Young children, who easily learn complex language structures could presumably also learn math concepts with minimum effort and maximum pleasure and efficiency. If children are given the opportunity to learn the language of math at the same time as they are learning their native language, they would presumably have an intuitive sense of how to solve math problems, just as they have an intuitive understanding their native language.

Children effortlessly master the complexities of their cultural language in advance of their ability to evaluate grammatical constructs. They would not do well on tests to determine their knowledge of verb conjugations. A five-year-old might say in a conversation, "[It's a] Good thing we got here early," but she would not be able to tell you the grammatical rule that determined when a subject and verb in a clause can be omitted.

By the age of three, language has become intuitive, and yet, three-year-old could not answer questions about grammatical constructs on a standardized bubble test. By five, children can understand most conversations they hear and, although they might still make grammatical mistakes, they speak at nearly the same level as their parents. Yet, they could not articulate the rules of when they can drop a subject and verb in a complex sentence they might use such as, "Good thing we stopped here early."

Students are not asked to sort out the grammatical constructs they use so naturally. A five could not identify the present perfect form of a verb such as, "They've already left for school" or past participle such as, "They left for school.". It is not until the tween years, when children have over a dozen years' experience with their native language do teachers break down the grammatical constructs discuss nouns, conjugation of verbs, compound and complex sentences.

### The easy route

There are many clichés suggesting that taking the easy route creates a weak character, and more difficulty in the long term. These clichés, if true in other realms, do not apply to learning language which is easily learned infancy and early childhood. The relevant question that applies to this inquiry is whether the propensity of children to easily learn language includes easy acquisition of the language of math.

If we would provide young children with a rich language experience that includes mathematical concepts, could math be an enjoyable activity that works with young children's linguistic genius? Young children cannot solve higher-level math problems, but they could potentially begin the process of understanding the language of math. And this might be the time when there is the most potential for learning how math vocabulary and concepts relate to their lives.

We would not expect children to learn a foreign language from learning a few words and phrases from a non-native speaker. Nor would we expect them to extrapolate the subtleties of vocabulary or the complexities of sentence structure from learning a few rules. Likewise, children who do not have a solid foundation in mathematical thinking are likely to struggle with basic mathematical concepts even after a decade of math classes.

However, that does not mean that there will not be any effort needed as Ruthy Alon writes _Mindful Spontaneity:_

It is important to note that what is said here refers to avoiding the frustration of barren and unnecessary effort, in the context of acquiring skill. It is not said here that effort is unneeded in life in general, or that human activity does not need to include phases of effort.

The benefit of an early introduction to math will mean that effort will expand knowledge rather than practicing repetitive rote problems.

### The learning trajectory

The limited time frame for learning language by being part of a speech community is the basis for a proposed change in early math education. I will ask us to consider replacing the current curriculum of repetitive arithmetic, such adding with pebbles and rote worksheets, to an immersion method where children are introduced to math vocabulary and concepts of algebra, geometry and even calculus.

I hope to reexamine the assumption that learning math requires hard work and perseverance. I am asking us to explore the idea that children can learn math as effortlessly as they learn their native language. Naming mathematical properties gives children a way to communicate, just as naming objects and emotions. It follows that if children have early exposure to mathematics, they have the potential to progressively develop a more thorough understanding over time. And if we consider that it takes time and experience to understand the properties of the physical world, we might consider facilitating this knowledge with an early start by naming these properties.

### My cousin's story

I was chatting with my cousin Bev, who comes to town a couple of times a year. On a recent visit, she graciously listened to my hypothesis that math, like language is easiest to learn in early childhood. I described the idea that children learn math concepts by listening in on family conversations. She shared her family experience which supported this idea.

Bev has three daughters. Her first daughter Lilly struggled with math. Year after year, Bev worked with Lilly on math homework at the dinner table while Bev's younger daughters played nearby. They were listening in which gave them an immersion experience in the vocabulary and concepts of math. Neither of Bev's younger daughters needed help with their math homework and they went on to take advanced math courses in college. They had apparently picked up the vocabulary and concepts without effort.

### The proactive approach

Researchers have not studied the acquisition of math as a language. However, since the currently implemented programs are not sufficiently addressing the problem of low competency in U.S. students, we could consider a different approach, a proactive approach with a basis in language and brain science.

# Chapter 7  
New Approaches to Teaching Mathematics

###  Shifting from repetitive basic math

"Awareness gives us the capacity for judgment, differentiation, generalization, the capacity for abstract thought, imagination, and much more." - Moshe Feldenkrais, _Awareness through Movement_

Rather than thinking that math needs to be dumbed down for young children we could give them the advantages of early exposure to high-level mathematical vocabulary and concepts because this is a time in their lives when they will absorb any language within earshot and evaluate it for its place in their web of knowledge.

This shift from an emphasis on repetitive rote arithmetic to an exploration of language and discovery would convey the message that the goal of education is the pleasure of learning without the potential for failure and the anxiety of knowing that there is a lot to lose.

Children would hear the vocabulary of math, and the vocabulary of fields where math is applied, such as physics, engineering, computer science and finance and it might even be a good idea if children can be shown how to compute with variables. Using language as a comparison basis, if children are given the opportunity to hear discussions that include mathematical thinking, they start the analysis of mathematical vocabulary and concepts as they hear them.

If math were taught as a language children would not be expected to do computations or regurgitate lessons, but rather, they would be shown the interesting properties of numbers and strategies for solving mathematical problems, which is not overwhelming to young children, as it can be for teens worried about a test on Friday.

"Since the children are freed from competition, and they do not work for praise or rewards, learning becomes its own true reward." - Maria Montessori

Introducing math as a language is also more representative of "real-world" mathematics where math is a balance of variables. The math of building a bridge, for example, balances relative strength of materials, cost, aesthetics, and so on. Even in accounting, the same numbers are used differently for making internal decisions and filing reports to the IRS.

### Bringing a speech community to children

If we consider math a language, we must provide a speech community to give children a foundation in math that focuses on an exploration of the physical phenomenon that children intuitively understand. Children have an opportunity to engage their curiosity and expand their awareness of high-level math vocabulary and conceptual ideas to better prepare them for future proficiency. This method allows for learning the language of math at the optimal time to learn language and acknowledges that conceptual ideas need "time to cook."

### High-octane math

The focus of a high-octane math program is the sharing of knowledge and ideas. The objective is to pass down the highest-level knowledge from one generation to the next. In this model conceptual math and its applications in science, engineering and technology would be an ongoing part of a child's life.

Young children seek to understand serious subjects and they continually ask questions if they are not discouraged. Toddlers are famous for asking "Why?" and "How?" Parents answer questions to the best of their ability but are often constrained by time, their own knowledge base, or their level of stress and anxiety.

Since it is critical that children hear about mathematical ideas from people who find them interesting and relevant alternatives to a family and school speech community are necessary.

Hart and Risley in _Meaningful Differences_ write:

By the age of 34 - 36 months, the children were also talking and using numbers of different words very similar to the averages of their parents.

The ability to learn mathematical terminology might be as genetically programmed in young children as learning spoken language. Children might like to learn to identify shapes including the tetrahedron, rhombus and double helix just as they like to learn the names of animals. Children might also be interested in learning to identify parts of a triangle and distinguish the structural properties of different triangle shapes. They might be interested in how triangles are used architecture to make bridges and other structures stronger.

Most mathematical vocabulary and concepts can be related to children's experiences. This relevance forms the foundation on which mathematics can be built. The application of mathematical principles to their lived experience is a means of engaging children's natural curiosity in mathematical phenomenon. This is not to suggest that children solve geometric problems, parrot concepts, or memorize facts for a test but is a means to engage children's natural curiosity in mathematical phenomenon.

They should not be held accountable for what they do not understand which is what the testing culture requires. Students would not be expected to do calculations, or regurgitate information, as is often the case in school math programs.

There will be initial misinterpretations, but continual repetition, variations, and perspectives will correct initial misunderstandings in the same way that children's grammatical errors and work themselves out over time.

### Curators of math content

There is no need to reinvent the wheel for teaching high-level conceptual math. Teachers and administrators could become curators of media content that students watch and discuss either in class or remotely.

A program could be assembled the many videos on the internet that explain and math concepts in cartoon form which otherwise might seem abstract (read pointless) and confusing. Cartoon-like videos often have entertainment value which would increase the likelihood of retention in memory. These videos are often listed under "physics of," such as the physics of jumping or the physics of flight. There are no calculations required to understand concepts.

An example of a high-octane math lesson might be a lesson in mechanical advantage. Most young children have experienced situations that are examples of mechanical advantage, such as playing on a teeter-totter. The youngest children can figure out how to balance a teeter-totter when the person at one end is bigger than the person on the other end. This is the time to introduce the idea of fulcrum and the relationships of other variables, force, load, and equilibrium.

Children can also figure out whether a small hammer or a large hammer would have more force and they might be more open to the ideas of leverage as young children than they will be when leverage problems come up than in algebra class.

Alison Gopnik Andrew N. Meltzoff, and Patricia Kuhl write in _The Scientist in the Crib:_

Preschool children have brains that are literally more active, more connected, and much more flexible than ours.

It would seem to be an advantage if children are introduced to math terminology and concepts when their brains are at their most active, connected, and flexible. With an understanding of the terms as variables in an equation, children would have foundation that will help them in later years. They will also be able to transfer their knowledge of math terms to other contexts, such as leveraging assets.

Part of early learning curricula includes learning about how the tilt of the earth causes the seasons. There are many other examples of dependencies where for every input there is on output. This is the time to introduce the term function and its mathematical notation, f(x). Ice cream for dessert is a function of finishing dinner. Any kid can understand that.

Other discussion possibilities include tracking of inventory, where inventory of an item is a function of sales volume, or compromises at the grocery store where food purchases are a function of food budget. Photographers could discuss the ratio of aperture opening in the camera to the amount of light.

Presenting many dependencies as functions of two variables might prevent that blank stare where students' eyes glaze over when encountering the notation f(x). The concept of one variable being the function of another is ubiquitous in everyday living but students who have not had exposure to the term 'function' from early childhood when their "brains are literally more active more connected and much more flexible than ours", will see the symbolic representation f(x) as incomprehensible.

Even the concept of limit, the basis of calculus, can be explained to young children in terms that they can relate to. Using the imagery of a soccer game, A soccer player controls the ball from the 50 yard line to the 40 to the 30. Then, imaging this is the internet, there is a buffering pause where the buffer symbol interrupts the play for a second. The play picks up at the 20 yard line and the player continues to the make a goal kick. The limit is the point the player approaches during the buffered second closing in between 30 and 20.

Math can be taught as an exploration without the expectation that children do calculations or remember the specifics of the lesson. By teaching math in this way, children could learn the vocabulary as part of their native language. They will associate math with the "real world," and later, learn to calculate using variables.

### The math of owning a car

While train schedules and container ships and other usual fare of algebra classes are irrelevant to t(w)eens, the math of owning a car might grab their attention. There are an endless number of features, financing options, and cost of maintenance to compare. Most have a mathematical component. Unlike many math problems, this is one that t(w)eens can relate to and one that is likely to be of use. The idea that students can leave school without this base of knowledge is unconscionable.

### The STEM Express is coming

The STEM Express is a mobile school that brings activities to young children with the goal of enhancing their knowledge base and providing a foundation for future academic achievement. Children are introduced to mathematical ideas in the context of science, technology, engineering and math applications as they are expressed in their lives; such as the physics of jumping over a rock.

The STEM Express is modeled after the Gymnastics Express, a mobile gymnastics school that brings young children a fun gymnastics experience in a non-competitive environment. Most pre and elementary schools as well as recreation programs do not have in-house expertise in gymnastics. The Gymnastics Express is designed to fill this void bringing gymnasts with a van full of equipment to young children. The program is designed to enhance children's physical development and awareness by giving young children a chance to climb, jump, hop, roll, balance and flip in a safe padded, supervised environment.

The STEM Express, like the Gymnastics Express, is a self-contained mobile unit that fills a void in experiential learning. The STEM Express is available to daycare, preschools, elementary schools, parks and playgrounds. Specialists bring hands-on projects and video presentations to children. The emphasis is to provide children with a foundational experience designed to engage their curiosity.

The STEM Express starts by teaching the vocabulary of the concepts that children instinctively understand. Toddlers kick with more force when they are angry, they run to the end of the block to beat their parents, and they spend considerable time jumping in puddles. They continually and endlessly explore the properties of the physical world.

Just as childhood is the best time to learn words like cat and dog, it follows, and this is the premise of this essay, that it is also the best time to introduce high-level mathematical vocabulary and algorithms starting with math principles children instinctively know. These include, as mentioned in the previous paragraph, the vocabulary of mechanical advantage and output force, relationships of speed, time and distance, and the effects of force and acceleration in the displacement of water. The hope is that familiarization with STEM vocabulary and concepts from the age when learning language is as instinctive as breathing will help children succeed in in later years.

STEM Express lessons are designed to spark interest and curiosity. They are intended to give children experiences and skills to inspire self-confidence and connection with their inner wisdom.

The STEM Express contrasts with the usual early childhood programs of mathematics which are largely focused on repetitive procedural problems. Traditional early education programs promote the idea of mastering arithmetic before moving on to more "difficult" mathematical concepts.

The STEM Express specialistsdo not bring tests or worksheets. Progress is not measured or compared with other students. Children are not held accountable for concepts they do not understand.

The STEM Express is one possible way to bring a program of high-level math, I will call it high-octane math, to young children.

### Baby night school

Meetups to provide surrogate dinner conversation are another possible way to provide back-up reinforcements and provide babies with a speech community of math specialists. Volunteers with an interest in math might be willing to join parent meet ups to give babies and young children the benefit of listening to conversations that included mathematical vocabulary and concepts.

Volunteers could directly engage with the youngest children, or alternately, engage with one another in math-related conversations and let babies listen in. Videos on the internet which demonstrate mathematical principles could provide the jumping off point for conversation. There are studies that suggest the videos alone are not enough.

Volunteers from the community could talk about topics that have a mathematical component such as investment banking, aviation, carpentry, programming, or anything else in their comfort zone. Pairs or groups of volunteers could "talk shop" at these meet ups to give babies the opportunity to hear conversations that include a mathematical component. Student volunteers might be willing to collaborate on math homework at these meetups while children are having a snack. The approach is, "Isn't this interesting," not, "Remember this for a test on Friday."

### Pee wee math

"Knowledge is acquired by formulating explanations and testing them against reality" – Pinker, _Enlightenment Now_

When I see young children on the ball fields playing organized sports, I think, "Why isn't there a pee-wee math program?" It is possible that children would not have so much difficulty with math if there were math programs using this model.

On every pee wee baseball team, there are parents who know enough about baseball, to coach young children and set the groundwork for future skill development. Parents provide enthusiasm and prodding which is necessary to activate children to run when they are supposed to run, stay standing at their designated position when the action is somewhere else, or even pay attention when they are at bat. At the end of each season, every child gets a trophy. And every child imagines that he or she will play in the big leagues.

### Music and math

"Ah, music," he said, wiping his eyes. "A magic beyond all we do here!" \- J. K. Rowling, _Harry Potter and the Sorcerer's Stone_

Music is a great entryway to discussions of mathematical concepts. The popular songs of preschool children reflect their interest in numbers and properties of the physical world. Toddlers love to sing counting songs such as, "One, two, buckle my shoe," "Five little monkeys jumping on the bed," "Sing a song of sixpence," and "99 bottles of beer on the wall."

Preschool children also like songs about motion such as "The wheels of the bus." Song time might be the best time for children to hear about the mathematical properties of wheels. The terms, such as circle, cylinder, volume, area, radius, circumference, and pi, are likely more interesting to young children when integrated with song time than to teenagers trying to get through algebra class. Exposure to these concepts also introduces the idea of related variables, and the idea of calculating new information from information already available.

Children might have an intuitive sense of whether a small wheel would have to turn more times than a large wheel to go the same distance. Demonstrations on how to calculate how many times a wheel turns to go a set distance, or comparisons of how many revolutions are necessary to go to a set distance with different size wheels, could be an early introduction to the mathematics of circles.

### Backup reinforcements

Back in the U.S. children who do not have exposure to mathematical thinking through their home environment will not likely experience this exposure in daycare, preschools and elementary schools. It is unlikely that schools will hire dedicated math teachers in daycares and preschools anytime soon.

In the meantime, community leaders and volunteers in non-profit organizations could provide surrogate high-level dinner conversations to babies and toddlers. Providing backup reinforcements to teach the youngest children would require a slight tweaking of our current system that provides help for students who have fallen behind. There is an army of volunteers who help students with their homework. Many have math-related skills that they do not share when they are helping students with arithmetic homework.

These volunteers might find their time makes a more substantial contribution by using their expertise proactively by teaching the vocabulary and concepts of high-level math to young children who are at the critical age for learning language

### Using calculators more effectively

Since children love devices and live in a culture of devices, calculators might be the back-door approach for young children to learn math vocabulary and concepts. Calculators might also help with memorization of math facts. Children might not mind filling out repetitious worksheets if they can use their calculators instead of depending on memory. The focus could be on learning to use the calculator to solve math problems.

Graphing calculators could be used to allow young children to try more advanced problems. Children would be able to see the pictures they make with different equations and variables. The idea is to correlate vocabulary such as linear and exponential growth with images. Young children might like to experiment with formulas and functions such as compound interest or quadratic equations. Of course, they would not be expected to solve problems using these functions, but an early visual reference might help students feel less afraid when these functions come up later in math class. Again, this is an exploration of ideas, not a task or something it is possible to fail at.

### Worksheets revisited

"In order for the child to dare to experiment and make mistakes, he or she needs to feel secure, to know that someone is watching who has the power to put things right again." -Ruthy Alon, _Mindful Spontaneity_

Taking guidance from the way children learn language, worksheets could possibly be used more effectively. As children learn language, they make grammatical errors, but in listening to others, they auto-correct. By contrast, when children are learning math, teachers mark the wrong answers with red checkmarks; they tally the number of incorrect answers and assign a judgmental grade.

Switching the intention of assignments from evaluation to learning would mean providing the correct answers, or steps to getting the correct answers, on worksheets rather than a mark to indicate a wrong answer. Providing correct answers eliminates the fear of shame attached to wrong answers. Children would understand that the important thing is to learn.

In this way, children would have a chance to auto-correct as they do when they are learning language. They would not be so quick to judge their own ability or see math as something to avoid. This is particularly relevant when students are afraid they will never get it, and worried they will never reach the end point.

The concern of educators is that worksheets would become a slop-fest where students don't even bother to try. This assumption assumes that scores and grades are a motivation to work harder. However, it is possible scores and grades have a negative effect on learning, as the failure of NCLB, Race to The Top and Common Core suggests.

### Big Bird and the Count upping their game

PBS has engaging science programs for adults and could also be a wonderful venue to teach higher-level math and science to toddlers. For example, Big Bird cannot fly, but he could explain how other birds fly. He could explain Bernoulli's equation where the shape of an airplane wing, like the shape of a bird's wing, plus momentum, allows a 450-ton airplane to lift off the ground and fly.

Gonzo, the vulture, unlike the species he is modeled after, does not fly either, but he could tell children how to figure out how long it will take to fly to Paris, given the distance, speed at which he flies. Kermit could discuss the math and physics of jumping. Miss Piggy could discuss the thrust and momentum of her karate chops.

The Count loves to count. He also shows children simple arithmetic like adding single-digit numbers

Greetings, it is I the Count! They call me the Count because I love to count things! HA HA HA!!!

Perhaps he could do a lot more than count. It would be helpful to children if he would help children become acquainted with the decimal system.

The Count might explain higher-level numerical ideas. He could even introduce sine and cosine because of their visual similarity to amusement park rides. This is not intended to be fully comprehended, rather an introduction to the vocabulary in the same way the Sesame Street provides children the vocabulary of the realm of emotions and complex social interactions.

The cookie monster could teach algebra. He might say, for example, "Cookie Monster has two cookies. Cookie Monster had four cookies yesterday. How many cookies did Cookie Monster eat?" Cookie Monster provides the answer, of course, "Cookie Monster ate two cookies." Cookie Monster could write the algebraic notation x=4-2. If this were a regular segment using different numbers, or variables, young children would come to understand the concept of solving an algebraic equation and not assume, as some students do, that algebra is another place to fail.

###  Elementary school math does not have to be disorganized and patronizing

The goal is for children to perform well because they understand the material on a deeper level; they are not just learning it for the test." Brown, PBS Parents

Children in the country of Singapore score highest of any country on standardized tests. In contrast to the American "Reform math" programs that bypassed mathematicians in designing the school math curricula, the country of Singapore initiated a math program developed by mathematicians. There are no assumptions about which demographic groups are capable of learning "regular math." The focus is on awareness of the thought process to master skills. From the beginning there is an organized logical program for optimal learning with the goal of solving complex, non-routine problems that often require a number of different strategies.

The Singapore math program takes into consideration that math facts and memorized algorithms are useless without grounding in experiential knowledge. This program takes students step-by-step in a logical sequence through experiential, and then pictorial (showing pictures of the experiential) and then to the abstract. Students are given the vocabulary at the experiential stage. Students don't spend years memorizing math facts and turning in worksheets of rote problems.

In Singapore math each component of topic is broken down into a carefully thought out, sequential activity. Children learn the decimal system when they first start school with Unifix interlocking cubes.. Unifix cubes are fun toys that snap together, but sequential activities using Unifix cubes which use tactile, visual and verbal involvement help children gain an intuitive sense of the decimal system. Teachers explain, teach the mathematical vocabulary of the activity. American children are adding with pebbles when they first start school.

### Beating the odds after seven year of age

"Sixth graders in the richest school districts are four grade levels ahead of children in the poorest districts."

It is possible to teach math to students who didn't get an early start learning math vocabulary and concepts, but it takes a heroic commitment from schools, parents and students.

In a New York Times article, "Money, Race and Success", authors Motoko Rich, Amanda Cox, and Matthew Bloch, report, "Programs that beat the odds do exist, they write, such as one program in New Jersey:"

"In one school district that appears to have beaten the odds, Union City, N.J., students consistently performed about a third of a grade level above the national average on math and reading tests even though the median family income is just $37,000 and only 18 percent of parents have a bachelor's degree. About 95 percent of the students are Hispanic, and the vast majority of students qualify for free or reduced-price lunches.

Silvia Abbato, the district's superintendent, said she could not pinpoint any one action that had led to the better scores. She noted that the district uses federal funds to help pay for teachers to obtain graduate certifications as literacy specialists, and it sponsors biweekly parent nights with advice on homework help for children, nutrition and immigration status.

The district regularly revamps the curriculum and uses quick online tests to gauge where students need more help or whether teachers need to modify their approaches."

In this example, success comes from a supportive community and integrated individualized approach to helping each child. These students did not have the advantage of an early start, yet they performed a third of a grade level above national average. It is possible to learn a new language after the first seven years of life but requires significant effort.

### Here's what makes early immersion math great for kids

"[Math's] hidden tendrils pervade every aspect of modern life." - Ian Stewart, _From Here to Infinity_

Just as there are many levels of understanding of spoken language, such as our ability to recognize irony, decipher the importance depending on how things are phrased, and what goes unsaid; there are levels of understanding mathematics. This richness is often lost on students where math class is focused on rote worksheets.

Proficiency in mathematical thinking is empowering and can advance the realization of children's goals. It can provide choices, and a structure for decision-making. Competency in mathematical problem solving is becoming a job requirement in the global economy. Manufacturing has moved to programmed robotics. Sales and commerce are now data-driven. Jobs, that don't require math but provide a middle-class income are scarce.

Even living within one's means requires an understanding of math, particularly the ability to consider multiple variables when making decisions. Mathematical thinking helps clarify long-term implications, or possible repercussions of financial decisions

Doing the math is a helpful impulse-control mechanism. It is a mindfulness exercise to calculate over-all cost of purchases including the cost of maintenance. Many people have been thrown into financial turmoil over an "unexpected" car repair bill. Buyers must calculate cost because no car salesman will say, "Once you add the cost of a license, insurance, gas and maintenance to the monthly payment, you won't have money for your children or other bills."

We are all vulnerable to the psychological tools used by sales and marketing specialists who encourage us to spend more than we are able. We are bombarded with advertising that appeals to emotions like pride, guilt, or self-pity. I, myself, feel vulnerable to offers for "free" three-course dinners at upscale restaurants and "deserved" vacations. But then I restrain myself as I remember my neighbors who bought a timeshare in Mazatlán, which they soon regretted, at a "free" dinner.

### The gift of mathematical thinking

"For only those activities that are easy and pleasant will become part of a man's habitual life and will serve him at all times." - Moshe Feldenkrais, _Awareness through Movement_

The decline in the ability to learn math is often interpreted as students' deficiencies of character such as laziness or rebelliousness. Yet, while students may appear to be lazy or rebellious, the human brain is doing what it is designed to do which is to start relying on habitual established patterns instead of forming new ones. The outward manifestations of this is that children often decide it is not worth their valuable time to analyze new mathematical concepts.

Low scores lead to accusations of deficiencies in character which are destructive and unfair. Students who have difficulty with math might have more confidence if they understood the early advantage others have had, in the same way that immigrants to a new country understand their disadvantage in learning a new language. Children in vulnerable populations could be taught about the critical age for learning a new language and how that might have impacted their view of math. They might have compassion for themselves and others if they understood that their parents and teachers did not transmit critical knowledge because had also missed the critical time for learning the language of math.

By giving all children the opportunity to learn math at the optimal time for learning language, children could start with a more equal footing.

We can also have more empathy for children who did not get the opportunity to learn these important concepts when it would have been easier for them.

The gift of mathematical thinking is one of the most thoughtful gifts we can give our children. Shall we dance?

####

Thank you for considering the ideas in this guide. I hope you will take a moment to leave a review on Smashwords or your favorite retailer.

Evelyn

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#   
Recommended Links, Readings  
and Resources

### Links

You can find Dr. Patricia Kuhl's October 2010 TED Talk, "The Linguistic Genius of Babies," here:

https://www.youtube.com/watch?v=qRRiWg6wYXw

John Bennett's November 2011 TED Talk, "Why Math Instruction is Unnecessary," can be found here:

https://www.youtube.com/watch?v=xyowJZxrtbg

You can find the policy analysis from the Southern Regional Education Board (SREB) "Confronting the Fade-Out Debate: Children Flourish and Gains Do Last in High-Quality Pre-K Programs" here:

 https://www.sreb.org/sites/main/files/file-attachments/15e05_prekfadeoutdebate.pdf?1458327945

You can find Carol Lloyd's article "Does our approach to teaching math fail even the smartest kids?" here:

 http://www.greatschools.org/gk/articles/why-americas-smartest-students-fail-math/

You can find Montessori concepts here: <http://www.learningtreemontessori.com/philosophy/>

Emily Badger's Washington Post article "How the housing crisis left us more racially segregated" can be found here:  https://www.washingtonpost.com/news/wonk/wp/2015/05/08/how-the-housing-crisis-left-us-more-racially-segregated/?utm_term=.dbb931e5055e

The PEW research report on international math competency can be found here:  http://www.pewresearch.org/fact-tank/2015/02/02/u-s-students-improving-slowly-in-math-and-science-but-still-lagging-internationally/

The Programme for International Student Assessment (PISA) can be found here: <http://www.oecd.org/pisa/aboutpisa/>

For more information on critiques of "reform math," check out <http://wheresthemath.com/> and <https://www.youtube.com/watch?v=Tr1qee-bTZI>

James Milgram, Professor of Mathematics at Stanford University essay on the Cliff Mass Weather Blog can be found here:  http://cliffmass.blogspot.com/2013/06/failing-math-curriculum-in-seattle.html

You can find the Lila MacLellan's article "In Japan, Hundreds Of Thousands Of Young People Are Refusing To Leave Their Homes" here: http://www.huffingtonpost.com/entry/japanese-people-living-in-recluse_us_57e570a5e4b08d73b83121cc?

You can find the video of the North Carolina disciplinary incident here:  http://www.nbcnews.com/news/us-news/video-appears-show-cop-body-slamming-student-s-c-classroom-n451896

For information on for-profit colleges see:  http://www.huffingtonpost.com/davidhalperin/trump-university-a-scam-b_b_10237880.html

The SAT College Board "2013 College-Bound Seniors Total Group Profile Report" can be found here:  http://media.collegeboard.com/digitalServices/pdf/research/2013/TotalGroup-2013.pdf

Information on the school-to-prison pipeline can be found here:  https://www.aclu.org/issues/juvenile-justice/school-prison-pipeline

The "No Child Left Behind Act" can be found here: http://www2.ed.gov/nclb/overview/intro/execsumm.pdf

You can find Carol Loyd's quote here: "Does our approach to teaching math fail even the smartest kids?"  http://www.greatschools.org/gk/articles/why-americas-smartest-students-fail-math/

You can find information on the 2006 financial meltdown here: "A decade after the financial crisis, many Americans are still struggling to recover"  https://www.seattletimes.com/nation-world/a-decade-after-the-financial-crisis-many-americans-are-still-struggling-to-recover/

You can find information on Baby Einstein videos here:  http://content.time.com/time/health/article/0,8599,1650352,00.html

You can find information on Singapore Math here:  
 https://www.pbs.org/parents/education/math/math-tips-for-parents/whats-singapore-math/

You can find Stephen Colbert's quote here: "Stephen Colbert's 'Tinfoil Hat' Segment Explains GOP Conspiracy Theories"  http://www.huffingtonpost.com/entry/stephen-colberts-tinfoil-hat-explains-gop-conspiracy-theories_us_57beb7a1e4b04193420d99f3?section

### Books

Bruce Bower, "Language learning may begin before birth" _(Science News,_ 9 February 2013)

Ruthy Alon, _Mindful Spontaneity: Lessons in the Feldenkrais Method_ Introductions by Carl Ginsberg, PhD and Dr. Bernard Lake (1996)

Moshe Feldenkrais, Awareness through Movement (1972)

Alison Gopnick, Andrew N. Meltzoff, and Patricia Kuhl, The Scientist in the Crib: What Early Learning Tells Us About the Mind (2000)

Immanuel Kant, _Critique of Pure Reason_ (1781)

Maria Montessori, To Educate the Human Potential (1947), From Childhood to Adolescence (1948) and The Montessori Method (1912)

Ian Stewart, From Here to Infinity: A Guide to Today's Mathematics (1996)

J.K. Rowling, _Harry Potter and the Sorcerer's Stone_ (2001)

### Articles and publications

Cohen, Patricia, "Child Care Expansion Takes a Toll on Poorly Paid Workers" (NYTimes 12 July 2016)

Zachary A. Goldfarb, "These four charts show how the SAT favors rich, educated families" (Washington Post, 5 March 2014) Data from the SAT College Board "2013 College Bound Seniors Total Group Profile Report"

David Halperin, "Friends in High Places: Who Endorses America's Troubled For-Profit Colleges?" ( _Republic Report_ , 21 June 2016), "Good Riddance 6 Facts about Shutdown ITT Tech" _Republic Report_ , 6 September 2016)

Betty Hart and Todd R. Risley, "The Early Catastrophe: The 30 Million Word Gap by Age 3" ( _American Educator_ , Spring 2003)

The Institute of Medicine and the National Research Council, "Transforming the Workforce for Children Birth Through Age 8" (2015)

Dr. Patricia Kuhl, "Early Language Acquisition: Cracking the Speech Code" ( _Neuroscience_ Volume 5, November 2004), "Brain Mechanisms in Early Language Acquisition" ( _Neuron_ Volume 67, September 2010), and "Decoding How Babies Learn Language: Q&A with Patricia Kuhl" ( _Cognitive Neuroscience Society_ , 14 January 2013)

Motoko Rich, Amanda Cox, and Matthew Bloch, "Money, Race and Success: How Your School District Compares" ( _NYTimes_ , 19 April 2016)

Diane Ravitch, "The Common Core Costs Billions and Hurts Students" ( _NYTimes_ , 23 July 2016)

James Romm, "Beginning Greek, Again and Again" ( _NYTimes_ , 2 January 2016)

Gillian White "The Empty Promises of For-Profit Colleges" ( _The Atlantic_ 15 September 2015)

R.L.E. Schwarzenberger, "The Language of Geometry" ( _Mathematical Spectrum_ Volume 4, Issue 2, 1972)

### Music

Mohombi, _Bumpy Ride_ album 2010

Ed Sheeran,"Thinking Out Loud" album 2014

Ice Cube, "What can I do?" Single 1999, _Fuck da Police_ album 2015

Images

NASA: The reflection nebula NGC 1999 in the constellation Orion

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# Acknowledgements

I would like to thank friends, collaborators and family who contributed to the ideas in this publication.

Doctor Danny, who was able to communicate at a very early age, what was working and what was not

My writing partner Tessa Bennion for her insight and ideas

The students I tutored for the valuable insights you provided. I did not understand at the time the reason math seemed so difficult for you

The University of Washington Human Subjects Division where I had the opportunity to read publications that were submitted by researchers with their renewal applications

Mina and Danny for allowing me to use your pictures to grace the cover of this guide

Beverly for sharing the story of her family

Dave Coker at Smashwords for providing this platform

Thank you all so much.

# About the Author

Through my work at the Human Subjects Division of the Office of Research at the University of Washington, I was introduced to current research in brain science and language acquisition. By applying this research to my experience as a parent and math tutor, I came to understand why so many children can't seem to learn math, even though they had no trouble learning their native language, whose rules, or algorithms, are much more complex.

This method of teaching math differs from the usual way math is taught in our schools. I would love to hear what you think. Please take a moment to rate this guide on Smashwords or your favorite retailer.

Please visit my blog at:

https://gamechangerforkids.wordpress.com/

I would love to hear from you. Please leave comments on my blog:

evelyn.raiken@gmail.com.

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