“Math is not about numbers.”The first time my prealgebra students hear me say this, they stare at me with confusion in their eyes. I’m sure some are silently thinking “OMG, my parents signed me up to take a class with a crazy woman. Of course, everyone knows that language arts is about words and math is about numbers." |

Math is the science of patterns. Mathematicians study patterns and relationships. They use a system of structured logic to analyze these relationships which are then often described using one or more of the languages of mathematics. At its core, math is pure logic.

Most people think math is the study of numbers because elementary school math primarily focuses on a branch of math called arithmetic. While elementary students do get glimpses of a few other branches of math such as algebra and geometry, most of their math studies involve adding, subtracting, multiplying and dividing different types of real numbers.

Algebra is the branch of math focusing on equations with variables. Each equation is a complete sentence that can be directly translated into a colloquially spoken language such as English, Spanish or Hindi. The beauty of a sentence that is translated into the language of algebra is that it can then be transformed using clearly defined properties and identities, such as the Order of Operations or the Identity, Associative, and Commutative Properties of Addition and Multiplication. These transformed equations can answer questions and reveal new insights into the original statement.

Educators use the term “Algebra” in two ways that differ from the definition of “Algebra” as a branch of mathematics. Algebra is the name of the first high school-level math course. It is also included as the name of a Common Core domain. These different definitions of the same word often cause confusion.

Algebra is the branch of math focusing on equations with variables. Each equation is a complete sentence that can be directly translated into a colloquially spoken language such as English, Spanish or Hindi. The beauty of a sentence that is translated into the language of algebra is that it can then be transformed using clearly defined properties and identities, such as the Order of Operations or the Identity, Associative, and Commutative Properties of Addition and Multiplication. These transformed equations can answer questions and reveal new insights into the original statement.

Educators use the term “Algebra” in two ways that differ from the definition of “Algebra” as a branch of mathematics. Algebra is the name of the first high school-level math course. It is also included as the name of a Common Core domain. These different definitions of the same word often cause confusion.

The Common Core is a set of learning objectives for students in each grade. It is essentially a checklist of what students should learn in each grade. Each item on this checklist is called a standard. The standards are grouped by topics called domains.

One of the domains for students in grades K-5 is called “Operations & Algebraic Thinking.” The standards included in this domain are as follows:

- Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

- Represent and solve problems involving addition and subtraction
- Understand and apply properties of operations and relationship between addition and subtraction.
- Add and subtract within 20.
- Work with addition and subtraction equations.

- Represent and solve problems involving addition and subtraction.
- Add and subtract within 20.
- Work with groups of objects to gain foundations for multiplication.

- Represent and solve problems involving multiplication and division.
- Understand properties of multiplication and the relationship between multiplication and division.
- Multiply and divide within 100.
- Solve problems involving the four operations, and identify and explain patterns in arithmetic.

- Use the four operations with whole numbers to solve problems.
- Gain familiarity with factors and multiples.
- Generate and analyze patterns.

- Write and interpret numerical expressions.
- Analyze patterns and relationship.

When elementary school students are working on any of the above topics included in “Operations and Algebraic Thinking,” they will often report that they are learning algebra. Elementary school teachers often shorten the name of this domain to simply “algebra.” Enrichment math classes for elementary students may even expand on the standards to include the introduction of variables.

While these concepts are essential for preparing students to eventually take algebra, mastering these learning objectives is not equivalent to taking a high school level algebra course. Most students who successfully master all of these concepts will need two additional years of math studies to be prepared for high school algebra.

High school algebra is a full year course focusing primarily on algebra but normally also includes topics from geometry and statistics. While the topics covered in high school algebra have not changed significantly over the past several decades, the methods used to teach high school algebra have changed dramatically since the passage of No Child Left Behind (NCLB) in 2002, the adoption of the Common Core in 2010, and the passage of Every Child Succeeds Act (ESSA) in 2015.

Students still learn to solve and graph linear equations. They still perform operations with polynomials. They learn to factor, solve quadratic equations, and work with rational expressions and equations. A side-by-side comparison of the table of contents of a modern algebra book and one that was published in the 20th century would not reveal any dramatic changes. However, a side-by-side comparison of the content exposes dramatic changes.

Current algebra textbooks begin chapters with examples of how to solve specific types of problems. They are followed by short exercise sets students can complete by following the step-by-step examples. Most units conclude with a short test prep section.

These textbooks focus on HOW to use step-by-step procedures to complete problem sets. They provide no explanation of WHY certain techniques are used. Topics are presented as discrete modules rather than combined to provide increasing complexity. Training in logic is completely missing. Logic is the life giving blood of all mathematics. Logic is the unifying essence determining if a topic or process is a member of the set of mathematical systems.

Current algebra courses typically focus on how to solve specific problems likely to appear on standardized tests rather than on how to apply general problem-solving techniques that will train students to solve real-world problems outside the classroom. The most frequent questions I ask my algebra students are, “

During a recent class meeting, a student requested help by stating,

A rigorous algebra course trains students to think logically and to break down complex problems into discrete manageable steps. I do not allow my students to use ‘shortcut’ formulas that they cannot prove. I challenge my students to derive the quadratic equation by completing the square of a problem where all the coefficients are replaced by variables. Less than half of the class expressed confidence in their ability to achieve this feat. Several raised their hands multiple times to express their lack of confidence. Each time I responded by asking, “Do you think you could figure out just one more step toward isolating x?” A few that needed this encouragement for almost every line. In the end, everyone was able to arrive at the quadratic equation. They were amazed at their own ability to combine multiple algebraic properties to solve a problem unlike anything they had ever attempted before. They demonstrated that they had achieved my year-long learning objective: To be able to use algebraic properties and deductive reasoning to solve problems.

It is essential that students are properly prepared before enrolling in algebra. Elementary students must first master operations with real numbers including: adding, subtracting, multiplying, and dividing whole numbers, integers, fractions, decimals, and percents plus order of operations, exponents and square roots. A student who demonstrates proficiency in ALL of these topics is ready to more closely examine WHY these arithmetic procedures work and apply the same procedures to expressions containing variables. This bridge between arithmetic and algebra is prealgebra. Students must master prealgebra before enrolling in algebra.

The average student in the United States takes algebra I in 9th grade. To prepare for algebra, students take middle school math that is equivalent to prealgebra.

Many school districts offer an accelerated math option for their top math students. These students often take algebra in 8th grade. Some school districts offer a double accelerated math option that allows a few select students to take algebra in 7th grade. Schools rarely allow students to take algebra before 7th grade.

Being allowed to register for algebra is not necessarily the same as being mathematically ready to take algebra. Schools often offer a less strenuous version of algebra for high school students who are not adequately prepared. I believe that course placement should be based on mathematical readiness and maturity rather than age or grade in school.

Students are ready for prealgebra if they can consistently, rapidly, and correctly do all of the following:

- Add, subtract, multiply, and divide whole numbers.
- Add, subtract, multiply, and divide integers.
- Add, subtract, multiply, and divide fractions and mixed numbers, including finding the lowest common denominator and reducing fractions to their simplest forms.
- Add, subtract, multiply, and divide decimals.
- Solve problems involving percents.
- Convert between fractions, mixed numbers, decimals and percents.
- Raise whole numbers to exponents.
- Take the square root of perfect squares.
- Use the Order of Operations.
- Solve simple word problems.
- Demonstrate that they are mature enough to handle a serious math classroom environment, homework requirements, and textbook.

I frequently meet parents who want to push their children into prealgebra or algebra before they are ready. It is essential that students do not enroll in advanced math classes until they demonstrate that they have mastered all the content of prerequisite courses. Students who are advanced before they are ready will end up with ‘holes’ in their math education that will limit their long-term math development.

Students are ready for algebra if they can:

- Apply all prealgebra arithmetic requirements to expressions containing variables.
- Simplify and work with radicals.
- Understand and apply rules of exponents.
- Apply basic mathematical properties and definitions to transform algebraic expressions.
- Apply properties of equality to solve simple equations.
- Translate phrases and sentences into algebraic expressions and equations.

Each child is a unique individual who will develop at a unique pace. Don’t push your child to advance to a class before being adequately prepared or allow your child to be trapped in a class with little or no new content to learn. Talk to your child’s teacher in the spring about potential class placement for the following academic year. Do not wait until summer or after school starts. If you disagree with the proposed class placement, ask for an objective test of skills. Use the test results as a diagnostic tool to identify areas of weakness.

Parents of elementary school students should inquire about advanced math tracks. Although algebra is rarely offered to students in public schools before 7th grade, students must often begin preparing for accelerated math in 4th or 5th grade in order to eventually qualify for 7th or 8th grade algebra.

Be aware that students who do not take algebra in the earliest year offered by their school district rarely have the ability to later move up into this highest math track. Early enrollment in algebra not only allows students to take more advanced math in high school, but also allows them to be prepared to take advanced high school courses in physics, chemistry, biology, computer science, and economics. Equally important, it ensures that they will take classes with the most serious and studious students in the school district which greatly affects their choice of friends.

Parents who wish for special accommodations to allow their child to take algebra earlier than it is offered through their school district, should consult someone with experience working with children who are radically accelerated to insure that they consider not only the academic implications but also the social and emotional aspects of the proposed placement.

As parents, we need to open the doors to learning but we must not push a child through those doors without adequate preparation. Carl Gauss, who is sometimes referred to as The Prince of Mathematicians, once said,

Properly presented to prepared students, Algebra introduces them to this greatest enjoyment.

Accelerated Math: What Every Parent Should KnowFew parents of elementary school students are thinking about high school advanced placement options. Unfortunately, class placement as early as 4th or 5th grade may have a profound impact on a child’s ability to eventually take advanced placement classes in high school...Read More... | From Multiplication Minutes to Math MasteryMath is not about crunching numbers; true math is about pattern recognition and logic. This does not mean that mastering basic calculations is not important...Read More... |

One of the greatest joys of my life is cooking with my daughter. We preserve farm produce by canning, freezing and drying. We precook meals for our busy weekends. We explore new techniques and cuisines together. We text each other when we find great sales at the grocery store or need a second opinion on a new creation. These days I normally step back to the roles of sous chef and bottle washer, letting her take the lead. As I mince garlic, chop onions, and wash dishes, I marvel that the little girl who used to give me ‘too much little help’ has grown into an expert cook and baker. Her life skills contrast sharply with the boarders that I take in. Although they are about the same age, have similar educational backgrounds, and hold responsible professional jobs, their life skills are surprisingly lacking. How is it possible to live more than 20 years and not know how to even boil pasta, cook rice, use a broom, clean a bathroom, or adjust a thermostat? Why does learning to change an air filter, hang a picture, or change light bulbs invoke so much fear? |

As I’ve taught these young women to plan and cook meals, maintain a house, assemble furniture, hem pants, repair a car, and plant gardens, I’ve realized that they all share a common background that contributed to their lack of life skills. While their families focused on academic and professional achievement, their mothers never taught them to be self-sufficient, problem solving, independent thinkers. They have book-learning, transcripts, degrees, and job skills, but they haven’t learned how to apply these skills outside of their fields of study. They have not learned how to teach themselves new skills.

I see many of my younger students heading in this direction. Their parents rush them from one activity to another as they focus on their academic achievement. They make sure that they get every advantage to get the best grades and highest test scores. Somehow they assume that life skills like time management, organization, responsibility, and problem solving will magically appear with age. It won’t.

While people are often impressed by the stories of the ‘super geniuses’ that I’ve worked with who started college at ages 9, 10 or 11, it’s my daughter who provides the road map that I use to design our programs. It’s not just that she can bake a perfect loaf of sour dough bread, install a laminate floor, negotiate leases, sew the most comfortable PJs, create all our advertising, manage our website, explain the neurobiology of giftedness, diagram sentences, and conjugate Latin verbs. It’s that she taught herself how to do all those things. She took the initiative to read books, articles and research journals, consult with experts, experiment, evaluate results, and continue to pursue a topic until she was satisfied that she understood it. It’s her lack of fear of the unknown -- her confidence that she can learn anything -- that makes her my pride and joy.

This is what I want for my students. I don’t just want to teach them math. I want to teach them how to overcome their fear of the unknown. I want them to gain confidence that they can learn anything.

Some of this I can accomplish in the classroom. I do this by not just focusing on how to solve a particular problem but by training them to practice problem solving techniques. Why does this problem look so scary? Can you identify the worst part of it? Can you think about a way to address just this one issue? Take things one step at a time. Break down big problems into smaller parts. Don’t skip steps. These are all things I say over and over to my students as we work through different types of problems to develop different math skills. With my older students I also talk to them about applying these techniques to other areas of their life.

Sometimes well-meaning parents make this task difficult. Some assume that if they can’t understand the homework, then it must be too hard for their child. Others who are more confident in their math skills often provide ‘too much big help.’ Whether they argue that their child shouldn’t be expected to complete such difficult assignments, make excuses for incomplete or sloppy work, or walk them through assignments that they are supposed to complete on their own, they are all sending their children the same message. They are giving them votes of non-confidence. They are saying “I don’t expect you to be a self-sufficient, problem solving, independent thinker.”

Parents don’t intentionally plan to send this negative message or sabotage their child’s personal growth. As mothers, it's in our nature to protect our children from the unknown -- to shield or children from the "big, bad adult world," -- to maintain their innocence for as long as possible. But we must not forget that we can continue to do all of these things whilst still allowing for personal growth. Parenting, like all other facets of life, is about balance -- balancing "easy" with challenging -- balancing praise with discipline -- balancing fun with necessary -- balancing freedom with restrictions.

Our successes are only as great as our failures. The road to success is paved with cobblestones, roadblocks, and accidents. Yet, it is this journey that creates a truly successful individual empowered to take on the world.

]]>I see many of my younger students heading in this direction. Their parents rush them from one activity to another as they focus on their academic achievement. They make sure that they get every advantage to get the best grades and highest test scores. Somehow they assume that life skills like time management, organization, responsibility, and problem solving will magically appear with age. It won’t.

While people are often impressed by the stories of the ‘super geniuses’ that I’ve worked with who started college at ages 9, 10 or 11, it’s my daughter who provides the road map that I use to design our programs. It’s not just that she can bake a perfect loaf of sour dough bread, install a laminate floor, negotiate leases, sew the most comfortable PJs, create all our advertising, manage our website, explain the neurobiology of giftedness, diagram sentences, and conjugate Latin verbs. It’s that she taught herself how to do all those things. She took the initiative to read books, articles and research journals, consult with experts, experiment, evaluate results, and continue to pursue a topic until she was satisfied that she understood it. It’s her lack of fear of the unknown -- her confidence that she can learn anything -- that makes her my pride and joy.

This is what I want for my students. I don’t just want to teach them math. I want to teach them how to overcome their fear of the unknown. I want them to gain confidence that they can learn anything.

Some of this I can accomplish in the classroom. I do this by not just focusing on how to solve a particular problem but by training them to practice problem solving techniques. Why does this problem look so scary? Can you identify the worst part of it? Can you think about a way to address just this one issue? Take things one step at a time. Break down big problems into smaller parts. Don’t skip steps. These are all things I say over and over to my students as we work through different types of problems to develop different math skills. With my older students I also talk to them about applying these techniques to other areas of their life.

Sometimes well-meaning parents make this task difficult. Some assume that if they can’t understand the homework, then it must be too hard for their child. Others who are more confident in their math skills often provide ‘too much big help.’ Whether they argue that their child shouldn’t be expected to complete such difficult assignments, make excuses for incomplete or sloppy work, or walk them through assignments that they are supposed to complete on their own, they are all sending their children the same message. They are giving them votes of non-confidence. They are saying “I don’t expect you to be a self-sufficient, problem solving, independent thinker.”

Parents don’t intentionally plan to send this negative message or sabotage their child’s personal growth. As mothers, it's in our nature to protect our children from the unknown -- to shield or children from the "big, bad adult world," -- to maintain their innocence for as long as possible. But we must not forget that we can continue to do all of these things whilst still allowing for personal growth. Parenting, like all other facets of life, is about balance -- balancing "easy" with challenging -- balancing praise with discipline -- balancing fun with necessary -- balancing freedom with restrictions.

Our successes are only as great as our failures. The road to success is paved with cobblestones, roadblocks, and accidents. Yet, it is this journey that creates a truly successful individual empowered to take on the world.

The internet is full of attention grabbing headlines about this including:

**Chinese space station carrying TOXIC chemicals to CRASH into Europe or New York NEXT WEEK**

**Chinese space station: Tiangong-1 could crash into Earth and hit Europe next week**

And this one that sounds like a movie ad:

**China’s Out of Control Space Station Coming to a City Near You!**

Don’t Panic! Despite the dire warnings, you are not at risk of being obliterated by a space lab crashing through your roof. Most of the lab will burn up as it re-enters the Earth’s atmosphere with remaining debris scattering over a large area.

But scientists around the world are tracking Tiangong-1 and making models to predict when and where it will land. Here’s one article that explains how they are doing it:

https://www.livescience.com/62077-tiangong-re-entry-date.html

While the models being developed to pinpoint the re-entry and landing sites must consider an incredible number of variables and depend on sophisticated modeling tools, the basics of understanding the path of this or any other satellite or projectile only require an understanding of the polynomial functions that we’ve been studying in Algebra this month.

And this one that sounds like a movie ad:

Don’t Panic! Despite the dire warnings, you are not at risk of being obliterated by a space lab crashing through your roof. Most of the lab will burn up as it re-enters the Earth’s atmosphere with remaining debris scattering over a large area.

But scientists around the world are tracking Tiangong-1 and making models to predict when and where it will land. Here’s one article that explains how they are doing it:

https://www.livescience.com/62077-tiangong-re-entry-date.html

While the models being developed to pinpoint the re-entry and landing sites must consider an incredible number of variables and depend on sophisticated modeling tools, the basics of understanding the path of this or any other satellite or projectile only require an understanding of the polynomial functions that we’ve been studying in Algebra this month.

While in orbit the Tiangong-1 Space Lab, like all other satellites, travels in an ellipse. We won’t work with ellipses until Pre-Calculus, but the equation is just a two-variable version of the equations we’ve been working with this week:

Where

Although we haven’t studied this in class yet, you can try to graph an ellipse using the same technique of picking values for one variable and solving for the other then plotting the points. Try finding points. Then, graph these ellipses:

Look at your graphs and use the analysis tools we learned in class to think about what happens when you change the value of any of the constants in the equation.

Many things will make a satellite, such as the Tiangong-1 Space Lab, change course or slow down. Normally, ground control corrects the path by activating one or more engines on the satellite. In 2016, China’s space agency lost its connection to the Tiangong-1 Space Lab, so it hasn’t been able to correct its path for two years.

As a satellite slows down, its path becomes less elliptical and more circular. Mathematically, the values of

Notice that the first equation gave you a long, stretched out ellipse. Each subsequent equation gives you a rounder ellipse until you end up with a circle in the last equation. (Mathematicians call this degree of stretchiness “eccentricity” and is usually calculated using the variable *e*.)

Notice that if*A = B*, the equation for the ellipse become the same as the equation for a circle with : *C = r^2*

Notice that if

If a satellite traveling in a circle slows down further, its path become a spiral and eventually follows the path of a projectile.

This week we worked on solving equations in the form

Substituting these variables gives us:

Since the altitude of any object on the ground is zero, we can find the time it takes for a projectile to land by solving the following equation:

This is the standard equation used to figure out how long a baseball or rocket will be airborne before it lands on the ground. Then the time can be used to find the distance traveled by putting it into a simple equation that normally looks like this:

Where:

We can use this equation for debris from Tiangong-1 Space Lab after it re-enters the Earth’s atmosphere.

Last year, in Pre-Algebra we played with the equation for the Force Due to Gravity:

Last year, in Pre-Algebra we played with the equation for the Force Due to Gravity:

We used this to confirm the acceleration due to gravity close to the surface of the Earth and to understand why there is very little gravitational force in the space station. When we get to Calculus we’ll be able to work with cases where *g* is a variable this is constantly changing over time.

Using the quadratic equations that we learned in class, it seems like the path of the Tiangong-1 Space Lab has three distinct parts. But we can see that the circle is just a special type of ellipse.

To see how projectile motion and orbiting are related think about this:

- Imagine that you throw a baseball. It will travel some distance following the path of a parabola and land somewhere on the ground.
- Imagine that you threw the same baseball with more force. It will travel further before landing on the ground.
- Now imagine that you are superman. You throw the ball so hard that the distance it travels before landing on the ground is greater than the radius of the Earth. There is no ground for it to land on. It will keep falling toward the Earth but never land. It goes into orbit.

In “Hidden Figures” Katherine Goble wrestled with the problem of how the path of a rocket would change from an elliptical orbit to a parabolic fall to Earth. Rewatch the movie and to find her “aha” moment when she realized that these two paths could be described with one equation. That equation involves trigonometry and polar coordinates. You can see Ms. Goble work out the mathematics of this in the movie. The unifying equation is:

I know what you’re going to ask me and the answers are:

“No, we can’t spend our entire next class meeting playing with these equations. You don’t yet have that math background to derive all of them. BUT we can spend part of the class period playing with projectile motion since it ties into the quadratic functions that we worked on.”

“Yes, some of these equations are the ones that Dylan derived and wrote on the window last year. As sad as it was to erase his work, I’ve very proud that you were able to add the derivation of the quadratic equation to the window last week.”

“Yes,I recommend that you spend some of your time off of school doing your own research on this.” I suggest the following:

http://www.physicsclassroom.com/class/circles/Lesson-4/Circular-Motion-Principles-for-Satellites

Rewatch the movie Hidden Figures.

Logic, often translated as thought or reason, builds the foundation for critical thinking skills. It is essential for success in higher level math, computer science, philosophy, science, engineering problem solving and decision-making. Logic demands that each statement be supported by reason. In logic puzzles, each conclusion can be supported by a “because” statement. At higher levels, students begin to use ‘if…then…” statements to support their conclusions. Logic puzzles require students to practice using both deductive and inductive reasoning skills. Deductive reasoning is the practice of using information from a larger set of information to understand other sets of related information. Inductive reasoning is the practice of using specific data to draw a much larger conclusion. These skills will prove indispensable as children emerge in their professional and academic lives. |

In the United States, formal training in logic is often limited to writing formal proofs in high school geometry classes. Yet, logic can easily be introduced to younger students through fun games and puzzles.

The key to teaching students logic through games and puzzles is to train them to consistently explain “why” they are able to make each of their moves. I train them to precede each move with a statement in the form of:

*“I know ________________ because ______________________.”*

Students are not allowed to make moves if they “think” something is true or if something “could” be true.

I always start with Binary Puzzles because the rules are very simple. Each puzzle is a square array with some cells containing 1s or 0s. The goal is to fill in the empty cells using the following rules:

Students are not allowed to make moves if they “think” something is true or if something “could” be true.

I always start with Binary Puzzles because the rules are very simple. Each puzzle is a square array with some cells containing 1s or 0s. The goal is to fill in the empty cells using the following rules:

In order to discuss our example puzzle, we will label each column and row so we can easily identify each cell. So in the example below, there is a 1 in cell Aa and a 0 in cell Bd.

Notice how this statement is supported by one of the rules.

Look at the updated puzzle. What other cells can you find based on the rules of the puzzle?

You should be able to make the following statements:

Look at the updated puzzle. What other cells can you find based on the rules of the puzzle?

You should be able to make the following statements:

“I know that cell Ab is a 0 BECAUSE rule #3 says that we can’t have three 1s next to each other in any row." “I know that cells Af and Df are 0s BECAUSE rule #3 says that we can’t have three 1s next to each other in any column.” “I know that cells Bb, Be, Ca, Cd, Ec, and Fe are 1s BECAUSE rule #4 says we can’t have three 0s next to each other in any row." |

Update the puzzle for each of these statements then use the additional information to try to fill in more spaces.

Based on this updated puzzle you should be able to make the following statements:

Based on this updated puzzle you should be able to make the following statements:

“I know that cell Ae contains a 1 BECAUSE rule #1 says that each row must have the same numbers of 1s as 0s and there are already three 0s in that row.’“I know that cells Ba and Ce contain 0s BECAUSE rule #1 says that each row must have the same numbers of 1s as 0s and there are already three 1s in those rows.’ |

After filling in these cells, you should be able to easily finish the puzzle.

Review each rule to make sure that your puzzle is correct:

- Confirm that each row contains three 0s and three 1s.
- Confirm that each column contains three 0s and three 1s.
- Scan each row and column to make sure that there are no instances where three 0s or three 1s are next to each other.

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 20 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 20 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 9 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 9 "8x8" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 5 "10x10" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 5 "10x10" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

The first time I saw him, he reminded me so much of Q. The way he rolled his head back when he was thinking. His lack of eye contact when speaking. His abnormally precise language, his obsession with numbers, especially very large numbers resulting from doubling, irrational numbers and those resulting from functions unknown to any mathematician other than himself. I played with him the way I’ve played with countless other children I’ve met like this. I created patterns and let him expand on the series. I invented functions using made up notation to see what he would do with them. His mother, anxious that he would make a good impression, encouraged him to look at me and answer my questions. “Wait” I hush her, almost in a whisper. “He’s thinking.”

I will never tire of watching these children think. They go into their own worlds. They don’t make eye contact. It’s not like they are avoiding eye contact; it’s that they just don’t see me or anything in this world for that matter. I can almost hear the wheels turning in their heads. I see them suspended between two worlds – the physical one that I occupy, and the sacred realm accessible only through the portal controlled by their own minds.

Their solutions to my challenges are never based on standard algorithms. I don’t expect them to be. In fact, I present problems that I’m sure they’ve never seen. It’s how their minds work that I’m studying. Can they explain their answer to me? I ask deeper questions to help them more clearly define their logic.

I’m no longer asking him questions to determine whether or not I’ll take him on as a student. I’ve already decided that. I want to see how he reacts to me and the problems I give him. I want to see if his mother trusts me enough to let us play our game without interference. He can talk math with me forever, but I know that I don’t really exist to him. I am just the source of the fascinating puzzles. His mother is also just a sounding board for the product of his mental exercises. At this point, I’m sure the endless monologues just exhaust her but soon they will also bewilder her when he crosses over into topics beyond her ability to comprehend. In time I will teach her to take her cues from his tone of voice and facial expressions to know when to smile and nod like she’s trying to converse with someone who’s speaking a foreign language.

I flip through my mental Rolodex as I think about where to place him. D is a perfect personality match but he is at least 3 years ahead. C would be perfect but she is 2 years behind. E would be perfect but his parents already push him so much that I’m considering dropping him. I will help any child explore their passions but I won’t be party to pushing a child just for bragging rights.

I could tutor him. We would have fun. But I know that there is something he needs more than math instruction. His mother doesn’t know what I know yet; if I don’t intervene, in four years, I will be holding her hands and listening to her cry about a crisis that she can’t yet imagine.

Some people would say he has Asperger’s or is ‘on the spectrum’. In my day, we would have called him ‘quirky.’ I’m not a psychologist so I don’t label him. To me, he is just a little boy who loves math. I know I can use his love of math to help him develop social skills that he doesn’t realize he needs. I also know that I will have to prepare him to deal with the college classroom before he is old enough to get a driver’s permit.

There was a time when I thought that this quirky social awkwardness must be tied to the same gene that determined mathematical genius. I believed that their social limitations had to be accepted as an immutable part of their personalities. I remember the joy of finding other boys like this for my son.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade. I’ve participated in dozens of meetings like the one above.

Their solutions to my challenges are never based on standard algorithms. I don’t expect them to be. In fact, I present problems that I’m sure they’ve never seen. It’s how their minds work that I’m studying. Can they explain their answer to me? I ask deeper questions to help them more clearly define their logic.

I’m no longer asking him questions to determine whether or not I’ll take him on as a student. I’ve already decided that. I want to see how he reacts to me and the problems I give him. I want to see if his mother trusts me enough to let us play our game without interference. He can talk math with me forever, but I know that I don’t really exist to him. I am just the source of the fascinating puzzles. His mother is also just a sounding board for the product of his mental exercises. At this point, I’m sure the endless monologues just exhaust her but soon they will also bewilder her when he crosses over into topics beyond her ability to comprehend. In time I will teach her to take her cues from his tone of voice and facial expressions to know when to smile and nod like she’s trying to converse with someone who’s speaking a foreign language.

I flip through my mental Rolodex as I think about where to place him. D is a perfect personality match but he is at least 3 years ahead. C would be perfect but she is 2 years behind. E would be perfect but his parents already push him so much that I’m considering dropping him. I will help any child explore their passions but I won’t be party to pushing a child just for bragging rights.

I could tutor him. We would have fun. But I know that there is something he needs more than math instruction. His mother doesn’t know what I know yet; if I don’t intervene, in four years, I will be holding her hands and listening to her cry about a crisis that she can’t yet imagine.

Some people would say he has Asperger’s or is ‘on the spectrum’. In my day, we would have called him ‘quirky.’ I’m not a psychologist so I don’t label him. To me, he is just a little boy who loves math. I know I can use his love of math to help him develop social skills that he doesn’t realize he needs. I also know that I will have to prepare him to deal with the college classroom before he is old enough to get a driver’s permit.

There was a time when I thought that this quirky social awkwardness must be tied to the same gene that determined mathematical genius. I believed that their social limitations had to be accepted as an immutable part of their personalities. I remember the joy of finding other boys like this for my son.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade. I’ve participated in dozens of meetings like the one above.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade.

I started HEROES as the mother of a child like this. After reading all the research and speaking to all the living experts I still wanted to meet another mother of a child like mine. I wanted to look at her in the eye and ask her if her child eventually grew up to be a productive, happy adult. I wanted to see that her grown child was OK when he grew up. I also wanted to be the mentor to other mothers who shared the same anxiety over their children.

Over the past decade, I’ve met thousands of mothers of brilliant children. I’ve leant my shoulder to hundreds of them as they cried; at times shedding tears of joy; at other times tears of despair. I’ve been enchanted by children making new friends at conferences. I’ve mourned the decision to commit a child to a psychiatric facility. I’ve celebrated with pre-teens who won a math completion against college students. I’ve grieved for a child hospitalized for eating disorders. I’ve congratulated teens who’ve earned patents. I’ve counseled teens locked up in juvenile detention.

Why do some become rising stars and others end up on suicide watch? There is no simple answer to that question but I’ve observed some critical variables.

Balance is probably the trickiest part of being a parent. In the movie Gifted, Mary enjoys the perfect balance between academic stimulation and social integration. Unfortunately, life is not as simple as a movie. Nevertheless, there is an important lesson to learn from Frank Adler’s battle with his mother over Mary’s education. Geniuses need academic stimulation. They need guidance. Mary didn’t learn differential equations on her own. Someone introduced her to the concepts of limits and infinity and the meaning of delta and epsilon. Geniuses also need love and companionship. They need friends who connect with them not just for what they know but for who they are. As Mary says, ‘Frank loved me before I was a genius.’

If your child does not find school adequately challenging then you may need to look for outside resources that will reinvigorate the thirst for learning. But it is important to not go overboard. I’ve met many children who used to love math until they burned out over too many tutors, competitions, and classes. Remember, it is more important to nurture the love of learning than to accelerate the rate of learning.

Over the past decade, I’ve met thousands of mothers of brilliant children. I’ve leant my shoulder to hundreds of them as they cried; at times shedding tears of joy; at other times tears of despair. I’ve been enchanted by children making new friends at conferences. I’ve mourned the decision to commit a child to a psychiatric facility. I’ve celebrated with pre-teens who won a math completion against college students. I’ve grieved for a child hospitalized for eating disorders. I’ve congratulated teens who’ve earned patents. I’ve counseled teens locked up in juvenile detention.

Why do some become rising stars and others end up on suicide watch? There is no simple answer to that question but I’ve observed some critical variables.

Balance is probably the trickiest part of being a parent. In the movie Gifted, Mary enjoys the perfect balance between academic stimulation and social integration. Unfortunately, life is not as simple as a movie. Nevertheless, there is an important lesson to learn from Frank Adler’s battle with his mother over Mary’s education. Geniuses need academic stimulation. They need guidance. Mary didn’t learn differential equations on her own. Someone introduced her to the concepts of limits and infinity and the meaning of delta and epsilon. Geniuses also need love and companionship. They need friends who connect with them not just for what they know but for who they are. As Mary says, ‘Frank loved me before I was a genius.’

If your child does not find school adequately challenging then you may need to look for outside resources that will reinvigorate the thirst for learning. But it is important to not go overboard. I’ve met many children who used to love math until they burned out over too many tutors, competitions, and classes. Remember, it is more important to nurture the love of learning than to accelerate the rate of learning.

It is more important to nurture the love of learning than to accelerate the rate of learning.

If your child is not connecting socially at school then you may also need to look into additional resources. Contrary to popular belief, brilliance does not equate to social awkwardness or isolation. The most amazing children I’ve met were not only nationally or internationally recognized as brilliant but also popular and possessed sophisticated senses of humor. Most of the children at HEROES are happy and well rounded. They also play soccer, compete in fencing, enjoy drama, play musical instruments and often rush from class to friends’ birthday parties.

Other children need help to find other children close to their age with similar abilities and interests. I’ve worked with dozens of children who were diagnosed with ADHD or Asperger’s Spectrum Disorders who were able to fully engage in programs with other children with similar abilities and interests. When I look at my class of students, I don’t see the ADHD or the ASD. I see children who are enjoying learning and the camaraderie of friends mutually working to solve problems and meet challenges. I also note that the monologues have given way to genuine conversation and debate and that eye contact naturally developed as classmates became friends.

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Other children need help to find other children close to their age with similar abilities and interests. I’ve worked with dozens of children who were diagnosed with ADHD or Asperger’s Spectrum Disorders who were able to fully engage in programs with other children with similar abilities and interests. When I look at my class of students, I don’t see the ADHD or the ASD. I see children who are enjoying learning and the camaraderie of friends mutually working to solve problems and meet challenges. I also note that the monologues have given way to genuine conversation and debate and that eye contact naturally developed as classmates became friends.

Before each math class, I check and restock all the supplies my students will need: wide ruled paper, colored pencils, crayons, and pens. Pens? Not Pencils? Yes, pens not pencils – black and purple pens to be precise.

I know this is unconventional but there’s a method to my madness. Think about it: Why do we use pencils for math? Normally this question is met with a blank stare since the use of pencils for math is so universal that most have never considered questioning this convention. The obvious answer, of course, is to erase our mistakes but, why do we want to erase our mistakes when we know that we learn from our mistakes?

I know this is unconventional but there’s a method to my madness. Think about it: Why do we use pencils for math? Normally this question is met with a blank stare since the use of pencils for math is so universal that most have never considered questioning this convention. The obvious answer, of course, is to erase our mistakes but, why do we want to erase our mistakes when we know that we learn from our mistakes?

I teach my students to embrace mistakes rather than erase their mistakes. My students self-grade with purple pens. I teach them to not just mark a problem right or wrong but to annotate their work with notes on their mistakes. Was it a concept that they did not understand? Was it an arithmetic mistake? My students must write a note about every mistake they find on their paper. After a few pages of work, they look back at their purple notes find a pattern. Most days, most students are making only one or two types of mistakes over and over. By highlighting their mistakes, each student is able to focus on specific individual learning goals.

Students are usually surprised by what they find out about themselves by highlighting their mistakes. They learn that:

Neatness matters – The number #1 cause for getting the wrong answer is not being able to read their own handwriting.

Basic arithmetic matters – Students often dismiss ‘careless’ arithmetic mistakes but they quickly realize that these errors matter when they actually have to calculate an answer rather than simply select from a set of multiple choice options.

Once students realize the importance of neatness they understand some of my classroom rules which include:

Telling the children is one thing, getting them to adopt these practices is quite another. In my classroom these rules are enforced by The Crumple Monster. Any page that does not meet my standards is removed, crumpled, placed in the recycling bin and replaced by a fresh sheet of paper. The Crumple Monster destroys any and all messy work. It is rare for a student to not have at least one page destroyed by The Crumple Monster on the first day. In fact I normally fill two commercial sized recycling bins on the first weekend of class.

How do the kids react to these tactics? I thought they might think I’m mean but they almost universally think I’m funny. They love writing with the purple pens and joke with each other that The Crumple Monster is going to get them.

Students are usually surprised by what they find out about themselves by highlighting their mistakes. They learn that:

Neatness matters – The number #1 cause for getting the wrong answer is not being able to read their own handwriting.

Basic arithmetic matters – Students often dismiss ‘careless’ arithmetic mistakes but they quickly realize that these errors matter when they actually have to calculate an answer rather than simply select from a set of multiple choice options.

Once students realize the importance of neatness they understand some of my classroom rules which include:

- All work must be written between, not across, the lines on the paper. Most new students seem to have never noticed that notebook paper is covered by a series of parallel horizontal lines. They certainly don’t demonstrate any understanding of the purpose of these lines.
- Work must begin at the top left corner and continue down the paper with writing going from left to right. I am no longer shocked to find that students literally do not know which way is up when it comes to paper. I flip and turn paper over on the students’ desks so that it faces the correct way. I demonstrate on my classroom poster by randomly placing the words “If I write all over the paper like this you can’t understand what I wrote.” This usually gets a few laughs while making the point.

Telling the children is one thing, getting them to adopt these practices is quite another. In my classroom these rules are enforced by The Crumple Monster. Any page that does not meet my standards is removed, crumpled, placed in the recycling bin and replaced by a fresh sheet of paper. The Crumple Monster destroys any and all messy work. It is rare for a student to not have at least one page destroyed by The Crumple Monster on the first day. In fact I normally fill two commercial sized recycling bins on the first weekend of class.

How do the kids react to these tactics? I thought they might think I’m mean but they almost universally think I’m funny. They love writing with the purple pens and joke with each other that The Crumple Monster is going to get them.

They eventually stop getting problems wrong because they can’t read their own handwriting. In doing so, they are taking their first steps to learning how to develop study skills that they will need for higher level math.

The bottom line is that they do learn to write to my standards. They eventually stop getting problems wrong because they can’t read their own handwriting. In doing so, they are taking their first steps to learning how to develop study skills that they will need for higher level math.

After students learn to complete their work so that it can be read, they can learn to add notes so that their math practice transforms from mindless, repetitive tasks that must be completed simply to earn a grade into meaningful exercises that they can use to learn from their mistakes. Only after their work is legible does it make sense to teach them how to take useful notes and keep an organized notebook.

Students do not magically acquire organizational skills as they get older. They learn organizational skills when they realize that they are beneficial. Whether it is writing legibly, writing notes or keeping a notebook, students must see the benefit of these skills before we can expect them to consistently adopt them.

]]>After students learn to complete their work so that it can be read, they can learn to add notes so that their math practice transforms from mindless, repetitive tasks that must be completed simply to earn a grade into meaningful exercises that they can use to learn from their mistakes. Only after their work is legible does it make sense to teach them how to take useful notes and keep an organized notebook.

Students do not magically acquire organizational skills as they get older. They learn organizational skills when they realize that they are beneficial. Whether it is writing legibly, writing notes or keeping a notebook, students must see the benefit of these skills before we can expect them to consistently adopt them.