**THE COUNTRY OF SINGAPORE VALIDATES LEARNING THEORY.** Singapore has scored at or very near the top in all math comparisons since 1995. But it wasn’t always like that. In 1984, Singapore’s students placed 16th out of 26 nations. The dramatic rise in achievement is the result of the country of Singapore developing its own math textbooks that: 1) Provide many experiences with hands-on objects before the abstract symbols and procedures are introduced; 2) Have a “less is more” philosophy, with thin textbooks that cover relatively few topics; 3) Require students to achieve mastery before moving on. The Math Whisperer program follows these guidelines, adding entry for students of any grade level among other features.

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Math understanding is a necessary tool in order to achieve these inalienable rights. The years our children spend in school are preparation for their adult lives. With today’s children having the possibility of living to age 100 and beyond, even if they focus the first 25 years of their lives in school, that leaves 75 years of adulthood beyond school.

Of course we want our kids to live well their entire lives. What they learn in school is a large part of their preparation for adulthood.

The capability of earning a good living is an important piece of the entire puzzle of how to live well for 75 years. And both common sense and data from the U.S. Department of Labor show that the more math a person knows, the more money they are likely to earn.

Math is critical in understanding savings, investment opportunities and pitfalls. Math is critical in understanding interest rates, which affects credit cards, car loans, house loans.

Today’s children will truly need to plan for a retirement. The planning involved is a math problem of Algebra II level.

We need to fix the math education system in this country for the sake of our children. In the meantime, parents can take a positive role by using my effective and efficient lessons to get your students on track for a successful adult life.

]]>First, imagine a world in which medical research was confined to small studies conducted by newly graduated doctors. As an example, a doctor would conduct a study on a treatment for colon cancer that he developed on 15 of his patients over two years, funded with small grants from the American Medical Association. Then he would present his findings at the annual AMA Conference. All presentations would be of a similar nature – small sample size studies of large medical issues. How far would medical science advance if this were the process for developing treatments and medicines?

In the medical establishment, there are systems in place for large-scale studies with oversight. The pharmaceutical companies invest enormous amounts of money into medicines, and want to recoup their investments. There is oversight for patients provided by the FDA, as well as the legal malpractice system. In addition, there is an element of drama involved in curing an individual, separating conjoined twins, or a finding a cure for a group of people suffering from the same condition. Heart-warming vignettes of smiling patients and their loved ones can be televised on the nightly news. In summary, there is a system of large-scale studies leading to treatments, along with patient protection.

This is a system that can run without leadership. Market forces and immediate suffering that begs to be alleviated are sufficient to drive this system forward. The continual breakthroughs in alleviating disease and suffering are proof that the medical system works.

Now consider the math education system in the United States. There is no system in place that fosters improvement. There are no effective checks and balances. There is no one person or group in charge. Over thirty years of data, from international comparisons to measures of achievement gaps, show that the math education system is absolutely stagnant. For example, international comparisons of 15 year olds, the Programme for International Student Assessment (PISA), show a decline in U.S. average scores over the last five tests, from 2000 through 2015. At the same time, the top scores of other countries have risen. The achievement gaps between whites and blacks, and between whites and Hispanics have remained almost constant over thirty years, in 4^{th} and 8^{th} grades as measured by NAEP, and in 12^{th} grade as measured by the SAT.

Thirty years spans an entire career, meaning that statistically no teachers or administrators have seen a classroom without an achievement gap. Statistically, no teacher or principal or school district superintendent, or parent, has seen world-class math education.

The equivalent of the pharmaceutical industry in math education would be the publishers of math curriculum. However, the publishers are not required to conduct any statistical studies showing their materials are effective. They pander to the math leadership, invest minimal amounts of money in their materials, mostly rehashing old lessons and updating with new photos. There is no blame here – they create what will sell. But they have no incentive to create materials that actually raise student achievement because the customers are not demanding it. There is no group that effectively monitors what materials work better than others, and no equivalent of a malpractice legal group that enforces effectiveness.

No Child Left Behind legislation can be related to the FDA in that it provided legal consequences for failure. However, because the implementation of improvement was left to the existing math establishment, it was doomed to failure. From the Secretary of Education down to the classroom teacher, our country put people in charge with no track records of improving achievement.

In education, there is a history of confusing experience with expertise. Our society uses years of experience and titles to confer respect on educators. Despite all the data that has been collected and is publicly available, educators at all levels get a pass on their own grades. Can you imagine that situation in the medical profession? Would you take your child with a life-threatening illness to a doctor with a track record below average, if that data were easily available?

On state and national levels we are treating people with track records of mediocrity as experts, and letting them make policy, distribute enormous amounts of money, and act as gatekeepers on all levels for innovators and the disruptors the field needs. Instead we should be listening to those educators who have actually improved student achievement above the norm.

There are instances wherein teachers and schools have made significant progress in closing the gaps. If our country is seriously interested in closing these gaps, these people with track records of success should be leading the national effort. Instead, in education we reward continued lack of progress with continued leadership roles and all the perks that go with that.

This is a field without effective leadership. No one is really in charge of raising student achievement. And from departments of education at all levels, any track record of raising student achievement above the norm is not considered as a necessary ingredient of earning the job. Experience, titles, and political abilities are the tickets to leadership positions. As an example, a former school district superintendent with a mediocre track record is now a U.S. Senator touted as an education expert. University professors in education are in powerful positions of influence. It is tradition for university professors in education to have some background in teaching of at least a couple of years. However, there is no requirement or tradition of selecting professors of future teachers from the small pool of teachers who have actually raised student achievement above the norm. Another source of leadership is the professional organization of the field, the National Council of Teachers of Mathematics. Their leadership is often comprised of former teachers of AP Calculus who rose through the ranks to district leadership positions. If they have raised achievement above the norm, it isn’t obvious.

There is a solution to raising math achievement and closing the achievement gap. The country of Singapore provides one example. After mediocre PISA scores, in the mid-1980’s they developed their own curriculum. It focuses on a very few key concepts, taught in great depth, with the progression of students using hands-on objects, then drawing pictures, and then using abstract notation. Singapore always scores in the top three in international comparisons now.

I have found similar results with middle and high school students who are several years behind grade level. In every case, these students have gaps in critical building blocks in a solid math foundation. For example, in general, these students cannot add simple fractions. They often think 4 – 5 = 1. Frequently, they do not understand what a variable, such as “x” means. There is no way a student with these gaps can be successful in Algebra 1. However, filling these gaps is easy with lessons that use the three-step progression of hands-on objects, followed by pictures, followed by abstract notation. Filling these gaps leads to the students gaining multiple grade levels within one school year.

Today’s math teachers and their leaders did not need this progression. They were successful (enough) with beginning at stage three, abstract notation. I would argue that they would have been more successful had they had all three steps. But the big point is that today’s teachers have not seen math taught properly, have not been trained to teach math well, are evaluated by principals and others who also have not seen math taught well. This is a system doomed to stagnation, just like the data shows.

Improving achievement in math will take:

- Leadership at the national and state levels. State departments of education can insist that their state math leaders, typically with titles like “Math Supervisor” have proven track records of raising achievement in at one previous job. Districts can impose the same requirements.
- Math curriculum needs a complete overhaul. Students require a solid foundation in relatively few critical building blocks, possibly only twenty. These critical building blocks MUST be taught well, the way education theory and the country of Singapore, have shown is effective, with a three step progression of hands-on-objects, followed by pictures, and finally abstract notation. (This is not to say that Standards are not important, but they are inaccessible to students who do not have a solid foundation.)
- Students cannot be allowed to progress until they have mastery over grade level critical building blocks. Our research indicates there are about two per grade level, beginning in second grade.
- Teachers and all administrators must be accountable to ensure that each and every one of their students achieve mastery over grade level appropriate critical building blocks.
- Teachers and all administrators must be supported in all ways so that they can accomplish this.

Over 45 million students are in American schools, and at least half of them are having difficulties in math. There are consequences in terms of high school graduation rates, college completion, career choices, whether students grow up to require government benefits or contribute to the tax base, and the overall economic competitiveness of the U.S. It will take leadership that has expertise to modify our math education system into an effective system that serves our students and our country.

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There is no way those students can understand algebra. Algebra books are full of endless variations of problems like: Simplify 3x^{2} + 6x^{2} + 4 – 5.

The bad news is that subtraction is taught in first and second grades. So middle and high school students who think that 4 – 5 = 1 have been missing a critical element of understanding subtraction for years. The good news is that reversing this misconception about subtraction can be accomplished fairly easily.

However, it is almost never as easy as just telling the student: 4 – 5 does not equal 1 and the correct answer is -1. Or that subtraction is read from left to right. Or that subtraction is not commutative like addition is. These are the standard “fixes” that teachers and parents who notice the student’s problem try. If you are the parent of a child who thinks that 4 – 5 = 1you can see this for yourself by saying one of those standard “fixes.” Then come back to your child in a few weeks with a similar problem, like 2 – 3 = ? or 4 – 6 = ? and see for yourself how effective just telling was.

The good news is that just a few activities and some practice about subtraction will fix this problem forever. Middle and high school teachers aren’t trained to do this, but my Subtraction lessons do accomplish this lifetime understanding.

]]>Randi’s parents were both in prison, and she lived with an aunt. She wore jewelry that looked like razor blades. She was disrespectful to teachers on many occasions. Being staffed in math class added to the picture of a troubled girl with an uncertain future.

Pythagorus’ theorem was the topic of a two week unit, and as usual, Randi was not keeping up. (This is the theorem that connects the lengths of the sides and the hypotenuse of a right triangle, a^{2 }+ b^{2 }= c^{2 }) I was able to spend some time working with Randi individually. We cut out several right triangles and squares to visualize the theorem. We drew pictures that illustrated the theorem. We used graph paper to better show the lengths of the sides of the triangles. Suddenly everything clicked for Randi. She got it.

In fact, she was able to figure out how to take the square root of a number from these experiences. For example, she could see that the square root of 64 must be 8 from one of the examples.

Randi is an example of an __honest learner__. When she understood something, she truly understood it. When she didn’t fully understand something, she just shut down.

Her positive experiences with Pythagorus’ theorem showed her that she could actually be good at math, and she came to expect to understand. The paraprofessional person assigned to Randi couldn’t keep up with her!

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My friend Darcy was one of those students who always put her hand up to answer the teachers’ questions, zipped through the homework and got A’s. She was admitted to a prestigious college and majored in physics. Upon graduation, she got a job as a geophysicist for an oil company. After 30 years of consistently earning a very good salary, she will retire with a six figure retirement savings. Sounds like a success story, doesn’t it?

Digging deeper, however, it turns out Darcy really wanted to be an astronomer. She spent two summers working at an observatory as an undergraduate, and absolutely loved it. But she did not feel her math background was sufficient to pursue a doctorate, which was required to be an astronomer. While she appreciates the salary of a geophysicist, which has allowed her to raise her children as a single parent and live a comfortable life, she has never enjoyed the work. It’s a job to her.

How could it happen that a person who did well in math in high school, and was able to earn a degree in physics from a highly regarded college not feel confident in her ability in math?! How could this lack of confidence cause her to give up her dream and settle for a job she can only tolerate?

The reason is that the way math was taught to Darcy did not involve true understanding. It was largely “monkey see, monkey do.” The ability to mimic the steps the teacher demonstrated was considered to be learning. As an example, the ability to add fractions with different denominators and consistently get the correct answer does not equate to an understanding of what the symbols mean; this is a fifth grade example. The situation just compounds as the years roll on. Darcy is an **honest learner** who recognized that she did not have true understanding. True understanding was not what she was offered in her school experiences.

Of the approximately 25% of students who are able to excel with the “monkey see, monkey do” system that exists in most of our schools, fewer than half are able to make the leap to real understanding.

The way for all students to really understand math is to give them a solid foundation in the very few critical concepts. This solid foundation consists of the three step process described by learning theorists such as Piaget. First the student has experiences with concrete objects, then moves to pictures, and then to the symbols of math. Darcy was taught the way most American students are taught – with the third step of symbols only. The three step method of teaching would have benefited Darcy and allowed her to follow her true professional passion.

I can relate to Darcy because I also “learned” math with Step 3 only, symbols not connected to the real world, and went on to earn a masters degree in physics. I worked with Darcy as a geophysicist myself.

]]>As is the case with most middle school students who are struggling in math, fractions are a real problem. Early in our first meeting together I asked Sam to solve this problem:

^{1}⁄_{2} + ^{1}⁄_{3} = ?

Sam answered ^{2}⁄_{5}

To Sam (as to many, many struggling students) the way to add fractions is to add numerators and then to add denominators.

They think that ^{a}⁄_{b} + ^{c}⁄_{d} = ^{a+c}⁄_{b+d}

To Sam, those fractions were just meaningless symbols. Otherwise, he could have seen that his answer couldn’t possibly be correct, as his answer of ^{2}⁄_{5} is less than ½. But to Sam, fractions (and most of math in general) was just meaningless numbers and symbols.

To help Sam understand fractions, I pulled out my handy-dandy fraction pieces. These are often plastic circles and pie pieces. Students like Sam need a concrete model, so that the symbols can make sense. Piaget was right, in my experience. The transition from concrete objects, where math is connected to the real world, to pictures, and then to abstract notation really works. Sam was able to figure out for himself that the only way to add fractions was using this formula, which made sense:

^{a}⁄_{b} + ^{c}⁄_{d} = ^{a+c}⁄_{b}

For the next few weeks, we worked on more complex problems, and Sam was cured!

Sam went on to get a B in 8th grade math the following school year!! A couple of years later I moved and my new house had oak floors that needed refinishing. I called Sam’s mom to see if he might want more tutoring, in exchange for refinishing my new floors. Sam’s mom reported that Sam was doing great in high school math, building on his success in 8th grade. So that one summer of tutoring got Sam on track for success! (I had to pay for the refinishing of the floors, but it felt great nonetheless.)

]]>First, imagine a world in which medical research was confined to small studies conducted by newly graduated doctors. As an example, a doctor would conduct a study on a treatment for colon cancer that he developed on 15 of his patients over two years, funded with small grants from the American Medical Association. Then he would present his findings at the annual AMA Conference. All presentations would be of a similar nature – small sample size studies of large medical issues. How far would medical science advance if this were the process for developing treatments and medicines?

In the medical establishment, there are systems in place for large-scale studies with oversight. The pharmaceutical companies invest enormous amounts of money into medicines, and want to recoup their investments. There is oversight for patients provided by the FDA, as well as the legal malpractice system. In addition, there is an element of drama involved in curing an individual, separating conjoined twins, or a finding a cure for a group of people suffering from the same condition. Heart-warming vignettes of smiling patients and their loved ones can be televised on the nightly news. In summary, there is a system of large-scale studies leading to treatments, along with patient protection.

This is a system that can run without leadership. Market forces and immediate suffering that begs to be alleviated are sufficient to drive this system forward. The continual breakthroughs in alleviating disease and suffering are proof that the medical system works.

Now consider the math education system in the United States. There is no system in place that fosters improvement. There are no effective checks and balances. There is no one person or group in charge. Over thirty years of data, from international comparisons to measures of achievement gaps, show that the math education system is absolutely stagnant. For example, international comparisons of 15 year olds, the Programme for International Student Assessment (PISA), show a decline in U.S. average scores over the last six tests, from 2000 through 2015. At the same time, the top scores of other countries have risen. The achievement gaps between whites and blacks, and between whites and Hispanics have remained almost constant over thirty years, in 4^{th} and 8^{th} grades as measured by NAEP, and in 12^{th} grade as measured by the SAT.

Thirty years spans an entire career, meaning that statistically no teachers or administrators have seen a classroom without an achievement gap. Statistically, no teacher or principal or school district superintendent, or parent, has seen world-class math education.

The equivalent of the pharmaceutical industry in math education would be the publishers of math curriculum. However, the publishers are not required to conduct any statistical studies showing their materials are effective. They pander to the math leadership, invest minimal amounts of money in their materials, mostly rehashing old lessons and updating with new photos. There is no blame here – they create what will sell. But they have no incentive to create materials that actually raise student achievement because the customers are not demanding it. There is no group that effectively monitors what materials work better than others, and no equivalent of a malpractice legal group that enforces effectiveness.

No Child Left Behind legislation can be related to the FDA in that it provided legal consequences for failure. However, because the implementation of improvement was left to the existing math establishment, it was doomed to failure. From the Secretary of Education down to the classroom teacher, our country put people in charge with no track records of improving achievement.

The NCTM acts as a gatekeeper in terms of what is published and what is presented in the conferences. It would be helpful if they would devote space for disruptive programs that raise student achievement and close the achievement gap. Instead, as the annual conference verified, they play it small, continuing with the status quo of thirty years of mediocrity.

In education, there is a history of confusing experience with expertise. Our society uses years of experience and titles to confer respect on educators. Despite all the data that has been collected and is publicly available, educators at all levels get a pass on their own grades. Can you imagine that situation in the medical profession? Would you take your child with a life-threatening illness to a doctor with a track record below average, if that data were easily available?

On state and national levels we are treating people with track records of mediocrity as experts, and letting them make policy, distribute enormous amounts of money, and act as gatekeepers on all levels for innovators and the disruptors the field needs. Instead we should be listening to those educators who have actually improved student achievement above the norm. And partly due to NCLB, there is enough publically available data to measure the expertise of former educators who purport to be experts.

There are instances wherein teachers and schools have made significant progress in closing the gaps. If our country is seriously interested in closing these gaps, these people with track records of success should be leading the national effort. Instead, in education we reward continued lack of progress with continued leadership roles and all the perks that go with that.

This is a field without effective leadership. No one is really in charge of raising student achievement. And from departments of education at all levels, any track record of raising student achievement above the norm is not considered as a necessary ingredient of earning the job. Experience, titles, and political abilities are the tickets to leadership positions. As an example, a former school district superintendent with a mediocre track record is now a U.S. Senator touted as an education expert. University professors in education are in powerful positions of influence. It is tradition for university professors in education to have some background in teaching of at least a couple of years. However, there is no requirement or tradition of selecting professors of future teachers from the small pool of teachers who have actually raised student achievement above the norm. Another source of leadership is the professional organization of the field, the National Council of Teachers of Mathematics. Their leadership is often comprised of former teachers of AP Calculus who rose through the ranks to district leadership positions. If they have raised achievement above the norm, it isn’t obvious.

There is a solution to raising math achievement and closing the achievement gap. The country of Singapore provides one example. After mediocre PISA scores, in the mid-1980’s they developed their own curriculum. It focuses on a very few key concepts, taught in great depth, with the progression of students using hands-on objects, then drawing pictures, and then using abstract notation. Singapore always scores in the top three in international comparisons now.

I have found similar results with middle and high school students who are several years behind grade level. In every case, these students have gaps in critical building blocks in a solid math foundation. For example, in general, these students cannot add simple fractions. They often think 4 – 5 = 1. Frequently, they do not understand what a variable, such as “x” means. There is no way a student with these gaps can be successful in Algebra 1. However, filling these gaps is easy with lessons that use the three-step progression of hands-on objects, followed by pictures, followed by abstract notation. Filling these gaps leads to the students gaining multiple grade levels within one school year.

Today’s math teachers and their leaders did not need this progression. They were successful (enough) with beginning at stage three, abstract notation. I would argue that they would have been more successful had they had all three steps. But the big point is that today’s teachers have not seen math taught properly, have not been trained to teach math well, are evaluated by principals and others who also have not seen math taught well. This is a system doomed to stagnation, just like the data shows.

Whether math leaders want to follow this three step progression or not, we need to have large-scale studies that measure whether the program they use is improving achievement or not. Just as medicines are not developed on fifteen people, solutions to raising achievement will not come from samples of this size. Yet many talks at the NCTM conference were of just this size of students. State test scores and nationally accepted assessments such as the Northwest Association MAPS tests can be used to measure student achievement.

Over 45 million students are in American schools, and at least half of them are having difficulties in math. There are consequences in terms of high school graduation rates, college completion, career choices, whether students grow up to require government benefits or contribute to the tax base, and the overall economic competitiveness of the U.S. It will take leadership that has expertise to modify our math education system into an effective system that serves our students and our country.

]]>Here is what I offer that is unique:

I have a track record of raising student achievement way past the norm. That means I know how it is done. In all respect to math educators with lots of titles, unless they have done this specifically, they know less than I do on the subject of teaching math effectively.

Like most educators, my goal is to help kids learn and achieve. However, unlike most educators, I am beholden to no one in the education system. Therefore I can speak freely and frankly.

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