The post About 1 in 3 HS Seniors Prepared for College Math, NAEP Shows appeared first on Education News.

]]>According to the latest scores from the National Assessment of Educational Progress (NAEP), only 37% of American 12th-graders are academically prepared for college math and reading. These numbers mark a dip from two years early when an estimated 39% of high-schoolers were prepared academically.

The biggest declines in proficiency came at the bottom tier, with growth in the share of students “below basic” in their abilities. In 2013, 35% tested at “below basic” in math, whereas that number has increased to 38% today. This marks the first drop in math scores in a decade. In reading, the average score was 287 out of 500, considerably lower than the average score of 292 in 1992.

Furthermore, the average scores among students in the bottom 10th percentile, as reported by Lauren Camera of U.S. News, dropped precipitously by four points in math and six points in reading. The reading scores for these students hit its lowest level since the test began its assessment of students’ abilities.

“These numbers aren’t going the way we want,” said Bill Bushaw, the executive director of the National Assessment Governing Board, the organization that released the scores. “We just have to redouble our efforts to prepare our students to close opportunity gaps.” The Education Department has also urged educators to double down on their efforts to prepare students for college effectively.

Policymakers and educators worry that students’ lack of preparedness hampers their college education. Unprepared students who go to college often burn through their financial aid and waste time taking remedial classes that do not earn credits toward a degree.

The results of the report were demographically split as well. In reading and math, Asian students performed the best, with around 48% of them scoring above proficiency levels. White students scored next best, while black and Hispanic students scored at the lowest levels in reading and math. Only 12% of Hispanic and 7% of black students tested as either proficient or above in math, notes Leslie Brody of the Wall Street Journal.

The report did not contain all bad news, however. High school graduation rates are rising, and 42% of test-takers said they had been accepted into a four-year college. Additionally, the dropout rate has improved for every racial and ethnic group. The report also found students did worse on these tests if their parents had not received a high school education, a phenomenon that disproportionately affects students of color.

Additionally, officials at the Department of Education are cautioning against extrapolating bleak conclusions from these results. Despite sounding concern, the acting commissioner of the National Center for Education Statistics, Peggy Carr, said that the drop in scores among students may be because more students are taking advanced level coursework, which is much more challenging. Moreover, since dropout rates are declining, the test was given to low-performing students who historically would not have even been in class, thus depressing the results.

The results of the 2015 assessment are based on a nationally representative sample of thousands of 12th-grade students from 740 schools, including private institutions.

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]]>The post Girls Still More Math-Anxious Than Boys, Study Says appeared first on Education News.

]]>According to a new study, more girls have negative feelings about math, and those emotions can result in “mathematics anxiety.”

But researchers at the University of Missouri, the University of California-Irvine, and the University of Glasgow in Scotland announced that they believe several issues that do not include math performance contribute to higher mathematics anxiety in more girls than boys.

“We analyzed student performance in 15-year-olds from around the world, along with socio-economic indicators in more than 60 countries and economic regions, including the U.S. and the United Kingdom,” said Dr. David Geary, Curators Professor of Psychological Sciences at the University of Missouri College of Arts and Science.

The research showed that girls’ math anxiety was not connected to the level of engagement their mothers had in science, technology, engineering, and math careers. It was also not related to the inequality of genders in the countries studied by the researchers, writes PsychCentral’s Janice Wood.

In more gender-equal and developed countries, the gender difference in math anxiety was larger. Also, boys’ and girls’ math performance was higher overall.

The study found that in 59% of the countries that were analyzed, the differences in gender anxiety were over twice the number of gender differences in math performance. These statistics, say the scientists, indicate that factors other than performance are the cause of higher math anxiety in girls than boys.

Geary says the study points to the fact that gender differences in the areas of mathematics anxiety and performance are complex.

Mathematics anxiety is defined as “negative feelings experienced during the preparation of and engagement in math activities,” writes the BBC.

The Glasgow University School of Maths and Statistics Professor Dr. Libert Vittert said that math anxiety can affect a student’s future job prospects.

Vittert added that she was told by a teacher when she was about 14 that it would be a good idea to stop taking math classes because she was unable to understand the subject. Dr. Vittert pushed on to receive a degree in pure mathematics from MIT and now has her Ph.D. She believes it is important to keep girls interested in STEM subjects.

In some cases, parents have an expectation that boys will do well in math and other STEM subjects. This assumption does, however, create an underlying mindset in girls that they are not good mathematicians, says Snow McDiggon of Parent Herald.

The fact is that women are underrepresented in many STEM fields, writes Jennifer Harrison for Gadgette. And even though many women are excellent mathematicians, females many times feel more anxious about math.

Highlighting role models in STEM-related fields is one way to start helping girls feel more empowered in mathematics. Also, parents and teachers alike can begin to be intentional in their encouragement of their females students’ and daughters’ performance in STEM classes.

But PsychCentral quoted Stoet, who said:

“Policies to attract more girls and women into subjects such as computer science, physics, and engineering have largely failed. It is fair to say that nobody knows what will actually attract more girls into these subjects. Policies and programs to change the gender balance in non-organic STEM subjects have just not worked.”

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]]>The post Stanford Report Touts Benefits of Visual Element to Math Education appeared first on Education News.

]]>Researchers at Stanford University have released a report titled “Seeing As Understanding: The Importance of Visual Mathematics for our Brain and Learning” that aims to dispel the notion that visual math, such as pictures, finger-counting, and diagrams, are only for lower-level tasks, and that higher-level math deals exclusively in symbols, notations, and words. They present evidence to suggest that visual mathematics may help students of all levels see, understand, and extend mathematical ideas.

Generally speaking, students who prefer visual thinking are regarded as having special needs, and children grow up thinking that counting on their fingers is an immature approach to mathematics. However, several mathematics organizations such as the National Council for the Teaching of Mathematics (NCTM) and the Mathematical Association of America (MAA) have long advocated for visual representation of mathematics in the classroom. Stanford’s new study reaffirms these groups’ advocacy and urges the necessity of fostering visual thinking.

Scientists argue that the human brain is comprised of “distributed networks,” and when humans handle knowledge, different areas of the brain are activated and communicate with one another. When we study mathematics, brain activity is distributed through many different networks, including two visual pathways. The failure to exploit these pathways through visual mathematics potentially hampers students’ mathematical abilities.

Notably, researchers found that students with structural disadvantages such as low-incomes, lower-literacy rates, etc., perform just as well as their more advantaged peers after a 15-minute session with visual exercises. The researchers emphasized the importance of students learning numerical knowledge through linear representations and visuals. According to the report, the dorsal visual pathway in the brain is the core region for representing the knowledge of quantity.

Additionally, a yet-to-be published study from researchers at Stanford demonstrates that children between the ages of 8 and 14 are developing part of the ventral visual pathway, an important brain “network.” This development indicates that as children learn, the visual processing parts of their brain become more interactive. If this interaction is not stimulated by visual activities, parts of children’s brains will not reach their full potential.

A section of the report is devoted to exploring finger-counting. There is a specific region of the brain, the somatosensory finger area, that is dedicated to the perception and representation of fingers. Often, when doing mathematical problems, our brain’s “finger-area” is stimulated whether or not we are using our fingers. The researchers urge mathematics educators to take advantage of this “finger-activity”; children who develop proficiency counting on their fingers will further brian development and promote future mathematics success. Regrettably, argue the researchers, most educators discourage finger-counting.

Despite the evidence, millions of students in the United States do not engage mathematics through visualization and representation. Most students approach it as a numeric and symbolic subject. The evidence, however, gathered by the report will help students and educators to “understand the impact of visualizing and seeing to all levels of mathematics, and suggests an urgent need for change in the ways mathematics is offered to learners.”

The full report is available online.

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]]>The post Report Sheds Light on Adults’ Use of Education in the US appeared first on Education News.

]]>The National Center for Education Statistics (NCES), the primary federal entity used for analyzing and reporting data having to do with education in the United States, has released the results from its Program for the International Assessment of Adult Competencies (PIAAC). This program is a large-scale study of adult skills and life experiences relating to education and employment.

The PIAAC conducted surveys of 3,660 adults ranging from 16 to 74 over the course of a year, measuring these individuals’ levels of literacy, problem-solving skills, and numeracy, a metric used to evaluate basic mathematical and computational skills.

In literacy, American adults perform at a rate consistent with the international norm. Additionally, the United States has a larger percentage of adults performing at the top and the bottom of literacy skills compared with other countries.

In numeracy and in problem-solving skills, the United States as a whole performed below the average. The United States features a smaller percentage of individuals at the top levels in numeracy and a larger percentage of adults at the bottom in problem-solving skills than other places evaluated by the PIAAC.

Specifically relating to the United States, American adults who perform at the top proficiency level in literacy, in numeracy, and in problem-solving skills are those aged between 25 – 34, rather than those in other age intervals.

A strong performance in literacy and numeracy is indicative of employment. In literacy, 15% of employed individuals performed at top literacy levels, while 12% of employed adults performed that well in numeracy. Adults who are unemployed or out of the labor force performed at much lower levels in literacy and numeracy.

Unsurprisingly, 75% of unemployed U.S. adults lack a high school accreditation; of these individuals, a third performed at the lowest level of literacy. Among these undereducated Americans, white Americans outperformed Hispanic and black Americans in literacy, in numeracy, and in problem-solving. Unemployed Americans performed no worse than the unemployed of other countries.

Among adults between 16 – 34, there is a strong correlation between one’s education and one’s performance in the workforce. Generally speaking, the higher level of education completed, the higher an adult would perform at top proficiency levels in all three of the areas. These statistics correlate with race. Much smaller percentages of black and Hispanic young adults performed in the top proficiency levels than their white peers. This disparity bespeaks an inequality of resources and opportunities available to young people in communities of color.

Interestingly, the correlation between education level and performance tapers off as age increases. For example, there were no measurable differences between adults ages between 66 – 74 and performing at the highest proficiency levels, who had a Bachelor’s degree or an advanced professional or graduate degree. As mentioned, the level of degree attainment correlated with performance among young Americans. These statistics indicate the changing nature of American education and underscore the necessity of a college degree in contemporary America.

The full report of the findings can be found online, and the data is of interest to anyone who wants a better understanding of Americans’ levels of basic competencies and how these relate to issues of age, education, employment, and race.

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]]>The post Louisiana Voucher Program Students Struggle on Math, Study Shows appeared first on Education News.

]]>Louisiana’s private school voucher program has shown widely-varying results in a study suggesting that students in the program scored substantially lower than public school students in math during the first year of the plan.

Researchers at Tulane University stated in the report that students did better on math performance in their second year of the program, but were still below their public school peers, reports the Associated Press.

The results of the study were confined to grades three through six, but a majority of students enter the voucher program at other grade levels. The researchers added that the examination did not reveal whether the findings would be the same for students who joined the plan the first year they began school in kindergarten.

The program was established to provide tuition for some low- or moderate- income students whose only choice would have been to attend low-performing schools. First a pilot program in New Orleans, the plan was extended across the state in 2012 by lawmakers based on a push by former Gov. Bobby Jindal.

There are currently over 7,000 young people involved in the project. While Jindal and other supporters saw the program as a way for students to leave poor schools, opponents have asked how effective the vouchers are and if the diverted money is harming public schools that are in need of funding.

The Education Research Alliance for New Orleans at Tulane and the School Choice Demonstration Project at the University of Arkansas jointly released the study. The researchers said reading scores were lower for voucher-school children than the scores of their public school counterparts, but not as dramatically as math scores.

The focus of the report was on students who had attended public schools and took 2011-2012 state standardized tests before they enrolled in the voucher program in 2012-2013 school year. The program was formerly known as the Louisiana Scholarship Program.

“Our estimates indicate that an LSP scholarship user who was performing at roughly the 50th percentile at baseline fell 24 percentile points below their control group counterparts,” the report said. The gap narrowed to 13 percentile points in math in the second year. There was an eight-point difference in reading the first year but reading scores improved in the second year to a point where they were not significantly different, statistically, from the control group, the report said.

The Louisiana voucher program is the fifth-largest in the country, writes Lauren Camera for US News and World Report. But another paper published two months ago found that voucher-school students had lowered math, social studies, reading, and science scores. The possibility of a student achieving a failing score grew by 24% to 50%.

Other states are considering similar programs, and the private school scholarship sector is becoming a reality in states across the nation. Advocates of the programs say the number of students using vouchers rose by 130% since 2008-2009.

The voucher program in Louisiana extends private school options to students enrolled in a school graded C, D, or F, or who are beginning kindergarten. Voucher kids take the same state exams as their peers in public schools, says The Times-Picayune’s Danielle Dreilinger.

Some of the explanations put forth to account for these unprecedented results include that private schools might not have been equipped to educate kids from disadvantaged families; that schools’ curricula might not have been the same as the state’s mathematics standards; Louisiana’s program being larger than programs that have experienced positive results but were smaller; and that more expensive and prestigious private schools might have had a better chance of providing more diverse support for kids at risk often do not accept vouchers.

Will Sentell of The Advocate said the results are sure to have an impact on debates that will take place during the regular 2016 legislative session. The deliberations will likely concern putting restrictions on the voucher program, which costs the state roughly $42 million a year.

“We must remember all scholarship program students previously attended failing and underperforming schools,” said Ann Duplessis, president of the pro-voucher group Louisiana Federation for Children and a former state senator from New Orleans.

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]]>The post A Closer Reading of ‘The Math Revolution’: Get on Board or Get Left Behind appeared first on Education News.

]]>By Barry Garelick

Peg Tyre, author of “The Good School” and “The Trouble With Boys”, is well regarded by many, including those who write about education. But I believe she missed the point in her latest article in The Atlantic. The story explores why there has been a surge in the number of teenagers who have excelled in advanced math topics as evidenced by increasing numbers of award winners in prestigious math competitions such as the International Math Olympiad. She describes what those students are doing differently than the students who do not make it to such airy heights. In my opinion, her article amounts to an advertisement that tells parents to get on board or get left behind.

**A One-sided View of the Extracurricular Schools: Where Conceptual Understanding Rules the Roost**

Tyre focuses on the extracurricular schools that these students attend, one of which is the Russian School. Students start early in such schools—as early as first or second grade—and continue on. (Other venues include the website “Art of Problem Solving” which she mentions.) In discussing the Russian School, Tyre talks with the founder of the school about his experiences with his kids when they were in second grade in Newton, Massachusetts in 1997.

“I’d look over their homework, and what I was seeing, it didn’t look like they were being taught math,” recalls Rifkin, who speaks emphatically, with a heavy Russian accent. “I’d say to my children, ‘Forget the rules! Just think!’ And they’d say, ‘That’s not how they teach it here. That’s not what the teacher wants us to do.’”That year, she and Irina Khavinson, a gifted math teacher she knew, founded the Russian School around her dining-room table.

Although this makes for a compelling story, we really don’t know what his kids were being taught. We only know that they were being taught to do something step by step, and Rifkin admonishes them to forget the rules and just think. The average reader, having been told by countless newspaper articles that depict traditionally-taught math as rote memorization with no understanding, may assume that this is what was happening. Many such readers have therefore been led to believe that conceptualizing mathematical problems is something you can do with little instruction or foundational skills and memorization— a pervasive theme of the article that Tyre, as a highly skilled writer, manages to keep understated.

She accomplishes this by first acknowledging that fluency is important and also by describing the prevailing disputes about how best to teach math:

“

Fiery battles have been waged for decades over what gets taught, in what order, why, and how. Broadly speaking, there have been two opposing camps. On one side are those who favor conceptual knowledge—understanding how math relates to the world—over rote memorization and what they call “drill and kill.” (Some well-respected math-instruction gurus say that memorizing anything in math is counterproductive and stifles the love of learning.) On the other side are those who say memorization of multiplication tables and the like is necessary for efficient computation. They say teaching students the rules and procedures that govern math forms the bedrock of good instruction and sophisticated mathematical thinking. They bristle at the phrasedrill and killand prefer to call it simply ‘practice’.”

It would have been informative if she had identified the “well-respected math-instruction gurus” who claim that memorization of procedures as well as addition and multiplication facts obscures student understanding. It would also have been extremely valuable and instructive if she had asked the opinions of the heads of these various extracurricular schools where they stood on the matter. As the paragraph stands, however, it is left to the reader to decide which one is correct. In the context of the rest of the article, the reader is led to believe that in the supposed dichotomy between procedure and understanding, understanding always should take precedence. (She makes this point explicitly in her book “The Good School” in the chapter on the teaching of mathematics.)

**And the Other Cherished Concept: Problem Solving**

Tyre talks also of “problem solving” which, like conceptual understanding, she casts as something that can be taught independent of foundational skills. Her disdain for such skills is embodied in the following statement:

“Sitting in a regular ninth-grade algebra class versus observing a middle-school problem-solving class is like watching kids get lectured on the basics of musical notation versus hearing them sing an aria from Tosca.”

Problem solving is much more complicated than that. One doesn’t learn to sing an aria from Tosca by doing just that. It is based on years and years of training in basic vocal skills. The majority of music lessons are about skills. Musicality is built up from mastery of the basics — and it is the same thing with math problem solving. Students are given instruction in solving basic types of problems such as distance/rate, mixture, work, number, coins, etc. Though frequently derided by reformers as not motivating students to solve them, nor giving any significant problem solving skills, these type of problems are an essential starting point. From there, students are given variants of the problems—well scaffolded so that they ramp up in difficulty—and eventually graduate to non-routine problems.

But for many educational experts and their followers, foundational skills play a minor role in learning how to solve problems. In their view, problem solving is taught by giving students open-ended, multi-answer problems and an insistence on different approaches. There is a belief that continued exposure to difficult problems for which students have had little or no prior knowledge builds up a problem solving *schema*, and provides them the motivation to learn what is needed to solve such problems on a just-in-time basis.

But math problems are not necessarily useful just because they may require outside-of-the-box insight and/or inspiration. Nor are they likely to result in a problem-solving “habit of mind.” Tyre falls prey to this fantasy, though it is highly unlikely that the Russian School and other schools operate in such manner.

**The Real Message: Let’s Help the Gifted**

Tyre’s solution to delivering better math education to more students is to get more students into these special schools—by better identifying gifted students and attending to their needs, particularly in low-income areas. She argues that No Child Left Behind, in fulfilling the noble goal of providing help to struggling learners, did so at the expense of those who could benefit from accelerated learning.

What this argument leaves out is that the programs in place to teach math in the lower grades have over the past two and a half decades slowly and steadily been influenced by reform-math agendas and do a poor job of teaching basic facts and skills despite the increased focus on helping the weaker performers. What Tyre proposes is to expand the number of students who go to such special schools to include those from low-income families. But this “let them eat cake” solution fails to help those low-income students who remain stuck in poorly taught math programs because they were “ungifted”.

Furthermore, it is not obvious how one would select gifted kids in the earliest grades for math potential. Math is not an all or nothing proposition. Success breeds interest and love, and success in the earliest stages of anything has more to do with the mechanics than some deep understanding or analysis. Finding and separating these kids is not the role of educators. They should not attempt to separate those with promise versus those who just work hard.

Tyre concludes with:

“Perhaps the moment is right for members of the advanced-math community, who have been so successful in developing young math minds, to step in and show more educators how it could be done.”

I would agree that perhaps the techniques used in these schools *could* be extremely effective – particularly the mastery of basic skills and facts which serve as the foundation on which students can build conceptual knowledge and problem solving skills. If they offer a proper (STEM-level) curriculum in class, push a little bit, and then provide advanced opportunities after school (in areas like math, robotics, poetry, etc.), then students will get what they need—thus fulfilling what she calls “a noble goal”. Kids in a STEM level curriculum can join the after-school programs and be successful later on. AMC math and the International Olympiad are competitions, not curriculum. The teacher should be the mentor and the one driving (and pushing) the learning process with a group of equal-level students. This should handle all required top-level learning. After-school should only be for “extra.”

If, however, the after-school program provides properly taught lower level instruction that students should have received in the first place, then it’s just a divergent form of tracking. Given the direction math education has been going for the past twenty five years, it is not hard to guess how Tyre’s article will be interpreted.

———-

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long-term substitute in a high school and middle school in California: “Confessions of a 21st Century Math Teacher.”

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]]>The post Bedtime Math App Shows Results, Newark Encourages Use appeared first on Education News.

]]>According to a study published this week in the journal *Science, *one app truly does help elementary school children learn the subject of math.

Researchers from the University of Chicago looked at a demographically diverse group of first-graders and their parents, totaling close to 600, from across Chicago. One group was allowed access to the iPad app “Bedtime Math,” which uses stories and sound effects with math problems children can solve with their parents. Meanwhile, the control group was given a reading app that offered similar stories but no math component.

By the end of the school year, researchers had discovered that the Bedtime Math app did help students perform better in math, which they say could be an important asset to parents who are unsure of their own math skills. Students who used the app on a frequent basis were typically around three months ahead of their peers in math achievement compared to the students who only used the reading app.

In an interview with Eric Westervelt for NPR, University of Chicago professor and one of the paper’s lead authors Sian L. Beilock said:

“We’ve shown that, when parents interact with their kids and talk with them about math, that really impacts what kids learn. We were interested in this because it really is a no-frills app, an easy way for parents to interact with their kids, to talk with their kids about math. It’s not an app that they use by themselves. And we thought that that potentially had promise in terms of what math knowledge kids gained.”

Beilock said the key was for parents to talk to their children about math, which should be looked on as part of the bedtime routine, writes Adrian Cho for *Science*.

The app, created two years ago by astrophysicist-and-mom Laura Overdeck, is now being promoted through a partnership between Overdeck and the Newark Public School system, the largest public school district in the state of New Jersey.

The district is putting the app to use as part of its efforts to close the achievement gap between high-income and low-income students, in addition to the confusion that students across the state hold concerning the subject of math in general.

Overdeck spoke to NJ Advance Media, saying that this is the first time partnering with a school district. She hopes it will show that math can be fun to do at home.

Six elementary schools in the district are participating in a pilot program using the app and are encouraging parents of children in kindergarten through the second grade to download it. They suggest that parents use the app on a voluntary basis for five minutes at a time with their children before bedtime for a few days a week. Officials plan to survey parents in order to see how the program is working.

According to the district’s Special Assistant of Math PreK-5 supervisor Darlene DeVries, the program will be encouraged within all 40 Newark elementary schools by next fall.

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]]>The post Poster Children of Math Education appeared first on Education News.

]]>By Barry Garelick

I had a discussion about math education recently with a principal of a middle school where I had once worked. I think highly of him and enjoy our discussions even though we may not see eye to eye. In our last discussion he expressed the following view:

“I think we have spent too many years teaching kids to ‘do math’ and not nearly enough time teaching true number sense. In education, we often swing too far in the opposite direction. I recall learning to divide fractions by ‘invert and multiply’, but no one ever taught me why that works. Just one example of doing vs. understanding.”

The invert and multiply example has for years served as the poster child for the reform math movement. It is used as evidence that traditionally-taught math is math taught wrong because it is presented as a bunch of tricks, relying on rote memorization with no conceptual understanding or connections to other concepts — students should see that math makes sense.

Before I get too far into this, let me say that I believe that students should be taught why the invert and multiply rule for fractional division works, and I have done so in classes that I have taught. I will also say that the accusations about traditionally-taught math are in large part based on mischaracterizations. I have talked about this at length here, and here, so won’t go into further detail on it except to say that when I (and many others I know) was educated in the 50’s and 60’s math was taught with understanding and connections, not merely as rote memorization.

**In the Interest of Full Disclosure and a Slight Digression**

In the interest of full disclosure, let me say that like the principal, I was not taught why invert and multiply works. As I told the principal when he brought the subject up, I did not find out how it worked until about 15 years ago, despite my having majored in mathematics.

We were first shown how to divide fractions using the common denominator method (see Figure 1).

We were also shown with diagrams what happens when a 5-inch piece of ribbon cut into inch lengths was used to show how works (See Figure 2). The method shown in my (and most) arithmetic books at that time stopped short of explaining the math behind why the divisor is inverted and then multiplied. We were only shown that in all cases—whole or mixed numbers divided by a fraction, or the division of two proper fractions like —multiplying the dividend by the reciprocal of the divisor produced the same answer as the common denominator method, or counting intervals as in the picture above. Thus, the invert and multiply procedure was extended to apply to all fraction divisions by virtue of the pattern we were seeing, but without the mathematical explanation behind it. (The math behind it is explained in this short video put together by John Mighton, the Canadian mathematician who founded the JUMP Math program that is gaining popularity in Canada.)

Although I was not instructed as to the math behind why the method works, the explanations I have described above illustrate what the various fractional divisions represent: answers the question “How many ’s are contained in 5?” and tells how many ’s are contained in . I might not have known why the invert and multiply rule worked, but I did know what the fractional division represented and how it was used to solve problems.

For sixth graders, such procedural understanding is a good start into what fractional division is. For fractions such as , the explanation for the invert and multiply rule is easier to convey — and understand — once the student has the algebraic tools by which to do so. Until that time, however, some teachers explain that the reason “invert and multiply” works is because “dividing by a number is the same as multiplying by its reciprocal” (inverse operations). It is similar to “subtracting a number is the same as adding its opposite”; also inverse operations.

Usually in seventh grade, students have learned the essentials for solving simple equations such as , which can then be extended to fractions to explain why the rule works. For example, an equation like can be solved by dividing both sides by . Since dividing by is done to leave with a coefficient of 1, students are taught that this goal is also achieved by multiplying by its reciprocal since the product is 1. Both sides are then multiplied by :

, and

… which shows that the invert and multiply procedure is equivalent to dividing by in the original problem.

**When Understanding is Part and Parcel to Procedure — and When it is Not**

The educational arena has been dominated by the fetish of understanding for more than 100 years. The prevailing group-think amongst the educational establishment and math reform movement is the fear that students will be “doing” math but not “knowing” math just as the principal I know had expressed to me.

Rote (i.e., non-understanding) learning is pretty hard to accomplish with elementary whole number math. The very learning of procedures is itself informative of meaning, and the repetitious use of them conveys understanding to the user. When learning to add and subtract, students make the connection between “I have 2 apples and got 3 more; how many do I know have?” and . Similarly, multiplication is understood so that “3 apples are in each bag, and there are 4 bags; how many apples in all?” can readily be represented by , and it is not difficult for the student to make the connection.

Unlike whole number operations, however, the conceptual underpinnings of fractional division are not part and parcel to the procedure. Even with an algebraic explanation, some kids will get it and some will not. Those who do get it may or may not remember why it works. While students undergoing instruction under the prevalent interpretations of Common Core may be able to recite an explanation they have been told, that is not the test of effectiveness of a math program. Their understanding is “rote understanding”. What matters to me is whether a student knows what fractional division represents. If a student can solve the problem, “How many oz servings of yogurt are in a oz container” by dividing by , and that this tells us how many are in , then I judge that student to have sufficient understanding. A student who has that understanding but does not know why the invert and multiply rule works is not at any significant disadvantage in solving fractional division problems.

**What the Poster Child Hath Wrought**

The “invert and multiply” example has served as a poster child for the reform math movement, and in their minds constitutes proof that traditionally-taught math is nothing more than the memorization of basic computational skills. Such skills are mistakenly viewed as rote learning and totally devoid of meaning. This is a gross mischaracterization. Liping Ma, author of “Knowing and Teaching Elementary Mathematics” and who taught elementary math in China, states that in the U.S. math in the lower grades is considered to be solely “‘basic computational skills’ … equivalent to an inferior cognitive activity such as rote learning.” (Ma, 2013)

In many US classrooms today, students must demonstrate an “understanding” of computational procedures before they are allowed to use standard algorithms. Such topsy-turvy approaches to math education have been around for more than two decades, but the interpretation and implementation of Common Core have made them more popular. To compensate for what reformers believe is a lack of understanding, the teaching of mathematics has been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

In so doing, students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding. But this “stop and explain” approach to understanding undermines what the reformers want to achieve in the first place. It is “rote understanding”: an out-loud articulation of meaning in every stage that is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.

The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education. Their views and philosophies are taken as faith by school administrations, school districts and many teachers – teachers who have been indoctrinated in schools of education that teach these methods.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. In so doing, the math reformers have unwittingly created a poster child in which “understanding” foundational math is not even “doing” math.

**References**

Buswell, Guy T., William A. Brownell, Irene Saubel. (1955) “Arithmetic We Need; Grade 6”; Ginn and Company.

Ma, Liping. (2013) *A Critique of the Structure of U.S. Elementary School Mathematics.* AMS, Notices; Vol. 60., No. 10 (DOI: http://dx.doi.org/10.1090/noti1054)

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Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long-term substitute in a high school and middle school in California: “Confessions of a 21st Century Math Teacher.”

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]]>Online retail giant Amazon has announced the launch of a national movement meant to put an end to the fear that many students in the United States feel when they set foot in a math classroom.

The “With Math I Can” initiative was created in an effort to have parents push a “growth mindset” rather than a “fixed mindset,” asking them to sign and shift to such instruction within the classroom. The movement’s website offers additional free resources helping users to determine exactly what a “growth mindset” entails, lesson plans, and more, writes Jason Del Ray for ReCode.

Created by a division of Amazon devoted to offering tech-based resources for K-12 education, the goal of the initiative is to change how students feel about math, as more than half of young adults do not consider themselves to be good at the subject, according to a survey by Change the Equation. That same survey found 93% of participants agree that good math skills are necessary to get ahead in life.

Amazon is looking to replace the thoughts of “I’m not good at math,” with “I will learn from my mistakes.” The company would like to see students adopt a growth mindset that focuses on the learning process rather than on concrete results that stem from solving individual problems.

The company will not create an ad campaign for the website, www.withmathican.org, although there are plans to push the program through social media and through an informational video, reports Ángel González for *The Seattle Times*.

A number of nonprofits devoted to education were brought into the program by Amazon, including the National Council of Teachers of Mathematics, ASCD (formerly known as the Association for Supervision and Curriculum Development) and Character Lab. Two school districts in California and one in New Jersey are also on board.

Amazon has not disclosed how much it cost to set up the program.

Rohit Agarwal, General Manager of Amazon K-12 Education, says the change came about after discovering that only 44% of low-income students attain a basic understanding of math while in school. Even after the launch of TenMarks, a math platform acquired by the company in 2013, the education team for Amazon said that despite seeing progress being made by students who used the program, they still found students who said, “I’m not good at math.” According to Agarwal, teachers who used TenMarks helped to inspire the new initiative.

“We believe that the attitude that it’s okay not to be good at math is just becoming too common,” Agarwal said in an interview. “Developing good math skills is essential to success at life.”

The program comes at a time when government officials and technology industrialists are expressing concerns about the adaptability of the US workforce as manufacturing jobs are stepping aside to create room for positions that hold a stronger focus on complex math and science skills.

Just last week President Obama proposed spending more than $4 billion across the next three years in order to increase student exposure to computer science.

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]]>The post Traditional Math: The Exception or the Rule? appeared first on Education News.

]]>By Barry Garelick

For those of you who have been following my writing, you are aware that I teach math in middle and high school and that I am extremely interested in the most effective ways of teaching the subject. I majored in mathematics and am pursuing the teaching of it as a second career after having retired five years ago. I balance my teaching by writing articles that address the problems with math education in the U.S. You also know that I was educated in the 50’s and 60’s, and thus obtained my math education via what is called “traditional” math teaching.

Because of my educational background and my beliefs in how math should and should not be taught, I often find myself engaged in the following dialogue, either in person or on the internet:

Someone: The traditional method of teaching math failed many students.

Me: The traditional method seemed to work well for me and many others I know.

Someone: You’re the exception.

Regarding my claim of traditional math working for me and others, I am mindful of the advice given by David Didau (author of “What if Everything you Know About Education is Wrong?”) who points out the following with respect to educational debates:

“If, in the face of contradictory evidence, we make the claim that a particular practice ‘works for me and my students’, then we are in danger of adopting an unfalsifiable position… We can insulate ourselves from logic and reason and instead trust to faith that we know what’s best for our students and who can prove us wrong?”

I will say in my defense, however, that “You’re the exception” is not much of an argument either. It is usually offered with either anecdotal evidence or none at all, and is based on a largely mischaracterized view of what traditional math is and was. Here’s but one example I saw recently in a promotional video explaining “Why is Math different now?” Starting at 0:35 seconds we hear:

“Decades of research shows [that] teaching people just to memorize algorithms and execute procedures hasn’t worked. It’s worked for a small group of people, but for most, they don’t like math, they don’t enjoy math, they don’t think they’re good at math, they have a really hard time with it because no one taught them to understand the concepts and why they’re doing what they’re doing. No one taught them how to think; we just taught them how to do, and execute; and this misguided idea that we just need to get to the answer as quickly as possible.”

Like many people who make this argument, he leaves unidentified the decades of research that he says support his claim — we are to take only his statements as evidence. I, on the other hand, do have some test data that sheds some light on the effectiveness of traditional methods, both textbooks and instructional quality. Specifically, test scores from the Iowa Tests of Basic Skills (for grades 3 through 8 ) and the ITED (high school grades) have been documented from the early 40’s through the 80’s for the State of Iowa.

The scores (in all subject areas, not just math) show a steady increase from the 40’s to about 1965, and then a dramatic decline from 1965 to the mid-70’s. (See Fig 1 below.) Indiana and Minnesota also showed this same pattern of ITBS scores as noted by Bishop (1989).

One conclusion that can be drawn from these test scores is that the method of education in effect during that period appeared to be working. And by definition, whatever was working during that time period was not failing. That the math could have been made more challenging and covered more topics in the early grades does not negate the fact that the method was effective. While some may argue that standardized tests scores do not measure true knowledge or “authentic” problem solving skills, the rise of the ITBS scores during this period has been of considerable interest to various researchers for some time (including Dan Koretz in a study he wrote for the Congressional Budget Office (1986), and Bishop (1989)).

These data raise the question of why there was a decline in test scores starting in the mid-60′s. One of the more popular explanations offered is that the population of test takers starting around that time began to include more minority students, resulting in a dilution effect. That argument fails to mention that the population of test takers in Iowa, Minnesota and Indiana remained primarily white, which has been noted by Bishop (1989) and Murray (1992). Specifically, the U.S. Census of 1950 shows that the population in Iowa was 99.2 percent white, declining by 0.7 percentage points to 98.5 percent white by 1980. Similarly, the populations of Minnesota and Indiana were 99 and 95.5 percent white in 1950, dropping respectively to 98.2 and 92.8 by 1970 (Hobbes, 2002).

With the above discussion as introduction, I will further clarify some of the issues associated with traditional math versus the reform versions of same. I will use the term “reform math” to refer what has replaced traditional forms of math education. It is also referred to as “progressive math.”

**Traditional Math and its Mischaracterization**

The term “traditional math” itself is confusing, and is sometimes referred to as “conventionally taught math”. The confusion comes about because traditional methods vary over time. Textbooks considered traditional for the last ten years, for example, are quite different than textbooks in earlier eras. For purposes of this discussion, I would like to confine the term “traditional math” as used in the U.S. to the methods and textbooks in use during the 30’s through the early 60’s. Such methods included topics presented in logical sequential fashion, building upon memorization of math facts and foundational standard algorithms and problem solving procedures.

The traditional model has been mischaracterized as relying on rote memorization rather than conceptual understanding. Calling the traditional approach “skills based,” math reformers deride it and claim that it teaches students only how to follow the teacher’s direction in solving routine problems, but does not teach students how to think critically or to apply prior knowledge to solve problems in new situations.

In light of this, I thought it might be interesting to look at some of the books used in previous eras that have been described as “having failed thousands of students”. Many, if not most, of the math books from the 30’s through part of the 60’s were written by the math reformers of those times. It makes the most sense to start with the series I had in elementary school: *Arithmetic We Need.* The reason is because not only is it from the 50’s, but also one of the authors was William A. Brownell, considered a leader of the math reform movement from the 30’s through the early 60’s. Today’s reformers also hold Brownell in high regard, including the prolific education critic Alfie Kohn, who talks about him in his book *The Schools Our Children Deserve* (Kohn, 1999).

In arguing why traditional math is ineffective, Kohn states “students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for *x*, but the traditional approach leaves them clueless about the significance of what they’re doing. Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.” He then turns to Brownell to bolster his argument that students under traditional math were not successful in quantitative thinking: “[For that] one needs a fund of meanings, not a myriad of ‘automatic responses’. . . . Drill does not develop meanings. Repetition does not lead to understandings.”

Figure 2 is taken from the 6^{th} grade book of *Arithmetic We Need*. The discussion of decimals in the text comes after the basics of decimals and their operations have been explained. The discussion in Figure 2 is geared to students who wish to explore underlying concepts further via an alternative method and comes with questions designed to guide students to an understanding of what happens when decimals are multiplied. (An earlier discussion in the book provided a general explanation of why the decimal point is placed by moving it to the left of the leftmost number in the product as many places as there are decimal places in both factors together.)

Figure 3 below is taken from the same book and provides a discussion of fractions related to a “number line.” The discussion and pictures represent the idea of fraction as number alongside whole numbers on a continuous line — an important concept and one which reformers say was missing in traditional math teaching. The discussion also put “improper” fractions in context — they are fractions like other fractions.

Figure 4 shows an approach taken in *Arithmetic We Need*, Grade 3 (Buswell et al, 1955b) to explain the role of place value in subtraction. The student is given instruction on what they are to demonstrate and then asked to solve four problems in the three ways shown in the diagrams. It is important to note that this particular method occurs after students have learned and mastered the standard algorithm for subtraction.

The series contained many exercises and drills including mental math exercises. Such drills might appear to run counter to Brownell’s arguments for math being more than computation and “meaningless drills,” but their inclusion ensured that mastery of math facts and basic procedures was not lost. Also, the books contained many word problems that demonstrated how the various math concepts and procedures are used to solve a variety of problem types.

Other books from previous eras were also similarly written—most authors were the math reformers of their day—and provide many counter-examples to the mischaracterization that traditional math consisted only of disconnected ideas, rote memorization, and no understanding. Examples from these books can be found in these particular articles.

Textbooks from previous eras included the alternate methods for addition and subtraction and other procedures, which have been the subject of many objections to reform math as well as how the Common Core is being implemented. Under reform math (and now under prevalent interpretations/implementations of Common Core), the standard algorithms for basic math operations are typically introduced last, after the student has been instructed to use these alternative strategies. In traditional math texts, alternate methods were introduced *after* students mastered the standard algorithms and procedures. The books did not insist that students use alternative methods and exercises, and using such methods was limited. After that, it was up to the student whether to use it or not, which means it served more as a side dish than the main dish it has now become in many U.S. classrooms.

Over the past few decades, reform math has been supplanting traditional methods of math education, mostly in grades K-6 and to a lesser extent in grades 7 and 8. In general, reform math promotes a teaching approach in which understanding and process dominate. As discussed above, teaching standard algorithms are delayed in the belief that learning those first will eclipse any understanding of what is going on when such procedures are followed. The result, reformers believe, will be students “doing but not knowing math”. By understanding how the tool works before being given the tool, reformers believe that when students get to more difficult and higher level math problems, they will be “thinking like mathematicians” and that conceptual understanding — more than procedural understanding and fluency — will guide their mathematical proficiency.

Reformers view with skepticism and disdain the idea that procedural fluency is part and parcel to understanding. Therefore, it is not unusual to find that in grades K-2, mental math and number sense are emphasized before students are fluent with procedures and number facts. Mastery of basic math facts is sought by using techniques that rely on patterns and techniques such as “making tens” as workarounds to straight memorization. Schools and districts are quick to tell parents—both suspicious and unsuspecting—that such circumvention strategies are part of a deeper understanding of math facts as opposed to the “mind-numbing” and “interest-killing” approach to math which in the past “failed thousands of students.”

These beliefs have led teachers and schools to issue warnings to parents to *not* teach their children the standard method at home because it would interfere with the student’s learning. For example, at the Jaworek School in Marlborough, Massachusetts, parents were given the following advice in the end of year newsletter: “Do not teach your child the “standard algorithm” for computations until he or she has learned it in school.” (See also the advice to parents for Grades 1 and 4 given by the Covington Elementary School (Covington, Washington) and the article by Tom Loveless of Brookings Institution on this trend.)

In addition these and other approaches to teaching math, the general pedagogical approach and classroom set-up is also different. Whole class and teacher-led explicit instruction (and even teacher-led discovery) has given way to what the education establishment believes is superior: students working in groups in a collaborative learning environment. (A more thorough discussion of the “symptoms” of reform math approaches is found here.)

**Variations of the “Exception” Argument**

The prevalence of reform math over the past few decades has heated up the rhetoric against traditional math. Such rhetoric includes the “You’re the exception” argument, of course, but there are other variations of this. Here are three of the most popular variations. Such arguments occur at school board meetings, in casual conversations, on the internet, and —disturbingly — in newspapers and on television.

**If traditional math teaching were effective, the U.S. would be at the top of the world in math.**

This argument ignores that in countries doing well on such international tests, students learn math mainly via traditional means — and over the past two decades, increasing numbers of students in the U.S. have learned math using the reform-based methods. Reformers are quick to point out that Japan and perhaps other Asian countries actually use reform methods, ignoring the fact that many students are enrolled in “cram schools” (called Juku in Japan) which use the drilling techniques and memorization held in high disdain by reformers.

The argument also fails to consider that traditional math can also be taught poorly. There have always been good and bad teachers, as well as factors other than curriculum and pedagogy that influence the data. In order for such arguments to work, one would have to evaluate how achievement/scores vary when factors such as teaching, socioeconomic levels and other variables are held constant and when pedagogy or curriculum changes. Studies have been conducted that examine how math is taught in specific areas of North America, as well as looking at the common traits of high-performing systems across the world. They indicate that when both conventional and non-conventional (i.e., reform) math are taught by well-trained teachers, students learning under traditional mathematics instruction show much higher achievement than those learning under the reform math methodology. (Stokke, 2015; see https://www.cdhowe.org/pdf/commentary_427.pdf)

**If traditional math worked, the knowledge learned in school would stay with us.**

That people do not maintain proficiency in math as they age says less about traditional or reform math than about the way in which a population’s knowledge and skill base is maintained over a lifetime. It is not evidence of failure of traditional math. The results of not using math on a consistent basis can also be seen in a study conducted by OECD. In the study, people from ages 16-65 in over twenty countries, including the U.S., were given the same exam consisting of math computations and word problems. According to the study, “the percentage of U.S. adults between 55 and 65 years old who scored at the highest proficiency level (4/5) …was not significantly different than the international average for this age group. (Goodman, et al., 2013).” These findings can be used in tandem with the first argument above since people in the U.S. in the 55 to 65 age group learned math via traditional math teaching—and the differences in proficiencies between the U.S. and other countries is not significant.

**Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.**

This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It goes something like this: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma. Few went to college. Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”

First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it. Also, I don’t know that most students *must *take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.

Secondly, while students only had to take two years of math to graduate, and algebra was not a requirement as it is now, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems. In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

Some blame the “changing demographics” on the decrease in proficiency, but this overlooks variables like poor curriculum and the reform-based approach to math which views memorization “workarounds” as deep understanding. Also frequently overlooked is the fact that students in low income families who make up the “changing demographic” cited in such arguments do not have access to tutoring or learning centers, while students in more affluent areas are not held hostage — dare I say “tracked”? — to poor curricula and dubious pedagogical practices.

**What’s Next?**

The debate over traditional versus reform-based math has been going on for some time—for so long, in fact, that some on the reform side are saying that there’s nothing to discuss, it’s boring, just let teachers teach. I agree that we should let teachers teach, and that parents be given choices of what type of math they want their children to have. That doesn’t appear to be happening any time soon.

I believe that the debate should continue and that there is plenty to discuss. People may choose to use the information I’ve presented here — or persist in ignoring it. I don’t expect that I’ve changed anyone’s mind about anything, but I am always hopeful that there are some exceptions.

I also do not think that I am alone in drawing a distinction between reform and traditional modes of math teaching. While traditional math can be taught properly as well as badly, I believe that poor teaching is inherent in most if not all reform math programs. I base this on having seen good teachers required to follow programs that present content poorly, lack a coherent logical sequence and rely on questionable pedagogies.

I would like to see studies conducted to document how U.S. students who do well in math and science and pursue STEM majors and careers are learning math. The chances are fairly good that such investigations would show that in K-8, many students are getting support at home, from tutors, or from the many learning centers that are springing up all over the U.S. at rapid rates. Since tutors and learning centers (and parents) tend to use traditional methods for teaching math, I somehow doubt that the clientele are exceptions to some ill-defined rule. In my view, as well as the view of many parents and teachers I’ve met, there are few exceptions to the educational damage reform math programs have caused, even when such programs are taught “well.”

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**Barry Garelick** has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long-term substitute in a high school and middle school in California: “Confessions of a 21st Century Math Teacher.“

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**References**

Bishop, John. 1989.* **Is the Test Score Decline Responsible for the Productivity Growth Decline?* The American Economic Review (Vol. 79, No. 1)

Buswell, Guy T., William A. Brownell, Irene Saubel. (1955a) “Arithmetic We Need; Grade 3”; Ginn and Company.

Buswell, Guy T., William A. Brownell, Irene Saubel. (1955b) “Arithmetic We Need; Grade 6”; Ginn and Company.

Clark, John R., Charlotte W. Junge, Harold E. Moser. (1952). “Growth in Arithmetic, Grade 6; Teacher’s Edition”; Harcourt, Brace & World, Inc.

Congressional Budget Office. 1986. *Trends in Educational Achievement.* Prepared by Daniel Koretz of Congressional Budget Office’s Human Resources and Community Development Division. Congress of the United States. Available at: http://www.cbo.gov/ftpdocs/59xx/doc5965/doc11b-Entire.pdf

Goodman, M., Finnegan, R., Mohadjer, L., Krenzke, T., and Hogan, J. (2013). Literacy, Numeracy, and Problem Solving in Technology-Rich Environments Among U.S. Adults: Results from the Program for the International Assessment of Adult Competencies 2012: First Look (NCES 2014-008). U.S. Department of Education. Washington, DC: National Center for Education Statistics. Retrieved Dec. 20, 2015 from http://nces.ed.gov/pubsearch .

Hobbes, Frank and Stoops, N. 2002. *“Demographic Trends of the 20th Century”*. U.S. Census Bureau. Washington DC. November.

Kohn, Alfie. (1999). “Getting the 3 R’s Right” in *The Schools Our Children Deserve* (Boston: Houghton Mifflin)

Murray, Charles. 1992. *What’s Really Behind the SAT-Score Decline?* , Public

Interest, 106 (1992: Winter) p.32

Stokke, Anna (2015). *What to Do about Canada’s Declining Math Scores. *Commentary No. 427. C. D. Howe Institute; Toronto, Ontario; Canada. Retrieved Dec. 20, 2015 from https://www.cdhowe.org/pdf/commentary_427.pdf .

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