Sunday, December 4, 2016

Standard algorithms, distractions

Part 1 Standard Algorithms
Alice Crary and W. Stephen Wilson conclude that reform math has swept out the traditional math. In the New York Times 2013, they wrote, "Today, the emphasis of most math instruction is on numerical reasoning (i.e., reform math's new jargon). This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms. Crary and Wilson point out, "The standard algorithms are either de-emphasize to students or withheld from them entirely." Moreover, "The staunchest supporters of reform math are math teachers and faculty at schools of education."  Now you know the reason that reform ideas persist in our classrooms. 

In reform math, reasoning is much more important that learning content knowledge. But it isn't. Problem-solving depends on knowing stuff. To perform arithmetic and algebra well, students must know in long-term memory factual and procedural knowledge. But, to reformers, facts don't matter much. 

The ideas, skills, and uses of arithmetic or algebra are not meant to teach students creative reasoning. Students do not take calculus to improve their creativity. Crary & Wilson explains that in all disciplines, including mathematics, science, and history, "Children need to master bodies of fact, and not merely reason independently." 

Indeed, learning facts in biology, chemistry, and physics or history "do not stunt students' growth [creativity, imagination, or curiousity] and prevent them from thinking for themselves." The same goes for arithmetic. To do arithmetic well means to know facts and standard procedures, which requires memorization and practice-practice-practice. Guess which nations soar far above U.S. kids in problem-solving? The more "rote" learners in East Asia. Knowledge enables problem-solving. Critical thinking is a product of knowledge, as are innovation and creativity. We are not all equally creative or innovative. Problem-solving is domain-specific. All the reasoning in the world will not help you solve a trig problem unless you know the trig.

Today, all we hear is critical thinking this, critical thinking that, but without a solid mathematical knowledge base, critical thinking (i.e., problem-solving in mathematics) is a moot point, but not to the reformers who believe that students can become good problem solvers without knowing basics (i.e, core arithmetic). The reform idea violates a fundamental cognitive science finding that critical thinking is domain-specific.

Part 2 Distractions
Textbooks are filled with colorful graphics and pictures that distract students from learning math. Likewise, covering the walls with colorful posters, sitting in small groups where kids face each other, and using gadgets (e.g., graphing calculators, laptops, tablets, etc.) distract students from learning. Consequently, students learn less. It is a conundrum in U.S. classrooms. Distractions can cause learning gaps. 

Similarly, when students are introduced to multiple strategies to do multiplication, such as arrays, areas models, latices, and partial products, cognitive overload in novices is often produced. The working memory is limited, so when extras are tossed into the mix, such as multiple strategies for doing math or writing explanations, novices often become confused and learn less.

Clark and Feldon ("Five Common but Questionable Principles of Multimedia Learning") conclude "Multimedia does not increase student learning beyond any other media including live teachers." Extraordinary teachers can produce amazing students, but multimedia won't. Personalized learning, adaptive software, and blended learning lack evidence of effectiveness, so students learn less. The hype is not evidence. 

©2016 LT/ThinkAlgebra 

Friday, November 11, 2016

Problem Solving in Mathematics

Sweller, Clark, & Kirschner explain that problem-solving in math should be taught through carefully sequenced worked examples, not general problem-solving skills or strategies that Polya advocated. The skills approach has not worked well in classrooms, but the content knowledge approach has.
“Many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge. The best-known exposition of this view was provided by PĆ³lya (1957). He discussed a range of general problem-solving strategies, such as encouraging mathematics students to think of a related problem and then solve the current problem by analogy or to think of a simpler problem and then extrapolate to the current problem. Nevertheless, in over a half-century, no systematic body of evidence demonstrating the effectiveness of any general problem-solving strategies has emerged.” 

Note. The quotes are from “Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics” Sweller, Clark, & Kirschner (Doceamus, November 2010).

Sweller, Clark, & Kirschner point out that general problem-solving skills independent of content are not supported by evidence. Common Core with its Standards for Mathematical Practice advocates that math should be taught via general problem-solving skills. The pedagogy is wrong.
“Recent ‘reform’ curricula both ignore the absence of supporting data and completely misunderstand the role of problem-solving in cognition. If the argument goes, we are not really teaching people mathematics but rather are teaching them some form of general problem solving, then the mathematical content can be reduced in importance. According to this argument, we can teach students how to solve problems in general, and that will make them good mathematicians able to discover novel solutions irrespective of the content.” The argument is not true!

“Whereas a lack of empirical evidence supporting the teaching of general problem-solving strategies in mathematics is telling, there is ample empirical evidence of the validity of the worked-example effect.

Practicing problem-solving strategies independent of worked examples doesn't work. Students learn little arithmetic and algebra. 
“Domain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies. There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem-solving without reference to worked examples (Paas & van Gog, 2006).” 

Students are novices, not little mathematicians. They need to learn content to support problem-solving.  
“Studying worked examples is a form of direct, explicit instruction that is vital in all curriculum areas, especially areas that many students find difficult and that are critical to modern societies. Mathematics is such a discipline. Minimal instructional guidance in mathematics leads to minimal learning (Kirschner, Sweller, & Clark, 2006).

Note. When I read Polya's book, I noticed that every example of problem-solving required knowing content knowledge. In short, without specific content knowledge (concepts, procedures, and applications) in long-term memory, you cannot solve problems in mathematics. In other words, you cannot solve a trig problem without knowing some basic trig. 

©2016 LT/ThinkAlgebra

Tuesday, November 8, 2016

Achievement Inequalities

Children are not the same. "Ability varies widely," says Charles Murray (Real Education). So, let's stop pretending that all kids are the same and can learn the same math. Children need different math curricula based on their cognitive abilities--not the same curriculum as in Common Core and state standards. In contrast, Nobel-prize winning Physicist Richard Feynman writes, "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities." Education increases inequality! 

"Fairness as the equal treatment does not produce fairness as equal outcomes," writes Thomas Sowell (Dismantling America). Some kids have more cognitive horsepower than others. Some kids do better at math than other kids. Some kids run faster than others. Some kids play the piano better than others. Some kids are more persistent than others, ad infinitum. Inequalities abound everywhere. It is unfortunate, says Sowell, that "virtually any disparity in outcomes is almost automatically blamed on discrimination." It's not discrimination! Sowell suggests jokingly that "tests discriminate against students who don’t study." 

Even if it were possible to equalize school resources, then there would still be an achievement gap in schools based on standardized tests. It seems that inputs from the family and community, not just school inputs (e.g. expenditures, facilities, teacher quality, etc.), account for much of the achievement gap. We do not live in Lake Wobegon where all the children are above average. In fact, half are below the median in intelligence. I think we can make progress in math performance, but there will always be inequalities no matter what is done. Also, the fact that math is taught poorly in many schools has been a major factor. Consequently, the result has been very slow, incremental progress or flat learning, which I consider, unacceptable. 

The top math students have skilled instruction and excellent practice with feedback, but they also have higher levels of cognitive horsepower than average kids. However, cognitive ability, by itself, is not enough to become a good math student. Kids need to practice to automate fundamentals, which requires effort and purposeful practice. Indeed, normal kids can learn arithmetic and do algebra. Also, school outputs, such as test scores, graduation rates, college readiness, etc. will always vary.

©2016 LT/   

Saturday, July 9, 2016

Long Division

Standard Algorithm for Division: Grade 3
1. Teach the steps and convergent strategies for the standard algorithm. I assume that you know the steps. Do not teach the partial quotient method. 
2. Make sure students have automated the steps. Drill for skill. I assume students have auto recall of multiplication facts.
3. Lastly, explain the reasons why the algorithm works. The illustration below will help you help students understand it better. Repeated subtraction and the distributive property are fundamental ideas in the algorithm.  Students do not need to write an explanation of why the algorithm works. 

Standard Algorithm for Division. My comments..

The method is repeated subtraction. First 15(20), then (15)(2), or 15(20 + 2). "The distributive axiom helps shorten the number of steps. Without it you would have to subtract 15 from 334 twenty-two times," explains Mary P. Dolciani [1].

Do not use other methods to verify an answer. Use the regular check.

To verify the answer, do the following 3rd-grade check: 
15(20 + 2) + 4 = the Dividend
300 + 30 + 4 = 334
334 = 334 True
(OR: 15 x 22 + 4)

In the 70s and 80s, many public, parochial, and private schools had taught the standard division algorithm in the second semester of 3rd grade. Multiplication and word problems dominated the 1st semester. Division and word problems dominated the 2nd semester. For example, the 1997 California 3rd-grade math standards included long-division. Some schools still do!

The old California standards, which were adopted in December of 1997, got it right. By the end of 3rd grade, students should know well (be able to apply and perform efficiently) the four standard algorithms for whole numbers (addition, subtraction, multiplication, and long division).   

[1] Dolciani & Wooton, Book 1, Modern Algebra Structure And Method, 1973 Revised Edition. This is one of the best Algerba-1 textbooks ever.  It also covers trig and vectors, which were used to solve problems.

Note. The properties of numbers give math structure.  

©2016 LT/ThinkAlgebra

Tuesday, May 17, 2016


Only a meager 25% of 12th-grade students are proficient in math in a national test, which indicates that math has been taught poorly in K-12. US math has been in trouble for decades! Most top-down, imposed math reforms, mandates, and policies haven't worked well, including reform math, Common Core, state standards-testing-accountability, test prep, inclusion, No Child Left Behind Act (replaced by Every Child Succeeds Act), technology hype, equalizing-downward crusades, self-esteem movement, untested fads (often called innovations), inferior minimal guidance methods (group work), etc. 

My views often challenge conventional thinking in education. 
1. Some say the NAEP (National Assessment of Educational Progress) proficiency levels (4th, 8th, 12th) are set too high when the underlying math problem has been low expectations via a substandard curriculum and inferior instructional methodsStudents are not learning nearly enough background knowledge (prerequisites). 
2The reality is that kids need to learn more math, especially standard arithmetic and algebra, much earlier than previous generations because of high-tech job demands.
3. Our kids are smart enough to do this. Knowledgeable teachers can fuse algebra to standard arithmetic as children learn arithmetic beginning in the 1st grade. 

4. Educators seem to ignore the cognitive science of learning such as the role of working memory and long-term memory, cognitive load, etc. Mathematical thinking and problem-solving come from mathematical knowledge (skills, ideas, uses) automated in long-term memory, not thin air.
5. The perception of parents and the actual performance of students are extremely disconnected, according to the PTA's Parents 2016.
6. Students enrolled at community colleges, sadly, often lack basic arithmetic and algebra knowledge (skills, ideas, uses); consequently, they are placed in remedial math classes to make up for the content they didn't learn in K-12. For example, 74 to 88% of the high school students that applied to Pima Community College from the nine Tucson-area school districts were placed in remedial math (PCC 2014). The remedial students were products of reform math.

7. Most children can succeed in math class when they are given focused, well thought out, achievable learning (performance) objectives within a logical, hierarchical sequence (Gagne); however, the objectives should fit the child's achievement or performance level, such as low, average, and advanced. Instead of a reasonable and flexible levels approach with different materials (curricular learning objectives) and actual math teachers, not reform math educators from schools of education, we have substituted inclusion policies in elementary schools with a generalist who attempts to differentiate instruction, which doesn't work well. Consequently, arithmetic and algebra have been taught poorly. Also, the one-size-fits-all stance of Common Core or state standards has been a flawed practice. Lastly, behavioral learning objectives should be performance-centered (Robert Mager). They must be Specific, Measurable, and Achievable (SMA). 

American math standards are not world-class.
The popular minimal guidance methods during instruction are inferior.
Reform math does not work.

Common Core or state standards are not world-class and put our students at a disadvantage starting in 1st grade. Inferior achievement in math doesn't start in high school or middle school. The difficulty begins in early elementary school. 

All the math results of the 2015 NAEP (National Assessment of Educational Progresstest are now available, including the 12th-grade, and they are not good. Math proficiency for 4th-8th-12th: 40%-33%-25%, respectively. What else do we know?

  1. Math achievement in 2015 was lower than in 2013.
  2. The percentage of weak students (below the Basic level) in math was much larger.
  3. Common Core revived inferior reform math. (JJmom2 writes, "Common Core threw out common sense and made simple calculations into ridiculously difficult problems, which really held kids back from moving on.") 
  4. Common Core standards and state standards are not world-class or for STEM students. They are a stumbling block for many students. (The problems of inadequate content, inferior instructional methods, and low expectations start in the 1st grade.) 
(Quote: JJmom2 comment is from the Answer Sheet, Washington Post blog by V. Strauss, 4-30-16; Nation's Report Card 2015 NAEP) 

We expect too little from elementary, middle, and high school students, and it shows on both national and international tests. A student does not need to be gifted or a genius to learn Algebra-1 with trig in middle school, AP Calculus in high school, or grasp some key algebra ideas in 1st grade. Average kids can learn when they are properly taught the prerequisitesAlgebra-1 with trig is a middle school subject when students are prepared well in elementary school, but most are not. Indeed, Common Core disregarded the recommendations of the National Mathematics Advisory Panel (2008) for getting more kids ready for Algebra-1 by middle school. Common Core intentionally delays Algebra 1 to high school, which is indefensible. It is not for STEM students.

Under Common Core, the 2015 NAEP (National Assessment of Educational Progressmath scores for 4th-, 8th-, and 12th-grade students were lower in 2015 than in 2013. (NAEP scores are often called the Nation's Report Card.) Also, in the PTA's Parents 2016 report, 90% of parents think their child is achieving at or above grade level in mathematics, but only 40% are proficient by the 4th grade (NAEP). The perception of parents and the actual performance of students are extremely disconnected. By the 8th grade, student proficiency drops to 33% in mathematics. [Note. 87% of Hispanic parents think their child is achieving at or above grade level in mathematics when only 26% are proficient in 4th grade. The percentage drops to 19% by the 8th grade. Black students are even lower.] (Sources: Nation's Report Card, 2015; PTA Parents 2016)

The "at or above proficient" math trend (2015 NAEP), from the 4th to 8th to 12th grades, has been a free-fall: 40% to 33% to 25%, respectively. The trajectory is negative, that is, the performance of students is distressingly bad. Ze've Wurman points out, “The only plausible explanation for such an unprecedented broad national decline is the Common Core."

Common Core revived constructivist reform math and its inferior minimal-guidance approaches, which had dominated the 90s: problem-solving, project-based, discovery, inquiry-based, etc. The methods were not efficient or effective. Students stumbled over simple arithmetic (Kirschner, Sweller, ClarkWhy Minimal Guidance During Instruction Does Not Work, 2006).

For decades, we have overemphasized pedagogy at the expense of mathematical content. The idea of pedagogy over content traces back to the 1989 NCTM reform math movement, which has been revived through Common Core. Indeed, the Common Core math standards are often interpreted through the lens of reform math, which makes simple calculations complex. Simply, our kids don't know enough math content, which has been confirmed through both international (TIMSS, PISA) and national tests (NAEP).

According to international tests, American kids have lagged behind in mathematics for decades. In fact, most state standards, including Common Core, have not been world-class(One exception was the 1997 California Math Standards. There were others.) Also, the math reforms over the years have not worked well. Indeed, evidence-lacking fads (often advertised and sold as innovations and reforms by special interest groups, as if new means better) have flourished in US education. The truth is that math has been taught badly. 
Our kids are mediocre in math because we made them that way. I agree with Ze'Ve Wurman. Common Core has added to a downward trend in math achievement. State standards are often interpreted as reform math. (NAEP 2013, 2015; Wurman's quote from Breitbart )

NAEP 2015: 12th-Grade Math

Only 25 Percent!
The NAEP 2015 results for 12th-grade mathematics (public + private schools) are not good. Only 25% of 12th-grade students were "proficient or above" in mathematics.

The average score dropped one point since 2013 and is nearly the same as it was in 2005. In short, the math reforms over the years haven't changed the dynamic--not NCTM reform math, not NCLB, not Common Core, not technology, not inclusion and other policies, etc. My conclusion is that there is something terribly wrong with the math reforms. In brief, the math reforms aren't supported by scientific evidence.

Note1. In NAEP math, the percentage of 12th-grade students who scored below the Basic achievement level increased from 35% in 2013 to 38% in 2015. In short, weak students are worse off. Not good!

Note2. Some say that the NAEP math scores for the proficient achievement at the 4th-, 8th-, and 12th-grade levels are set too high; however, I think, American expectations for students in math are often too low. IndeedAmerican students underperform on national and international tests. Furthermore, the underperformance starts in 1st grade when students are not taught standard arithmetic. Our math standards are not at the Asian level.    

NAEP 2015: 12th Grade College Readiness Estimate
The 2015 NAEP's 37-Percent Estimate for College Readiness
Math scores often indicate academic preparedness for college. The NAEP governing board determines that only 37% of the 12th-grade students were "academically prepared for college."

Note. According to a new metric from the NAEP, only 37% of 12th-grade students are college-ready in math and reading. To be college-ready, 12th-grade students should score at least 163 out of 300 in math and 302 out of 500 in reading. Furthermore, the percentages of students academically prepared for college in math and reading (both 37%) were lower than in 2013: 39% math and 38% reading.

Remember: The main selling point of Common Core was that it would make all students college and career ready, a truly extraordinary claim that lacks evidence. 
NoteThe math proficiency level in 12th-grade NAEP is not the same as its college readiness metric, which includes community college and is more in line with ACT, SAT, college outcomes, placement in college-bearing courses (not "admissions" to post-secondary institutions), etc. College readiness means ready for College Algebra, a course that is similar in content to high school Algebra-2. College Algebra is often the only required math course at community colleges.  

 The underlying strength of a math program, I think, can be inferred from the Advanced Benchmark levels of the Trends in International Mathematics and Science Study (TIMSS). 
At the 8th-grade level, 48% of Singapore students reached or exceeded the Advanced Benchmark compared to 7% of US students. At the 4th-grade level, 43% of Singapore students reached or exceeded the Advanced Benchmark compared to 13% of US students. Clearly, a huge number of Singapore students learned content significantly above their grade level. Indeed, grade levels are not always an acceptable gauge of achievement. In state testing, saying a student is at grade level or proficient in mathematics is pointless. What is consequential is the math content (ideas-skills-uses) the student knows and applies.

The TIMSS Advanced Benchmarks show that American math programs are 
weak both in curriculum and instructionAsian students lean toward rote learning and drill for skill. On average, Asian nations rank the highest in math achievement on international tests. Asian students know much more math knowledge (ideas, skills, uses) than American students at all grade levels. Contrary to popular belief, the Asian "rote-leaning" nations also rank the highest in creative problem-solving skills (2014 PISA - Programme for International Student Assessment for 15-year olds). Indeed, there is a strong link between gaining mathematical knowledge in long-term memory over time and solving problems (mathematical thinking). Also, there is a cultural difference. Asian parents regard education, especially mathematics, the highest priority while many American parents do not. American students grossly underperform in math.] Note. Highly-regarded Finland has been displaced by Asian nations in the 2014 PISA Problem Solving Test and the 2015 PISA math and reading tests for 15-year olds. Also, Finnish 4th and 8th graders were about the same as American students (TIMSS).
(Advanced Benchmark: TIMSS 2011; Creative Problem Solving: PISA 2014; NAEP 2013, 2015; Note. The 2015 TIMSS results will be announced in November 2016)


Saturday, April 2, 2016

Cultivating Young Math Talent

To me, Common Core math, aka state standards, and Every Student Succeeds Act does not establish realistic goals because the kids who walk through the school door vary widely in math ability, knowledge, and motivation. Early exposure and practice, I believe, are a step in the right direction for developing achievement, but, likewise, to obtain better achievement requires ability, determination, and drill-for-skill. Unfortunately, US educators frequently downplay ability, trivialize drill-for-skill development, and focus too much on standardized testing. In the real world, some kids are better at math than others. That's life! Moreover, the one-size-fits-all dogma of Common Core and state standards is an inferior approach to math education. 

Also, I believe the math education most kids get is full of gaps. If you get a 75% correct on a test, then you miss 25% of what you need to know, says Salma Kahn. "Concepts build on one another." Kids need a "good grasp of basics" to acquire a higher-quality achievement.  But, many students never master the fundamentals of standard arithmetic. Instead, they are taught doses of test prep; (i.e., bits and pieces) and reform math, which sidelines or delays the standard algorithms and substitutes several, more complicated ways to perform simple arithmetic. Indeed, Common Core or state standards are often interpreted through the lens of reform math. 

Focus on Developing Achievement
In education, everything seems to be about race and "illusions of fairness" rather than excellence and achievement. The focus should be on cultivating talent and developing achievement, but it isn't, regardless of the rhetoric and hype. Instead, there has been an anti-intellectualism movement in our culture. No Child Left Behind, now Every Student Succeeds Act, mantras such as Algebra for All, College for All, and ideas such as inclusion and one-size-fits-all are not what they seem to be. Also, included are Common Core (linked to testing and so-called Mathematical Practices), most math reforms (reform math), minimal guidance methods (discovery), group work, technology (i.e., the silver bullet), etc. All of these are smokescreens and part of an anti-intellectualism culture. Indeed, they are not credible because they are not supported by scientific evidence. I am not sure we can fix an anti-intellectualism culture that prescribes progressive ideas.

"We need nerds because they make the world a better place," writes David Hopkins. We need smart kids. We need smart, minority kids, too, but, in my opinion, the people in charge of our educational system (i.e., the progressives), seem bent on suppressing excellence and achievement, regardless of their deceiving rhetoric. "The Common Core standards initiative is part of the progressive push to centralize education, says the Heritage Foundation in a new report," writes Dr. Susan Berry. Note. Common Core has been renamed "state standards" with little change. 

Proper instruction should increase differences, and, therefore, inequalities, explains Richard Feynman, Noble-prize winner in Physics. (Progressives: Well, we can't have that! It's racist.) “Equalizing downward by lowering those at the top” is an illusion of fairness that hurts bright blacks and Hispanics, observes Thomas Sowell (Dismantling America, 2010). Many low-income kids are smart and can be high achievers in math, but they seldom get the accelerated instruction that meets their needs. Our best students in math should be tracked starting in early elementary school. (Progressives: Well, we can't have that! It's racist.). In my opinion, inclusion policies and other progressive social policies in our schools interfere with the academic advancement of bright black and Hispanic students by lowering those at the top so that one size fits all. 

I think it is absurd to deny better students algebra in middle school or fast track math in elementary school. If we can develop talent in music at an early age, then we can do it in mathematics, too. "Talent matters but motivation [commitment and passion] may matter more," writes Daisy Yuhas (Scientific American, Think Like A Genius). To develop ability in mathematics requires content acceleration and task commitment. Gaining knowledge is key to becoming proficient and an expert. American schools rarely fast-track mathematical ability. Opportunities for high-achieving students are often limited. Excellence has not been the focus of schooling for decades. Tracking, if it is implemented correctly, can help all achievers, including black and Hispanic students, by establishing manageable achievement benchmarks and code of excellence in our schools.

High-Achieving Students Need Acceleration
"Today, researchers, policymakers, and teachers pay little to no attention to high-achieving students.... Many such students spend their days in school unchallenged--relearning material they have already mastered ... In academics, so far only in mathematics do we have reliable ways to detect potential talent early on ... High achievers may have exceptional task commitment, meaning they are willing to engage in study and practice that, though not necessarily enjoyable, is instrumental to improvement ... Acceleration significantly boosts both achievement and motivation in gifted students ... Schools hardly ever use acceleration strategies, yet acceleration should be a key part of gifted education." explain Subotnik, Olszewski-Kubilius, & Worrell (Scientific American, Think Like A Genius, November/December 2012). High-achieving math students need acceleration that advances them, not enrichment. 

Scores Are Flat And Going Down
In reform math, the standard algorithm has been denigrated. For example, students are taught 3 or 4 nonstandard ways to multiply numbers, a reform math pedagogy that had not worked in the past. The multiple ways often confuse students, overload working memory, and alienate parents. Consequently, fewer students master the standard algorithm. The TIMSS (International) test scores show that US students are not improving like students in some other nations. Moreover, the 4th- and 8th-grade 2015 National Assessment for Educational Progress, or NAEP, math scores are worse than the scores from 2013. Tom Loveless (Brown Center Report) points out, “The bad news is that there also is no evidence that CCSS [Common Core State Standards] has made much of a difference during a six-year period of stagnant NAEP scores.” Apparently, what we consider good teaching and good practices are inadequate, inferior, and seriously flawed—e.g., reform math, minimal guidance, one-size-fits-all, Common Core, group work, test prep, technology hype, and inclusion, just to name a few.  

Math & Music
Many Asian parents guide their children into math and music. From the beginning, children are taught that math is important. Asian parents also push early musical training (playing piano or violin) because they believe that learning to play a musical instrument helps with learning math. Incidentally, musical talent can be spotted in kids aged 2 to 4. Also, Asian parents frequently teach basic arithmetic (e.g., addition facts) to their preschool children. Even after their children have started regular school (1st grade), parents continue math lessons at home using workbooks. Students practice math at school and at home. Also, after school, many Asian students go to cram schools to prepare for consequential exams. For example, in Singapore, the 6th-grade primary (math) exit exam determines which secondary school (7th-12th) the child can attend, aka tracking.

In Singapore, tracking students starts in the 1st grade with a pull-out program for incoming students who lack numeracy skills. (Note. Public school in Singapore starts in 1st grade.) The catch-up program lasts for two years and works well. Tracking starts again in the 4th grade when math becomes harder. Some students are placed in a different textbook with a separate teacher. [Note. The Singapore curriculum and instructional methods are not perfect in my opinion. The curriculum relies too much on bar models rather than on writing equations to model problem situations. The early curriculum should include number lines and negative numbers in 1st grade. Moreover, 1st graders should learn functions (input-output model), built tables of values, and graph linear equations in Q-I. Students should do more measuring in 1st grade: mass in grams, liquid volume in milliliters and liters, solid volume in cubic centimeters, length in centimeters and meters. Also, in 1st grade, the curriculum should include perimeters of squares and rectangles (addition) and areas of squares and rectangles (counting).]  

US Math Achievement Is Stalled
Curriculum & Instruction Are Weak!
Math achievement has been stalled for years. US students, on average, are mediocre compared to their peers in some other nations (TIMSS). Indeed, on international tests, our students are not achieving as fast as students in some other countries. The latest indicator is that the 2015 National Assessment for Educational Progress (NAEP) math scores for 4th and 8th graders are lower than in 2013. As stated above, the math curriculum, instructional methods, and progressive social policies are seriously flawed. 

The strength of a math program is found at the Advanced Benchmark level of TIMSS, an international test. At the 8th grade level, 48% of Singapore students reached or exceeded the Advanced Benchmark compared to 7% of US students. At the 4th grade level, 43% of Singapore students reached or exceeded the Advanced Benchmark compared to 13% of US students. Roughly half of the Singapore students learn content significantly above their grade level. The TIMSS Advanced Benchmarks show that American math programs are weak both in curriculum and instruction. The rote-leaning kids in Asian nations not only master math but also dominate in problem-solving. [Advanced Benchmark data from 2011 TIMSS; PISA]  

Good instruction should increase differences.
Richard Feynman, a Noble-prize winner in Physics, writes about the ethics of equality in education, “In education, you increase differences. If someone’s good at something, you try to develop his ability, which results in differences or inequalities. So if education increases inequality, is this ethical?” (“Surely You’re Joking, Mr. Feynman” by Richard Feynman, 1985) The idea that instruction should increase differences conflicts sharply with the prevailing progressive ideology of inclusion and sameness and with Common Core's one-size-fits-all tenet, aka every student gets the same instruction regardless of the achievement level. Indeed, "equalizing downward by lowering those at the top," in the name of fairness, is an "illusion of fairness" and a twisted idea. (Note. "equalizing downward..." and "illusion of fairness" by Thomas Sowell)

Many failed ideas and assumptions, practices and pedagogies, and theories and conjectures in education don’t get junked overnight. They hang around for decades and decades. Some are repackaged and start anew, such as the resurgence of reform math. Similarly, like the failed programs of the past, the newest trends and fads (often called innovations) are not supported by the cognitive science of learning. In short, the claims of effectiveness lack scientific evidence. 

Squandering Talent
Starting in elementary school, our best students in math and science are not only underserved and underfunded, but they are pushed to the back burner and left to fend for themselves as if they don’t exist. Much of our potential talent is squandered in the American system. Many low-income kids are smart and can be high achievers in math, but they seldom get accelerated instruction that meets their needs. Parents of high achieving kids should be outraged. 

Tom Loveless, a Brookings Institute researcher, writes, "You need to cultivate talent over time in mathematics."  Indeed, high-achieving minority students should be tracked into advanced groups, starting in lower elementary school. Ther best math students of all races should be tracked. Jill Barshay (Hechinger Report, Education by the Numbers, March 28, 2016) writes, "Loveless's research raises an age-old question of whether excellence is sacrificed by well-intended efforts to promote equity." The fairness policies, which are illusions of fairness, hurt low-income black and Hispanic kids in urban schools. Black and Hispanic kids have every right to be in advanced and honors classes. But, their talent must be spotted early enough and cultivated through tracking. Unfortunately, most American educators believe that tracking exacerbates inequality, even if it means denying high-achieving minority students the opportunity and boost they need. I don't believe that tracking--when done properly--increases inequality or makes students feel bad. I have been hearing these excuses for over 40 years. Even 1st-grade students know which kids are brainy. It is absurd to deny better students algebra in middle school or fast track math in elementary school.

We need to cultivate talent at very young ages. The American excuse has been that practice to cultivate talent takes away from childhood. It's baloney. Also, the cultivation of young talent requires exceptional instruction. However, high-achieving, mathy kids are not likely to get the level of math instruction needed in elementary schools and many middle schools. Furthermore, students who do not take a high-quality, traditional Algebra-1 course in middle school, as defined by the National Mathematics Advisory Panel (2008), will likely be locked out of advanced math and science courses, such as algebra-based physics, in high school. Math talent needs to be identified, trained, and developed early on. If we can develop talent in music at a very young age, then we can do it in mathematics, too. 

Some school districts are making stupid decisions based on Common Core’s one-size-fits-all mantra. One is kicking Algebra-1 out of middle schools, such as in the San Francisco Unified School District, according to Ana Tintocalis of KQED News. The SFUSD justifies its decision for two unverified reasons. One is that Common Core pushes algebra to 9th grade, which is a ruse. Common Core allows content to move up and down. The second is that tracking students, based on a student’s achievement in math, is wrong and a matter of social justice. No, the reason is that tracking doesn’t fit the entrenched progressive tenet (ideology) of “sameness,” which, Thomas Sowell (Dismantling America, 2010) has described as an illusion of fairness.

Inclusion policies lower those at the top.
Tracking students is not wrong! Not tracking students is equivalent to “equalizing downward by lowering those at the top,” which is a twisted tactic in many public schools and a “crazy idea taught in schools of education,” writes Thomas Sowell (Dismantling America, 2010). Tom Loveless (2016 Brown Center Report on American Education) writes, “Recent research indicates that high-achieving students may benefit from tracking…. Tracking is significantly correlated with performance on AP tests, which holds true for black, Hispanic, and white subgroups.” 

Absurdity Reigns
It is absurd to deny better students algebra in middle school or fast track math in elementary school. The idea arises from the bent mentality of a one-size-fits-all dogma in Common Core and state standards. Both high achievers in math and low achievers are tossed together in elementary school classrooms (inclusion policy). The same goes for middle school math classes. All the kids, regardless of ability and achievement, get the same math content and work in groups (inclusion). In 9th grade, both high achievers and low achievers are in the same algebra class. Furthermore, in Common Core classrooms, students don’t need to drill-for-skill to learn math, which is idiocy raised to the nth power, if that was possible. The best math kids, starting in 1st grade, need a separate curriculum (acceleration) taught by an algebra teacher to cultivate their ability, which is malleable. 

Opinion (Real World Problems)
The math reformists say that Common Core teaches math differently. The new math approach requires discussion of so-called real-world problems in small groups (inclusion) and reasoning that leads to deeper understanding. For example, students voice their opinions when comparing two advertisements from competitors (aka real-world problems) or the pros and cons of social issues. Students argue and defend their views based on the numbers, percentages, probabilities, averages, charts, graphs, and text of the ads (the given information). But, students have no way to figure out the correctness of the claims suggested in the ads. They don't have enough information. All ads are bent and misleading. Bias is everywhere, and it is easy to lie with statistics! "Data analysis is rarely simple and straightforward. It may be possible to draw more than one conclusion," writes Sherry Seethaler (Lies, Damned Lies, and Science, 2009). Furthermore, children do not know how to do costs benefits analysis. 

Given a real-world problem in school, children are not expected to produce information (data) to find patterns that will help them make better choices and reduce risk. In fact, K-8 children probably do not understand risk-reward covariance. They don't realize that the possible choices are often false dichotomies. In my opinion, the so-called deep understanding that allegedly comes from the discussion in groups is merely juvenescent opinion and shallow thinking. In many cases, so-called real-world problems are oversimplified, superficial, and not the real world at all. Perhaps, it is better to teach kids personal finance and basic economics. 

Moreover, many important decisions youth and adults make are subjective. Real decision making deals with uncertainty. There are risks. Often, we downplay or ignore the risks. Which college should I attend? Which major should I select? If I major in x, will I be able to find a job when I have a degree? Will I be able to pay off student loans? Should I get a degree online or attend a traditional university?  Which smartphone should I buy? Which carrier is the best? Which car should I buy? Should I buy a used car or a new car? Should I pay cash or finance the car? Should I finance a new TV and furniture? Should I rent an apartment or buy a house? Is product x better than product y? 

Incidentally, math, itself, is not a matter of opinion; it is a matter of fact. 

Real World

The reality is that most high school graduates are not college-ready. And, there is no evidence that Common Core’s one-size-fits-all approach will magically make them college-ready, not even for community college. 

©2016 LT/  
First Draft, To Be Revised
Please excuse typos and errors.