Tuesday, February 23, 2016


Dolciani, Brown, et al.
©1981 HMC
As an educator, I have been fighting bad ideas for decades. One is constructivist-based reform math. Another is the unwarranted assault on algebra and math in general (anti-knowledge). Still, another is the Common Core's one-size-fits-all dogma. There are more, such as minimal guidance during instruction, group work, calculator use, trendy fads, technology hype, Common Core's equating of college-ready with career-ready, unneeded yearly standardized testing, NCLB and its successor, the new science standards that put process over content, and so on.

The one-size-fits-all idea is an inefficient, backward recipe for teaching kids mathematics. Common Core (i.e., state standards) and its testing baggage, along with technology hype, untested fads, evidence-lacking math reforms, and inefficient minimal guidance methods during instruction, are not the solutions for our math woes. The world economy is flat. It is in neutral. Likewise, US education is neutral; it has been flat (middling) for decades.  There should be no excuse for very slow growth. 

American schooling (i.e., curriculum and instruction), along with mandates, policies, fads, and just plain wrong ideas, has not leapfrogged our students from the mediocre levels compared to some other nations. Also, it has been an embarrassment and a disgrace that the US ignores many of its future math and science students in urban areas. (We need these kids!) American schooling is locked in neutral. It often squanders future talent that is required for vital STEM areas.

"Illusion of Fairness"

Bad Ideas Flourish in Education
College-Ready = Career-Ready (Achieve --> Common Core)

In 2008, Achieve laid the framework for what was to become Common Core in a document called Math Works.

College-Ready is the same as Career-Ready in the one-size-fits-all scheme of Achieve and Common Core. The state standards, which were based primarily on Common Core with some adjustments, are the same for all studentsThus, under Common Core's one-size-fits-all ideology and advanced math keystone, it doesn't matter whether the student attends a 4-year university, a 2-year community college, a vocational school, or goes straight into the workforce without a postsecondary training. All students get the same, a misguided idea that started with Achieve.

The maneuver is truly a stupid idea. The stratagem can be traced to Achieve, special interests (technology, financial, etc.), elites, and other influential people (e.g., Coleman, Gates, Duncan, et al.) who wanted to redesign schooling to fit their skewed vision without public input. The backbone of the skewed vision was that all students needed advanced math. In fact, Common Core equated advanced math to career readiness, an imprudent, shortsighted idea. In the real world, all students don't need advanced mathematics, but some do. In my opinion, students should not be required to learn advanced math to get a high school diploma. Unfortunately, many students have not been given an option. Algebra 2 is a high school graduation requirement in Arizona and other states. Uniformity and other progressive "illusions of fairness" have trumped common sense and distorted or delayed the education of our youth. (Note. For more on "illusions of fairness," refer to Thomas Sowell's Dismantling America.)

1. Low-Achieving Students
 Most high school seniors are not ready for community college, but we have known this for decades. The K-12 math curriculum (reform math) and its entrenched teaching methods are mostly to blame. Beginning in the 1st grade, students need to master standard arithmetic, not reform math, to prepare for algebra in middle school. Calculators should be banned. From the get-go, young students must memorize facts, learn place value, perform standard algorithms, and apply the arithmetic rules for operations and equations. For decades, American math programs, especially reform math programs, have failed our students. Many students are low-achievers because we have made them that way. Kids are novices and need to drill for skill.

2. High-Achieving Students
The best students in urban elementary and middle schools are often neglected and grossly underfunded. We have wasted talent by not developing it at all levels. In fact, many high achievers have not been identified, yet they are in almost every classroom. The best math students, 1st through 5th grade, need a different (accelerated) math curriculum via a pull-out program that is taught by an algebra teacher. These students can learn much more math and learn it faster than typical students, but they are often held back by pointless "sameness" policies, in particular through the grouping of students of various achievement levels and abilities into the same math class (inclusion). Often talent and ability are squandered in this system. Ordinary classroom teachers and so-called "math educators" from progressive schools of education are not prepared to teach these kids. The idea that the better students can fend for themselves in an inclusive environment is idiotic!   LT 2-26-16

(Note. Please excuse typos and other errors on this page, including grammar. The order of the topics is randomThe post is in the first-draft form.)

I partially agree with Andrew Hacker (The Math Myth, 2016) that the one-size-fits-all approach, rooted in Common Core and state standards, needs to be booted. Furthermore, students should not be required to pass Algebra-2, which is the case in many states, to get a high school diploma. There should be an alternative path for the students who do not plan to go to community college and who want to enter the workforce immediately after high school graduation. That said, I have misgivings about Harker's workaround solution of statistics and quantitative reasoning that is independent of math. There is no evidence to support it. Likewise, "Algebra for All" and similar foolish mantras do not work in the real world. "Laptops for All" (the technology mantra) and other fads will not change the narrative that 2/3 of high school graduates are not ready for college because of math deficiencies.

Therefore, I propose at least three different levels of  math and science curricula for students who intend to 
(1) attend a community college (associate's degree), 
(2) attend a four-year university (bachelor's degree), and 
(3) go into the workforce or get a certificate from a vocational training school, and then go into the workforce. 
The proposal leads to awarding different levels of high school diplomas, such as the three degrees in Texas. Texas was one of a few states that had not adopted Common Core.

Algebra is good for you! When was the last time you heard that? Indeed, algebra is powerful and fundamental mathematics and should be studied and celebrated. If you have learned standard arithmetic well, then you can learn algebra. Be sure to take an algebra-based physics course and learn to apply algebra to the physical world and to unlock the secrets of the universe. WOW! Algebra is also an excellent problem-solving tool. It teaches students to solve problems in a structured, organized, step-by-step manner by breaking down a large problem into smaller components. It teaches students to write and solve equations that model the real world. Also, Learning algebra is an essential foundation for learning higher-level math, such as trig, precalculus, and calculus.

Learning algebra well should not depend on a graphing calculator. Unfortunately, the main reason students struggle with algebra is that K-6 standard arithmetic was replaced by reform constructivist math, which had promoted the use of calculators as early as kindergarten and trivialized the mastery of standard arithmetic. Reform math via Common Core and state standards remains popular today. Unfortunately, education has a history of bad ideas; evidence-lacking reforms, fixes, and fads; and multiple theories that don't work. David Hume once wrote, "A wise man proportions his belief to the evidence." Sometimes, the evidence doesn't matter much in education circles. Schools often adopt programs, technologies, and trending fads without substantial proof that they work. Most don't work as expected! 

Math & Science Scores Count!

What we ignore in education is that school quality eventually impacts economic growth! Students must accumulate knowledge and skills. Eric A. Hanushek says that the cognitive skills of the labor force can be measured by math and science scores. He writes, "The relationship between math and science test scores and growth is extraordinarily important." In schooling, we have been substituting quantity for quality. We need quality, which is harder to reach. Kids don't need more years of low-quality schooling, says Hanushek. Low-quality schooling has led to the community college conflict. Hanushek points out, "Many American students arrive at college unprepared for the coursework ahead of them and, therefore, have to take non-credit remedial classes." 

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The Community College Problem is a K-12 Problem 
Nearly 68% of incoming students to a community college are underprepared and not college ready, but it is not the fault of algebra or the community college placement tests. The root of the problem, which is K-12 math, has not been addressed as the solution. IQ is never mentioned. Unfortunately, the remedial programs at the community college level have not worked well. "How can community colleges increase the completion rate?" may be the wrong question. There is a host of reasons for students not finishing degree programs. The primary factors of dropping out of or delaying community college have been cost, balancing a working schedule (job) with college classes, personal problems that pop up, and inadequate preparation (readiness). The main reason that students drop out of college is the escalating cost. More students are enrolling in community colleges than ever before, and 2/3 of them are not college-ready.  Also, capable students need to take tougher courses in high school, especially in mathematics and science. In Delaware, for example, two of the recommendations to reduce remediation rates in college put the spotlight on high school:
(1) school districts need to reduce remediation rates by providing targeted interventions before the 11th grade and 
(2) school districts need to prepare students who enter the 12th grade ready to succeed in either precalculus or calculus. 

The problem is that not all students need advanced math. Usually, Algebra 1 (called Elementary Algebra in community college) is all that is needed, if that. Also, students need to review and practice for the math placement test. The math students should have mastered must be linked to the math that is needed in the student's chosen career path to earn an associate's degree and certificate at a community college. In my opinion, Algebra 1 is usually enough to start, but it is not enough for STEM. STEM programs require advanced math (Algebra 2 or above), and they are the most competitive to get in. Put simply, the specific field of study should determine the math requirements. I would like to see an Algebra 1 placement test as the base at the community college associate's degree level, even for nonmathematical careers. An Algebra 2 or advanced math placement test should be available for those in the STEM areas. If a student needs advanced math later on because of changes in his career path, then he can take it down the road. Students who plan to transfer credits from the community college to a university must abide by the university's placement tests. (Note. An art school issues a bachelor's degree in landscaping. I told the curriculum director that students need trig, but they don't need a STEM trig course. They need a trig course that relates directly to their job. The course can be taken concurrently within their regular curriculum.)

In Arizona and most states, high schools often dump underprepared students into community colleges. Consequently, some community colleges have attempted to replace college algebra, a workaround, with excellent sounding courses, such as statistics and quantitative reasoning, to meet the vague criteria, which is a band-aid solution so that underprepared students can move into credit courses more quickly. Put simply, community colleges are passing the buck just as the public schools had done. The disconnect between high school and community college didn't happen overnight; it had taken decades of bad instruction and fads. Today, 86% of incoming students think they are academically prepared for community college, but a whopping 68% of them are not, according to the Expectations 2016 report. In short, the student's feel-good perceptions finally have collided head-on with cold reality. The idea that "one can be anything one wants to be" has become a cruel joke for some students. I think students can prepare for placement tests, but very few did because they had done very well in high school (grade inflation). Students took low-level, mislabeled "college-prep" courses in high school and got high marks, but the so-called college-prep courses were in name only, not in content. The students had been bamboozled.  High schools must tell students the truth about their progress or lack thereof and implement proper interventions long before 11th grade.  

Suggestions for Community College Students
You failed the math placement test. It is not a catastrophe. Avoid the remedial math trap [pre-algebra, elementary algebra (Algebra 1), intermediate algebra (Algebra 2) sequence] found at many community colleges. Instead, your goal should be to test out of remedial studies to start degree courses. In short, you need to study for the math placement test! Change your attitude, take time off, get a job to pay for a tutor, get a practice book to guide you (e.g., Math for the Accuplacer by Bob Miller), find practice tests online, study, study, study, practice-practice-practice (drill for skill). A good tutor can help you over the rough spots. Don't spend money you don't have. 
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Algebra has not caused the college dropout problem, but Andrew Hacker (The Math Myth, 2016) seems to think that it has contributed significantly to the dropout problem and believes that algebra has been a well-entrenched irrational barrier for many students. Unfortunately, Hacker left out a huge package of interrelated, multivariable factors that accounts for dropouts, so his conclusions are suspect, misleading, or bent. Harker was a professor of mathematics according to the inside cover of his book (The Math Myth, 2016); however, he did not have any degrees in mathematics. On page 4, Harker writes, "I'm not a mathematician in that I have no degrees in the discipline." What credible mathematics department at an institution of higher learning would award Hacker a professorship in mathematics without at least a Ph.D. in mathematics? Hacker had no math degree or credentials for such a position. Even though I find some of Harker's ideas intriguing, his solution to replace higher-level math with statistics and quantitative reasoning lack evidence. The root of the algebra problem is that kids don't know standard arithmetic and have been weaned on calculators. Arithmetic isn't taught well in the US. It is fixable, however. Don't blame algebra. Blame reform math taught in our K-12 schools and those who promote it.

Let me put this in perspective. The primary factors of dropping out of college are cost, balancing a working schedule (job) with college classes, personal problems that pop up, and inadequate preparation. The main reason that students drop out of college is the escalating cost. Most students take six years to graduate with a bachelor's degree rather than four years at a public university. [Incidentally, in 1980, the number one reason for dropouts was failing grades (College Board), not cost.] These same factors influence community college dropouts. It often takes more than two years to complete an associate's degree. Many students are placed in remedial math due to the poor teaching of standard math in K-12. Indeed, remedial math slows progress to completing a degree, but it is not the catastrophe as Hacker and others have asserted. There are many factors or variables that slow degree completion other than remedial classes in math, writing, and reading. Perhaps, the most consequential variable is financial.

One of Hacker's main point has been based on the utility of algebra. Hardly anyone uses quadratic equations in their coursework or profession. No kidding! Hacker's replacement for algebra, especially in the nonmathematical areas, is called quantitative reasoning, such as calculating and comparing mortgages in ads, car deals, etc. Our product is better than others. BUY IT! No kidding! Indeed, almost all of us rely on data analysis and quantitative reasoning done by other people and purported experts. We seem to trust the viewpoint of others, even though those people may have vested interests, biased agendas, and hidden motives; e.g., retail sales of mutual funds that have high fees and commissions. We tend to overlook the law of averages and are susceptible to confirmation bias and anecdotal (evidence).  In short, the average consumer doesn't think much. Also, it is difficult for ordinary people to weed through conflicting data from statistical studies. Coffee is good for you. Coffee is not good for you. Moreover, correlation is often implied as causation. Indeed, lying with statistics is easy. 

Charles Wheelan (Naked Statistics, 2103) points out, "Although the field of statistics is rooted in mathematics, and mathematics is exact, the use of statistics to describe complex phenomena is not exact." There is plenty of wiggle room, he says. Averages are uncertain numbers. Also, Sam L. Savage, a  professor of statistics, writes that nearly half of his graduate students often get basic statistical ideas wrong. He reminds us that a "statistical model is not the truth. It is a lie to help you get your point across." If a student compares two ads with charts, graphs, tables, and other information to find a good deal and concludes that the 2nd ad is the better deal, then how does the student know that it is the better deal? He doesn't. Also, the likelihood of typical high school or college student learning statistics well is far-fetched and contrived. In contrast, well-prepared students can learn algebra to get into college and avoid remedial math. (Aside. In math, you learn that something that is infinite can add up to something that that is finite. How can that be?  It is called convergence, a calculus idea; however, the concept of convergence can be taught using guess and check in long division.)

We often base our beliefs, decisions, choices, and judgments on anecdotal evidence, the flaw of averages, incorrect assumptions, fads, trends, and junk science. We are not good at logic (computers are) and frequently engage in confirmation bias. Taking a course in statistics or quantitative reasoning, as a substitute for Algebra or higher math is not going to change the narrative (no evidence). In my opinion, a quantitative reasoning course might make you more aware of the issues, but not necessarily more competent. As an adult with a busy family life, you will not likely pull out your old college textbook on quantitative reasoning and calculator. Also, I have misgivings about reasoning that is independent of math knowledge and skills, which seems to be the theme in quant-reasoning.

In short, Hacker argues that we should not require subjects or topics that would not be useful in everyday life or a chosen vocation. Hacker pounces on algebra as the main culprit that stops students in their tracks and wipes out their dreams. Really? He is biased. [Algebra is good for you. Take an algebra-based physics course in high school to learn to apply algebra and understand how the universe works. WOW!]

Perhaps, the main reasons that students do not advance or earn a degree at a community college are financial, poor study habits and attitudes, and not being college-ready academically in math, reading, and writing (preparation). On the other hand, Harker writes, "In no way is this book anti-mathematics," but he gives conditions. Likewise, he writes, "I want to affirm that basic algebra is necessary for everyone. I use it every day myself." But how does he use algebra? Hacker's example below is very elementary (school).

Hacker, who is a professor of political science, is not a mathematician and does not have a degree in mathematics at any level. Also, political science is not science. Hacker claims he uses algebra all the time as a social scientist, but his example is a simple proportion and solving for x (Hacker's example: 20 is to 13 as 35 is to x). Furthermore, he claims it is elementary algebra and solving for x, but simple proportions are an essential part of an elementary school arithmetic curriculum. Also, I have taught trig ratios, linear equations, and ratio/proportion equations to 4th- and 5th-grade students who were well prepared in standard arithmetic. (Note. Using statistics doesn't make you a scientist. Moreover, data can usually be interpreted in more than one way.)

Harker argues that most people don't use algebra in their vocation, so why study algebra or higher math beyond algebra? If I were to use this kind of bent logic, then the same would be said of poetry and literature, languages such as Latin or Chinese, music, art, physics, chemistry, history, and so forth. Why study any of these disciplines? Good grief, what a dull world it would be without the broad spectrum of the liberal arts, which includes math, science, the humanities, and the fine arts.

Real World: Students who want to attend a community college to get a two-year degree must do well in algebra, even if the student has selected a nonmathematical field. If incoming students are placed in remedial algebra classes at a community college, then the standard math in K-12 has not been taught well. Furthermore, the student's attitude, effort, and study habits can play a crucial role, too.

In fact, students are not asked to learn algebra for its utility or to be creative. Indeed, Algebra is a stepping stone to higher math and opens the way the world and universe works, but it also teaches students to solve problems in a structured, organized, step-by-step manner by breaking down a large problem into smaller parts. It teaches students how to write equations and solve them based on a problem (modeling). Algebra does not prevent students from discovering their talent, as Harker implies, and it is not primarily a vehicle for students to find their creative niche in life, although, for some, it is. Learning factual and procedural knowledge does not inhibit problem-solving in mathematics. In fact, it enables and enhances problem-solving.

Vocational Training: If you had wanted to become a cosmetologist, then you should graduate from high school, earn a post-secondary certificate from a cosmetology vocational training school to start your "dream job" in 12 months, but you will have to know some chemistry and pay off student loans.

Many students and adults do not like math because it is abstract, which makes it harder to understand than other subjects, yet, it is a powerful thinking tool that unlocks the secrets of the universe and how the world works. However, one cannot do much mathematical thinking without a mastery of factual and procedural knowledge (i.e., prerequisite content in long-term memory). Being able to do math makes you smart. Remember this: Future employers want to hire young people who are smart, who have demonstrated their knowledge and skills in solving problems, and who are willing to work hard within the organization.

On the one hand, Hacker says that all kids should take algebra in high school, which I think is a good idea, but then he says that algebra is the number one academic reason for college dropouts. Hacker acknowledged that there are many reasons why kids drop out of community college, but his sharp criticism zoomed in on algebra. I disagree with his conclusion. He seems to cherry-pick studies and use anecdotal accounts and testimonials that confirm his bias and agenda. (Note. Hacker provides no valid evidence that students will become more literate or competent in math using a quantitative reasoning approach, which is his alternative to algebra.)

The fundamental problem is that kids have been graduating from high school with a weak background in math. Incoming community college students were taught NCTM reform math using calculators and minimal guidance methods during instruction. Consequently, arithmetic and algebra had not been taught well in schools for decades under the reform math agenda. And, it continues today via Common Core and state standards. Many kids coming into community college don't have the prerequisites for math, reading, and writing. It is a clear signal that something is terribly wrong in K-12 math instruction.

Hitting a wall at algebra in community college is not caused by the algebra. It had originated from a lack of prerequisite knowledge and skills in K-12. Put simply, many high-school students are not prepared to take algebra at a community college, but it is not the fault of algebra. It is the fault of how math had been taught in the US. While most kids are not "mathy," it does not mean that those typical students cannot learn arithmetic and algebra at an acceptable level, but it necessitates an attitude change, memorization, effort, practice, persistence, and good teaching.

Hacker suggests alternatives to algebra, such as statistics and quantitative reasoning, but he defines quantitative reasoning as being independent of math content. It doesn't work. (Again, no evidence.)

Substituting statistics or quantitative reasoning for knowledge and skills in math lacks evidence.
"What I need is real algebra content to think about! It is pretty hard to understand algebra without doing algebra. I need problem sets to practice." The same holds for standard arithmetic!

The so-called quantitative reasoning approach is mostly calculator-based, not knowledge-based. Quantitative reasoning is often an opinion about issues and frequently involves learning a keying sequence on a calculator, let's say, to calculate compound interest or credit card debt. However, according to the cognitive science of learning, reasoning without sufficient content knowledge in long-term memory doesn't work. Also, to truly understand statistics, calculus is needed. It is easy to lie with statistics. The quantitative reasoning is reasoning independent of knowing the math. In short, one does not have to know much actual math to do quantitative reasoning, just key the right sequences into a calculator and make conclusions. Really? There is little valid evidence that quantitative reasoning makes a student more competent numerically. There is little evidence that the conclusions drawn are nothing more than contrived opinions.

In my opinion, Hacker cherry-picked scientists who don't use calculus that much. (Aside. See, even scientists don't use calculus!) Did he ask a physicist or chemist? Apparently, not. At the same time, he says that math is not a liberal arts subject, but he is flat wrong. Really? In fact, math and science are fundamental disciplines of the liberal arts curriculum at a good university, together with the humanities and arts.

Hacker writes, “All of us should get arithmetic under our belts in elementary school [K-5].” He is correct; however, the current reform math methods of instruction inhibit or delay the mastery of standard arithmetic. Hacker does not mention the reform math catastrophe. For decades, it has been the biggest problem in math education even to this day. Most kids are taught reform math and do not master standard arithmetic in K-5. Also, Hacker says that arithmetic is addition, subtraction, multiplication, division, followed by fractions, decimals, percentages, and ratios, and the statistics we encounter in our everyday lives.” Again, he is correct. He asserts that “arithmetic is all that’s needed to interpret charts in the Wall Street Journal or graphs in The Economist, as well as public documents and corporate reports.” He calls it adult arithmetic. There are flaws in the approach. First, there is no evidence to support it.

Hacker thinks the one-size-fits-all model is wrong. I agree! While some kids don't need to learn trig or precalculus, others do. He says that Common Core (along with ACT and SAT) inflicts precalculus on students, yet there are no precalculus standards in Common Core. I agree that there should be alternatives to higher-level-math for students who plan to enter the workforce after high school. Arizona and other states now require Algebra-2 (or its equivalent) for high school graduation, which is another bad idea in a series of bad ideas implemented over the years.

The quantitative reasoning approach to mathematics is firmly in the constructivist reform math camp, which has diminished the importance of mathematical knowledge and skills in long-term memory for problem-solving and understanding. The approach is an inadequate substitute for problem-solving and conceptual understanding. The idea that students can do quantitative reasoning without knowing much arithmetic or algebra and make the "right" decision is naive.  Students are asked to apply the math they don't know well or can't do and to work with equations and concepts they don't understand. Even when taught, most students (and adults) do not understand exponential equationsStudents often learn calculator keying sequences of formulas (e.g., continuous compound interest), but learning a keying sequence is not the same as learning to solve an exponential equation to find an answer to a problem. Mathematics is not about opinion; it is about facts. Facts are needed to empower thinking skills. Quantitative reasoning, it seems, is more issue-centered and often yields to someone's opinion, viewpoint, or choice with some numbers (statistics), charts, and graphs  (visuals) tossed in for believability. Math and quantitative reasoning are not coequal, as Harker implies. Moreover, it is easy to lie with statistics, opinion polls, etc. To advance, however, students must learn mathematics starting with standard arithmetic in 1st grade. In some sense, quantitative reasoning might be called critical thinking, but critical thinking is difficult to measure.

Higher Math As a Requirement for a High School Diploma Is Another Dumb Idea.

Applying the natural log (ln)

In college courses of quantitative reasoning, students are not required to solve exponential equations or use equation solving techniques such as "applying the natural log to both sides," so they do not develop a firm understanding of exponential equations and their functioning in these courses. The exponential equation (left) is from a high school Algebra-2 course, which has about the same content as a College Algebra course. Nevertheless, the content covered often varies in College Algebra and Algebra-2 courses.

Being able to solve this equation should not be a requirement for regular high school graduation. Likewise, the SAT or ACT should not be the exit test in high school, but it is in some states. Special interest groups win again! In short, Algebra-2 should not be a required course for a high school diploma, but it is in Arizona and other states. It is a dumb idea! Incidentally, the calculation step (ln 18 + 5)/3, which is the last step of the procedure to solve the equation, had been done on a slide rule. Today, the calculation is done on a calculator. (Note. ln18 ≈ 2.890, and then add 5 and divide by 3 to get x ≈ 2.630.)

Here is an algebra equation from a college-level text on quantitative reasoning: x - 4 = 6, an equation type I used to teach to 1st graders to solve by adding 4 to each side. Can you believe it? Another example is solve the equation 7w = 3s + 5 for s. Indeed, students should be able to solve formulas for a specific variable, but why not use existing formulas such as d =(1/2) gt^2, solve for t, from the free fall equation?

Quantitative Reasoning math courses are for students who majored in nonmathematical fields, such as education, fine arts, humanities, etc. It may not be a viable alternative. Even if nonmathematical majors had taken a quantitative reasoning math course in college, as adults, they are not going to reach for their calculators and old college textbooks to figure out their credit card debt, inflation models, mortgage payments, or write a spreadsheet to calculate their income tax or to track investments. It makes no sense! Also, Teachers need to learn to teach standard arithmetic, not to take courses in quant-reasoning.

In other words, almost all of us, not just those in the nonmathematical fields, depend on (trust) the word of others; e.g., the bank, credit card company, an insurance company, government data, etc. Often, experts use flawed averages and make incorrect assumptions, etc. It is scary!  For example, the Long-Term Capital Management (LTCM), a large hedge fund partly founded by Merton and Scholes, who were 1997 Nobel Prize winners in Economics for their new method to determine the value of derivatives, decided to leverage its investments in "neutral" derivatives, not realizing that these would increase risk. They weren't neutral. Enter the Black Swan.

Richard Feynman at CalTech:
 If it disagrees with experiment,
 it's wrong."

LTCM started to run into trouble in 1998 and later crashed into a "worldwide financial crisis." How could experts mess up so badly? Of course, "depending on others," even the so-called financial or economic experts or pundits, is not sound quantitative reasoning. Real science teaches us to be skeptical. Scientific minds search for valid evidence, but, too often, the evidence is absent or scant.

From the early 2000s on, the housing bubble built up. Enter the Black Swan. Government policies by trustworthy people created much of the mess. Also, the economic model or theory used did not account for a sudden decline in home prices. (Housing prices can't go down, the experts said!) The theory or model didn't work in the real world. The experts thought it would, and it brought down the house, literally. It is an example of well-educated and well-intentioned people had made stupid decisions that nearly gutted the middle class and its assets. In contrast, real scientists test theories to make sure they work in the real world. If they don't agree with experiment, then they are wrong, says Richard Feynman. (Note. The real-world physics experiments don't destroy peoples' lives while the economic experiments pushed on our society through government monetary policies have.) Unfortunately, the classroom has become an experimental ground zero for education reformists, government policies, elites, and special interests. ("It sounds like a good idea, so let's try it to see what happens" attitude permeates education. Most ideas (often called innovations) are fads that don't work because there is no supportive evidence.)

What separates real science from the rest is the replication of results. Scientific people do not accept results if other scientists cannot replicate them. Unfortunately, the scientific skepticism, repeatability, and logic do not carry over to education. "Science does not prove anything true--all it does is get rid of false views and, thusly, gets closer to the truth," writes Nigel Warburton of Karl Popper's falsification idea (A Little History of Philosophy, 2011). There is nothing like this in education.

Thinking is hard work, takes effort, and requires sufficient knowledge in long-term memory, says Daniel T. Willingham, a cognitive scientist. It is our nature to avoid thinking as much as we can. And, so we do. We often depend on the viewpoints of others. Also, we believe things that others believe, and we believe attractive people, celebrities, and influential people, says Willingham. We let politicians dictate education policy. None of these people are educators or experts. Likewise, we must be skeptical of claims that one curriculum is better than another. Whenever I hear an education program is research-based, I run the other way. In data analysis, it is usually possible to interpret the data in different ways. Mortgage A is better than Mortgage B. My banker showed me figures. Really? Your banker wants a commission for selling you a mortgage so be skeptical.

Richard E. Nisbett (Mindware - Tools For Smart Thinking, 2016) nails it: "I compared people's reasoning to scientific, statistical, and logical standards and found large classes of judgments to be systematically mistaken. Inferences frequently violate principles statistic, economic, logic, and basic scientific methodology . . . .  Many of our beliefs are often fundamentally flawed or sorely mistaken
. . . .  We are overly influenced by anecdotal evidence . . . . We operate as if we thought the law of large numbers also applied to small numbers." In inference making, Nisbett, himself, says that he sometimes violates these principles. It's the way our brains work. Note. Most studies in education lack quality and repeatability.

Daniel T. Willingham (When Can You Trust The Experts, 2012) writes, "We have reasonable good tests to measure content knowledge. We don't have good tests to measure students' analytical abilities, creativity, enthusiasm, wisdom, attitudes toward learning . . . . If someone approaches you with a curriculum that he claims will boost kids' creativity, you might ask yourself how he knows whether or not it works . . . . Critical thinking is so difficult to measure."

Let me close with a quote from Jordan Ellenberg (How Not To Be Wrong - The Power of Mathematical Thinking): "It is pretty hard to understand mathematics without doing some mathematics." Quantitative reasoning is not the same as "doing" mathematics.

This post is a work in progress. Some changes have already been made. 2-23-16, 2-25-16, 2-28-16
Kailey, 6th grade

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Friday, February 5, 2016

Standard Arithmetic Curriculum

ThinkAlgebra strongly recommends the teaching of a "standard arithmetic" curriculum that focuses on the critical foundations of algebra as stated by the National Mathematics Advisory Panel (2008), not a version of NCTM "reform math"--via Common Core or state standards--that does not. Students need a standard arithmetic curriculum to advance, not a reform math curriculum.

The teaching of standard arithmetic isn't going back to something that did not work in the past; it is boosting and revitalizing something that did work. Indeed, content knowledge in arithmetic and algebra (aka skills, ideas, uses) is the foundation for doing the math. Knowledge in long-term memory enables clear thinking, problem-solving, and creating.

The "understanding" of math is in the "doing" of math. For example, in 1st grade, students begin to understand the idea of addition (and perimeter) by calculating the perimeters of squares, rectangles, and other polygons using memorized facts and the efficient procedures of standard arithmetic. The number line is helpful at first. 
Elementary students also need science, geography, music,
art, history, etc.
But students need more than knowledge of standard mathematics; they also need knowledge of history, geography, science, art, and music--subjects that are often shortchanged in elementary school. Moreover, students need more than mere exposure to these subjects; they need knowledge in long-term memory. Understanding in reading requires broad knowledge in long-term memory. Reading comprehension, vocabulary development, and reading-to-learn open up the child's future. Put simply, knowledge is the key to clear thinking. 
Teaching kids "how to think" as a generalized skill has been a failed methodology in education for decades. We need to make sure students have gained sufficient domain knowledge upon which to think.  Kevin Ashton (How To Fly A Horse) writes, "Creation is a result--a place thinking may lead. Before we can know how to create, we must know how to think. Having ideas is not the same thing as being creative. Creation is execution [work], not inspiration. We are not all equally creative, just as we are not all equally gifted orators or athletes. But we can all create." The quality of one's thinking is a function of one's knowledge, and the quality of one's creativity is a function of one's thinking. The base of thinking well and, therefore, creating, is knowledge! The crux of the matter is that many educators do not think knowledge in long-term memory is that important. They are dead wrong! Downplaying knowledge is contrary to the fundamentals of the cognitive science of learning. (Read the next paragraph on reform math via Common Core.) 

Reform Math via Common-Core-Influenced State Standards
Slows the Learning of Standard Arithmetic.

SB, a journalist for TH, reports on Common Core’s use in a local school district (January 23, 2016). She writes, “Multiple strategies, versus a single algorithm, are taught. Common Core expects students to conceptually understand math. Students, for instance, are not taught rote memorization of multiplication tables. BH, the district’s PreK-12 mathematics coordinator, said students instead are taught reasoning and conceptual understanding to be fluent in multiplication facts.” Note. Reform math doesn't get kids to the level they need to be for success in algebra-1 in 8th grade or sooner. Not all kids will get there, but the students who learn math faster than others should, but most do not under a Common-Core-based curriculum. Also, in mathematics, the beginner is often overloaded cognitively with the extras of reform math (e.g., multiple strategies, making drawings, writing explanations, etc.) and with prep for mandatory testing (Every Child Succeeds Act); consequently, the student makes slow progress. Students should not waste time on the area model or strategy for division or multiplication, and most of the other strategies. In the modern classroom of group work, reform math, and test prep, there is little time for the teaching of standard arithmetic, which, for the reformers, is not a high priority. Also, other subjects are pushed aside or undervalued, such as science, geography, library time, music, history, art, etc. 

Really? Frankly, many of the so-called math reforms reported by journalist SB above are fads or practices with sparse evidence. The Common Core way sounds so inviting and appealing: Kids can reason the facts into memory. All you need is understanding. No need for rote memorization and all that old school stuff. No more drill for skill. Practice is out. It sounds too good to be true, and it is! On average, the math reforms have not worked as expected. The reason is that most math reforms have been based on anecdotal evidence and faddish ideas. That is, most reforms were not anchored in valid scientific evidence. The reformers seem to ignore the science of learning. The math reforms are unsatisfactory for teaching standard arithmetic to young children, yet they persist. They often disregard the cognitive science of learning and the major role of long-term memory in learning new ideas and problem-solving. The "multiple strategies" approach often confuses novices, increases cognitive load, and alienates parents. In the real world, students learn new stuff from old stuff they already know. Hence, knowledge in long-term memory is the key to learning more math, faster; it enables clear thinking and problem-solving (i.e., critical thinking in math). Consequently, gaining essential factual and efficient procedural knowledge in long-term memory as quickly as possible should be a primary goal of math education, but, apparently, it isn't.

The consequence: math achievement has been stalled for decades. Recent data corroborate a continuing trend of poor math achievement. The 2015 NAEP math scores for 4th and 8th-grade students are lower than in 2013. According to the latest ACT scores, most students are still ill-prepared for college. It is disappointing, but not surprising. (NCTM math reforms since the early 90s combined with NCLB test-based accountability reforms since the beginning of 2000s have led to substandard math achievement. So have fads, progressive policies of sameness, and weak teacher training.) Nothing new. Diane Ravitch writes in her blog (1-31-16), "The bad part about ESSA [the new Every Child Succeeds Act] is that it preserves the mindset of NCLB, a mindset that says that standards, testing, and accountability are the keys to student success. They are not." 
In its 2008 report, the National Mathematics Advisory Panel (NMAP) stated the Critical Foundations of AlgebraDr. William Quirk, Ph.D. in mathematics, noted that some of the critical foundations are missing from Common Core, now state standards. They included the automatic recall of single-digit math facts, the automatic execution of the standard algorithms, the proficiency with fractions, including decimals, percentages, and negative fractions, and the geometry formulas to analyze the properties of 2- and 3-dimensional shapes (perimeter, area, volume, surface area). Also, the NMAP made clear that "practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees up working memory for more complex aspects of problem-solving." That is, the NMAP acknowledged a fundamental idea of the science of learning. As expected, the main stumbling blocks in algebra are the automatic recall of number facts, the automatic execution of the standard algorithms, and the fraction equivalents and operations, including decimals and percentages. In summary, a primary reason students stumble in algebra is that they are weak in standard arithmetic.
Other Comments
Reformers believe that kids can "reason" the multiplication facts into memory. No need for rote memorization, which is Old School. The reformers are wrong, of course. It’s illogical to think this way. Rote does not preclude some level of conceptual understanding. The concept of multiplication is simple and easily shown on a number line, and the meaning of single-digit facts, such as 2 x 5, which is two sets of 5 in each set, is also uncomplicated for 2nd graders to grasp. Indeed, students should learn essential facts and step-by-step procedures by repetition, which is drill for skill.

The people who wrote the standards, policymakers, so-called “math educators,” professors from schools of education, and many others who advocate the reforms in math have little knowledge of the science of learningReadiness for an idea or topic is not a matter of age but a function of mastering prerequisites in long-term memory. Students learn new ideas that are linked to old ideas they already know. They must memorize the times tables and work with the standard algorithms to advance to the next level. They need to learn to reduce fractions to their lowest terms, add numbers with different signs, line up decimal points, and so on. In short, students must master standard arithmetic to advance to the next level, which is algebra in middle school. Reform math via Common Core doesn't get them there. 

The Common Core Way slows the learning of standard arithmetic and leads to increased cognitive load. The "multiple strategies" approach in Common Core or state standards confuses students, alienates parents, increases cognitive load, and makes learning standard arithmetic more elusive and needlessly complicated. To move forward, students must master standard arithmetic first. However, under the yoke of Common Core, students' minds are cluttered with multiple convoluted strategies to do simple arithmetic, which stalls achievement. In reform math, the standard algorithm has been put on the back burner, which is contrary to the science of learning and the advice of mathematicians, e.g., W. Stephen Wilson, H.H. Wu, James Milgram, and many others, like Sandra Stotsky, who was on the National Mathematics Advisory Panel (2008) and validation committee. 

The "understanding" of arithmetic is in the "doing" of arithmetic and includes skills, ideas, and uses. Indeed, to do arithmetic well, students must be competent in calculating the standard algorithms first. Moreover, students need to grasp when to add, subtract, multiply, and divide, which is pattern recognition. Also, learning standard arithmetic requires three primary ingredients: (1) competence in computational skills(2) key ideas of K-6 mathematics (rules, fractions, variables, equations, functions, negative numbers, graphs), and (3) uses of mathematics (applications in science, business, finance, economics, tech, engineering; also area, volume, perimeter, etc.). [Notes. "Understanding grows gradually," writes the late mathematician Robert B. Davis (The Madison Project, 1957). Incidentally, my Teach Kids Algebra program (TKA) focuses mostly on the fundamental ideas of elementary school mathematics. The biggest drawback is that students do not learn enough standard arithmetic in their regular classrooms. It has been a decades-old problem. Moreover, the influence of the classroom teacher has dwindled over the decades due to reforms and fads imposed on schools. Teachers have little say in curriculum or instruction. They are often told to teach to the testThey did not create the mess in education and should not be blamed for it. Put simply, bad ideas and fads are often imposed on teachers who have little say.]

In Common Core, rote learning, which is learning by repetition, has been replaced by conceptual understanding and, apparently, reasoning, both of which are difficult to pin down or measure (quantify). Indeed, how do reasoning and understanding make students "fluent" in the times tables or the standard algorithm? It's a leap of faith. They don't! The assumption from reform math is contrary to the science of learning. Furthermore, “fluent” or “fluency” in reform math via Common Core or state standards is not explicitly defined. What does fluent mean in this context?

Rote learning means learning by repetition and is a fundamental technique for learning anything. The concept of multiplication and the meaning of the multiplication facts (3 x 5 means three sets of five in each set or a total of 15) are easy to grasp when practiced well. Also, the standard algorithm always works and is easy to learn when the single-digit multiplication facts are automated in long-term memory. Learning multiplication or standard arithmetic well requires ample practice, that is, drill for skill. The notion of math reforms is that reform pedagogy and group work are much more important than "mathematical substance." The teacher doesn't teach; the teacher facilitates. Long-term memory knowledge is not that important, say the reformists. The reformers are wrong! 

Memorizing the single-digit multiplication facts (i.e., automating them in long-term memory) makes them instantly available for learning new content (factual and procedural knowledge), new concepts, and the standard algorithms. They enable problem-solving. Instead of memorizing the facts, Common Core wants kids to "reason the facts," that is, calculate them from known facts, which, presumedly, they had memorized. It is illogical. Indeed, a child cannot "reason something" without first gaining sufficient knowledge in long-term memory. Also, calculating the multiplication facts (as needed) wastes time and needlessly clutters and reduces memory space needed for problem-solving and learning new content. (Note. "Working Memory" space is very limited.) In short, the so-called Common Core way often ignores the intrinsic relationship between working memory and long-term memory and the critical importance of gaining factual and standard procedural knowledge in long-term memory to enable problem-solving, understanding, and illumination.

Indeed, it is fantasy, even bizarre to believe or postulate that those students through reasoning and understanding will acquire fluency, which is to "get" the times table into long-term memory. Consequently, most kids are not good at calculating (i.e., the skills, ideas, uses of arithmetic). Facts and efficient procedures must be automated to do arithmetic well. A visually-moderated sequence (VMS) of steps, if practiced enough, becomes automatic in long-term memory. Memory is better (and faster) than generating single-digit math facts by calculating (reasoning the facts) or using an inefficient area model to multiply or divide, and so on. 

Lastly, Richard E. Nisbett (Mindware) points out, "Although the unconscious mind [long-term memory] can compose a symphony and solve a mathematical problem that's been around for centuries, it can't multiply 173 by 19." The conscious mind [working memory] can do arithmetic, but it follows the rules automated in the unconscious mind. Nisbett points out, "The unconscious mind operates according to rules. I know the rules of multiplication, I know the number 173 and 19 are in my head, I know I must multiply 3 by 9, save the 7 and carry the 2, and so forth. I can check that what's available in my consciousness is consistent with the rules that I know to be appropriate. But none of this can be taken to mean that I am aware of the process by which multiplication is carried out." Nisbett sums up, "Don't assume that you know why you think what you think or do what you do.1-30-16 To Be Revised
Special Interests, Philanthropists, & Government Rule Education
The latest technologies, innovations, fads, and reforms have not energized rapid gain in K-12 math and science achievement.
 Moreover, they will not revive rapid economic growth either, asserts economist Robert J. Samuelson, who says that our economic growth lags behind regardless of innovation. (Economic growth was about 2% in 2015 compared to China's growth of nearly 7%.) Our education system has stalled for decades and so has economic growth. Are the two strongly correlated or is it merely a coincidence? Also, it appears that special interests (school publishers, test makers, tech companies, etc.) have highjacked control of K-12 education with the assistance of government mandates and money ($$$$$) from both government and philanthropists who think they know how to fix education. The government at all levels (federal, state, and school district) and philanthropists (e.g., Gates, Zuckerberg, et al.) are often dead wrong. Just as in government, education has become a giant bureaucracy empowered by special interests. 
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