Tuesday, September 22, 2015

Random Thoughts #4

Random Thoughts #4 (in no particular order)

Our kids are locked into a 
Common Core test culture! 

The Decline of Academics 
The Rise of Anti-Intellectualism 

In the US, sports trump academics, and jocks outclass geeks by a light year. Frequently, the decisions people make are about 30% rational and 70% emotional, says Jim Clifton, CEO Gallup. In fact, our feeling-based society often devalues individual academic achievement and academics in general. In many classrooms, excelling is not cool. "Reading is for losers. Math is for geeks," explains Greg Gutfeld (Not Cool). Scientific evidence is viewed as just another opinion, which, perhaps, partially explains why reformists seem to ignore the Science of Learning, 2015.

Also, there's a lot of talk about good schools, improving education, especially by using the latest technology, and so forth, but the narrative is mostly about good intentions and untested fads (aka innovations), such as Common Core. Feel-good education policies, fads, and notions originate from good intentions, I'm sure, but most innovations (notions, fads, etc.) fail because they lack solid evidence of effectiveness. Indeed, evidence or facts don't seem to matter much to progressive reformists who have an unrealistic passion for the new (aka innovation) and an irrational hostility for the old. Many widespread, favored classroom practices of today are not supported by evidence and are among the least effective, yet reform math people don't seen to care.

But "new" doesn't necessarily mean "better." In education, we have been spending billions and billions on innovations (reforms, fads, etc.)--especially on the latest technology--and hoping for the best. Over the decades, increasing technology use in the classroom has not turned into better student achievement. In fact, many popular and trendy reforms or fads in education are counterproductive. In contrast, some of the old stuff (i.e., old school), such as standard arithmetic and explicit instruction with worked examples are very effective when taught well. It is not enough to know some math, which is a good start; it is also important to know how to apply what you know in math. 

[Aside. This post consists of random, often contrarian thoughts in no particular order. It is in rough draft form, so please excuse typos and errors. I repeat myself, a lot. Latest additions or updates: 10-11-15]

"The best way to know if an idea is right is to see if it predicts the future," writes Steve Pinker (Harvard). The fallacy of many education policies and innovations (fads) is that they start as good intentions (feeling-based), not via the science of learning (cognitive science). We often rely on the ability of so-called experts to predict the future, but the presumed experts often make policies and claims that fall flat because they are supported by ideology or beliefs, not valid evidence.

Furthermore, most teachers were not taught the cognitive science of learning, which is an "evidence-based core of what educators should know about learning," such as the critical role of practice to push knowledge into long-term memory, or the "understanding of new ideas via examples," or the fact that children are novices and don't think like experienced adults, etc. (Quotes/Ideas: The Science of Learning, 2015)

Moreover, I am not surprised that more technology use in schools has been linked to lower test scores, according to the OECD, but this will not slow the tech stampede into our schools at an enormous cost, with little value to actual student achievement. Tech is not the silver bullet. (I include calculators, smartboards, software, etc. as technology.) Also, I cringe every time a policy or notion is advanced and publicized as "for the kids," which--when put under scrutiny--is more "for the adults or special interests." [Aside. The OECD implements PISA, an international test for 15-year olds. According to the OECD, one weakness is, "U.S. students have particular problems with mathematical literacy tasks where the students have to use the mathematics they [should] have learned in a well-founded manner." In short, too many students don't know basic arithmetic skills, such as "using the number pi in calculations." Our students are incredibly shortchanged, not only in basic math knowledge, both factual and procedural but also in being able to apply or utilize that knowledge.]

Our kids are locked in a Common Core test culture (Click). 
Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Equally wrong is expecting students to do critical thinking without sufficient background knowledge in long-term memory. Sadly, schooling has been entrenched in accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. This approach may be okay for business, but K-12 schooling is not a business, and it is not okay. 

"I am a novice, not a pint-size mathematician."
We overload the working memory of beginners with extras. 

Children learning arithmetic or algebra should not be using calculators or over-burdened with questionable and annoying extras, such as indeterminate "deep" understanding, confusing and inefficient multiple models/strategies (as stressed in reform math), unrealistic Common Core Mathematical Practices, unjustifiable group work/collaboration, far-fetched, misguided real-world problems, time-wasting discovery/inquiry activities, or paragraph writing. All these extras are from reform math.  

[Aside. The "extras" are based on adult "thinking," not in the cognitive science of learning.Students are novices. They are not experts; they are not peer math teachers; they are not writers of math; they are not little mathematicians; they are not miniature adults. "Novices and experts cannot think in all the same ways (The Science of Learning, 2015)." 

We teach reform math via Common Core instead of traditional arithmetic. Reform math people oppose standard algorithms and substitute many different, inefficient, non-standard alternatives as the primary methods of calculation. Consequently, there is little time left to spend on standard algorithms. In short, many students do not automate basic K-6 arithmetic, which is necessary for a valid algebra course.  

Children are beginners, not pint-size mathematicians.
"Don't expect novices to learn by doing what experts do," writes cognitive scientist Daniel Willingham (Why Don't Students...). The reasons are simple. Kids lack both background knowledge and experience to do anything even remotely close to what mathematicians and scientists do. Furthermore, children do not think like experienced adults. They are not little mathematicians, junior scientists, or little adults. A good example of flawed thinking--that kids should emulate what experts do--is the Standards for Mathematical Practices, which are lodged in progressive constructivism and are the silent backbone of Common Core reform math. Willingham explains, "There are significant differences between how experts and novices think." Consequently, instructing students to be creative, pint-sized mathematicians, that is, to emulate what mathematicians do, seems rather pointless. It is just another empty-headed idea that does not agree with cognitive science.  

Multiplication Facts should not be calculated as needed; they should be memorized.
In contrast to Common Core, students should first learn and practice the essentials of standard arithmetic for automaticity and solve routine problems first, not wordy, complicated word problems with extra information and certainly not far-fetched real-world problems, which are often championed by Common Core, even though students lack sufficient background knowledge and experience. For math facts, "Memory is more reliable than calculations (The Science of Learning, 2015).The multiplication math facts, for example, are implanted in the standard algorithms for multiplication and long division. Math facts should be memorized and repeatedly used over a period to stick in long term memory, not calculated as needed, which wastes time, increases errors, clutters working memory, and inhibits fluency in using standard algorithms. New knowledge builds on old knowledge. The more math content you know, the more content you can learn and the faster you can learn it, says Daniel T. Willingham.

Held Captive
In education, we are held captive to bad ideas, counterproductive reforms, and untested innovations [e.g., accountability, metrics, benchmarks, performance indicators, inclusion, NCLB, sameness, Race to the Top, Common Core, standardized testing, the 4Cs (critical thinking, collaboration, communication, and creativity), mathematical practices, etc.], and many of us, as educators, have convinced ourselves that the current reform approach (via Common Core, standardized testing, NCLB, etc.) is probably okay for kids; however, we cannot logically justify the reasons that kids get a steady diet of test prep (hence, not much education) and that Common Core reform math is the same [one size] for all students, without regard to abilities or achievement, which, in my opinion, is equalizing downward.

The road to mediocrity, decline and failure is paved with good intentions; feeling-based policies, mandates, reforms, notions, and trendy, evidence-lacking fads (often called innovations). I think, education, especially the latest vision of math curriculum and instruction, has been on this road before. I think teachers and parents can disrupt the most recent vision (reform math), which isn't new because it started in 1989 with the NCTM reform math era. Repackaging old failures as innovations seems commonplace in education. 

The Re-definition
Under Common Core, math education has undergone a "re-definition" that focuses on real-life or real-world problems; hence, it required group work and calculators early on and diminished knowledge of standard arithmetic. Indeed, Common Core's expectation is a calculator dependent and dominated math curriculum. But H. Wu, mathematician (UC-Berkeley), refutes this narrow perspective. We should not "think of mathematics exclusively as a tool for solving real-world problems." Mathematics is a complex system, an "edifice," says Wu. It is a symbolic language in which the "symbols and equations of mathematics express not just ideas but the relations between ideas," writes Leonard Mlodinow (The Upright Thinkers). Note. "re-definition" is Wu's term regarding the New Math, but it is also applicable to the reform math Era of the NCTM then and Common Core now. 

Wrong Message: "I wasn't good at math either."
Sports do not operate in isolation, says Amanda Ripley (The Smartest Kids in the World): "Combined with less rigorous material, higher rates of child poverty and lower levels of teacher selectivity and training, the glorification of sports chipped away at the academic drive among US kids." Many kids believe math is merely one of several competing options and not high up on the list. Math is more abstract and, therefore, difficult than other subjects; consequently, many students avoid math, limiting their future. Many kids believe that they will get better at reading by practicing, but not in math. "You are either good at math, or you are not," which is a counterproductive belief not supported by cognitive science. Indeed, most kids, I think, can learn standard arithmetic and algebra well, but they have to work at it (drive) and be persistent (conscientiousness). And, according to Carol Dweck (Growth Mindset), we as educators and parents need to establish a proper mindset and stop telling kids: 
(1) Not everybody is good at math. Just do your best. 
(2) That's OK; maybe math is not one of your strengths. 
(3) Don't worry, you'll get it if you keep trying.
(4) Great Effort! You tried your best.
(5) I wasn't good at math either. 
Another conundrum is that modern reform math via Common Core is typically taught, not standard arithmetic. We need to prioritize math content and streamline the curriculum so that only essentials are taught and learned to automation. Not everything is important, but standard arithmetic is.  [Aside. As computer use increases, we need skilled workers who are reliable and competent, that is, we need "workers who are smarter, better trained, and more conscientious," writes Tyler Cowen (Average Is Over). "The premium is on conscientiousness." But, this is not the narrative we are fed.]

Arithmetic That Is Arithmetic: 1912
8th Grade Exam 1912 (The Arithmetic Part), Bullitt County Schools, KY
The arithmetic taught in 1912 is harder, in my opinion than the arithmetic taught in the late 20th century under NCTM reform math standards or, today, under Common Core reform math standards.

In 1912, kids did arithmetic using standard algorithms and paper pencil. No calculators, of course. In short, they were taught to calculate quickly, recognize key problem types, and apply straightforward arithmetic to solve questions.

[Aside. Here is a mental arithmetic question for 3rd/4th graders from Ray's Intellectual Arithmetic (1877), a 140 page textbook for 3rd/4th grade combined: If 12 peaches are worth 84 apples and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches? Indeed, a 140 page textbook for two grade levels is a novel idea compared to today's 4th grade 500-page enVisionMath.

Understanding Is a Matter of Degree.
A child's understanding of something is not the same as an adult's. A 1st grader's understanding of place value is not the same as a 5th grader's, etc.  
Understanding should be inferred via a student's ability to solve arithmetic questions, that is, by doing arithmetic, and not on selecting a nonstandard, "understanding-type" algorithm, or making a drawing, or writing an explanation, which are arguable points of reform math, such as in Common Core. And, to do arithmetic well presupposes that the student knows arithmetic well through study, memorization, and practice (drill for skill). Standard arithmetic knowledge in long-term memory, both factual, conceptual, and efficient procedural, is imperative, yet, kids, today, are not always required to master standard arithmetic. For example, standard algorithms are often delayed, marginalized, or not practiced enough. Typically, what is taught under the yoke of Common Core is the latest revision of NCTM reform math--a progressive ideology of sameness or equalizing downward and an ed theory of constructivism--via inefficient minimal guidance methods, such as discovery, or project, or problem-solving learning in group work; complicated, cumbersome multiple models or strategies to do simple arithmetic; so-called real-world questions that require calculators, etc. Below is an example of a 5th-grade parents guide to Common Core.

The Common Core Brand of K-12 Reform Math
#3. Multiply 5.3 by 2.4 using the area model (5th Grade Quiz). Show our work. 
[Aside. Unfortunately, educators are told to teach multiple models [many ways] to do simple arithmetic, not the standard algorithms that are efficient, easy to learn, and always work. Standard algorithms should be taught first, not put on the back burner. The area model shown below is total nonsense. When would a student use an area model to calculate products? It is pointless, useless, and ridiculous.] 
Screenshot above from my Math Notes in 2014: http://thinkalgebra.blogspot.com/2014/07/multiplemodels.html 

The New SAT Continues the Common Core Reform Math Brand.
The new SAT (2016) is a product of Common Core. It has nearly twice as many calculator questions as non-calculator questions. The heavy use of graphing calculators reflects the overall Common Core scheme: Let's concentrate school math on solving real-world problems so that kids use calculators. Calculator use in elementary school, as early as kindergarten, dates back to the failed NCTM reform math standards of 1989. The new SAT of 2016, which locksteps to K-12 Common Core, also overemphasizes data analysis, probability, and statistics. Students must rely on TI-84 graphing calculators for these topics, especially the statistics functions. The inclusion of these topics (and others) is another tactic used by reformists to defend calculator use among young students, even if their arithmetic and algebra knowledge and skills are weak.

The NEW SAT question type (#16) is, in my opinion, a 7th-grade pre-algebra level question, not a high school level. Note. A few of my Title I fifth graders in my Teach Kids Algebra program could figure this out, too. A well-prepared 7th-grade pre-algebra student should find the answer simply by examining the graph. No calculator is needed, just knowledge. The answer has to be either C or D because both have a y-intercept at -4. A quick "rise to run" check (1 to 3) means the slope is 1/3, not 3. In short, no calculator is needed, so why is this a SAT calculator-allowed question?  I can only guess, but, apparently, under Common Core reform math, the expectation is that most high school students will not gain sufficient knowledge of algebra fundamentals to figure this out without using a graphing calculator. It This is yet another example of dumbing down the math. The NEW SAT has nearly twice as many "calculator allowed" questions as "no calculator allowed" questions. It is cause for alarm! (Question Source: Kaplan 2016 SAT)
[Late Note. Perhaps, the question is considered a nonroutine problem because you need to know stuff, such as the y-intercept, slope, linear equation form, etc. and know how to figure these out from a visual as you apply the concepts to develop an equation in y = mx + b form. Gee that is knowing and using what you know. And it starts with knowledge.]

In my opinion, this calculator problem (#16) clearly illustrates the sharp difference between knowing basic math in long-term memory (and applying it) and Common Core's expectation of a calculator-dependent-dominated math curriculum. Unlike their peers in top-performing nations, American students are lost without calculators. They don't know math. They can't do simple calculations, such as 2.54 x 1000 or -7 + (13/17) + 7 without reaching for the calculator. (Answers: 2540; the fraction 13/17) Over the years, calculators have dumbed down math content via US reform math programs. [Note. For -7 + (13/17) + 7, the calculator spits out .7647058824. What does that mean? The student didn't know that -7 and 7 are opposites (inverses) and equal zero when added via the commutative and associative rules-7 + 7 = 0 or 7 + -7 = 0.]

[Aside. Common Core, like NCTM reform math, believes that real-world problem solving should be the focal point for K-12 school mathematics. What a terrible idea! Of course, this reform ideology "justifies" the frequent use of calculators, especially graphing calculators. What is lost? Fundamental math ideas and skills (factual, conceptual, and procedural knowledge) that are not linked directly to the real world problems are marginalized. With a calculator, it is now possible to calculate problems without knowing much math in long-term memory, so why memorize, or practice, or drill for skill? It is nonsense, of course. It is not the way novices master mathematics. Reform math, in my opinion, is an anti-knowledge approach

A similar anti-knowledge approach is found at the university level. For example, in Harvard Calculus, the student can pass calculus by using a graphing calculator and without knowing much algebra, says H. Wu, a mathematician at UC-Berkeley, who is anti-reform. It is a terrible idea! Also, students can pass a College Algebra course by using a graphing calculator. (FYI: Many College Algebra courses are about the same as a good high school Algebra 2 course.) Why should we be concerned about the reform math in K-12, Harvard Calculus, AP Calculus, etc.? Our kids, including many of our best kids, are weak in math fundamentals because we made them that way under the facade of reform math, first with NCTM, and now with Common Core. Arithmetic is no longer arithmetic, algebra is no longer algebra, and calculus is no longer calculus. Like most high school math courses, AP Calculus relies heavily on graphing calculators, which is one reason that a growing number of universities and colleges don't accept AP Calculus for college credit, not even a 5. Another reason is that AP skips important content. Also, there are no proofs required in AP calculus. In short, AP is simply not up to the university level, not only in calculus but also in several other subjects. End]

Indeed, calculators have dumbed down content in mathematics. In short, the graphing calculator has replaced knowledge and skills. Our kids are weak in math, and we made them that way. H. Wu adds that under reform math, there is "a serious lack of essential technical facility—the ability  to undertake numerical and algebraic calculation with fluency and accuracy." Common Core follows the same reform trend and avoids or delays standard algorithms.

Pattern Recognition
Pattern recognition comes from the experience of doing and studying many types of math questions. "Oh, this is a percent of change problem." Understanding the problem (pattern recognition) is much more important than understanding "why" an algorithm works, and I think, this is what reform math people and Common Core fail to grasp, but G. Polya (How to Solve it) did. He lectured, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." Polya points out: "[Give students] plenty of opportunity for imitation and practice.... You learn to do problems by doing them." Pattern recognition is the key to solving math questions. Understanding a standard algorithm is ancillary because understanding develops slowly. Understanding does not produce mastery; practice does!

Students need to learn patterns for different types of problems. Learning patterns requires substantial imitation, practice, and experience. Pattern recognition is the key to problem-solving, observes Ray Kurzweil, MIT, How to Create a Mind. Moreover, kids are novices and do not need to learn multiple model stuff or do discovery lessons in groups. As Richard Feynman once exclaimed: Why do elementary school children learn or practice stuff that has little value? Furthermore, K-6 teachers are generalists and don’t know enough math content to teach standard arithmetic well (or Common Core reform math) contends Dr. H. Wu in a recent article. Additionally, parents are confused and baffled because they cannot understand the convoluted reform math or what is going on. Likewise, kids are confused, frustrated, and don't get it.

Speed, Efficiency, & the Correct Answer
Furthermore, I have always thought that the purpose of learning standard arithmetic was to solve questions as efficiently and quickly as possible, but not according to reform math (via Common Core), which stresses learning many different ways to solve the same problem, most of which are time-consuming, confusing, and more complicated than standard arithmetic. When Common Core is interpreted as reform math, the standard algorithm is merely one of many ways to solve math questions, and it is often pushed to the back burner to focus on what I call the “understanding” ways to do math, which include drawings, writing explanations, nonstandard "understanding" procedures, invented algorithms, intermediate algorithms, etc. I think most of this "understanding stuff" is math education for teachers rather than standard arithmetic for kids. We should not be training kids to be little math teachers or young mathematicians.

Unfortunately, children seldom practice for mastery the best strategies, which typically are old-school arithmetic, such as the standard algorithms. The importance of standard algorithms has been marginalized by reformers, even though the intrinsic merits and fundamental importance of automating standard algorithms for novices have been substantiated by many mathematicians, including W. Stephen Wilson, H. H. Wu, and so on. While math questions can often be solved in different ways, teachers should emphasize speed and efficiency to get the right answer. In short, novice students should learn the most efficient ways to solve arithmetic questions from the get go. Don't clutter the minds of beginners with a bunch of non-standard algorithms (many ways) or ask them to make drawings or write explanations. Instead, teach students pattern recognition of problem types and calculating using standard algorithms first and later some tricks (shortcuts).  

Unfortunately, students are often asked to make drawings, or write explanations, or use inefficient algorithms or models that purport to show their understanding, such as the area model for multiplication or the partial quotient method for a division, etc. But, I think, this should not be the reason to study arithmetic. Usually, straightforward standard arithmetic or algebra is the best strategy, but it has been marginalized in Common Core and early reform math programs. Indeed, the “many ways” for “understanding” do not stand up to the scrutiny of cognitive science or even common sense. Why make arithmetic harder than it is?

I agree to disagree. 
Wagner & Dintersmith (Most Likely to Succeed) focus on preparing kids for the innovation era as if the 20th century were not an innovation era. It is what I think: We are not producing enough home-grown talent. Our exploding tech companies are short on STEM talent; hence, for decades, major high-tech companies have imported foreign talent because many could not find enough home-grown talent. Furthermore, many high-tech companies locate branches where the talent is: Asian nations. And, while our STEM graduate schools are simply the best in the world--at least for now--they still attract a host of international students. Some stay in the US to work while a growing number return to their native roots.

I agree with Wagner & Dintersmith that the business model and standardized test approach are counterproductive and should be discontinued. But, sadly, Dintersmith repackages progressive child-centered-ideology that failed in the past—child driven discussion and child-centered assessment, etc. Lectures (explaining with worked examples) are out and replaced by a project-based approach in which teachers are facilitators, not academic leaders who know the content. Dintersmith's approach is anti-knowledge. He thinks content [knowledge] people are old school.

I agree with Dintersmith that children should master core academic content, but I do not agree with his approach, which is child-centered-project-based learning. The progressive ideology of the four Cs (critical thinking, collaboration, communication, and creativity) has displaced memorization, repetition, and practice. In short, gaining knowledge in long-term memory is not that important. Frankly, kids don’t study arithmetic because it will make them more creative or collaborative, and so on. Ordinary thinking makes them creative. Kids study arithmetic to learn it (automate it) and to use it to solve problems.

Sameness Ideology
Progressives [aka liberals] postulate that fairness and equality should dominate education, not individual achievement or excellence. Thus, in my opinion, sameness or uniformity has become the mantra of the education business, which is test-driven. Indeed, Common Core is the centerpiece of the test-based reforms, which drives both curriculum and assessment and makes lofty promises of college and career readiness without evidence. Kids are told to follow their passion in college, but many kids end up without a job,  crushing student debt and live with their parents.

Math Talent is not being developed.
"It's Your Brain That Count!" 
I am not sure we know what to do to change the culture in low-income Title 1 schools. The liberal answer has been to put more money into these schools (e.g., Title 1 funds). But, we have neglected the most important reform, which is equalizing funding. But, even if this would happen, I am skeptical that there would be leapfrog-type improvements. I am sure of one thing, however.  I have found that teaching algebra lessons in the early grades (1-5) in a low-income Title 1 school works well when I explicitly teach complex content through worked examples with practice sheets I make up. And, even though these kids had come from low-income families, I discovered a lot of math talent distributed among them. It’s there, but it just isn’t being developed through test-based accountability reforms such as Common Core’s EnableNY. I agree with Amanda Ripley, who asserts, "Poor kids could learn more than they were learning." But, I disagree with the narrative from   Diane Ravitch and others that the main problem in our schools is poverty. In Finland, Heikki Vuorinen says the opposite. He is quoted by Amanda Ripley: "Wealth doesn't mean a thing. It's your brain that counts." Vuorinen's message conflicts with the popular poverty narrative found in the US. The Finns didn't wait until poverty was cured to change their educational system.

Smart low-income kids cannot get to real Algebra 1 in middle school because Common Core reform math pushes Algebra 1 to high school. Also, CC is not set up for STEM. When I taught my algebra enrichment program (Teach Kids Algebra--TKA), I found many bright minority students, but, under the grip of Common Core in which everyone gets the same, they won't get to real Algebra 1 in middle school. No students will.

What does fluency mean in Common Core or to progressive reformists? It doesn’t mean practicing the standard algorithms for automation. To the Common Core people, fluency means to do something in many different ways, which allegedly implies a deeper understanding.  In Common Core, as I understand it, the standard algorithm is merely one of the many ways, often not the preferred way. That said, standard algorithms are not practiced for mastery. The fact that standard multiplication algorithm is pushed from 3rd to 5th grade validates the motives of the writers that the standard algorithms are just not all that important, which is a shift away from standard arithmetic. All those cognitive models, or strategies or many ways in Common Core's EngageNY scripted curriculum apparently are much more important because they allegedly demonstrate fluency and deeper understanding, even though evidence supporting such claims is lacking when placed under scrutiny. In contrast, standard arithmetic, when explained well and taught for automation (fluency), has always worked well for most kids.

Coming Soon

©2015 LT/ThinkAlgebra.org

Credits: Caitlin

Sunday, September 13, 2015

What has happened to math education in the US? 
We chase after test scores instead of credible academic achievement and continue a sameness ideology via Common Core. Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Also, we expect kids to do critical thinking without sufficient background knowledge in long-term memory, even though, according to cognitive science, factual knowledge in long-term memory must precede higher-thinking skills. 

Minimal teacher-guided instruction (such as in reform math) is ineffective and inefficient, and contributes to substandard achievement, according to researchers Kirschner, Sweller, & Clark. Kids cannot solve math problems on a near-empty math tank (long-term memory). For example, critical thinking will not help a student solve a trig problem without specific, prerequisite trig knowledge in long-term memory and practical experience working similar trig problems.

Immanuel Kant sums it up this way: "Thoughts [critical thinking] without content [solid background knowledge in long-term memory] are empty."

Inferior Methods of Instruction Dominate
The decline of arithmetic skills "is the fault of" popular minimal teacher guidance instructional methods, such as "discovery learning," explains Anna Stokke of the D. Howe Institute.  Stokke, a professor at the University of Winnipeg's department of mathematics and statistics, writes, "You know what's the worst kind of instruction? The kind of instruction that makes kids feel stupid. And that's what a lot of that discovery stuff does; their working memory gets overloaded, they're confused. That's bad instruction." Moira MacDonald summarizes the report this way: "The report puts a good deal of the blame on discovery or experimental learning approaches [aka reform math] that encourage students to explore different ways to solve math problems instead of using a single standard algorithm and often promote concrete tools such as drawing pictures, or using blocks or tiles to represent math concepts. The idea is students will gain a deeper understanding of math and be better equipped to apply it to a variety of situations [aka Reform math]. What happens is students' working memories get overwhelmed...." Cognitive overload hinders learning. 

The minimal teacher guidance methods, which are found in trendy reform math programs, and now in Common Core, don't work. Inferior (minimal guidance) instruction contributes to inferior achievement. Teachers should focus on instructional "techniques that are actually going to work," that is, stress strong teacher guidance methods (explicit instruction), explain Kirschner, Sweller, & Clark, and should teach the standard algorithms starting in 1st grade--not the "many ways" (i.e., strategies or models) in reform math that, in my opinion, impede achievement. The way math has been taught for decades, including the Common Core test reform era--such as the early use of calculators; overuse of manipulatives, drawings, and strategies to represent concepts; minimal guidance methods; slow pace, time-consuming activities and nonstandard procedures; teaching to the test, and a spiraling curriculum--explains the inadequate achievement in basic arithmetic. Kids stumble over standard arithmetic because it is not taught well. What has been taught is a complicated variant of arithmetic called reform math. It looks like standard arithmetic, but it isn't the real thing!

Moreover, when a student reads a word problem, the student should recognize the problem type. Indeed, pattern recognition is very important in solving math problems. "Oh, that's an area problem, just multiply these numbers (234 and 63) since A = lw,  and I'll get the answer in square meters. It will take me less than 15 seconds." The student's method is clear and straightforward. That's what kids need to be able to do, and do very quickly. Doing something fast implies fluency.

What about the latest technology; e.g., tablets, laptops, etc.? 
The Organization for Economic Cooperation and Development (OECD) points out recent research by  Leonid Bershidsky that suggests computers are likely an obstacle to learning. He writes, "The OECD study found that the use of computers was negatively correlated with improvements in student performance on math tests." Consequently, the hell-bent drive in the US to use computers, laptops, or tablets in the classroom, even graphing calculators, may be the wrong direction. Larry Cuban (Stanford) keeps telling us that the latest technology has not been the answer to our achievement woes.  In my opinion, students should be able to do basic arithmetic without a calculator and basic algebra without a graphing calculator. 

(Aside. Perhaps, we ought to bring back the slide rule. "Multiplication just required lining up two numbers and reading the scale," writes Cliff Stoll, physicist. "Multiplication simplifies to sums; division becomes subtraction," etc. You can't go wrong. Slide rules were used extensively for calculations in mathematics, chemistry, physics, engineering, etc. Today, a slide rule costs $20-$40, if you can find one, compared to $140 graphing calculator, or make one yourself. At the least, kids will learn to read scales, figure out the order of magnitudes, and mentally keep track of the decimal point and the sign (+ or -) of the answer. Incidentally, the slide rule put a man on the moon. FYI: The math of slide rules is logarithms. To multiply, add logs; to divide, subtract logs. In short, the slide rule works because the logarithm of the product is the sum of the logarithms: log (ab) = log a + log b. Click for an online simulated slide rule)
log (ab) = log a + log b

Ability Matters [A Lot]
The reality is that kids with higher academic ability (let's say, IQs of at least 110), are faster learners of academic material, such as math, science, literature, history, reading vocabulary and comprehension, etc., than kids with lower academic ability; hence, kids with higher academic ability should receive differentiated instruction (more rigorous and challenging academic work that is faster paced or accelerated and much more complex), but under the sameness ideology of Common Core, many don’t or merely get enrichment, and, consequently, they languish in boredom and stagnate academically in regular [inclusion-type] classrooms. For the most part, educators and policymakers tend to ignore the best students. In contrast, it has been my experience that the best students need just as much teacher attention as other students, perhaps more. I have often noted that advanced kids don't need enrichment, which keeps them where they are; they need acceleration options to move forward as fast as they can go. Some of the best are invited to Johns Hopkins summer youth academic program. In the real world, the "sameness ideology" brings down the achievement of all students, not just the advanced kids. The averaging of test scores often masks a school's overall poor achievement. Averages often mislead. We need to see a distribution of scores, not an average.   

Academically Gifted
Will Fitzhugh (The Concord Review) writes about gifted kids, “I thought of this the other day when I read about students in summer programs at the Johns Hopkins Institute for the Advancement of Academic Youth in Baltimore. In The Boston Globe the article said: “Students from 21 states and 15 foreign countries—some as young as seventh grade—devour full-year high school courses in the arts, history, math, science and languages in only three weeks. For a rare few, a standard nine-month curriculum is absorbed in seven days.” The academically gifted need an entirely different curriculum. Richard Rusczyk (the Art of Problem Solving) says that AP calculus is for well-prepared, average students. It is not up to the university level, does not stress proofs, and depends too much on graphing calculators. Furthermore, it's too easy and boring (too slow of a pace) for the mathematically gifted.

Aside. First Draft. It is not an essay. Please excuse typos and errors. Additions are frequently made. The last changes were made on 9-20-15. 11-12-154

Charles Murray (Real Education) identifies the "sameness" flaw in the standards approach (Common Core is the latest example) this way: “To demand that students meet standards that have been set without regard to their academic ability is wrong and cruel to the children who are unable to meet those standards." Under Common Core, cut scores for passing are intentionally set high so that most students fail. Murray explains, In large groups of children, academic achievement is tied to academic ability. No pedagogical strategy, no improvement in teacher training, no increase in homework, no reduction in class size can break that connection. Even the best schools will have children who do not perform at grade level.In many schools, the averaging of test scores provides cover for the poor performing students. Charles Wheelan (Naked Statistics) points out, "Students with different abilities or backgrounds may also learn at different rates. Some students will grasp the materials faster than others for reasons that have nothing to do with the quality of teaching. Our attempts to identify the 'best schools' can be ridiculously misleading."

Kids are locked into a Common Core test culture!
Evgeny Morozov (To Save Everything, Click) points to another basic flaw of standards-based education such as Common Core, "Schools concentrate all their efforts on improving test scores, even if children learn much less as a result." For decades, we have been chasing after test scores, fruitlessly trying to copy or match Singapore, South Korea, Finland, Shanghai, and other nations, and, as a consequence, we have locked ourselves into an entrenched test culture rather than an achievement culture. And, that's wrong. "Teaching to the test" is a lousy curriculum and an inexcusable way to teach math to novices. Skills, such as critical thinking or the "Standards for Mathematical Practice" (SMPs) have pushed essential content knowledge aside as mere background noise, even though factual knowledge in long-term memory must precede skill. John Ewing, Executive Director of the American Mathematical Society and President of Math for America, writes, "Test scores can be increased without increasing student learning." In short, Ewing means that you cannot substitute test scores for achievement, that is, "Test scores are not the same as achievement."

Today, the goals of education, I think, seem to focus on (1) getting higher test scores and (2) sameness, rather than academic excellence, which is gaining (making) knowledge. Sure, there are usual platitudes found in mission statements, such as "citizenship, character, lifelong learning, career readiness, environmental awareness, respect for diversity," including the progressive four Cs of "critical thinking, collaboration, communication, and creativity." In contrast, Will Fitzhugh (The Concord Review) astutely points out that gaining (i.e., making) knowledge must be one of the goals of education, yet "making knowledge, the foundation of learning," has been displaced by so-called higher thinking skills. The test-based changes (reforms) in education are very costly with only exiguous improvements. In fact, the reforms have locked us into a test culture. (Aside. Incidentally, there are certain groups of people who have higher IQs than the rest of us.)

Naturally, "We want kids to be creative, to be good collaborators, to be good critical thinkers. But our ability to measure these qualities is quite limited," writes Daniel Willingham, a cognitive scientist. Indeed, many popular classrooms practices are based on ideology or ingrained beliefs, not scientific investigation. We can't do scientific investigations without measurements. So, how do we know our theories in education are correct? We don't! The popular 21st-century practices of teaching kids to be "little scientists" or "little mathematicians" without a solid knowledge base in those subjects have no scientific basis. The 21st Century practices are a skills approach, not a knowledge approach. Consequently, the reality is, "Children will be unable to reason well if their knowledge base is poor."

Thoughts without knowledge are empty [Kant].
The idea of doing valid critical thinking after reading a few paragraphs (a popular Common Core skill or technique) without solid background knowledge in that topic is not based on science and does not carry over into adulthood as many decisions are made by emotion and beliefs, not logical or critical thinking. Teachers are told to teach abstract skills or strategies, such as finding the main idea in a paragraph, yet teaching skills in reading cannot replace background knowledge and content. In arithmetic, efficient procedural knowledge [computational skills] is based on factual knowledge; they go together. But, the stress in Common Core reform math has been on thinking skills outlined in the  Standards for Mathematical Practices (SMPs), not on the automation of essential factual and procedural knowledge that enables problem-solving. Repeatedly, I have written that kids are not little mathematicians; they are novices who need to master essential content knowledge. But, typical interpretations of Common Core as reform math don't do that. 

Will Fitzhugh writes, "Skills have taken the place of content. Content, after all, can be such a pain." He cautions, “But let’s also remember that one of the goals of education must be the acquisition of knowledge,” even for its sake. Fitzhugh points out, “To make knowledge, which is the foundation of learning, it is necessary to apply thought to information, to think about the facts that have been gathered, and this is work only an individual can do.” In short, the Internet does not make knowledge, human beings make knowledge. Also, remember, the more you know, the more you can learn and the faster you can learn it, says Daniel Willingham. Knowledge builds knowledge.

The idea that kids should work in groups to foster collaboration later on in the workplace is far-fetched and not supported by science. The experts are wrong to make grand claims without supporting evidence, but they do it all the time to validate, albeit falsely, their beliefs. The claims made by Common Core, such as college and career readiness, have no basis in science, either. Willingham (When Can You Trust the Experts?) writes, "In education, there are no federal or state laws protecting consumers from bad educational practices." In education, there are no real experts, he says. Furthermore,  always be skeptical of claims made by people who assert that their opinions or practices are researched-based.

The explicit teaching of standard arithmetic is not backward thinking; it works for most students, but not all. Students need to automate essential facts and efficient procedures through memorization and practice, even drill for skill, to move forward. If there is little in the tank (long-term memory), the student won't travel very far. It is basic cognitive science. Knowledge, both factual and procedural, in long-term memory, enables higher-level thinking. Mastering standard arithmetic--not reform math (many ways) strategies--primes the brain for algebra. 

Dr. Katharine Beals (Out in Left Field blog) astutely observes that the current reform approach substitutes math education skills for efficient math skills. The curriculum and instructional methods taught in the classroom come right out of ed schools, and they are not the standard arithmetic (factual and standard procedural knowledge) that kids need to automate to advance to algebra. She writes, “Is the goal to teach kids how to do the math, or how to be math teachers?" I have often said that kids are not little mathematicians; they are novices. In my opinion, novices don’t need to "play with" inefficient, alternative strategies (models, nonstandard algorithms, etc.), explain or write explanations for their answers, or draw diagrams (visuals) to model, explain, or prove their answers. Kids are novices; they need to practice the core of standard arithmetic, both facts and standard procedures (skills). In contrast to Common Core, what matters in solving a multi-step word problem is a straightforward flow of standard calculations (a logical order), which is the thinking process of the student. Nothing more is needed. In solving word problems, kids should not be thinking about models, diagrams, or drawings, but about numbers and operations, that is, the actual math, itself. Students need to abstract the numbers that are essential to do the calculations in the right order to get an answer. 

Below is a "math education" example from Essentials of Mathematics for Elementary Teachers (Musser, Burger, & Peterson). It shows three different addition algorithms, and then asks a question, which method shows the best understanding of place value? Justify.
Common Core reform math, just like NCTM reform math before it, is about showing understanding through an algorithm (or drawing) and justifying it. It substitutes math education skills for essential math skills the student needs to automate to move forward. Common wrongly Core assumes that the algorithm or model (strategy) that shows understanding is the best algorithm or model (strategy) to use, which is nonsense. The implication is that Trevor's method, which is the standard algorithm, shows the least understanding of place value, but the argument assumes that Trevor doesn't know place value, which is a weak argument. Maybe, Trevor's understanding of place value far exceeds that of Nick and Courtney. Understanding should not tie to a certain algorithm or a drawing; it is an unconscious idea in long-term memory. Common Core reform math focuses on using nonstandard or invented procedures or visuals that allegedly "show" understanding as if that is the same as actual "student understanding." Well, it's not!

Understanding is "part nonverbal and part unconscious," says mathematician David Ruelle (The Mathematician's Brain); hence, "the processes of mathematical thought are difficult to analyze." Ruelle reminds us that mathematics is a matter of using knowledge from long-term memory, not of opinion. You cannot solve problems in mathematics with an empty knowledge tank. In contrast to Common Core, the standard algorithm is efficient, tried-and-tested, and always works, and it relies on the memorization of single-digit math facts. Knowing the math facts in long-term memory for auto recall precedes procedural skill, says Daniel T. Willingham (Why Don't Students...). Willingham, a cognitive scientist, stresses, "Factual knowledge [in long-term memory] must precede skill."

We are cheating our students with "sameness" and other popular progressive ideas and programs!
Today, everything seems to be about race, closing gaps, sameness,  fairness, Common Core and testing, equalizing downwardetc. rather than gaining knowledge and improving individual academic achievement (excellence), so I don't know where education is going, certainly not to a good place. Improving scores on tests, based on "so-called" higher standards (aka test-based reform), should not be the goal, yet it seems to be the fundamental driving force in curriculum and instruction today. Test prep, which strongly influences the curriculum, leads to myopic learning if that. We have been down the road of test-based reform before (2001 NCLB act), and it failed. Today, we are back on the same road, often repeating the mistakes of the past and expecting a different outcome. We are cheating students. They need a broad-based, liberal arts curriculum. 

Regrettably, progressive (aka liberal) reformists, supported by big money, the powers that be (government), pundits, and others--all of whom know little about education, cognitive science, and learning--make up most of the rules and policies in education and impose their control, liberal ideas, mandates, and academic fads, such as government sponsored Common Core, on schools, teachers, and children. Common Core has become a ruse for more government encroachment and a recipe for continued lackluster achievement. Common Core, with its baggage and clutter, drives high-stakes testing and the costly technology boom, but it lacks valid and clear evidence for its vague claims of college and career readiness and of leapfrogging improvement in math and reading. In fact, it ignores STEM. The latest technology in the classroom, observes Larry Cuban (Stanford), both then and now, has not worked as a solution to our education woes. Indeed, we do not learn from our mistakes of the past; we just repeat them. Will Fitzhugh laments that kids aren't taught history content. History content knowledge, along with knowledge from other domains, such as science and math, has been replaced by higher level thinking skills. Higher level thinking skills won't help much in solving a trig equation without knowing and applying trig content, and so on. Facts [knowledge] must precede skills. 

[Note. Daniel Willingham, a professor of psychology at the University of Virginia, says that he doesn't think teachers are dumb; he thinks their training is dumb. He writes, "But the problem in American education is not dumb teachers. The problem is dumb teacher training." Willingham explains, "Teachers are smart enough, but you need more than smarts to teach well. You need to know your subject, and you need to know how to help children learn it. That’s where research on American teachers raises concerns." He continues, "Teachers themselves know that their training focuses too much on high-level theory and not enough on nuts-and-bolts matters of teaching." I should point out that those high-level theories have not worked in the classroom.] (Source: Willingham's OptEd in NYTimes, 9-8-15) 

E. D. Hirsch Jr. writes, ""The conclusion of cognitive research concerning skills is this: Broad knowledge of many domains is the only foundation for wide-ranging problem-solving and critical-thinking skills; hence, broad knowledge [i.e., the liberal arts, such as math, science, history, literature, language, art, music, etc.) is what the schools, media, and policies should try to impart....Teaching strategies (skills) instead of knowledge has only yielded an enormous waste of school time."  In summary, Hirsch writes, "Abstract skills falter without a foundation of content supporting them." It is the situation we have today, not only in history, but also in math, science, and other key academic disciplines, such as literature. So-called higher thinking skills trump content knowledge. Hirsch is saying that it doesn't work that way.  

Many Low-income Kids Are at the Bottom
In low-income elementary schools, the ELA test scores are often very low, usually under 25% proficient, but the math scores are even lower. On the surface, it appears that low-income kids are not learning much, which could be true. What is more likely is that many low-income students often do not have achievable learning objectives. Do they have the funding and resources needed? I don't know. The liberal answer has been to pour more money into these schools (Title 1 funds, etc.), but has this changed the narrative? The rich schools can afford the add-ons, but the poor schools can't.  [Data Reference: Wilmington, Delaware, Smarter Balanced 2015 scores]

Indeed, the federal and state governments, the school district, a prevailing progressive [liberal] ideology of sameness, and the Common Core's one-size-fits-all-test-based-accountability scheme have failed these kids. For example, in Wilmington, DE, a city elementary school (SH), in which 81% of the students are low-income, scored 15.7 % proficient in math and 20.9% in ELA, while a suburban elementary school (NS) in the same school district, in which 3.8% of students are low-income, scored 77.1% proficient in math and 87.5% in ELA. (Don't jump to conclusions or confuse correlation with causation.) Under our current education system, most of SH students have little chance to catch up to their peers at NS, not in education or in life, although, I hope there will be many exceptions. Some of these students think they are dumb because they keep failing the test. Charles Wheelan (Naked Statistics) explains, "There are schools with extremely disadvantaged populations in which teachers may be doing a remarkable job, but the student test scores will still be low--albeit not nearly as low as they would have been if the teachers had not been doing a good job."

School averages are often untrustworthy.
An elementary school that has several high scoring students can easily skew the school's average to 50% proficiency or above to cover up the lower scores of other students. A good example is one of the highest ranked school districts in Arizona, partly based on test scores, high school graduation rates (92%), AP participation (23%), and other factors. But how well are all students educated? In 2014, 80% of its high school graduates that applied at Pima Community College were placed in remedial math courses. The flaw of averages shows that the success of some students masks the failure of others. In other words, averages are often misleading.  If the education in this top-rated school district were exceptional, then its percentage of remedial math students at community college in 2014 would not be 80%. Liberal reformers say that Common Core will fix this with the claim of 0% remedial, which is an idiotic assertion because there is not a shred of evidence to support it. Furthermore, high school graduation rates are often artificially inflated with online "credit recovery" programs. Also, there is rampant grade inflation in K-12. [Aside. I do not know the two-year graduation rate at Pima Community College, but the 4-year graduation rate at the University of Arizona is 40%. The truth hurts! "SAT scores for the Class of 2015 were the lowest since the test was revised and re-normed in 2005," writes Carol Burris. The SAT scores are lower because of modern reforms, says Burris, not because more students are taking the test.

Has Common Core contributed to the achievement gap? Are its goals too lofty or unachievable for some kids? Can some low-income students score just as high as other students? I would answer 'yes' to these questions with some caveats, of course. In the real world, for many students, there is no fairness in funding or excellence in their education, just dependency on government schools and government programs. Poor schools tend to remain poor.

The Common Core Way is roughly the same approach as before (2001 NCLB era), which was, write some standards, impose them on schools, write some tests and make them high-stakes for schools, teachers, and students, then see what happens. The plan failed miserably. So, what did we do? We wrote new standards and now expect different results. How simpleminded! We are repeating the mistakes of the past! The Common Core Way is backward thinking. Sadly, schooling has been entrenched in a culture of accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. It may be okay for business, but education is not a business, and it is not okay! NCLB (No Child Left Behind), Common Core, high-stakes standardized testing (ranking/grading schools-teachers-students), Race to the Top (RttT), and wavers are some of the recent upshots of this egregious (education) culture. Muller observes, "Under NCLB, scores on standardized tests are the numerical metric by which success and failure are judged." In the mad, mad world of education, schools, kids, and teachers are defined by a test score, which is truly an outrageous idea. It is extreme. It shows that education has run amok!  

The "One Size Fits All" Common Core model doesn't work in the real world because kids coming through the school door are not the same. Students vary widely in industriousness and persistence, numeracy, language and vocabulary, attitude and beliefs, resources, and academic ability. What is the point of giving a Common Core test, knowing that 70% are likely to fail? Why feed kids the same curriculum and the same tests? Some kids are at a level that calls for a more challenging or accelerated math curriculum while other kids are at a level that requires a more basic curriculum, and so on. Our present system of sameness and inclusion is absurd. John Hattie writes, “Levels-based curricula with clear milestones, targets or expectations, which are then aligned with the assessment system, are more likely to have an impact on student learning than year-based curricula." Kids need achievable learning objectives, and, because academic ability and background vary widely, the learning objectives and tests should vary accordingly.

Aside. For example, American Red Cross uses levels-based swimming lessons and tests, which are based on clearly defined levels. There are achievable learning objectives for beginning swimming, and so on for each course. The ARC doesn't give the intermediate level swimming test to swimmers in its beginner's course. No one would pass.

Why do some schools score so much higher than other schools?
Is it the teachers? Probably not, because they are products of the same liberal ideology disseminated by progressive schools of education. Perhaps, it is the students. It is a hard pill to swallow. "Children differ in their ability to learn academic material" because they differ in their abilities and backgrounds, writes Charles Murray (Real Education), but American education practices and policies ignore this fact and brand Murray a racist, which he is not. Perhaps, it is the testing because standardized tests use the dreaded bell curve, which means half the students are below average. Adding to our education woes, we have been diverting billions and billions and billions of dollars into goals that are unattainable, all in the name of sameness, while neglecting academic excellence (gaining/making knowledge), which should be the top objective in education, but our leaders give it lip service. Indeed, Common Core, linked to standardized testing, and technology, are just some of the latest academic fads. By fads, I mean, not backed by valid evidence!

So what can be done?
We can dump Common Core and its testing and restore the explicit teaching of standard arithmetic, along with parts of algebra, geometry, and measurement so that most kids, not all kids, reach a level to handle pre-algebra by 7th grade. Some students will not be ready for a valid pre-algebra course in 7th grade or even 8th grade. Other students will need pre-algebra spread over two years (7th & 8th grade), and still others will take pre-algebra in 7th grade and be ready for a valid Algebra 1 course by 8th grade. Ideas, such as algebra-for-all, college-for-all, or laptops-for-all are silly and nonsense. In addition to a school culture that impedes progress, we have a curriculum problem and an instructional problem.

Many widespread, favored classroom practices are not supported by evidence and are among the least effective. Kirschner, Sweller, & Clark (2006) point out that minimal guidance methods, in which the teacher is a facilitator of learning, are typically the least effective classroom practices. These constructivist-based minimal guidance practices, which are taught in ed school, have many names, including such favorites as inquiry-based, discovery learning, problem-based, etc. Some teachers are convinced that inquiry- or discovery-based learning, which favors group work, is the best way for kids to learn math, but it is not true. In short, reforms should be based on cognitive science, not popularity, intuition, or ideology. Current cognitive research supports strong, direct teacher guidance, not the teacher as facilitator. "Direct instruction involving considerable guidance, including examples, resulted in vastly more learning than discovery," write Kirschner, Sweller, & Clark.

Kirschner, Sweller, & Clark explain, "After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to intermediate learners. Direct instructional guidance is defined as providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture. Learning, in turn, is defined as a change in long-term memory." Kids are novices, not experts, so they need straight-forward teacher guidance and encouragement, especially with new, more complex content. Furthermore, kids don't need to work in groups, which is often counterproductive. Kevin Ashton (To Fly A Horse) points out, "Working individually is more productive than working in groups." Ashton goes against popular group think (the status quo).

Factual Knowledge Must Precede Skill
You don't get to higher-level thinking except through lower-level thinking (knowing and applying) in long-term memory. It is a major concept in cognitive science that is ignored by current progressive reformers because it doesn't fit their ideology. Guess who is in charge of education? Progressive reformists! 

Cognitive scientist Daniel Willingham often says that understanding is vague and difficult to assess. He also stresses that "factual knowledge must precede skill." Willingham continues, "If factual knowledge makes cognitive processes work better, the obvious implication is that we must help children learn background knowledge." Knowledge, not strategies, should be the focus in elementary school. Students should not be taught to be little mathematicians or little scientists. They are novices, not experts. They need to know facts to think. 

Michael E. Martinez (Future Bright ) points out, "Research supports a view of intelligence as both lower-order and higher-order. The mind's ability to engage in higher-level operations must in some way rest on a foundation of lower level functions. Higher-level reasoning and problem-solving draw, in turn, on the mind's software--the accumulated content-rich knowledge acquired through experience."

Draft 1
Please excuse typos and errors. 
Model: Chloe 

©2015 LT, ThinkAlgebra.org