Wednesday, February 25, 2015

Strong Guided Instruction

Children need strong, teacher-guided instruction. 
Minimal-guided instruction has been an epic flop. 

Traditional arithmetic works well when taught well. Students become better at mental math because they have memorize basic number facts. Furthermore, the standard algorithm always works. We keep forgetting that little kids are novices. They don't think like adults. Children need to memorize and practice to put mathematical knowledge, both factual and procedural, into long-term memory for instant use in problem solving. Students cannot do mathematics without knowing some mathematics. Also, understanding is a slow process. It does not produce competency, practice does. Do not expect instant understanding or hold kids back because their understanding is partial or incomplete. Furthermore, Jason Zimba, one of the two major writers of CC math standards, explains, "The standards also allow for approaches in which the standard algorithm is instructed in grade 1, and in which only a single algorithm is taught for each operation."  Note. Isaac Newton invented a fast way to calculate answers to physics problems, called calculus. It always worked (i.e., it was consistent with experimental data), but he didn't understand why the calculus worked; it just did. The "why" would take another 200 years. 3-9-15  

Explicit teaching, which uses a carefully-planned sequence of worked examples, let's say in math, works well for almost all students. Students learn concepts through examples, lots of practice, and repetition, says Zig Engelmann. However, since the 60s, teacher-led instruction has been called "old school" or the opposite of “good” teaching. Explicit, teacher-led instruction—using examples, practice, and repetition—“contradicts much of what educators are taught to believe about good teaching,” writes J. E. Stone (Clear Teaching).  

Stone says that explicit teaching [the teacher is the academic leader that leads instruction by explaining examples on the board, etc.] has not been popular in K-8 schools, not because it didn't work but because it goes against progressive reform ideology taught in schools of education. The Progressive Era revolution of the 60s affected education by attacking teacher-led exercises, scripted lessons, skill grouping, choral responding, repetition, etc., says Stone. “Thus, education professors and theorists denigrate teacher-led practice as ‘drill and kill,’ its high expectations as ‘developmentally inappropriate,’ and its emphasis on building a solid foundation of skills as ‘rote learning’.” Kids have not been taught a solid foundation of arithmetic for decades and decades. 

Today we have teachers as facilitators, not academic leaders; mainstreaming (inclusion); a weak, incoherent, downgraded curriculum; low expectations for students; popular reform methods of instruction (i.e., minimal guided, not teacher-led) that do not work; reforms such as Common Core, intrinsically linked to standardized testing; etc. Education is no longer a "work hard and achieve" narrative; it is a political, money-driven narrative.    

Note. I have quoted this study [Kirchner-Sweller-Clark (Why Minimal Guidance During Instruction Does Not Work...)] since it first appeared in Educational Psychologist in 2006. The instructional methods in classrooms across the US--mostly group work activities with minimal teacher guidance or no teacher guidance--have failed our students for decades. The minimal guidance instructional methods (e.g., discovery, constructivist, problem-based, inquiry-based, etc.), which are championed in schools of education, extend to Common Core. They are part of the progressive movement in education, starting with Dewey. Kids do a lot of group work, use manipulatives, etc. Their desks are in groups of 3 or 4, so kids face each other. The teacher is not the academic leader in the classroom. The teacher's role has diminished to a "facilitator" of learning. In short, the teacher no longer teaches. 

Kirchner-Sweller-Clark (Why Minimal Guidance During Instruction Does Not Work...) write, "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described."

Kirchner-Sweller-Clark write, "Cognitive load. Sweller and others (Mayer, 2001; Paas, Renkl, & Sweller, 2003, 2004; Sweller, 1999, 2004; Winn, 2003) noted that despite the alleged advantages of unguided environments to help students to derive meaning from learning materials, cognitive load theory suggests that the free exploration of a highly complex environment may generate a heavy working memory load that is detrimental to learning. This suggestion is particularly important in the case of novice learners, who lack proper schemas to integrate the new information with their prior knowledge."

© 2015 LT, ThinkAlgebra,org

Tuesday, February 3, 2015


Random Ramblings #1 by a Contrarian: In No Particular Order. Even some rants!
First Draft (Please excuse errors, typos, etc. Changes are made frequently.)
I am not ready to post it, yet, here it is, totally unorganized. Whenever an idea pops into my mind, I write. I am a contrarian. And, I often quote others and repeat myself (a lot). 

The test-based NCLB law has failed. In many schools, the curriculum has focused mostly on test-prep. (If it is not in Common Core, then skip it.) Sadly, the government funds government created failures--from Head Start, to Title 1, to NCLB, etc. Common Core is a political narrative, which is advanced and controlled by an elite--not an education narrative. Common Core is intrinsically linked to test-based reform (NCLB), which, in my opinion, downgrades children to numbers [test scores]. A child is not a data point. The main assumption, under NCLB, was that test-based reforms would work with consequences built in (sanctions and punishments). The fundamental assumption was wrong. 

Sort kids into homogeneous math sections by
achievement starting in 1st grade.
Aside. Walk into most any elementary school classroom and you will find students with a wide range of ability and achievement--fast and slow learners together. For decades the elementary school classroom is like the old, inefficient one-room schoolhouse. The status-quo solution has been differentiated instruction within the same classroom (more inefficiency), but the best solution is common sense, that of putting faster learners together (tracking) with an accelerated, challenging math curriculum. ["Oh, we can't do that." It goes against the progressive ideology of fairness, but what's fair about holding our best students back?Common Core means that everyone gets the same, which is like the old inefficient one-room schoolhouse. The caveat is that kids are not the same. Putting high achievers in math and low achievers in the same math classroom has been a recipe for underachievement and a regression to mediocrity. The kids who learn math faster get bored and the kids who struggle stay behind. In my view, mainstreaming [inclusion] for math class has led to underperformance at all levels. In short, the system of heterogeneous classes for math is deeply flawed. Moreover, the Common Core standards, themselves, are flawed, especially when the curriculum distilled from them is taught as reform math. Starting in 1st grade, we should arrange kids into homogeneous math sections by achievement [knowledge] with different teachers. Don't worry about their self-esteem. Worry about their competency!

Aside. Traditional arithmetic, practice, and memorization to automate fundamentals [factual and efficient procedural knowledge] in arithmetic, starting no later than 1st grade, are not obsolete as many progressive [Common Core] ideologues or reformers claim. Actually, they are essential and of the highest priority!

Note. Kids in poor schools do not score as well as kids in rich schools. By averaging test scores, several high achievers can conceal the poor performance of the other students, says Diane Ravitch (...American School System). It is the "flaw of averages," writes Professor Sam L. Savage (The Flaw of Averages). Averages are uncertain numbers. They are unsound and unreliable. Furthermore, in rich schools, which have more resources, demanding parents insist on math acceleration, while bright kids in poor school districts don't get accelerated math instruction to prepare capable kids for real Algebra 1 in middle school. The expectations are lower. 

A typical first grader in my Title 1 Teach Kids Algebra
Enrichment Program builds function tables and graphs
linear equations in Q-I.

Therefore, smart low-income kids cannot get to real Algebra 1 in middle school because Common Core reform math is not set up for STEM. When I taught my algebra enrichment program (Teach Kids Algebra--TKA) to little kids, grades 1 to 5, in urban low-income Title 1 schools [as a guest teacher a couple years ago], 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.

TKA teaches how math works. LT
In fact, my elementary school TKA algebra Title 1 enrichment program was a casualty of Common Core when the school district diverted Title 1 funds to Common Core. What distresses me most is that Common Core reform math and its testing paradigm treat kids as if they all have the same cognitive ability, which is a "clean slate" idea. They don't. 

In my opinion, test prep and yearly Common Core standardized testing, which often start as early as 1st grade in many schools, do not benefit students. Instead, they narrow the curriculum and waste valuable instructional time. Students would be better off if teachers make their own achievement tests according to the students that walk through the school door. The "one-size fits all" paradigm of Common Core/NCLB does not fit reality. 

Aside. Paradigms, theories, or models are just that; they are not reality. For example, physicists postulated the Standard Model that  explains "what the world is and what holds it together," but the mathematical model is not reality; it is pure speculation and does not include gravity. As one physicist said, "We just made it up!" 

Another example: Piaget postulated a theory of cognitive development that occurs in specific stages, which we now know is wrong, yet it still greatly influences education today every time educators adhere to ideas such as "clean slate," or the content in math or science as being "developmentally inappropriate," or "everyone gets the same." Zig Engelmann (The Dreaded Standards) writes, "The [Common Core] standards clearly follow the Piagetian myth that children first manipulate then internalize the manipulations, which slowly grows into concert operations, and later into abstractions for formal operations." That's not the way it works, counters Engelmann. (Note: The quote about the standard model is from

Unfortunately, under the Common Core, schools are forced to invest heavily in testing and the technology used for the online testing, but how does this help kids? The testing obsession has resulted in "curriculum-trimming, test-prep, and educational disruption being visited upon on the poorest schools and districts." Under the one-size-fits-all Common Core, K-8 teachers are told to focus on reading and math, which are often taught as reform math, and not so much on science, geography, history, literature, essay writing, etc. Test prep, reform math, or test-based instruction/accountability is not the same as teaching essential math content that is carefully sequenced and coherent.(Note: Quote found on Diane Ravitch's blog by an unidentified reader.) 

Lastly, it has also been my experience that paying attention (attention span) requires sustained effort to optimize learning, but some students seem unwilling to make the effort needed. In my view, students in groups make attention more difficult. The lack of persistence, industriousness, or effort is a detrimental attitude in some students. Students need to come to school ready to learn. Furthermore, I recently read that teaching "grit is racist," which, in my opinion, is a destructive idea cooked up by progressive ideologues who have taken over education. 

Aside. Whatever happened to recess? Elementary school kids need to get away from their desks and run around. 

There is nothing wrong with conscientiousness, industriousness, or hard work in school or at home. To create is work, says Kevin Ashton (How to Fly A Horse: The Secret History of Creation, Invention, and Discovery). You need knowledge from long-term memory to operate on it in working memory. 

Welcome to the mad, mad world of math [mis]education!  Note: The NCLB act stipulated that all students would be "proficient" in math by 2014. Well, that has not worked! Common Core comes along and stipulates that all students will be "college/career" ready. Surely, you must be joking! It is the same old progressive rubbish from so-called experts who make foolish, wild, or unsupported claims. Under Common Core, kids are a test score, which is a pernicious idea.

"The only certainty is that nothing is certain."  (Pliney the Elder)

"Thoughts [i.e., critical thinking or problem solving in math] without content [organized knowledge] are empty." (Kant) 

"In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?", writes Richard P. Feynman, Nobel Prize in Physics. Actually, doing math starts the process of understanding, which grows gradually over the years with practice, repetition, and experience, explains Feynman, who used to say that you do not understand anything until practiced. "Foundation skills need to be routinized so that the mind is free to think," explains Professor George Farkas in a recent study.

"Surveying the many unsuccessful and hugely expensive attempts at school reform in our past, historians Tyack and Cuban observed the same mistakes being repeated over and over again: top-down remedies, grandiose claims about the latest technology, disdain for teachers." It irks me that teachers are always blamed for poor academic performance when there are many other factors, such as a top-down, flawed curriculum, single parent and/or low-income families, etc. (Source: American Scholar, "School Reform Fails the Test," by Mike Rose) 

We have yet to establish a logically sequenced, coherent learning hierarchy for K-6 arithmetic that synchronizes arithmetic to algebra (equality, variables, building x-y function tables based on a relationship between two sets of numbers), graphing functions (x,y), writing equations that are solutions to word problems, etc.) and to applications (e.g., measurement and geometry), and so on--all starting in 1st grade. The hierarchy should consist of logically sequenced, coherent behavioral objectives [Mager] that are very specific, measurable, and manageable. In short, teach skills in a sequence of objectives of increasing complexity [Lemov, et al.]." In addition, teachers need to know what proficiency looks like in context. Moreover, I recommend that educators follow Gagne's Nine Steps of Instruction, which include explicit instruction and practice to mastery. But, to do this well, the teacher needs a logically sequenced, coherent learning hierarchy. Each learning objective should state or show a clear context for proficiency. Today's math textbooks are not designed this way. The first six are below.   
The First 6 of Gagne's 9 Steps of Instruction 
4. present the material using worked examples, 5. provide guided practice in class, 6. elicit performance (practice on your own: homework), 7. provide feedback, 8. test performance, 9... 

After looking at the latest NAEP math scores, it seems to me that there are too many scores at or near the bottom and too few scores at or near the top, which brings down the average score in math achievement. [NAEP tests are US government tests given every two years to a sampling of grades 4 and 8, sometimes 12.] Out of a scale score of 500, the average in math is 241 in 4th and 284 in 8th. Averages are misleading because they are uncertain numbers. Claims or policies based on averages are often wild, even silly. Moreover, one cannot [and should not] extrapolate beyond the known data because the predictions, which are speculations without firm evidence, are unreliable, a key idea I used to teach to elementary kids in Science--A Process Approach in the late 60s. Yet, Common Core forecasts (claims) that all high school graduates will be college and career ready [without remediation], which is nonsense. Diane Ravitch (Reign of Error) observes that high school graduation rates and tests scores are at a historic high, but incremental progress does not mean much when most high school graduates place in remedial math courses at community colleges. In short, many kids, especially low-income K-12 urban students, are not learning much. The problem starts in 1st grade. It is clear, at least to me, that reform math curriculum, test-based accountability, minimal teacher guidance methods, and fads have not worked in K-12. In my opinion, Common Core reform math continues many of the same policies that have failed in the past. 

Furthermore, Doug Lemov, et al. (Practice Perfect) write, "As cognitive scientists like Daniel Willingham have shown, it's all but impossible to have higher-order thinking without strongly established skills and lots of knowledge of facts." Lemov stresses the automation of fundamentals (drilling) to unlock creativity. He also points out that students need to "automate skills to free participants' cognition to be more creative and to free up processing capacity." He explains, "With practice you'll get stronger results if you spend your time practicing the most important things." Which key math skills (20%) do students need to practice 80% of the time for automation (Lemov, Woolway, & Yezzi) to prepare for more complex math, such as algebra-one? Why do you think the late Richard Feynman, a Nobel Prize winner in physics, who also excelled in mathematics, practiced calculus problems in every spare moment, which drove his wife crazy? For kids [novices], the learning and practice of fundamentals over and over again [automation] leads to new insights, connections, intuitions, and fresh ideas. Practice increases the ability to solve problems.

"Foundation skills need to be routinized so that the mind is free to think," explains Professor George Farkas in a recent study. In short, constant review of fundamentals is important. "The value of practice begins at mastery," write Lemov, et al. "It is critically important to "build up layers of related automated skills so that participants can do complex tasks without actively thinking about them." The idea is to automate fundamentals. "Many types of higher-order thinking are in fact founded on and require rote learning." 

Arne Duncan, who heads the US Department of Education, says we should drop No Child Left Behind and replace it with an updated federal law that keeps yearly testing to track student progress and rate teachers, which implies that Common Core will continue to be the basis of instruction and testing. Furthermore, I know of no top-performing nation that gives tests yearly and uses them to evaluate teachers and punish schools. The federal culture of "education control" won't end until we eliminate the department and reassign control back to the classroom teacher. Note. In the Netherlands, students spend 200 days in school and the teachers decide how to section students and use that time because there are no regulations specifying class size, test based accountability, etc. Teachers in the lower secondary schools make 94% of the decisions. (Source: The Hechinger Report) I won't see it in my lifetime where teachers run the schools and adjust the curriculum to fit the students who walk through the classroom door with minimal, if any, government oversight.     

Frankly, we should drop Arne Duncan and his department, the testing, NCLB, and Common Core reform math. Instead, we should empower individual schools and teachers to design a curriculum that matches the students that walk through the door and ask the same teachers to write tests that specify clearly defined "behavioral objectives" (Mager, Gagne) to assess progress in their classrooms. Lemov, et al. say that the objectives must be focused, measurable, and manageable. We already have federal tests (NAEP), which are given every 2 years to a cross-section of US 4th and 8th grade students, sometimes 12 grade. Kick out substandard standards. The problem is the curriculum (content actually taught in the classroom) and how it has been taught (often using inefficient instructional methods). The curriculum should be a logical, hierarchical sequence, starting with K-6 arithmetic, which flows smoothly from one grade level to the next and in which capable students are prepared for Algebra 1 by 8th grade, which implies tracking: the curriculum would not be the same for all students. For example, Singapore pulls out weak math students for math class [tracking] in the 1st and 2nd grades in an attempt to catch them up and again in the 4th-6th grades when math becomes more complex. Singapore uses the same standards, but adjusts the curriculum to the students using different materials and teachers.         

The one-size-fits-all Common Core reform math standards ignore STEM and are not up to the Asian Level. Moreover, the new standards do not take into account that cognitive ability varies widely among students; consequently, they "equalize downward by lowering those at the top," says Thomas Sowell (Dismantling America). In short, the one-size-fits-all model does not fit reality. Why do smart people make such dumb and foolish decisions for our kids? Apparently, we are in an age in which facts don't matter. One cannot perform higher-level operations without having strong prior knowledge in long-term memory. This is basic cognitive science, which Common Core trivializes.  Furthermore, traditional arithmetic, memorization, and practice are not obsolete; they are essential. 

If teachers had a rigorous math curriculum (a learning hierarchy) that is focused, logical, coherent, and world class), coupled with strong instructional guidance and support, then kids would more likely learn the arithmetic and algebra they need to know. Unfortunately, instead of tried-and-true traditional arithmetic, teachers are told to implement test prep and reform math using minimal teacher guidance methods that don't work. It's called Common Core reform math.   

It's 2015! What's different? Nothing! American children are undereducated. If the purpose of schooling is college/career readiness and test taking (Common Core's main goals), then our kids are shortchanged in other academic subjects, the arts, P.E., foreign languages, etc. But, in my view, they are also shortchanged in basic math. My 2010 analysis showed that Common Core's math deficiencies start in 1st grade when compared to the Singapore grades 1-6 math syllabi and the 2010 Core Knowledge K-8 math sequence. The Common Core reform math standards ignore STEM and are not up to the Asian level. Common Core starts by teaching reform math, not the traditional arithmetic that prepares capable students for algebra-one by middle school. Bad policies, myths, fads, pseudo science, and wrong assumptions permeate American schooling and don't seem to go away. For example, many educators still believe that increasing self esteem is required for learning, or that students can engage in meaningful critical thinking without proper context [extensive background knowledge], or that students can solve arithmetic problems without prior knowledge and mastery of fundamentals in long-term memory. Thoughts without content [organized knowledge] are empty (Kant). "For Kant, there is nothing contained in the concept of '7' and '5'  that makes the knowledge that adding them together will result in '12' immediately obvious or ineluctable. What he was getting at is perhaps easier to see if we consider larger numbers like for example, 38976 and 45204; their sum 84180 certainly does leap out at me (Brodie).” Thought requires content to operate on. 

If we want students to do higher-order thinking, then they need to start with lower-order, content-rich knowledge (i.e., lower level thinking). Intelligence involves both, starting with lower-order knowledge and practice in applying that knowledge. To move forward, students need to master (automate) fundamentals through practice, then more practice. There is no substitute for automaticity of factual and efficient procedural background knowledge in arithmetic. Practicing arithmetic to automation is important (Willingham). To be polite, let me say that Common Core reform math is out to lunch because mastery or automation is not the main goal. Standard algorithms for operations are put on the back burner and/or replaced with inefficient, alternative procedures.

I recently read an article that almost implies, perhaps unknowingly, that competency in reading, writing, and math no longer fit the changing educational needs of the 21st century. Apparently, teamwork and problem solving are more important skills for the workplace, which are "Practices" listed or inferred in Common Core. Yet, students need substantial knowledge in long-term memory to do problem solving, but Common Core doesn't focus on mastery of fundamentals, not in arithmetic. It focuses on reform math "strategies." Moreover, understanding comes through practice, but, unfortunately, memorization and practice are downgraded in the way CC reform math has been implemented. My point is that no one will hire you if you cannot read, write, and do math well because most new jobs will be STEM or related to STEM. This means that students must demonstrate competency in reading, writing, and math, even at the community college level, to qualify for STEM-related associate degrees. Furthermore, no university worth its salt will admit you unless you are intelligent enough to earn a bachelor's degree. We don't talk about cognitive ability in education, even though it varies widely. The education solution for such variation is called differentiated instruction within the classroom, which is another bad idea. The one-size-fits-all Common Core model doesn't fit reality.

Michael E. Martinez (Future Bright) writes (Long Quote), "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 [practice]. Short-term memory not only holds information, it also works on information. It is one thing to remember the number 10 and 11, but another to multiply them together to get 110. Because the mind actively works on information, we must recognize a related memory function--working memory. Short-term memory and working memory are highly correlated." In short, one cannot perform higher-level operations without having strong prior knowledge in long-term memory. Common Core's emphasis on critical thinking without mastering foundational knowledge or without proper context is a major stumbling block and marginalizes cognitive science. In arithmetic and algebra, kids must memorize and practice to gain knowledge in long-term memory for use in working memory to solve problems. The formal process should start no later than 1st grade with addition, subtraction, multiplication as repeated addition, integer concepts (How many steps from -3 to 7 on a number line? or questions such as 4 + -6 = x), metric measurement (Draw a line 6 cm long), algebra concepts (Find the number x so that x + x - 4 = 9 + 3), and geometry (e.g., perimeters of polygons and areas of squares and rectangles, make a picture of a function by plotting points in Q-I).   

Reinventing The Wheel Is Stupid

Starting no later than 1st grade, kids don't need to reinvent the wheel in arithmetic, but, in my view, this seems to be the Common Core's hidden agenda. In contrast to Common Core, from the start of 1st grade, kids need to memorize math facts so the math becomes easier and much more fun. They need to be able to perform standard algorithms well--not toy with inefficient, alternative algorithms that seem to dominate Common Core. Also, young kids need to understand broad concepts, which are not that complicated, such as the abstract meaning of number by place value (27 = 2t + 7:  2tens + 7ones), how numbers are generated (n + 1 = n'), standard operations, how numbers behave (rules), how new ideas build on old ideas, and how arithmetic is used to solve problems. Common Core apparently seems to ignore the fact that little kids are novices not little mathematicians. 

Understanding does not produce mastery; practice does. Yet, conceptual understanding, not mastery of fundamentals [knowledge], has been an ingrained mantra of school philosophy in teaching math for decades. Common Core is the product of school culture, says Barbara Oakley. The school culture mantra of conceptual understanding trumps knowledge, and has been, in my view, a primary reason that our kids are lousy at arithmetic and algebra. Barbara Oakley writes (WSJ), "Achieving conceptual understanding doesn't mean true mastery. For that, you need practice." As a tutor, I cannot tell you how many times students have told me they understand the concepts, but have difficulty working the problems. Understanding concepts does not produce mastery; practice does, which reminds me of the late Richard Feynman who often said that you don't understand anything until practiced.

Moreover, I do not know why the Common Core cultists hold a grudge against traditional arithmetic, standard algorithms, and systematic explicit instruction--all of which require substantial practice to master. On the other hand, the Common Core reform cultists assert that kids must learn a revamped, untested variant of arithmetic using inefficient, alternative algorithms and minimal teacher guidance instructional methods--all packaged as the Common Core Way in which conceptual understanding, not knowledge, is stressed. Furthermore, over the decades, the power to make decisions has shifted from classroom teachers to the government, corporate, and financial elite. Teachers are unfairly maligned because test scores nearly remain flat (no matter which reform is implemented) because the reforms are based on erroneous assumptions that permeate schooling. Indeed, Common Core is just the latest untested reform imposed on the public schools and born in school culture, says Oakley.

In my view, the math content, itself, and the "reform" pedagogy that is often implied or directly written into the math standards are a mismatch for most students. In short, the CC math standards, themselves, are botched. CC reform math avoids STEM and is not up to the Asian level in the lower grades. One mathematician puts our kids two years behind by middle school. In short, CC reform math downgrades knowledge. Excuses are given, such as "implementation is uneven and flawed," or "not enough money or resources," or "lousy instructional materials (textbooks)," or "poorly trained teachers in the New Common Core Way," etc. Very few say that the Common Core, itself, is flawed and wrong for kids. 

Common Core should be junked. Dr. Sandra Stotsky says there are better standards that can easily replace Common Core. She writes, "Massachusetts once had standards that looked nothing like Common Core, were judged to be among the best in the country and have an empirical record of contributing to academic gains for all Bay State students." She is referring to results of NAEP and TIMSS testing. She should know; she was in charge. Common Core has no track record, while Core Knowledge K-8 math sequence and the Stotsky Massachusetts standards have a good track record. Moreover, the implementation of the Stotsky-era standards went smoothly. 

Frankly, teachers are "reformed" to death. The reforms are mostly inexpert fads (not evidence based, written by inexperienced people) and often called innovations, which, sadly, don't work in the real classroom. They don't scale up. The government, corporate, and financial elite have highjacked public education. We spend more on technology, get less learning, and often believe ideas that are factually wrong, as the late Jeanne Chall (Harvard Graduate School of Education) often pointed out. She believed in the "importance of direct, systematic instruction." She said that the 20th century was dominated by discovery approaches in spite of the research that supported a later theory called explicit teaching. Chall explained that the teacher should not be a child-centered facilitator but the academic leader that uses explicit, systematic instruction in the classroom. This is not only true for reading, but also for arithmetic. [I rephrased the idea of "power shift" from Losing Our Way by Bob Herbert. Chall Sources: Ravitch (...American School System), Wikipedia: Jeanne Chall.]    

Self-Esteem Bunk

Chua & Rubenfeld (The Triple Package) write, "Schools all over the United States made it their mission to instill self-esteem in their [students]...This means self-esteem has to precede achievement. Kids have to be given self-esteem before they've achieved anything." It's bunk. "Increasing self-esteem does not improve academic performance."  


Will Fitzhugh (The Concord Review) recently wrote about the increasing verbal and physical abuse of teachers by students in the UK. He continues, "In the United States as well, a number of fine teachers say that they are leaving the profession primarily because of the out-of-control attitudes and behavior of poorly-raised children who will not take any responsibility for their own education and don't seem to mind if they ruin educational chances of their peers." He points out, "As long as too many of us think education is the teacher's responsibility alone, we will have failed to understand what the job of learning requires of students, and we will be unable to make sense of the outcomes of our huge investments in education." Indeed, most of the problems I see in schools are due to poor parenting. Parents are not teaching their kids how to behave, succeed, and achieve in school or in life.  

Feynman & Einstein

Feynman wrote, If it disagrees with experiment, it’s wrong. In that simple statement is the key to science.  It doesn’t make a difference how beautiful your guess is, it doesn’t make a difference how smart you are, who made the guess, or what his name is.  If it disagrees with experiment, it’s wrong.” A major problem with research in education is that "experiments" with one group of students are not repeatable with other "similar" groups of students. Most of the experiments do not account for time on task or teacher knowledge-skill-experience-enthusiasm, etc. Moreover, claims of "statistically significant" or "research-based" should not imply better achievement in the classroom.  

Jonathan Wai, research scientist at Duke University, writes, "Feynman was a physicist, a field in which this statement uniformly applies. However, standards are different in various fields, and on of the fields in which this statement probably does not uniformly apply is education. There are certainly some education ideas that are backed by solid science, but there are many other ideas which have no scientific backing at all [e.g., group work, using calculators, inquiry-discovery activities, using manipulatives, minimal teacher guidance methods, etc.], but, for whatever reason, are still immensely popular." That's the main problem in education--popularity wins, even if students learn less. Many educators have accepted myths, fads, and pseudo science, including the fad that the "Common Core standards-standardized testing-technology" package, imposed on public schools and students by government, will magically make all kids college-and-career ready. It is total nonsense. 

Einstein wrote, "If you can't explain it simply, you don't understand it well enough." FYI: I can explain traditional arithmetic simply, but I cannot do that with Common Core reform math, which makes simple arithmetic more difficult and complicated than it actually is. 

Sage on the Stage

Research by Guido Schwerdt reports, "We find that teaching style matters for student achievement, but in the opposite direction than anticipated by conventional wisdom: an emphasis on lecture-style presentations (rather than problem-solving activities) is associated with an increase--not a decrease--in student achievement. This result implies that a shift to problem-solving instruction is more likely to adversely affect student learning than to improve it." In short, more lecturing, such as presenting worked examples and explaining how things work in math, which is what good math teachers do, translate into more learning in math. Lecturing is the opposite approach to the Common Core Way (group work, discovery learning, etc.). FYI: When I taught 1st to 5th graders algebra as a guest teacher, I lectured (wrote a lot of examples on the board and presented explanations). Then, students worked individually on practice sheets as I circulated around the room to give feedback and help. Kids in the 3rd to 5th grades took notes. 

What is Common Core Reform Math? 

The best explanation comes from Mock Turtle: "[We have] ... different branches of Arithmetic--Ambition, Distraction, Uglification, and Derision." Common Core reform math looks like arithmetic, but it is not the real thing. It is not traditional arithmetic that prepares students for algebra. Reform math is about solving math problems without knowing much math. The idea is that students can solve math problems by critical thinking, which is a stupid idea. Students need prior knowledge to solve math problems. Common Core reform math does not apply arithmetic to problem solving. Indeed, Common Core, which is taught as reform math, is a mismatch to the needs of most students, according to a study from the University of California at Irvine and Penn State, headed by Professor George Farkas. Professor George Farkas points out, “However, activities such as routine practice or drill, math worksheets, problems from textbooks and math on the chalkboard appear to be most effective, probably because they increase the automaticity of arithmetic. Foundation skills need to be routinized so that the mind is free to think.” 

1. This is a traditional arithmetic question for 3rd & 4th Graders in 1877. Today, elementary school kids don't know enough arithmetic to work this problem. Perhaps, the same is true for most middle school and some high school students, too. If 3rd or 4th graders could figure this out in 1877, why can't 3rd or 4th graders in 2015 do the same?
If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches?  

2. This is a 5th grade arithmetic quiz under Common Core's reform math. Reform math screws up arithmetic by insisting that students learn many inefficient, alternative ways to calculate. It is totally absurd! In short, Common Core reform math does not apply real arithmetic to problem solving. For novices, it makes arithmetic confusing and much more difficult than it is. 

1. Find 15.7 + 9.72 by decomposing the number by place value. Show your work.
2. Find 9.53 - 4.6 using a place value chart. Show your work.
3. Find 5.3 x 2.4 using an area model. Show your work.
4. Find 4.8 / 0.8 using a number line model. Show your work.
5. Find 3.6 / 12 using a bar model. Show your work.
Sources: Ray's New Intellectual Arithmetic (1877); Kaplan: A Parent's  Guide to the Common Core Grade 5 (2014)

I believe the way content is taught in America [as constructivist reform math] makes it unduly and excessively more difficult for no good reason. And, the quiz above clearly demonstrates it. 

Rebecca Goldstein (Plato At The Googolplex) writes, "All mortals are fallible, even the smartest among us, including the scientists. We are prey to cognitive lapses, some of them built into the very machinery of thinking, such as the statistical fallacies we are prone to commit." In short, many educators bullheadedly hang on to wrong theories, fads, myths, and instructional methods that aren't backed by real evidence or have been disproved by "countervailing evidence," which they fail to acknowledge. I see this in Common Core reform math. Why are economists, philanthropists, ed school professors, US Department of Education, influential corporations, and politicians making education policy? "Who are these people to tell" educators what and how to teach. They are experts who are not experts at all. I don't trust the so-called experts.   

The BIG Lie 
Education needs a large dose of reality. 
Kids vary widely in academic or cognitive ability, so why feed all students the same curriculum? It is as simple as that. Half the kids are below average in IQ. Kids with very low IQ simply cannot learn the complex content that kids of higher IQ can learn. Educators often preach to kids that they can be anything they want!. No they can't! It’s a BIG lie, says Charles Murray. The idea is part of an Utopian mindset. Other forms include “Everyone can go to college,” or “Everyone can be college/career ready under Common Core,” or  “Everyone can learn algebra,” etc. Such false narratives are filled with fatal flaws and illusions and go against current cognitive science. One of the Common Core test makers confessed that only 50% of the students will pass its mandated tests, but in NY state, only 30% of students passed the Core tests. Furthermore, in the new Common-Core-Aligned GED, the "pass rates plummeted." Incidentally, IQ or intelligence is not distributed by race or class. According to Chua & Rubenfeld (The Triple Package) point out, "There are black and Hispanic subgroups in the United Sates far outperforming many white and Asian subgroups. 

Memorization of Number Facts Is Essential, Starting in 1st Grade

Students should not calculate single-digit facts (i.e., via Common Core reform math strategies), they should memorize them so they are instantly usable in solving math problems. Price, Mazzocco, & Ansari (The Journal of Neuroscience), explain that students who memorize by rote single-digit math facts starting in 1st grade [traditional arithmetic] rather than always calculating them [Common Core reform math] become much better math students, especially when the math becomes more complex, and the benefits reach far into high school with significantly higher PSAT scores. Being able to retrieve 5 + 6 = 11 automatically from long-term memory is different from [and much better than] calculating it using a Common Core reform math strategy (e.g., 5 + 6 = 5 + 5 + 1 = 10 + 1 = 11), which is a different mental process that clutters working memory, leaving less space for problem solving.

It is a lie to think that any good teacher can teach anything well. Furthermore,  it is sad that "most of our teachers don't possess a deep working knowledge of any discipline, at least not in the way that good teaching demands," writes Jonathan Zimmerman (Why Is American Teaching So Bad?). This is especially true of elementary and middle school teachers. The schools of education are to blame. Teachers are not trained to teach, says Elizabeth Green (Building a Better Teacher). Teachers are weak in math and science. They don't understand the structure of math, how and why math works, how one idea connects to another, or how to teach these ideas coherently, etc. They just follow lousy textbooks or the Common Core materials handed to them.

It's The Test

Jonathan Zimmerman, an education historian, paraphrases Dana Goldstein (The Teacher Wars), "For centuries, Americans have simultaneously lauded teachers’ moral virtue and deplored their lack of adequate knowledge and skills." But this has changed. Today teachers are focused on test scores, test-prep (a narrowed curriculum)--not on teaching moral virtue, such as in the "McGuffey-Reader style curriculum" era when both basics and moral values were taught well. "Schools are now rewarded or penalized based on their students' performance on standardized tests." 
I am not a test score. I am a real person.
Education is reduced to a test score. Goldstein writes, "American teachers feel alienated from education policy making," which has been taken over by "politicians, corporate philanthropists, and economists." To make matters worse, Common Core has been imposed on teachers and students by government mandates and big money. The only way to teach arithmetic is the Common Core Way. It's bunk! 

Teach This, This Way, Period!

If I were still teaching, and if an administrator told me I had to teach this curriculum in only this way, then I would walk out of the classroom. This is the situation teachers face daily with Common Core reform math, which is couched in school culture. 

When I introduced algebra to little kids, grades 1-5, as a retired guest teacher, I created the curriculum, wrote the practice materials, and taught the content as I saw fit. I did not align the content with state standards. The algebra concepts helped kids understand arithmetic better and how important it was to automate number facts and standard algorithms in long term memory (knowledge). I went against the district policy that kids should work in groups. There were no calculators or manipulatives. Kids did not collaborate in groups, etc. I lectured (explained how math works through examples), asked questions, then circulated around the room to give feedback and help to individual students as they worked on the practice sheets I made. Kids in grades 3-5 also took notes. 


Learning knowledge, which is what education should be about, allows students to learn more knowledge faster, which is a basic concept in cognitive science. Will Fitzhugh (The Concord Review) writes, "E. D. Hirsch and others have shown that having knowledge is what makes it possible to gain more knowledge. And being able to gain more knowledge is really necessary in life." In short, the more you know, let's say in arithmetic, the more arithmetic you can learn and the faster you can learn it because new knowledge builds on old knowledge.  Learning traditional arithmetic in long-term memory, starting no later than 1st grade, which requires memorization, repetition, and lots of practice, will make students smarter, faster, and prepare capable students for algebra-one by middle school--if teachers follow the Core Knowledge K-8 math sequence. In contrast, Common Core eliminates STEM, junks traditional arithmetic, cursive writing, and pushes most algebra standards to high school. The focus of Common Core is test taking, not knowledge gaining. 

The Wisdom of Old Souls

In 1895 the "Committee of 15," sanctioned by the National Education Association (NEA), made recommendations for elementary schools (grades 1-8). Arithmetic should be taught beginning in 1st grade with 60 minutes of oral drill daily plus five textbook lessons a week to prepare students for geometry and physics in 7th grade and algebra in 8th grade. These recommendations make good sense even today, but, unfortunately the wisdom of old souls has been replaced by a progressive agenda of reforms (ideology) that doesn't work (e.g., Common Core taught as constructivist reform math, minimal teacher guidance instruction, etc.).

©2015 LT/ThinkAlgebra