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
Wednesday, February 25, 2015
Tuesday, February 3, 2015
Reflections
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).
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.
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.
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 ParticleAdventure.org.)
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.
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.
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."
Responsibility
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."
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.
Knowledge
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
ThinkAlgebra.org
Comments: ThinkAlgebra@cox.net
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.
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| Sort kids into homogeneous math sections by achievement starting in 1st grade. |
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.
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| A typical first grader in my Title 1 Teach Kids Algebra Enrichment Program builds function tables and graphs linear equations in Q-I. |
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| TKA teaches how math works. LT |
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 ParticleAdventure.org.)
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.
"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.
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| 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... |
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."
Responsibility
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
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.”
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."
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| I am not a test score. I am a real person. |
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.
Knowledge
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
ThinkAlgebra.org
Comments: ThinkAlgebra@cox.net
Friday, September 26, 2014
Outcomes
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| Merry Christmas Be Well! |
"Outcomes" is a collection of compelling and/or often contrary cogitations regarding Common Core and education. [Last changes made on December 24, 2014]
See my latest reflections and random thoughts about education:
Click Reflections.
Richard Feynman used to say that you don't understand anything until practiced. Even then, your "understanding" could be minimal not only because "understanding" is an indeterminate idea that is difficult to measure, but also because "understanding" develops slowly over years of study. Incidentally, "understanding" does not produce mastery; practice does! Furthermore, kids should not calculate single-digit number facts, they should memorize them starting in 1st grade, so they stick in long-term memory. If we want to jumpstart kids in math then students need to be taught traditional arithmetic starting in 1st grade--not Common Core reform math. Furthermore, we must kick out accountability testing. Note. Sometimes, Feynman would write equations on the board during a lecture on cutting-edge physics, then suddenly exclaim--this makes no sense. This is the way it works out, but I don't understand it.
Moreover, Paul Thomas (in his blog, The Becoming Radical) says that Common Core is the problem, which has drained billions from the public schools to pay testing and technology companies, money that should have been spent on essential school resources, textbooks, science equipment, etc. Moreover, as many writers have stated before, the new testing is designed to fail at least half of the kids. Why design tests that fail half of the kids? Thomas writes, "The Common Core advocacy is market-driven, benefiting those invested in its implementation. But ... [the] market success is at the expense of the very students who need our public schools." I should add that kids are taught [Common Core] reform math, not tried-and-true traditional arithmetic. Common Core reform math does not prepare students for algebra in middle school, puts capable students behind, and wastes valuable instructional time by teaching inefficient, alternative operations that lead nowhere. Common Core reform math looks like arithmetic, but it is not the real thing.
Common Core reform math claims it teaches understanding better than traditional arithmetic. I guess, it must be by magic because the claim has no basis in evidence. Common Core advocates predict that learning different ways to calculate will pay off when students get to higher math, but there is absolutely no proof that this is the case. In fact, the opposite is true. In contrast, traditional math, which involves the memorization of number facts, the fluent use of paper-pencil standard algorithms through practice, and the application of arithmetic to solve routine word problems, has been a tried-and-true method for centuries. Indeed, factual and efficient procedural knowledge in long-term memory are required to solve problems. When taught well, traditional arithmetic prepares students for algebra-one by middle school, especially if the Core Knowledge K-8 math sequence is taught well. In essence, Common Core methods make simple arithmetic difficult and more complicated than it is. This is The Common Core Way: Let's not simplify arithmetic; let's complicate it. Bad idea.
Common Core claims it emphasizes critical thinking, but, as Immanuel Kant wrote, "Thoughts [critical thinking] without content [knowledge] are empty...." The leaders or so-called masterminds in education lack wisdom, which cannot be taught. They also lack knowledge in mathematics and cognitive science. Hippocrates wrote, “There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance.” This is what we have in education today: ignorance and progressive ideology. Evidence does't matter.
What cognitive science research reports is that doing math starts the process of understanding it, which grows gradually over the years with practice, repetition, and experience. There are no short cuts. In contrast, the basic idea in Common Core reform math and 21st Century Skills is that children can become problem solvers without building a solid knowledge base in long-term memory, stating in 1st grade. How? Apparently, by working in groups to collaborate, by doing inquiry/discovery activities with minimal teacher guidance, and by learning inefficient, alternative methods to calculate (reform math). It's total nonsense, of course! Also, our best college students don't want to be teachers or go into STEM; they go into finance or other fields to make money. Many of the financial fields require calculus. Some higher-level college math courses are required in many majors--not just STEM. Finland's education system is often promoted as a model for the US. But, it should be noted that, according to the most recent math scores from TIMSS, Finnish 4th and 8th graders are about the same in math achievement as American 4th and 8th graders. Moreover, both Finnish and American students are left in the dust compared to the scores of students in Asian nations or regions. Moreover, Paul Thomas (in his blog, The Becoming Radical) says that Common Core is the problem, which has drained billions from the public schools to pay testing and technology companies, money that should have been spent on essential school resources, textbooks, science equipment, etc. Moreover, as many writers have stated before, the new testing is designed to fail at least half of the kids. Why design tests that fail half of the kids? Thomas writes, "The Common Core advocacy is market-driven, benefiting those invested in its implementation. But ... [the] market success is at the expense of the very students who need our public schools." I should add that kids are taught [Common Core] reform math, not tried-and-true traditional arithmetic. Common Core reform math does not prepare students for algebra in middle school, puts capable students behind, and wastes valuable instructional time by teaching inefficient, alternative operations that lead nowhere. Common Core reform math looks like arithmetic, but it is not the real thing.
Common Core reform math claims it teaches understanding better than traditional arithmetic. I guess, it must be by magic because the claim has no basis in evidence. Common Core advocates predict that learning different ways to calculate will pay off when students get to higher math, but there is absolutely no proof that this is the case. In fact, the opposite is true. In contrast, traditional math, which involves the memorization of number facts, the fluent use of paper-pencil standard algorithms through practice, and the application of arithmetic to solve routine word problems, has been a tried-and-true method for centuries. Indeed, factual and efficient procedural knowledge in long-term memory are required to solve problems. When taught well, traditional arithmetic prepares students for algebra-one by middle school, especially if the Core Knowledge K-8 math sequence is taught well. In essence, Common Core methods make simple arithmetic difficult and more complicated than it is. This is The Common Core Way: Let's not simplify arithmetic; let's complicate it. Bad idea.
Common Core claims it emphasizes critical thinking, but, as Immanuel Kant wrote, "Thoughts [critical thinking] without content [knowledge] are empty...." The leaders or so-called masterminds in education lack wisdom, which cannot be taught. They also lack knowledge in mathematics and cognitive science. Hippocrates wrote, “There are in fact two things, science and opinion; the former begets knowledge, the latter ignorance.” This is what we have in education today: ignorance and progressive ideology. Evidence does't matter.
Note. Please excuse typos, misspellings, grammatical errors, etc. I rearrange ideas and often digress. I am not a writer. I add, move, rewrite, delete, and repeat text. Please excuse redundancies. I write about Common Core, mathematics, education, science, reforms, and other odds and ends. LT, ThinkAlgebra.org. I can be reached at ThinkAlgebra@cox.net. It's not that our kids are dumb; it's that they aren't taught to master content like kids in top-performing nations. For decades, US math instruction has been lousy, and I fear that Common Core will make it worse.
Physical exercise and learning new things have impact on cognitive ability. The lack of recess, physical education, the arts and music has hurt kids. The lack of focus on knowledge and essential skills, especially in mathematics and science has hurt students. Ninety-minute periods in middle school and high school are absurd. Often, students are placed into higher level math classes or AP classes without the proper background knowledge, skills, and experience. Typical students don't take calculus to learn to be creative. They take calculus to learn calculus and how to apply it. It should also be noted that most of the students who take AP calculus don't develop insight or a good understanding of the math. It requires memorization, repetition, and extensive practice.
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| My Teach Kids Algebra (TKA) Class: 3rd Grade. Kids can learn more math than K-8 teachers are prepared to teach. |
In contrast, American kids [starting in 1st grade] simply are not taught to automate fundamentals, especially the memorization of number facts, the fluent use of standard algorithms, and the application of factual and efficient procedural knowledge to solve routine word problems (applications) to build a strong storehouse of factual and procedural knowledge in long-term memory to use for problem solving. To accomplish these, extensive practice and repetition are necessary, yet, for decades, memorization and practice have fallen out of favor in progressive classrooms. Instead of traditional arithmetic that advances capable kids to algebra-one by middle school, under the one-size-fits-all Common Core dictate, American kids are taught questionable, unverified, confusing, inefficient, alternative methods called reform math, along with test prep. Kids aren't educated; they are coached to take tests.
In 2009, I wrote that Common Core was not for STEM students. In short, Common Core math standards do not prepare kids for the many careers that involve STEM or parts of STEM. Moreover, starting in 1st grade, the new math standards are not world class and below many Asian benchmarks. Common Core has left high achievers in the gutter, which is another reason that Common Core and its testing regime should be tossed in the waste can of bad ideas. It makes little sense to punish achievers with a non-challenging curriculum. Through top-down, one-size-fits-all Common Core, laws, mandates, and regulations, the federal government, big money, such as Bill Gates and others, and powerful corporations, such as Pearson, are in control of education--not the states, not the school districts, and not the teachers. It is tragic that the American school system chases after mediocrity and equalizes downward.
Outcomes" is a collection of compelling and/or often contrary cogitations regarding Common Core, testing, and education. Understanding does not produce mastery; practice does. Early on, kids are not mastering basic arithmetic. For decades our best kids have been ignored and bored. Moreover, for decades, educators have been training kids to take tests, but that's not education. Equalizing performance (outcomes) is the central theme of Common Core. Talent and individualism cannot flourish under such an ideology, which focuses on group work, inquiry-discovery and other reform or progressive pedagogy. Even our best math students underperform under this regime. They are merely average, if that, compared to their peers in Singapore or South Korea.
Marc Tucker (Fixing Our National Accountability System), writes "High stakes tests in the top performing countries are used to hold students, not teachers, accountable, the obverse of what happens in the United States." Today's students don't study enough, read books enough, write essays enough, learn math and science enough, or exercise enough. Teachers need to stop inflating grades and using ineffective instructional methods, starting in 1st grade. Parents and teachers should stop fixating on a child's self-esteem and worry more about the child's competency.
There has been a shift of power in America "from ordinary Americans to the corporate and financial elite," says Bob Herbert ( Losing our Way). This is especially evident in education in which there is no equality of funding. Poor school districts stay poor and lack resources. The people making huge profits under Common Core are corporations, such as Apple, Microsoft, Pearson and other technology, publishing and testing companies, etc., along with powerful hedge fund managers and their investors. Furthermore, nearly every writer of the Common Core standards, almost all of whom were from special interest groups (e.g., ACT, College Board, Achieve, etc.), is cashing in--one way or another. These people want power, not truth. Common Core has never been shown to work, yet it is promoted by the US Department of Education and backed by government money, big money from Bill Gates who financed Common Core, and others. Many influential business people, politicians, governors, and powerful heads of institutions (e.g., AFT, NCTM, NEA, think tanks, academia, etc.) were asked to endorse Common Core--although they never read the document.
Our schools should not copy schools in other nations, such as Finland, South Korea, Singapore, etc. [Singapore math works in Singapore. In math, Singapore tracks less-able students in 1st and 2nd grade and again in 4th to 6th grade. The Singapore 6th grade exit test determines which "level" of secondary school the child attends next. In short, Singapore prudently tracks students. The system works well in Singapore, but for decades, the US has been diametrically opposed to tracking, which, I think is a mistake because kids vary widely in academic ability. Finland's international test scores (TIMSS) are about the same as US 4th and 8th graders. Also, only 28% of Finnish 9th graders correctly calculated 1/6 x 1/2 in research conducted by Dr. Olli Martio, University of Helsinki.]
The fact is that Finnish K-8 students are not any better in math than American K-8 students. In contrast, top-performing Asian nations, which have from about 40% to 50% of 4th and 8th grade students scoring at the Advanced Level in TIMSS, leave Finnish and American students in the dust. Only 7% of our 8th grade students reach the Advanced Level and only 4% of Finnish 8th graders make that level. In short, US & Finnish students underperform.
The so-called "transformative," progressive reforms in American education, such as test-based reform, have not leapfrogged our students, either. For decades there have been only modest gains, if that, not rapid improvement, says Diane Ravitch. Children have been trained to take tests, not educated, explains Linda Perlstein (Tested). We don't need to transform education; we need to restore it. The top-down, one-size-fits-all reform model [Common Core] of standards>standardized testing>data collecting>punishment) is the road of mediocrity, littered with bad ideas, ill-founded fads (often called innovations), and junk science. Teachers need to take back education. One major step forward is to teach arithmetic as arithmetic (memorizing number facts and learning standard algorithms, starting in 1st grade) and not as reform math, such as in Common Core with its "strategies" approach, which looks like arithmetic but isn't the real thing. Understanding does not produce mastery; practice does. Another major step is to bring back "McGuffey-Reader-style curriculum [from which] American kids learned not only the basics, but also values such as honesty and patriotism, and immigrant kids assimilated by learning our language, laws, and customs (Eagle Forum)."
Bad ideas, fads, myths, whims, and pseudo-science have corrupted traditional arithmetic and have impeded US math achievement for decades. Furthermore, test-based reforms do not foster quality. In fact, in my opinion, states have not been accountable. Statism has not provided most children with a high-quality education for decades. Children are still subjected to contentious top-down, one-size-fits-all reforms that have failed to produce rapid improvement. Children vary widely in academic ability, yet all students are fed the same Common Core curriculum in the name of fairness. In short, no child gets ahead. It makes no sense. It is a recipe for mediocrity.
The fact is that Finnish K-8 students are not any better in math than American K-8 students. In contrast, top-performing Asian nations, which have from about 40% to 50% of 4th and 8th grade students scoring at the Advanced Level in TIMSS, leave Finnish and American students in the dust. Only 7% of our 8th grade students reach the Advanced Level and only 4% of Finnish 8th graders make that level. In short, US & Finnish students underperform.
The so-called "transformative," progressive reforms in American education, such as test-based reform, have not leapfrogged our students, either. For decades there have been only modest gains, if that, not rapid improvement, says Diane Ravitch. Children have been trained to take tests, not educated, explains Linda Perlstein (Tested). We don't need to transform education; we need to restore it. The top-down, one-size-fits-all reform model [Common Core] of standards>standardized testing>data collecting>punishment) is the road of mediocrity, littered with bad ideas, ill-founded fads (often called innovations), and junk science. Teachers need to take back education. One major step forward is to teach arithmetic as arithmetic (memorizing number facts and learning standard algorithms, starting in 1st grade) and not as reform math, such as in Common Core with its "strategies" approach, which looks like arithmetic but isn't the real thing. Understanding does not produce mastery; practice does. Another major step is to bring back "McGuffey-Reader-style curriculum [from which] American kids learned not only the basics, but also values such as honesty and patriotism, and immigrant kids assimilated by learning our language, laws, and customs (Eagle Forum)."
Bad ideas, fads, myths, whims, and pseudo-science have corrupted traditional arithmetic and have impeded US math achievement for decades. Furthermore, test-based reforms do not foster quality. In fact, in my opinion, states have not been accountable. Statism has not provided most children with a high-quality education for decades. Children are still subjected to contentious top-down, one-size-fits-all reforms that have failed to produce rapid improvement. Children vary widely in academic ability, yet all students are fed the same Common Core curriculum in the name of fairness. In short, no child gets ahead. It makes no sense. It is a recipe for mediocrity.
Note. I recall a lesson (written in a university journal as a model) in which middle school kids in groups were discussing whether the 3D object on their desks was a cylinder, or a pyramid, or something else. And, the "researchers," who were students (education majors), just let the kidsgo on and on. What a waste of time! The discussions of the different groups were recorded, and a sample of one group's dialog (4 kids) was in the journal. ("I think Jane is right. It's a pyramid, and she's always right.) What irks me is that this lesson serves as a model for other teachers to follow. The idea is that a good teacher should watch the groups and not intervene. It's a bad idea. (Incidentally, the object was a cylinder. Any good first grade student would know that. When I find the journal in one of my boxes, I will print the entire conversation.)
Common Core: In my opinion, educators have been conned and left out in the cold. The "powers that be" always win. Teachers have been cut off from the decision making process. This is what happens when an oligarchy rules education. The cult-like obsession with Common Core and its excessive standardized testing is illogical and hurts kids. Teachers are caught in a quagmire. They do what they are told to do to keep their jobs. Instead of teaching essential knowledge that kids must know to advance, standardized testing (NCLB) has become the centerpiece of US education, and Common Core is right in the middle. In my opinion, these top-down reforms have little to do with improving education. The new standards, themselves, the way they are taught (as reform math), and excessive standardized testing are counterproductive. Diane Ravitch has characterized No Child Left Behind as test and punish, which is an abhorrent policy. Peter Green (Curmudgucation blog) writes, “Why do we have these policies that don’t make sense? Why does it seem like this system is set up to make schools fail? Why do states pass these laws that discourage people from becoming teachers?"
Note. First Grade Common Core: "Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13)." (Common Core 2014 website.) Notice that Common Core asks students to calculate single-digit sums, which is an epic error. Students should not calculate single-digits sums in working memory, they should automate them in long-term memory for instant use. Many of the strategies named above are a crutch and not needed. Read below.
➡ Students should not calculate single-digit facts (i.e., via 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, rather than always calculating them, become good math students, and the benefits reach far into high school with much higher PSAT scores. Being able to retrieve 5 + 6 = 11 automatically from long-term memory is different from [and better than] calculating it using a 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.
In my view, the strategies approach, starting in 1st grade to teach arithmetic, makes little sense, and many of the strategies, themselves, are only tangentially related to arithmetic, that is, Common Core reform math is not the real thing. For example, if a 1st grade child knows 6 + 6 = 12, then 6 + 7 is one more than 6 + 6 or 13 [because 7 is one more than 6]. While this type of reasoning helps kids, but I would not call it a strategy. I call it mathematical thinking that a 1st grader can do; however, before the child can do "one more than" reasoning, the child must first memorize [automate in long-term memory] 6 + 6, which is not the goal, as far as I can tell, in 1st grade Common Core. (In contrast, it is the goal in Singapore 1st grade.) My argument is that if a child can memorize 6 + 6 = 12, then should not the child also be able to memorize 6 + 7 = 13, 6 + 8 = 14, etc.? The facts are obvious on the number line. Why waste time on strategies or inefficient alternative algorithms. Why not just memorize the number facts and learn efficient standard algorithms. Ah, but this is not the "Common Core Way." Furthermore, Common Core claims that its standards are pedagogy free, which is a total lie. Just read the 1st grade standard above--it's the Common Core Way to teach math and it goes against cognitive science according to researchers Price, Mazzocco, & Ansari.] Indeed, the overemphasis on conceptual understanding in Common Core has pushed aside the vital importance of factual and procedural knowledge that leads to understanding. Understanding comes from extensive practice of efficient procedures, such as standard algorithms, which require the memorization of number facts for instant recall from long-term memory. [Note. In the very early 70s, the best textbook for 7th grade was Concepts, Skills, and Applications (Scott Foresman, Publisher), so the idea of conceptual understanding is not new. Is it possible to do math with limited conceptual understanding? Yes, Isaac Newton invented a calculation method that solved physics problems (calculus). The methods always worked (agreed with experimental observations), so he knew how to apply the calculus, but he did not know why his calculus worked. It just did. The conceptual basis for his method would take another 200 years. Maybe, we should remember that kids are not little mathematicians. Understanding grows slowly over the years, and it is difficult to measure.]
Frankly, strategies should never replace or hinder the memorization of basic facts or the learning of standard algorithms through practice in first grade or any grade. Memorization and practice to gain factual and procedural knowledge in long-term memory do not squelch creativity and learning, as many say. Common Core is the latest version of reform math that downgrades memorization and practice for mastery in the early grades, which is a primary reason that our kids suck at basic math. The strategies were part of NCTM math standards and reform math pedagogy of the 90s, which failed.
[However, these strategies are not new. You can find some of them in old elementary textbooks, such as those in the 1950s, as Barry Garelick has pointed out, but in the 50s, as Garelick explains, kids also memorized the facts and learned standard algorithms to do arithmetic. Today, kids don't. In my opinion, some of the reform math strategies are enrichment or involve unnecessary, often confusing calculations that clutter working memory. In short, the strategies don't focus on or drive students to automate basic arithmetic facts in long-term memory. Common Core kids come into 3rd grade and can't add 231 + 47, a calculation my Title 1 urban 1st graders did well in the early 80s. Under Common Core, the standard algorithms, which are a function of place value and automated number facts, have been pushed aside.]
Note. Typical first grade students can memorize the addition facts and learn the standard algorithm. This is what is done in Singapore. I taught 1st grade self-contained in the early 80s under a deseg order at an urban Title 1 school. The children were good at memorizing and at using what they learned to solve word problems. They were pleased with themselves. More recently (2011) I taught algebra to over 40 minority 1st graders at a city Title 1 school in Tucson. In each case, I made up my own math curriculum. Very young children can learn more than teachers are prepared to teach. We have known this since the 1950s, even before.
The Common Core strategies may look like arithmetic, but they are not the real thing. This is not smart arithmetic education. Why make something that is not difficult, difficult? Also, in Common Core, words like "understanding" and "fluency" are vague or ill-defined. The word "mastery" or "automation" of essential arithmetic procedures and facts in long-term memory for use in problem solving is not mentioned. The concepts of addition and subtraction are easy for 1st graders to understand at their level using a number line. Understanding, which is difficult to measure, says Daniel T. Willingham, a cognitive scientist, changes over time and grows slowly, and it seems to be a product of practice. Richard Feynman used to say that you don't understand anything until practiced. Feynman, a Nobel Prize winner in Physics, is saying that understanding and insight come through practice, lots of it, not instantly. Feynman practiced calculus at breakfast, on this way to work, at lunch, at home at night, etc. It drove his 2nd wife crazy and ended in divorce. He was brilliant, yet he took time to review and practice fundamentals, which is what all talented people do. They go back to fundamentals.
A lot of K-12 students have just bits and pieces of understanding in arithmetic, if that, and the same is true for algebra because math fundamentals were not practiced to mastery. Indeed, many students who take AP calculus, AP statistics, or AP Physics never truly understand it. Common Core reform math will not change this.
As I stated years ago, Common Core is a political document intrinsically linked to the government's crumbling No Child Left Behind law, intrusive standardized testing, questionable data collection, and ill-advised policies. Education is ruled by an oligarchy. Educators have been conned and left out in the cold. The "powers that be" always win. Teachers have been left out of the decision-making process. Teaching under Common Core has become a exasperating chore for many classroom teachers rather than a delight, which is a sad commentary on education today. Teachers have to put up with government rules, regulations, policies, including NCLB, and reform after reform after reform. In addition, a few bad apples, poor working conditions, low prestige, and low pay have marred the teaching profession. Then, there is also Common Core that drives some of our best teachers out of teaching and prompts many parents to homeschool their kids. I am fortunate that I am not a teacher in the Common Core era. I left full-time teaching when it was no longer fun, and, a couple years ago, I stopped tutoring algebra and precalculus.
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| 3rd Grade: Teach Kids Algebra |
But, in 2011, I founded a unique algebra program, which was built on arithmetic and designed for little kids as a part-time guest teacher until Title I funding went to Common Core. The school district classified me as a College Prep Assistant. I wrote and produced my own materials. Algebra comes out of arithmetic.
Outcomes" is a collection of compelling and/or often contrary cogitations regarding Common Core and education. I think we ought to bury NCLB, Race to the Top grants, wavers, Common Core, and other blather coming from "high-IQ idiots" of the ruling class, who pretend to know what's best for kids [even though they have never been classroom teachers]. They believe ordinary people are too stupid to know what is good for students; consequently, over the years, they have produced a flimsy education system that equalizes downward. Common Core reform math screws up arithmetic, doesn't measure up to the Asian level or prepare capable students for Algebra 1 in 7th or 8th grade. This is moving backwards, not forward. Smart does not mean wise or prudent. We need leadership that is wise, but where is it? Apparently, there is very little wisdom in the top-down model of education.
"The only reform that almost no one in authority wants to see enacted: equalizing [K-12] funding nationwide," points out William Deresiewicz. There should not be rich school districts and poor school districts based on property tax, which has created a "class" inequality we see today. Even with equalizing funding, inequalities will still exist. Some kids are better at math than others, etc. Academic ability, conscientiousness, and hard work count!
There is a group of people who "believe" they are smart enough to make sweeping decisions, or reforms, or changes in education and can control [take over] education through government and big money--even though they have never been teachers in a K-12 classroom (the real world). [Why are non-educators making K-12 ed policies?] These people, who often claim a higher status than the rest of us, are dead wrong! Unfortunately, there are many self-serving high-IQ idiots in charge. William Deresiewicz writes, "We do need experts, to be sure, but we also need them not to be in charge." The idea that the "affluent and powerful have merit" and the rest of us don't matter that much is stupidity. The education system has created an underclass. "The poor are poor because they are inferior," the progressive elite say. Some teachers believe this, too. Such an assumption is rubbish. Who are these people? They are often politicians, corporate philanthropists, professors, economists, etc., along with many federal, state, and local officials--people who never taught K-12. There are exceptions, of course.
Dana Goldstein (The Teacher Wars) writes, "American teachers feel alienated from education policy making." If teachers don't matter that much in making education policy, then kids don't matter either. "It's for the kids" mantra is the ultimate deception. Telling kids they can be anything they want to be is another deception. Common Core is yet another reform forced upon teachers and kids with consequences--from an increased number of standardized tests and practice tests, instructional time stolen for test prep, not enough time [or money] for art, music, field trips, etc., says Dana Goldstein.
Quick Fixes: Hidden Danger
Evgeny Morozov (To Save Everything, Click, The Folly of Techological Solutionism) asserts, "The ballyhoo over the potential of new technologies to disrupt education--[such as online courses, etc.]--is a case in point. Digital technologies might be a perfect solution to some problems, but those problems don't include education--not if by education we mean the development of the skills [knowledge] to think critically about any given issue." Furthermore, we have been bamboozled into believing that the best way to improve education is to put more and more technology [Internet, computers, iPads, calculators, laptops, cell phones, smart boards, etc.] into our classrooms. The "quick fix" idea has never worked in education. We have had digital technology in our classrooms since the late 80s, and our test scores in math have remained relatively flat. Blackboards have been replaced by costly smart boards. Common Core has thrown out traditional arithmetic and cursive writing. Cognitive science has been marginalized and replaced by junk science. Teachers are told to teach the "Common Core Way." It's pathetic! "Schools concentrate all their efforts on improving test scores even if children learn much less as a result," explains Morozov. Technology and test-based instruction will not fix education.
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| Junking traditional arithmetic and replacing it with Common Core reform math was an epic error. Common Core reform math frustrates and confuses students and ignores basic cognitive science. |
Amy Chua & Jed Rubenfeld (The Triple Package), point out, "In short, education--like hard work--is not an independent, but a dependent variable. It's not the explanatory factor; it's a behavior to be explained. Successful groups in America emphasize education for their children because it's the surest ladder to success." A stress on education WITH high expectations, discipline AND hard work, and impulse control are cultural behaviors. It seems that many Americans may have lost these, say Chua & Rubenfeld. "America failed the marshmallow test."
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| Veruca Salt (Julie Cole) in Willy Wonka I Want It Now! |
Bad ideas plague K-12 education.
The first really bad idea was to junk traditional arithmetic and replace it with reform math ideology, such as in Common Core, which, in my opinion, is a regression to more mediocrity.
Furthermore, Kids are not the same so why feed them the same curriculum? Kids vary widely in math ability, yet they are placed in the same math class under the pretense of fairness, which is ambiguous. The kids who are better in math are bored with CC reform math and don't advance. The one-size-fits-all dogma of Common Core supports inclusion, not tracking. In contrast, Singapore tracks incoming 1st grade students who are weak in number skills (a pull out program), but, in my opinion, Singapore should also track (pull out) the very best math students, starting in 1st grade. Schools in the US don't track young children, which, I think, is a blind position. Mike Petrilli asserts, "The greatest challenge facing America's schools today isn't the budget crisis, or standardized testing, or teacher quality. It's the enormous variation in the academic level of students coming into any given classroom." The enormous variation, which is caused by inclusion [fairness] policies, has been a recipe for mediocrity in math education. Cognitive ability to learn math well, which varies widely, has been ignored. A child's academic ability and effort matter in learning arithmetic, algebra, and higher mathematics well.
We are living in "an age of diminished expectations," explains Ross Bouthat (New York Times Columnist). In my opinion, this is especially true in education. We underestimate the math and science content kids can [and should] learn, and we have become complacent with underperformance, which we judge as normal. Furthermore, the lack of critical thinking by those in charge of education, from local school boards and politicians to state and federal government agencies and officials and big money people like Bill Gates, has produced many inept policies, fads, myths, and bad ideas. Educators need to break free from a "standards-testing-data" mind set that has gripped schooling for decades. We don't need to transform education; we need to restore it. This is the antithesis of Bill Gates, Common Core, which Gates financed, and the progressive agenda of the US Department of Education and special interest groups that have profited hugely from Common Core.
The outcomes of K-12 Common Core are unknowable, yet billions of dollars have been put into Common Core. The outcomes (college/career readiness) claimed by Common Core reformers are speculation as there is no evidence they are true. If the math standards are inadequate, that is, not up to world-class benchmarks, which is the case, then the curriculum based on the Common Core standards would be ineffectual in that most students will continue to underperform.
Our schools may not be in decline as Diane Ravitch argues; however, they lag far behind in math and science education. The "We'll Try Anything" approach has failed again and again. The latest assumption is that our students need more technology in the classroom to raise test scores. Wrong! We have had computers and calculators in classrooms since the early 80s, yet math test scores have remained roughly flat. Is there a correlation here? Most Asian schools are low tech. In addition, the parents of Asian students pay for tutors, math lessons, and after-school cram schools. Typical American parents don't.
Michael E. Martinez (Future Bright) writes, "We recognize that some children display uncommon aptitudes for music, swimming, chess, mathematics, science, or art. Indeed, the entire field of gifted and talented education is premised on an assumption that some children are marked with unusual aptitude--so much so that the standard curriculum is an impediment to the child's development." An inclusion approach (everyone gets the same) has been a recipe for mediocrity, yet, it is a commonplace in our schools.
Our kids are not dumb. Indeed, better instruction, a focused curriculum, and teacher quality are important, but, according to Will Fitzhugh (Concord Review), "The most important variable in student academic achievement is not teacher quality, but the student's academic work." It's not the teacher's fault that a child is lazy or doesn't come to school prepared and ready to learn. Early on, parents should provide the environment for achievement and "teach their kids how to be successful in school," says Larry Winget. Mike Petrilli asserts, "The greatest challenge facing America's schools today isn't the budget crisis, or standardized testing, or teacher quality. It's the enormous variation in the academic level of students coming into any given classroom." The enormous variation, which is caused by inclusion [fairness] policies, has been a recipe for mediocrity in math education.
We have separated math from key applications, especially applications in science. Morris Kline (1973), a mathematician, writes, "Mathematics is not an isolated, self-sufficient body of knowledge. It exists primary to help man understand and master the physical, the economic, and the social worlds. It serves ends and purposes."
Prove Its Worth!
In math, we should show students "what mathematics accomplishes," and one enormous venture (Science--A Process Approach or SAPA) did this in 1967. In the 60s, SAPA taught mathematics in science. [I was fortunate to be a K-6 SAPA teacher.] K-6 SAPA taught the math kids needed to understand and do the science. In fact, 4 of the 6 processes taught in 1st grade (SAPA, Part B) were math or math related, such as numbers (arithmetic), measurement (metric), graphing, and geometry. Alfred North Whitehead (1912) said that educators need to prove mathematics, that is, "prove its worth!" Most ES teachers are weak in both math and science, so they can't do this. Indeed, many MS and HS math teachers don't have a science background. How many math teachers can actually show what mathematics accomplishes to prove its worth? The math education conundrum that existed 100 years ago in Whitehead's era, exists today! Why learn something when educators cannot prove its worth?
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| My 4th Grade Algebra Classes Doing Science 2012-2013 School Year Photo by LT |
In my Teach Kids Algebra (TKA) program, 4th graders did a ball bounce experiment from 5 different heights, 3 trials per height, recorded data, calculated averages, graphed it, figured out the best fit line, the slope, wrote a linear equation, and tested interpolated and extrapolated values. There was class discussion on measurement error.
Real scientists don't set out to prove their theories or equations right; they set up experiments to prove they were wrong. In this sense, science self-corrects itself. In short, science progresses through theories being falsified, says Karl Popper (1902-1994).
Science is highly mathematical, but you would not know it when you read elementary school science textbooks which are practically math-less, according to the late Richard Feynman, Nobel Winning Physicist. Perhaps, we should think like Max Tegmark, physics professor at MIT (Our Mathematical Universe), who writes, "Everything in our world is purely mathematical--including you." He says the Universe is not only similar to mathematical structure; it is mathematical structure. The physical world is described by mathematics; our reality is mathematics. Mathematics describes nature very well, and it is for this reason that young children should learn it well. Lastly, the world was lucky that Isaac Newton decided not to become a farmer as his mother insisted.
(Note. In TKA, no calculators were used. Also, it should be noted that the two classes of 4th graders I taught were Title 1 kids in a poor urban school district, 95% Hispanic.)
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| Mr. Wizard (Don Herbert) 1950s. Girl: Rita |
Mr. Wizard provided science instruction every Saturday morning for boys and girls. Today, we have nothing like Mr. Wizard, nothing to inspire children to get interested in real science. (Note. With the advent of test-based instruction, science faded out of the curriculum in many schools. It was never a priority in the first place.) Photo: Rita wrote equations for area. She thought the area of the inner white circle was larger than the the area of the outer white circle, but both turned out to be the same (9pi). "The next time you want to be accurate about something, make no judgements. Instead, measure and calculate." It is easy to fool our minds and get things wrong, so measure and calculate. Good advice for every math and science student and adult.
Bad ideas, fads, myths, whims, and pseudo-science have
corrupted traditional arithmetic and impeded math achievement.
Insert
In the New Math (1970s), the teacher asks, why is 9 + 2 = 11? A typical student should say 9 + 1 is ten (make 10) and one more is 11. The teacher says, “Wrong. 9 + 2 = 9 + (1 + 1) by the definition of 2. Thus, by the associative law, it becomes (9 + 1) + 1; therefore, 9 + 1 is 10 by definition of 10, and 10 + 1 is 11 by the definition of 11.” The New Math used number properties and definitions to do arithmetic. It is correct arithmetic, but it is also tiresome and confusing. (Reference: Why Johnny Can’t Add, Chapter 2, 1973, by Morris Kline, mathematician)
Today, the Common Core reform math teacher asks, “What is 8 + 5?” One student might answer 13. Another student might say the same as 5 + 8. The teacher says, “No. You didn’t tell why it's 13.” The teacher continues, “It is 10 + 3 = 13! You add 2 to 8 to make 10 and subtract 2 from 5 to make 3 [a double calculation], thus, it is 10 + 3 or 13," making a total of three calculations that clutter working memory. The Make Ten strategy changes an expression into another expression comprised of more compatible numbers that yield the same answer. The mathematical principles are not explained. It makes arithmetic overly complex, confusing, and perplexing to little kids. Make Ten and other reform math strategies have been major failures since the early 90s. Reform math strategies, such as make ten, are tangential, and should not be the primary or first method taught to kids for doing arithmetic. Note. Isaac Newton, a polymath, invented calculus, a fast way to get the right answer to physics problems, but the conceptual underpinnings of his calculation method didn't come for another 200 years. Newton knew that his fast method always worked [It agreed with observations]. He knew how to apply it, but he didn't know why it worked. Isn't getting the right answer to a problem the main purpose for learning mathematics? It was good enough for Newton. Nobel Prize winning physicist Richard Feynman often remarked that you don't understand something until practiced. In short, the how comes before the why.
In short, instead of memorizing basic facts, "make 10" and other strategies become the method of adding single digits, which is not practical or useful with larger numbers. Make 10 is not essential and leads nowhere. It's enrichment, if that. It is often confusing to young children and counterproductive.
This "make ten" approach is tedious, confusing, and not practical. Kids don't need to explain or write reasons why 8 + 5 is 13. It just is. Check it on a number line. It is obvious.
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| 8 + 5 = 13. It is obvious! It needs to be memorized to stick in long-term memory. There is no need to explain it, or write out the reasons why it is true, or make a drawing. And, there is no need to Make 10! |
The fact needs to be memorized. Kids don't have to prove that 1 + 2 = 3 or that 2 + 1 = 3. It is obvious! We accept it as a fact without proof. Likewise, students should not struggle through clever, complicated manipulations to show or prove that 8 + 5 = 13. We accept it as fact (8 + 5 = 13) and memorize it. In my opinion, Make Ten and many other Common Core reform math strategies lead nowhere and inhibit learning. Sadly, the strategies or models are commonplace in Common Core reform math. (It is the Common Core way.) Consequently, our kids cannot add because the mastery of basic number facts is not the actual goal. Furthermore, standard algorithms, which are excellent for practicing basic facts, have been shoved aside early on in Common Core reform math. Therefore, kids do not master basic arithmetic to prepare for algebra-one in middle school. Single-digit math facts should not be calculated; they should be automated for instant use.
1. Richard Feynman once wrote, "If you ever hear yourself saying, 'I think I understand this,' that means you don't." (Quote found in The Sense of Sytle by Steven Pinker. The late Dr. Feynman was a Nobel Winner in Physics.)
2. "The better you know something, the less you remember about how hard it was to learn." Teachers must know math, be good at explaining it, and prove its worth, starting in 1st grade. Understanding does not produce mastery; practice does. In short, students need explicit instruction (not inquiry/discovery based instruction in groups), along with memorization, repetition, practice, and lots of word problems. (Quote: The Sense of Sytle by Steven Pinker.)
3. The world was lucky that Isaac Newton decided not to become a farmer like his mother insisted. He hated farming. He went back to school at age 17 and started to study mathematics.
4. Common Core reform math looks like arithmetic, but it is not the real thing. Reform math doesn't get our kids where they need to be.
5. Alfred North Whitehead (1912), writes, "There can be nothing more destructive of true education than to spend long hours in the acquirement of ideas and methods which lead nowhere....Quadratic equations are part of algebra and algebra is the intellectual instrument for rendering clear the quantitative aspects of the world." Whitehead defends quadratic equations as important part of every child's education because they explain how the universe works, such as in Newton's laws of motion etc.
Common Core presents a flimsy "version" of arithmetic that lacks vision and doesn't get our kids to where they need to be. W. Stephen Wilson (mathematician) says that many kids never get to traditional arithmetic, much less master it, which, for centuries, has been the solid background needed for algebra. The reform math methods and strategies, which Dr. Wilson and many other mathematicians advise against, lead nowhere. We sent men to the moon using the slide rule; today we send kids who used calculators and taught reform math since grade school to remedial math.
Morris Kline adds, "The average schoolteacher is obliged to follow a curriculum that is laid out for him. He would not be allowed to depart from the syllabus." Kline asserts, "All talk about modern society requiring a totally new kind of mathematics is sheer nonsense." Kline, a noted mathematician, wrote these things in 1973, which, uncannily, describes today's Common Core reform math uproar. Not much has changed. Dumb ideas last because they are repackaged again and again and sound great!
Bad ideas, such as reform math, minimal teacher guidance methods, group work, Common Core, NCLB, mandated testing, RttT grants, wavers, fads, myths, pseudo-science, etc., which have infiltrated education and dictated policy, have not produce bright, high achieving students. Our kids suck at math because we made them that way. The way we think about education has been contaminated with bad ideas, junk, fads, myths, and pseudo science. It's pathetic. Moreover, our kids don't study enough, read literature and books enough, write essays enough, learn math and science enough, or exercise enough. Furthermore, many students often lack industriousness or do just enough to get by, but this isn't the teacher's fault. This is not only a schooling problem, it is also a parenting problem. American parents gladly shell out money for video games, gadgets, sports and lessons, but not for tutors. In short, many parents give education lip service. Education is not the top priority in many families today--sports, the newest technology, celebrities, and entertainment are. It is called pop culture.
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Common Core reform math looks like arithmetic, but it is not the real thing! The outcomes are that it makes simple arithmetic difficult and doesn't focus on mastery. Under Common Core and the reform math programs that came before, students do not master basic arithmetic to prepare for algebra in middle school. "In reform math, students use different strategies [that] are inefficient and won't help students advance," writes Laurie H. Rogers (Betrayed).
Steven Strogatz (Mathematician at Cornell) writes, "If we only teach conceptual approaches to math [i.e., reform math] without developing skill at actually solving math problems, students will feel weak. Their mathematical powers will be flimsy. And if they don’t memorize anything, if they don’t know the basic facts of addition and multiplication or, later, geometry or still later, calculus, it becomes impossible for them to be creative." (Strogatz's quote from The Atlantic in an article by Jessica Lahey)
Barry Garelick, in an Education News article ("Mathematics Education: Being Outwitted by Stupidity") writes, "The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationalists on the panel, that the way mathematicians learn is to learn how to do it first and then figure out how it works later." Reform math is just the opposite and a reason our kids suck at math.
"Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe."
Understanding does not produce mastery; practice does.
In my opinion, Common Core math standards should be used only as guidelines and "applied judiciously rather than robotically." [But, they are not used as a guide; they are used as dogma.] Instead of Common Core's dogma, students need achievable objectives that should be reshaped or modified as needed by teachers because academic ability varies widely and the background knowledge in vocabulary and arithmetic of students who walk through the school door diverges significantly. Consequently, CC's doctrine of "one size fits all" and its mandated testing are unintelligent and misguided ideas. There have been many bad ideas imposed on classroom teachers without their input. Common Core is one of the latest. I don't think one can fix Common Core. Teachers should trust the sensible Core Knowledge K-8 math sequence as a guide to write their own instructional objectives and refocus on essential factual and efficient procedural knowledge. Students need to learn arithmetic, not reform math.
Note. I think educators take Common Core literally. It should not be dogma, and it should not be interpreted as ideology or doctrine and taught as reform math, which, I think, are bad ideas. The standards equalize downward; i.e., everyone gets the same or one size fits all--another bad idea. In contrast, teachers should be empowered to ignore CC and write achievable objectives for the students who walk through the school door. In addition, teachers ought to junk reform math and bring back standard algorithms, which require the memorization and automation of math facts, and ask kids to do a lot of word problems. This [mastery] process requires a lot of practice and starts in 1st grade. For centuries, old-fashioned arithmetic, when it was taught well, has proved to be a reliable and trustworthy preparation for algebra and higher math. So why toss it out or marginalize it? Downplaying crystallized intelligence [factual and procedural knowledge in long-term memory] in arithmetic has failed for decades, which is a primary reason that our students suck at math. Teachers have been bombarded by bad reform after bad reform. They have no input. Top-bottom reforms, such as Common Core, don't work.
Starting in 1st grade, Common Core seems to justify inefficient, alternative algorithms to do operations or drawings or writing to "show" or "explain" understanding (?) by arguing, in my view, that not all students can memorized math facts needed for the standard algorithms. Really? Yet, in my opinion and experience, almost all students are good at memorizing. Memorization of single-digit number facts has not been a problem in other nations, and it wasn't a problem in the early 80s, when I taught first grade. Today, memorization of essential number facts is a problem, say the Common Core reform math advocates, perhaps, because memorization and drill [practice] do not conform to progressive ideology and have fallen out of favor in most classrooms. But, I should point out that this is not a new problem. Kids haven't been mastering arithmetic for decades. They are not taught to think in terms of symbols that convey concepts. "Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe." Understanding evolves slowly and changes over time. What is more important, as Oakley points out, is to shift essential factual and efficient procedural knowledge into long-term memory through practice and practice and practice in order to maximize problem solving in math, especially word problems. You cannot apply something you don't know well.
Consequently, the outcome has not been good for crystallized intelligence in long-term memory, which is hugely important and prerequisite for problem solving in mathematics. Michael E. Martinez points out, "[It's] a mistake to downplay crystallized intelligence in favor of fluid intelligence." In short, you don't solve math problems without extensive math knowledge and practice. Downplaying crystallized intelligence in math has failed for decades. In short, kids are not taught to truly master traditional arithmetic. And, the same [downgrading traditional arithmetic] seems to be happening with Common Core reform math. Classroom teachers have been bombarded with reform after reform. Almost all were bad ideas and failed. Common Core reform math and the way it is taught do not match reality and ignore cognitive theory.
Michael E. Martinez (Future Bright) writes, "One domain of study might have been pivotal in the cultivation of fluid intelligence--mathematics....Early mathematics achievement now appears to have surprising power to predict student academic achievement in high school--both in mathematics and reading....Intelligence is...the product of education." It is a basic outcome of education when mathematics is taught well. My take on Common Core, however, is that the math is not taught well. In Common Core classrooms, what is taught looks like arithmetic, but it is not the real thing. Early on, little kids don't memorize math facts for mastery or focus on fast, efficient algorithms for basic operations (i,e., standard algorithms). Students can build fluid intelligence through problem solving in mathematics, which is the reason that word problems are so important, starting in 1st grade, but students must know basic arithmetic facts and standard algorithms to calculate answers to word problems. Furthermore, they must learn to recognize key patterns in word problems to know which operation or operations are needed (planning). All of this requires a lot of experience and practice, which has fallen out of favor in progressive classrooms. "As teachers teach, students become smarter," says Martinez. In short, teachers should teach directly, not facilitate.
Some kids, for whatever reasons, will always be better than others in math, but a widespread educational dogma taught in many ed schools is to equalize downward to narrow gaps, that is, let no child get ahead. However, kids are not the same in ability, so why feed them the same curriculum? It's a daft idea! We can never have equal outcomes. There will always be achievement gaps because good education creates inequalities, says the late Richard Feynman. Students who study more, practice more, pay attention in class, are more industrious, can delay gratification, and have an optimistic attitude create inequalities, etc. To excel in school mathematics, students need academic ability, realistic goals, and the persistence and conscientiousness to achieve through practice-practice-practice.
Dr. Richard Feynman ("Surely You're Joking, Mr. Feynman!" 1985) is a Nobel Winner in Physics. He writes, "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?" Feynman made this blunt statement at an ethics in education conference. We need another Richard Feynman today, but we don't have one. Stephen Hawking has popular books, but he was not a Nobel Prize winner like Feynman. A Feynman comes along once in a lifetime, like Einstein and Newton.
Common Core's one-size-fits-all and top-down creed and ideology cannot close or eliminate the achievement gaps. The Common Core is a political document written in stone--not a flexible, feasible educational blueprint for positive change in math education. And, NCLB testing, which is part of the Common Core Package, should be buried, too. We cannot reform education unless we empower teachers, says Dana Goldstein (The Teacher Wars). Common Core does just the opposite and does not permit variation. It is imposed on teachers and students by government.
Common Core math is simply another edition of reform math [repackaged], which has not worked in the past. It looks like arithmetic, but it isn't the real thing. Common Core is not traditional arithmetic, and it is not taught the same way traditional arithmetic is taught.
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| Common Core 3rd Grade Lesson Counting Up Method for Subtraction From Erick Erickson Diary |
Common Core is not proven, tested, or reliable. It is a test-based reform. Moreover, the one-size-fits-all Common Core reform math standards, in my opinion, have been thrust into our schools by the powers that be and big money. States adopted the new standards before the final version was written or the costs calculated--an epic error in judgement, critical thinking, and common sense. A major consequence is that our kids suck at basic arithmetic because they are not actually learning arithmetic for mastery, but this is nothing new. Dr. Richard Feynman used to remark that you don't understand anything until practiced. I don't know why Common Core people make arithmetic instruction substandard [not up to Asian benchmarks] or needlessly difficult. Under Common Core, kids often have to "demonstrate understanding" using visuals, which they draw.The approach is nonsense. Understanding varies from student to student because academic ability and instruction vary widely. There is one thing for sure. Insight or understanding is the result of achieving essential knowledge in long-term memory [mastery]. Truly, you don't understand something until practiced, practiced, and practiced.
You don't make drawings to do arithmetic.
Who does that? It's another bad idea!
(Below) No wonder our kids are weak in arithmetic. They are not taught the real thing, that is, traditional arithmetic. Also see MultipleModels.
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| Area drawing for multiplication of two decimals. It is total nonsense! You don't make drawings to do arithmetic. (Image: Kaplan, 5th Grade Common Core) |
Knowledge of traditional arithmetic built up in long-term memory (mastery) through lots of practice (memorization and repetition) fosters understanding. It is tried and true and trustworthy, so why have schools changed to Common Core, which is the latest "form" of reform math, when the outcomes have no basis in evidence? Large scale, top-down, abrupt changes (often called reforms or innovations) have not worked well in the US public school system. Teaching Common Core as repackaged reform math is just the latest of countless fads and bad ideas that have crept into education.
Some outcomes, however, are knowable. I was talking to a 3rd grade teacher who said the kids this school year were taught Common Core last year. The results are easy to predict. She said her incoming 3rd graders could not add or subtract larger numbers, have no sense of place value, have not automate single-digit math facts, know nothing of standard algorithms, etc. In short, the students cannot calculate to solve problems, which is the main purpose for studying arithmetic. What will become of these kids? They don’t know arithmetic, and our reform math programs made them that way. Incoming Common Core third graders don't know number facts and efficient procedures that 1st graders easily learn in Singapore. Our kids are about 2 years behind because reform math programs do not require kids to practice fundamentals to mastery; consequently, there is little understanding. Professor W. Stephen Wilson says that students don't get to actual arithmetic when taught constructivist reform math programs, such as 5th grade Investigations/TERC. He calls it "pre-arithmetic." Common Core follows the same road.
Common Core Reform Math
It looks like [or pretends to be] arithmetic, but it's not the real thing. And, it is hindering students from mastering essential arithmetic. Here is an example.
Tell how to make 10 when adding 8 + 5. The student writes that you can't. The teacher responds and writes the "correct" answer, "Yes you can. Take 2 from 5 and add it to 8 (8 + 2 = 10), then add 3. The item above was found on Joanne Jacobs blog.
It is not a joke as some "commenters" suggested on her blog. It is not the teacher's fault, either. US teachers are told to teach Common Core reform math, which, I think, is a waste of time for most students because there are many variations that can confuse beginners, such as breaking 8 into 2 + 2 + 2 + 2 and 5 into 2 + 1 + 1 + 1. Add the 2s to get 10, then add the 3 ones. This does not lead to memorizing math facts for auto recall, which is essential. Kids are novices, not little mathematicians or experts. They need to practice and memorize basic math facts and get good at using standard algorithms to calculate starting in 1st grade. Mastery should be the goal because mastery leads to understanding. Understanding does not produce mastery; practice does.
Other commenters say this type of problem (make 10) is commonplace in Singapore math in K and 1st grade. Wrong! The Singapore primary syllabus makes no mention of it. To the contrary, the syllabus specifies that kids should memorize math facts in grade 1 and practice standard [formal] algorithms--not demonstrate proficiency by making tens, which is a reform math strategy (pedagogy) found in Common Core. What does "demonstrate proficiency" mean? What does "proficiency" look like? In fact, Singapore public schools start with 1st grade, not K, so there are no Singapore math standards for K. Singaporean parents have the option to pay tuition for private K schools, but the academic quality varies widely at these schools. They are not dependable. Indeed, many students skip K and go straight into 1st grade when they are old enough. In addition, roughly 2/5 to 1/2, perhaps more, of the incoming 1st grade Singapore students do not have an English speaking environment at home, yet they learn to speak, read, and write English in school.
Note. Make 10 is an inefficient reform math strategy. It clutters up working memory. Instead of memorizing single-digit facts for auto recall, like Singapore 1st graders, US students are asked to make 10, that is, to calculate single-digit number facts, which eats up a lot of working memory space. It is not necessary or beneficial. Why practice make ten at all? Why not simply memorize 8 + 5 = 13 for auto recall? Kids certainly don't need to write an explanation that 8 + 5 are 13. 8 + 5 = 13 is a true equation because it is obvious by inspection. [The left side {13} and the right side {13} name the same point on the number line and therefore are equal by the transitive law of equality: 13 = 13. A weakness in teaching young children arithmetic is that they never learn why an equation is true or false.]
Children should not calculate basic math facts in working memory, which takes up mental space that should be used for problem solving; they should memorize them for auto recall from long-term memory starting in 1st grade. But, this is not happening in reform math. Early on, starting in 1st grade, American teachers teach strategies rather than drill math facts for auto recall, or practice place value using standard algorithms, or teach kids when to add, subtract, or multiply to solve word problems, or teach kids how to write equations that are solutions to word problems. The student cannot become efficient at using standard algorithms without automating math facts. The math that students learn today, if it can be called that, such as in Common Core, takes the form of constructivist reform math, which is flawed.
"In reform math, students use different strategies [that] are inefficient and won't help students advance," writes Laurie H. Rogers (Betrayed). The Common Core/NCLB/RttT reforms have many unintended consequences, that is, "huge collateral damage in the form of a narrowed curriculum, loss of classroom creativity, and the rise of teaching to the test," writes Andy Hargreaves. But, the major consequence is that our kids stink at basic arithmetic because they are not actually learning arithmetic for mastery. Mastery is gained through practice, which leads to understanding. As Darren Miller writes, "Understanding is key...[and] comes with the mastery gained through practice." Not the other way around. The late Dr. Richard Feynman used to remark that you don't understand anything until practiced.
Dr. W. Stephen Wilson, who teaches mathematics at Johns Hopkins University, according to his assessment of 5th Grade Investigations/TERC (constructivist reform math), asserts that kids never get to arithmetic. What they learn looks like arithmetic, but it isn’t the real thing. It's constructivist reform math. Kids should learn the real thing!
In my view, the "many levels" in the same math class, an overemphasis on group work and inquiry learning, calculator use, multiple fads, minimal guidance instructional methods, inefficient strategies for calculating, and making drawings, have led to widespread underachievement and unintended consequences. Asian kids are superior in math compared to American kids (TIMSS, PISA). It's not that Asian kids overachieve; it's that American kids underachieve. In the US, it has been a mistake not to focus on mastery of fundamentals of arithmetic, a bad idea that has been part of progressive ideology for decades. Instead of mastering fundamentals in the early grades like kids in top performing nations, US students are taught inefficient strategies or models that make simple arithmetic needlessly difficult. Also, kids in the early grades are not taught the standard algorithms, which are efficient and always work. In short, constructivist reform math screws up arithmetic. "In reform math, students use different strategies [that] are inefficient and won't help students advance," writes Laurie H. Rogers (Betrayed). Gaining conceptual understanding is not mastery. Practice (memorization and repetition) is needed for mastery and to move students forward. Students don't need complete understanding either. There is no such thing because understanding grows slowly over time.
Darren Miller writes, "So repeat and repeat and repeat until the repetition begets memorization. That’s what Mrs. Barton [Miller's 3rd Grade teacher] did until every one of her students knew the multiplication tables. Don’t allow a pet pedagogical theory to harm students’ ability to calculate. Teach them what works. Give them the most efficient tools out there." Common Core doesn't give kids the most efficient tools, but traditional arithmetic does.
Kids need achievable learning objectives, but the one-size fits-all Common Core doesn't not allow for variation, which is deeply disturbing. Moreover, the learning objectives need to be specific, observable, measurable, and adjustable to meet the needs of the students who walk through the school door.
The biggest problem I have tutoring high school students is that they will often say they understand the concepts but can't work the exercise problems, much less the word problems. This means they don't truly understand the concepts and/or have not mastered some of the important arithmetic and pre-algebra fundamentals (knowledge gaps) when they were in elementary and middle school. They are products of reform math.
Walter Mischel (The Marshmallow Test), a psychologist, writes, "The ability [of preschoolers] to delay immediate gratification for the sake of future consequences in an acquirable cognitive skill. This skill set is visible and measurable early in life and has profound long-term consequences for people's welfare and mental and physical health over the life span." Anyway, this is Mischel's theory, but there is little evidence that it is true. Even longitudinal studies don't account for unmeasured traits that may affect the outcome. The feedback from the adults was self-reported, so the data is hardly scientific. We know that the brain is malleable or capable of change, but malleability is an inexact idea. You cannot turn a student with average IQ into a Ph.D. in mathematics or physics no matter how much the student studies and practices. Rather, executive function can help create competent math students, which is a worthy goal, but brain plasticity and ability have limits. You can increase IQ up to a point, but it can also decrease. Smart people often make stupid decisions, says Mischel, and he is right. Think about it! Who controls education? {smart non-educators} No wonder education is a mess of bad ideas. Another outcome is that teachers who criticize Common Core on social media are often reprimanded and written up. Test-based reform failed under NCLB, yet Common Core is an intrinsic part of test-based reform. The outcome isn't going to change.
Elementary teachers spend too much time on understanding rather than on practicing basic arithmetic for mastery. Liana Heitin (Curriculum Matters) writes, "Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe." Mastery means instant recall from long-term memory that 5 x 5 = 25, which takes practice. You don't arrange beans in a 5 by 5 array to demonstrate that 5 x 5 = 25. That's a crutch, which can impede understanding, not mastery, says Oakley. She writes, "Having students stop to continually check and prove their understanding can actually impede their understanding, in the same way that continually focusing on every aspect of a golf swing can impede the swing." Mastery also implies that students have some understanding of the math and are able to apply it to solve problems. But understanding is a relative term and develops slowly over time. Unfortunately, reform math procedures in Common Core, such as counting up for subtraction needlessly complicate arithmetic and do not lead to understanding (no evidence) or the standard algorithm. In short, they are not practical; they are useless. Common Core justifies them by saying that not all students can memorized math facts needed for the standard algorithms. Yet, in my opinion, almost all students are good at memorizing. Memorization of single-digit number facts has not been a problem in other nations, and it wasn't a problem in the 80s, when I taught first grade. Suddenly, now, memorization of essential number facts is a problem.
Note. Many speak of equal opportunity, which is not a possibility because the students who come through the doors of different schools have different vocabulary and math backgrounds. Students vary widely in academic ability and knowledge; consequently, the curriculum should adjust to meet the needs of students, but Common Core does not allow variation. Kids need achievable learning objectives. But, many educators underestimate the content very young children can learn with explicit teaching, memorization, and practice, especially in essential arithmetic, starting in 1st grade.Instead of teaching traditional arithmetic using tried and true methods, Common Core tells teachers to use reform math methods. Thus, the outcomes will be totally different.
Notes.
Kids need achievable "behavioral learning objectives" (Robert Mager), and, because academic ability varies widely, the learning objectives should vary or be flexible from school to school, even from classroom to classroom, according to the students who walk through the school door. But, Common Core does not allow for variation. Since the 70s, educators have switched from behavioral learning objectives, which specify conditions and criteria for acceptable performance, to nonspecific or less specific objectives such as understands concepts, solves problems, acquires fluency, is proficient, or know from memory, which are unclear, ambiguous, and ill-defined. "Teachers rarely specify conditions and criteria in their objectives," says Pearson Education, so the learning objectives are fuzzy and vague. In math, teachers often say they teach "understanding," yet understanding is indeterminate without specifying the conditions and criteria. On the other hand, Mager's behavioral learning objectives specify the conditions and the criteria for acceptable performance. According to Mager, acceptable performance is observable and measurable and must be stated clearly--both conditions and criteria. Mager is not talking about rubrics, which are subjective. Without the conditions and the criteria for acceptable performance, teachers don't know what acceptable performance looks like. In short, there must be an observable behavior that is measurable. Also, under the Common Core, teachers are told to teach multiple strategies (constructivist reform math) rather than traditional arithmetic. But, in my view, they need to cut out the trivial and go straight to basic [traditional] arithmetic (the real thing). This is difficult with Common Core because it is a repackaged "strategies" approach and doesn't go straight to essential arithmetic. Consequently, teachers focus on strategies, not traditional arithmetic, which is built on concepts, basic facts, efficient algorithms, and routine applications. In Common Core, all students get the same, which is a shabby approach to education because academic ability varies widely. Common Core cannot narrow achievement gaps without equalizing downward. In short, most kids are not learning enough basic arithmetic that is essential for algebra-one in middle school. It's a conundrum. The effectiveness of Common Core or any set of standards taught as constructivist reform math has always been questioned and strongly criticized by mathematicians and scientists. In addition, the lack of word problems, applications, and arithmetic used in science in the US elementary school math curriculum have been grounds for concern since the 70s when Professor Richard Feynman read elementary school math and science textbooks and called them "lousy." Read more about Feynman and his conclusions on MultipleModels.
[On a personal note, I stayed in the classroom because teaching was fun and retired when it became a chore. I no longer tutor high school mathematics, such as precalculus, and, sadly, my Teach Kids Algebra guest teacher program has gone unfunded for the past two years because of district budget problems. Teaching under Common Core and a host of other government regulations, including NCLB, has become a chore for many classroom teachers rather than a delight, which is a sad commentary on education today. I feel fortunate that I am not a teacher in the Common Core era. It is clear, at least to me, that the wrong people are in charge of education; consequently, the public schools have been losing some of its best teachers who are fed up.]
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