We chase after test scores instead of credible academic achievement and continue a sameness ideology via Common Core. Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Also, we expect kids to do critical thinking without sufficient background knowledge in long-term memory, even though, according to cognitive science, factual knowledge in long-term memory must precede higher-thinking skills.
Minimal teacher-guided instruction (such as in reform math) is ineffective and inefficient, and contributes to substandard achievement, according to researchers Kirschner, Sweller, & Clark. Kids cannot solve math problems on a near-empty math tank (long-term memory). For example, critical thinking will not help a student solve a trig problem without specific, prerequisite trig knowledge in long-term memory and practical experience working similar trig problems.
Immanuel Kant sums it up this way: "Thoughts [critical thinking] without content [solid background knowledge in long-term memory] are empty."
Inferior Methods of Instruction Dominate
The decline of arithmetic skills "is the fault of" popular minimal teacher guidance instructional methods, such as "discovery learning," explains Anna Stokke of the D. Howe Institute. Stokke, a professor at the University of Winnipeg's department of mathematics and statistics, writes, "You know what's the worst kind of instruction? The kind of instruction that makes kids feel stupid. And that's what a lot of that discovery stuff does; their working memory gets overloaded, they're confused. That's bad instruction." Moira MacDonald summarizes the report this way: "The report puts a good deal of the blame on discovery or experimental learning approaches [aka reform math] that encourage students to explore different ways to solve math problems instead of using a single standard algorithm and often promote concrete tools such as drawing pictures, or using blocks or tiles to represent math concepts. The idea is students will gain a deeper understanding of math and be better equipped to apply it to a variety of situations [aka Reform math]. What happens is students' working memories get overwhelmed...." Cognitive overload hinders learning.
The minimal teacher guidance methods, which are found in trendy reform math programs, and now in Common Core, don't work. Inferior (minimal guidance) instruction contributes to inferior achievement. Teachers should focus on instructional "techniques that are actually going to work," that is, stress strong teacher guidance methods (explicit instruction), explain Kirschner, Sweller, & Clark, and should teach the standard algorithms starting in 1st grade--not the "many ways" (i.e., strategies or models) in reform math that, in my opinion, impede achievement. The way math has been taught for decades, including the Common Core test reform era--such as the early use of calculators; overuse of manipulatives, drawings, and strategies to represent concepts; minimal guidance methods; slow pace, time-consuming activities and nonstandard procedures; teaching to the test, and a spiraling curriculum--explains the inadequate achievement in basic arithmetic. Kids stumble over standard arithmetic because it is not taught well. What has been taught is a complicated variant of arithmetic called reform math. It looks like standard arithmetic, but it isn't the real thing!
Moreover, when a student reads a word problem, the student should recognize the problem type. Indeed, pattern recognition is very important in solving math problems. "Oh, that's an area problem, just multiply these numbers (234 and 63) since A = lw, and I'll get the answer in square meters. It will take me less than 15 seconds." The student's method is clear and straightforward. That's what kids need to be able to do, and do very quickly. Doing something fast implies fluency.
What about the latest technology; e.g., tablets, laptops, etc.?
The Organization for Economic Cooperation and Development (OECD) points out recent research by Leonid Bershidsky that suggests computers are likely an obstacle to learning. He writes, "The OECD study found that the use of computers was negatively correlated with improvements in student performance on math tests." Consequently, the hell-bent drive in the US to use computers, laptops, or tablets in the classroom, even graphing calculators, may be the wrong direction. Larry Cuban (Stanford) keeps telling us that the latest technology has not been the answer to our achievement woes. In my opinion, students should be able to do basic arithmetic without a calculator and basic algebra without a graphing calculator.
(Aside. Perhaps, we ought to bring back the slide rule. "Multiplication just required lining up two numbers and reading the scale," writes Cliff Stoll, physicist. "Multiplication simplifies to sums; division becomes subtraction," etc. You can't go wrong. Slide rules were used extensively for calculations in mathematics, chemistry, physics, engineering, etc. Today, a slide rule costs $20-$40, if you can find one, compared to $140 graphing calculator, or make one yourself. At the least, kids will learn to read scales, figure out the order of magnitudes, and mentally keep track of the decimal point and the sign (+ or -) of the answer. Incidentally, the slide rule put a man on the moon. FYI: The math of slide rules is logarithms. To multiply, add logs; to divide, subtract logs. In short, the slide rule works because the logarithm of the product is the sum of the logarithms: log (ab) = log a + log b. Click for an online simulated slide rule. )
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| log (ab) = log a + log b |
Ability Matters [A Lot]
The reality is that kids with higher academic ability (let's say, IQs of at least 110), are faster learners of academic material, such as math, science, literature, history, reading vocabulary and comprehension, etc., than kids with lower academic ability; hence, kids with higher academic ability should receive differentiated instruction (more rigorous and challenging academic work that is faster paced or accelerated and much more complex), but under the sameness ideology of Common Core, many don’t or merely get enrichment, and, consequently, they languish in boredom and stagnate academically in regular [inclusion-type] classrooms. For the most part, educators and policymakers tend to ignore the best students. In contrast, it has been my experience that the best students need just as much teacher attention as other students, perhaps more. I have often noted that advanced kids don't need enrichment, which keeps them where they are; they need acceleration options to move forward as fast as they can go. Some of the best are invited to Johns Hopkins summer youth academic program. In the real world, the "sameness ideology" brings down the achievement of all students, not just the advanced kids. The averaging of test scores often masks a school's overall poor achievement. Averages often mislead. We need to see a distribution of scores, not an average.
Academically Gifted
Will Fitzhugh (The Concord Review) writes about gifted kids, “I thought of this the other day when I read about students in summer programs at the Johns Hopkins Institute for the Advancement of Academic Youth in Baltimore. In The Boston Globe the article said: “Students from 21 states and 15 foreign countries—some as young as seventh grade—devour full-year high school courses in the arts, history, math, science and languages in only three weeks. For a rare few, a standard nine-month curriculum is absorbed in seven days.” The academically gifted need an entirely different curriculum. Richard Rusczyk (the Art of Problem Solving) says that AP calculus is for well-prepared, average students. It is not up to the university level, does not stress proofs, and depends too much on graphing calculators. Furthermore, it's too easy and boring (too slow of a pace) for the mathematically gifted.
Aside. First Draft. It is not an essay. Please excuse typos and errors. Additions are frequently made. The last changes were made on 9-20-15. 11-12-154
Charles Murray (Real Education) identifies the "sameness" flaw in the standards approach (Common Core is the latest example) this way: “To demand that students meet standards that have been set without regard to their academic ability is wrong and cruel to the children who are unable to meet those standards." Under Common Core, cut scores for passing are intentionally set high so that most students fail. Murray explains, “In large groups of children, academic achievement is tied to academic ability. No pedagogical strategy, no improvement in teacher training, no increase in homework, no reduction in class size can break that connection. Even the best schools will have children who do not perform at grade level.” In many schools, the averaging of test scores provides cover for the poor performing students. Charles Wheelan (Naked Statistics) points out, "Students with different abilities or backgrounds may also learn at different rates. Some students will grasp the materials faster than others for reasons that have nothing to do with the quality of teaching. Our attempts to identify the 'best schools' can be ridiculously misleading."
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| Kids are locked into a Common Core test culture! |
Naturally, "We want kids to be creative, to be good collaborators, to be good critical thinkers. But our ability to measure these qualities is quite limited," writes Daniel Willingham, a cognitive scientist. Indeed, many popular classrooms practices are based on ideology or ingrained beliefs, not scientific investigation. We can't do scientific investigations without measurements. So, how do we know our theories in education are correct? We don't! The popular 21st-century practices of teaching kids to be "little scientists" or "little mathematicians" without a solid knowledge base in those subjects have no scientific basis. The 21st Century practices are a skills approach, not a knowledge approach. Consequently, the reality is, "Children will be unable to reason well if their knowledge base is poor."
Thoughts without knowledge are empty [Kant].
The idea of doing valid critical thinking after reading a few paragraphs (a popular Common Core skill or technique) without solid background knowledge in that topic is not based on science and does not carry over into adulthood as many decisions are made by emotion and beliefs, not logical or critical thinking. Teachers are told to teach abstract skills or strategies, such as finding the main idea in a paragraph, yet teaching skills in reading cannot replace background knowledge and content. In arithmetic, efficient procedural knowledge [computational skills] is based on factual knowledge; they go together. But, the stress in Common Core reform math has been on thinking skills outlined in the Standards for Mathematical Practices (SMPs), not on the automation of essential factual and procedural knowledge that enables problem-solving. Repeatedly, I have written that kids are not little mathematicians; they are novices who need to master essential content knowledge. But, typical interpretations of Common Core as reform math don't do that.
Will Fitzhugh writes, "Skills have taken the place of content. Content, after all, can be such a pain." He cautions, “But let’s also remember that one of the goals of education must be the acquisition of knowledge,” even for its sake. Fitzhugh points out, “To make knowledge, which is the foundation of learning, it is necessary to apply thought to information, to think about the facts that have been gathered, and this is work only an individual can do.” In short, the Internet does not make knowledge, human beings make knowledge. Also, remember, the more you know, the more you can learn and the faster you can learn it, says Daniel Willingham. Knowledge builds knowledge.
The idea that kids should work in groups to foster collaboration later on in the workplace is far-fetched and not supported by science. The experts are wrong to make grand claims without supporting evidence, but they do it all the time to validate, albeit falsely, their beliefs. The claims made by Common Core, such as college and career readiness, have no basis in science, either. Willingham (When Can You Trust the Experts?) writes, "In education, there are no federal or state laws protecting consumers from bad educational practices." In education, there are no real experts, he says. Furthermore, always be skeptical of claims made by people who assert that their opinions or practices are researched-based.
The explicit teaching of standard arithmetic is not backward thinking; it works for most students, but not all. Students need to automate essential facts and efficient procedures through memorization and practice, even drill for skill, to move forward. If there is little in the tank (long-term memory), the student won't travel very far. It is basic cognitive science. Knowledge, both factual and procedural, in long-term memory, enables higher-level thinking. Mastering standard arithmetic--not reform math (many ways) strategies--primes the brain for algebra.
Dr. Katharine Beals (Out in Left Field blog) astutely observes that the current reform approach substitutes math education skills for efficient math skills. The curriculum and instructional methods taught in the classroom come right out of ed schools, and they are not the standard arithmetic (factual and standard procedural knowledge) that kids need to automate to advance to algebra. She writes, “Is the goal to teach kids how to do the math, or how to be math teachers?" I have often said that kids are not little mathematicians; they are novices. In my opinion, novices don’t need to "play with" inefficient, alternative strategies (models, nonstandard algorithms, etc.), explain or write explanations for their answers, or draw diagrams (visuals) to model, explain, or prove their answers. Kids are novices; they need to practice the core of standard arithmetic, both facts and standard procedures (skills). In contrast to Common Core, what matters in solving a multi-step word problem is a straightforward flow of standard calculations (a logical order), which is the thinking process of the student. Nothing more is needed. In solving word problems, kids should not be thinking about models, diagrams, or drawings, but about numbers and operations, that is, the actual math, itself. Students need to abstract the numbers that are essential to do the calculations in the right order to get an answer.
Below is a "math education" example from Essentials of Mathematics for Elementary Teachers (Musser, Burger, & Peterson). It shows three different addition algorithms, and then asks a question, which method shows the best understanding of place value? Justify.
Understanding is "part nonverbal and part unconscious," says mathematician David Ruelle (The Mathematician's Brain); hence, "the processes of mathematical thought are difficult to analyze." Ruelle reminds us that mathematics is a matter of using knowledge from long-term memory, not of opinion. You cannot solve problems in mathematics with an empty knowledge tank. In contrast to Common Core, the standard algorithm is efficient, tried-and-tested, and always works, and it relies on the memorization of single-digit math facts. Knowing the math facts in long-term memory for auto recall precedes procedural skill, says Daniel T. Willingham (Why Don't Students...). Willingham, a cognitive scientist, stresses, "Factual knowledge [in long-term memory] must precede skill."
We are cheating our students with "sameness" and other popular progressive ideas and programs!
Today, everything seems to be about race, closing gaps, sameness, fairness, Common Core and testing, equalizing downward, etc. rather than gaining knowledge and improving individual academic achievement (excellence), so I don't know where education is going, certainly not to a good place. Improving scores on tests, based on "so-called" higher standards (aka test-based reform), should not be the goal, yet it seems to be the fundamental driving force in curriculum and instruction today. Test prep, which strongly influences the curriculum, leads to myopic learning if that. We have been down the road of test-based reform before (2001 NCLB act), and it failed. Today, we are back on the same road, often repeating the mistakes of the past and expecting a different outcome. We are cheating students. They need a broad-based, liberal arts curriculum.
Regrettably, progressive (aka liberal) reformists, supported by big money, the powers that be (government), pundits, and others--all of whom know little about education, cognitive science, and learning--make up most of the rules and policies in education and impose their control, liberal ideas, mandates, and academic fads, such as government sponsored Common Core, on schools, teachers, and children. Common Core has become a ruse for more government encroachment and a recipe for continued lackluster achievement. Common Core, with its baggage and clutter, drives high-stakes testing and the costly technology boom, but it lacks valid and clear evidence for its vague claims of college and career readiness and of leapfrogging improvement in math and reading. In fact, it ignores STEM. The latest technology in the classroom, observes Larry Cuban (Stanford), both then and now, has not worked as a solution to our education woes. Indeed, we do not learn from our mistakes of the past; we just repeat them. Will Fitzhugh laments that kids aren't taught history content. History content knowledge, along with knowledge from other domains, such as science and math, has been replaced by higher level thinking skills. Higher level thinking skills won't help much in solving a trig equation without knowing and applying trig content, and so on. Facts [knowledge] must precede skills.
[Note. Daniel Willingham, a professor of psychology at the University of Virginia, says that he doesn't think teachers are dumb; he thinks their training is dumb. He writes, "But the problem in American education is not dumb teachers. The problem is dumb teacher training." Willingham explains, "Teachers are smart enough, but you need more than smarts to teach well. You need to know your subject, and you need to know how to help children learn it. That’s where research on American teachers raises concerns." He continues, "Teachers themselves know that their training focuses too much on high-level theory and not enough on nuts-and-bolts matters of teaching." I should point out that those high-level theories have not worked in the classroom.] (Source: Willingham's OptEd in NYTimes, 9-8-15)
E. D. Hirsch Jr. writes, ""The conclusion of cognitive research concerning skills is this: Broad knowledge of many domains is the only foundation for wide-ranging problem-solving and critical-thinking skills; hence, broad knowledge [i.e., the liberal arts, such as math, science, history, literature, language, art, music, etc.) is what the schools, media, and policies should try to impart....Teaching strategies (skills) instead of knowledge has only yielded an enormous waste of school time." In summary, Hirsch writes, "Abstract skills falter without a foundation of content supporting them." It is the situation we have today, not only in history, but also in math, science, and other key academic disciplines, such as literature. So-called higher thinking skills trump content knowledge. Hirsch is saying that it doesn't work that way.
Many Low-income Kids Are at the Bottom
In low-income elementary schools, the ELA test scores are often very low, usually under 25% proficient, but the math scores are even lower. On the surface, it appears that low-income kids are not learning much, which could be true. What is more likely is that many low-income students often do not have achievable learning objectives. Do they have the funding and resources needed? I don't know. The liberal answer has been to pour more money into these schools (Title 1 funds, etc.), but has this changed the narrative? The rich schools can afford the add-ons, but the poor schools can't. [Data Reference: Wilmington, Delaware, Smarter Balanced 2015 scores]
Indeed, the federal and state governments, the school district, a prevailing progressive [liberal] ideology of sameness, and the Common Core's one-size-fits-all-test-based-accountability scheme have failed these kids. For example, in Wilmington, DE, a city elementary school (SH), in which 81% of the students are low-income, scored 15.7 % proficient in math and 20.9% in ELA, while a suburban elementary school (NS) in the same school district, in which 3.8% of students are low-income, scored 77.1% proficient in math and 87.5% in ELA. (Don't jump to conclusions or confuse correlation with causation.) Under our current education system, most of SH students have little chance to catch up to their peers at NS, not in education or in life, although, I hope there will be many exceptions. Some of these students think they are dumb because they keep failing the test. Charles Wheelan (Naked Statistics) explains, "There are schools with extremely disadvantaged populations in which teachers may be doing a remarkable job, but the student test scores will still be low--albeit not nearly as low as they would have been if the teachers had not been doing a good job."
School averages are often untrustworthy.
An elementary school that has several high scoring students can easily skew the school's average to 50% proficiency or above to cover up the lower scores of other students. A good example is one of the highest ranked school districts in Arizona, partly based on test scores, high school graduation rates (92%), AP participation (23%), and other factors. But how well are all students educated? In 2014, 80% of its high school graduates that applied at Pima Community College were placed in remedial math courses. The flaw of averages shows that the success of some students masks the failure of others. In other words, averages are often misleading. If the education in this top-rated school district were exceptional, then its percentage of remedial math students at community college in 2014 would not be 80%. Liberal reformers say that Common Core will fix this with the claim of 0% remedial, which is an idiotic assertion because there is not a shred of evidence to support it. Furthermore, high school graduation rates are often artificially inflated with online "credit recovery" programs. Also, there is rampant grade inflation in K-12. [Aside. I do not know the two-year graduation rate at Pima Community College, but the 4-year graduation rate at the University of Arizona is 40%. The truth hurts! "SAT scores for the Class of 2015 were the lowest since the test was revised and re-normed in 2005," writes Carol Burris. The SAT scores are lower because of modern reforms, says Burris, not because more students are taking the test.]
Has Common Core contributed to the achievement gap? Are its goals too lofty or unachievable for some kids? Can some low-income students score just as high as other students? I would answer 'yes' to these questions with some caveats, of course. In the real world, for many students, there is no fairness in funding or excellence in their education, just dependency on government schools and government programs. Poor schools tend to remain poor.
The Common Core Way is roughly the same approach as before (2001 NCLB era), which was, write some standards, impose them on schools, write some tests and make them high-stakes for schools, teachers, and students, then see what happens. The plan failed miserably. So, what did we do? We wrote new standards and now expect different results. How simpleminded! We are repeating the mistakes of the past! The Common Core Way is backward thinking. Sadly, schooling has been entrenched in a culture of accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. It may be okay for business, but education is not a business, and it is not okay! NCLB (No Child Left Behind), Common Core, high-stakes standardized testing (ranking/grading schools-teachers-students), Race to the Top (RttT), and wavers are some of the recent upshots of this egregious (education) culture. Muller observes, "Under NCLB, scores on standardized tests are the numerical metric by which success and failure are judged." In the mad, mad world of education, schools, kids, and teachers are defined by a test score, which is truly an outrageous idea. It is extreme. It shows that education has run amok!
The "One Size Fits All" Common Core model doesn't work in the real world because kids coming through the school door are not the same. Students vary widely in industriousness and persistence, numeracy, language and vocabulary, attitude and beliefs, resources, and academic ability. What is the point of giving a Common Core test, knowing that 70% are likely to fail? Why feed kids the same curriculum and the same tests? Some kids are at a level that calls for a more challenging or accelerated math curriculum while other kids are at a level that requires a more basic curriculum, and so on. Our present system of sameness and inclusion is absurd. John Hattie writes, “Levels-based curricula with clear milestones, targets or expectations, which are then aligned with the assessment system, are more likely to have an impact on student learning than year-based curricula." Kids need achievable learning objectives, and, because academic ability and background vary widely, the learning objectives and tests should vary accordingly.
Aside. For example, American Red Cross uses levels-based swimming lessons and tests, which are based on clearly defined levels. There are achievable learning objectives for beginning swimming, and so on for each course. The ARC doesn't give the intermediate level swimming test to swimmers in its beginner's course. No one would pass.
Why do some schools score so much higher than other schools?
Is it the teachers? Probably not, because they are products of the same liberal ideology disseminated by progressive schools of education. Perhaps, it is the students. It is a hard pill to swallow. "Children differ in their ability to learn academic material" because they differ in their abilities and backgrounds, writes Charles Murray (Real Education), but American education practices and policies ignore this fact and brand Murray a racist, which he is not. Perhaps, it is the testing because standardized tests use the dreaded bell curve, which means half the students are below average. Adding to our education woes, we have been diverting billions and billions and billions of dollars into goals that are unattainable, all in the name of sameness, while neglecting academic excellence (gaining/making knowledge), which should be the top objective in education, but our leaders give it lip service. Indeed, Common Core, linked to standardized testing, and technology, are just some of the latest academic fads. By fads, I mean, not backed by valid evidence!
So what can be done?
We can dump Common Core and its testing and restore the explicit teaching of standard arithmetic, along with parts of algebra, geometry, and measurement so that most kids, not all kids, reach a level to handle pre-algebra by 7th grade. Some students will not be ready for a valid pre-algebra course in 7th grade or even 8th grade. Other students will need pre-algebra spread over two years (7th & 8th grade), and still others will take pre-algebra in 7th grade and be ready for a valid Algebra 1 course by 8th grade. Ideas, such as algebra-for-all, college-for-all, or laptops-for-all are silly and nonsense. In addition to a school culture that impedes progress, we have a curriculum problem and an instructional problem.
Many widespread, favored classroom practices are not supported by evidence and are among the least effective. Kirschner, Sweller, & Clark (2006) point out that minimal guidance methods, in which the teacher is a facilitator of learning, are typically the least effective classroom practices. These constructivist-based minimal guidance practices, which are taught in ed school, have many names, including such favorites as inquiry-based, discovery learning, problem-based, etc. Some teachers are convinced that inquiry- or discovery-based learning, which favors group work, is the best way for kids to learn math, but it is not true. In short, reforms should be based on cognitive science, not popularity, intuition, or ideology. Current cognitive research supports strong, direct teacher guidance, not the teacher as facilitator. "Direct instruction involving considerable guidance, including examples, resulted in vastly more learning than discovery," write Kirschner, Sweller, & Clark.
Kirschner, Sweller, & Clark explain, "After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to intermediate learners. Direct instructional guidance is defined as providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture. Learning, in turn, is defined as a change in long-term memory." Kids are novices, not experts, so they need straight-forward teacher guidance and encouragement, especially with new, more complex content. Furthermore, kids don't need to work in groups, which is often counterproductive. Kevin Ashton (To Fly A Horse) points out, "Working individually is more productive than working in groups." Ashton goes against popular group think (the status quo).
Factual Knowledge Must Precede Skill
You don't get to higher-level thinking except through lower-level thinking (knowing and applying) in long-term memory. It is a major concept in cognitive science that is ignored by current progressive reformers because it doesn't fit their ideology. Guess who is in charge of education? Progressive reformists!
Cognitive scientist Daniel Willingham often says that understanding is vague and difficult to assess. He also stresses that "factual knowledge must precede skill." Willingham continues, "If factual knowledge makes cognitive processes work better, the obvious implication is that we must help children learn background knowledge." Knowledge, not strategies, should be the focus in elementary school. Students should not be taught to be little mathematicians or little scientists. They are novices, not experts. They need to know facts to think.
Michael E. Martinez (Future Bright ) points out, "Research supports a view of intelligence as both lower-order and higher-order. The mind's ability to engage in higher-level operations must in some way rest on a foundation of lower level functions. Higher-level reasoning and problem-solving draw, in turn, on the mind's software--the accumulated content-rich knowledge acquired through experience."
Draft 1
Please excuse typos and errors.
Model: Chloe
©2015 LT, ThinkAlgebra.org
