 |
| Our kids are locked into a Common Core test culture! Click |
The Rise of Anti-Intellectualism
In the US, sports trump academics, and jocks outclass geeks by a light year. Frequently, the decisions people make are about 30% rational and 70% emotional, says Jim Clifton, CEO Gallup. In fact, our feeling-based society often devalues individual academic achievement and academics in general. In many classrooms, excelling is not cool. "Reading is for losers. Math is for geeks," explains Greg Gutfeld (Not Cool). Scientific evidence is viewed as just another opinion, which, perhaps, partially explains why reformists seem to ignore the Science of Learning, 2015.
Also, there's a lot of talk about good schools, improving education, especially by using the latest technology, and so forth, but the narrative is mostly about good intentions and untested fads (aka innovations), such as Common Core. Feel-good education policies, fads, and notions originate from good intentions, I'm sure, but most innovations (notions, fads, etc.) fail because they lack solid evidence of effectiveness. Indeed, evidence or facts don't seem to matter much to progressive reformists who have an unrealistic passion for the new (aka innovation) and an irrational hostility for the old. Many widespread, favored classroom practices of today are not supported by evidence and are among the least effective, yet reform math people don't seen to care.
But "new" doesn't necessarily mean "better." In education, we have been spending billions and billions on innovations (reforms, fads, etc.)--especially on the latest technology--and hoping for the best. Over the decades, increasing technology use in the classroom has not turned into better student achievement. In fact, many popular and trendy reforms or fads in education are counterproductive. In contrast, some of the old stuff (i.e., old school), such as standard arithmetic and explicit instruction with worked examples are very effective when taught well. It is not enough to know some math, which is a good start; it is also important to know how to apply what you know in math.
[Aside. This post consists of random, often contrarian thoughts in no particular order. It is in rough draft form, so please excuse typos and errors. I repeat myself, a lot. Latest additions or updates: 10-11-15]
"The best way to know if an idea is right is to see if it predicts the future," writes Steve Pinker (Harvard). The fallacy of many education policies and innovations (fads) is that they start as good intentions (feeling-based), not via the science of learning (cognitive science). We often rely on the ability of so-called experts to predict the future, but the presumed experts often make policies and claims that fall flat because they are supported by ideology or beliefs, not valid evidence.
Furthermore, most teachers were not taught the cognitive science of learning, which is an "evidence-based core of what educators should know about learning," such as the critical role of practice to push knowledge into long-term memory, or the "understanding of new ideas via examples," or the fact that children are novices and don't think like experienced adults, etc. (Quotes/Ideas: The Science of Learning, 2015)
Moreover, I am not surprised that more technology use in schools has been linked to lower test scores, according to the OECD, but this will not slow the tech stampede into our schools at an enormous cost, with little value to actual student achievement. Tech is not the silver bullet. (I include calculators, smartboards, software, etc. as technology.) Also, I cringe every time a policy or notion is advanced and publicized as "for the kids," which--when put under scrutiny--is more "for the adults or special interests." [Aside. The OECD implements PISA, an international test for 15-year olds. According to the OECD, one weakness is, "U.S. students have particular problems with mathematical literacy tasks where the students have to use the mathematics they [should] have learned in a well-founded manner." In short, too many students don't know basic arithmetic skills, such as "using the number pi in calculations." Our students are incredibly shortchanged, not only in basic math knowledge, both factual and procedural but also in being able to apply or utilize that knowledge.]
Our kids are locked in a Common Core test culture (Click).
Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Equally wrong is expecting students to do critical thinking without sufficient background knowledge in long-term memory. Sadly, schooling has been entrenched in accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. This approach may be okay for business, but K-12 schooling is not a business, and it is not okay.
 |
| "I am a novice, not a pint-size mathematician." |
Children learning arithmetic or algebra should not be using calculators or over-burdened with questionable and annoying extras, such as indeterminate "deep" understanding, confusing and inefficient multiple models/strategies (as stressed in reform math), unrealistic Common Core Mathematical Practices, unjustifiable group work/collaboration, far-fetched, misguided real-world problems, time-wasting discovery/inquiry activities, or paragraph writing. All these extras are from reform math.
[Aside. The "extras" are based on adult "thinking," not in the cognitive science of learning.] Students are novices. They are not experts; they are not peer math teachers; they are not writers of math; they are not little mathematicians; they are not miniature adults. "Novices and experts cannot think in all the same ways (The Science of Learning, 2015)."
We teach reform math via Common Core instead of traditional arithmetic. Reform math people oppose standard algorithms and substitute many different, inefficient, non-standard alternatives as the primary methods of calculation. Consequently, there is little time left to spend on standard algorithms. In short, many students do not automate basic K-6 arithmetic, which is necessary for a valid algebra course.
Children are beginners, not pint-size mathematicians.
"Don't expect novices to learn by doing what experts do," writes cognitive scientist Daniel Willingham (Why Don't Students...). The reasons are simple. Kids lack both background knowledge and experience to do anything even remotely close to what mathematicians and scientists do. Furthermore, children do not think like experienced adults. They are not little mathematicians, junior scientists, or little adults. A good example of flawed thinking--that kids should emulate what experts do--is the Standards for Mathematical Practices, which are lodged in progressive constructivism and are the silent backbone of Common Core reform math. Willingham explains, "There are significant differences between how experts and novices think." Consequently, instructing students to be creative, pint-sized mathematicians, that is, to emulate what mathematicians do, seems rather pointless. It is just another empty-headed idea that does not agree with cognitive science.
Multiplication Facts should not be calculated as needed; they should be memorized.
In contrast to Common Core, students should first learn and practice the essentials of standard arithmetic for automaticity and solve routine problems first, not wordy, complicated word problems with extra information and certainly not far-fetched real-world problems, which are often championed by Common Core, even though students lack sufficient background knowledge and experience. For math facts, "Memory is more reliable than calculations (The Science of Learning, 2015)." The multiplication math facts, for example, are implanted in the standard algorithms for multiplication and long division. Math facts should be memorized and repeatedly used over a period to stick in long term memory, not calculated as needed, which wastes time, increases errors, clutters working memory, and inhibits fluency in using standard algorithms. New knowledge builds on old knowledge. The more math content you know, the more content you can learn and the faster you can learn it, says Daniel T. Willingham.
Held Captive
In education, we are held captive to bad ideas, counterproductive reforms, and untested innovations [e.g., accountability, metrics, benchmarks, performance indicators, inclusion, NCLB, sameness, Race to the Top, Common Core, standardized testing, the 4Cs (critical thinking, collaboration, communication, and creativity), mathematical practices, etc.], and many of us, as educators, have convinced ourselves that the current reform approach (via Common Core, standardized testing, NCLB, etc.) is probably okay for kids; however, we cannot logically justify the reasons that kids get a steady diet of test prep (hence, not much education) and that Common Core reform math is the same [one size] for all students, without regard to abilities or achievement, which, in my opinion, is equalizing downward.
The road to mediocrity, decline and failure is paved with good intentions; feeling-based policies, mandates, reforms, notions, and trendy, evidence-lacking fads (often called innovations). I think, education, especially the latest vision of math curriculum and instruction, has been on this road before. I think teachers and parents can disrupt the most recent vision (reform math), which isn't new because it started in 1989 with the NCTM reform math era. Repackaging old failures as innovations seems commonplace in education.
The Re-definition
Under Common Core, math education has undergone a "re-definition" that focuses on real-life or real-world problems; hence, it required group work and calculators early on and diminished knowledge of standard arithmetic. Indeed, Common Core's expectation is a calculator dependent and dominated math curriculum. But H. Wu, mathematician (UC-Berkeley), refutes this narrow perspective. We should not "think of mathematics exclusively as a tool for solving real-world problems." Mathematics is a complex system, an "edifice," says Wu. It is a symbolic language in which the "symbols and equations of mathematics express not just ideas but the relations between ideas," writes Leonard Mlodinow (The Upright Thinkers). Note. "re-definition" is Wu's term regarding the New Math, but it is also applicable to the reform math Era of the NCTM then and Common Core now.
Wrong Message: "I wasn't good at math either."
Sports do not operate in isolation, says Amanda Ripley (The Smartest Kids in the World): "Combined with less rigorous material, higher rates of child poverty and lower levels of teacher selectivity and training, the glorification of sports chipped away at the academic drive among US kids." Many kids believe math is merely one of several competing options and not high up on the list. Math is more abstract and, therefore, difficult than other subjects; consequently, many students avoid math, limiting their future. Many kids believe that they will get better at reading by practicing, but not in math. "You are either good at math, or you are not," which is a counterproductive belief not supported by cognitive science. Indeed, most kids, I think, can learn standard arithmetic and algebra well, but they have to work at it (drive) and be persistent (conscientiousness). And, according to Carol Dweck (Growth Mindset), we as educators and parents need to establish a proper mindset and stop telling kids:
(1) Not everybody is good at math. Just do your best.
(2) That's OK; maybe math is not one of your strengths.
(3) Don't worry, you'll get it if you keep trying.
(4) Great Effort! You tried your best.
(5) I wasn't good at math either.
Another conundrum is that modern reform math via Common Core is typically taught, not standard arithmetic. We need to prioritize math content and streamline the curriculum so that only essentials are taught and learned to automation. Not everything is important, but standard arithmetic is. [Aside. As computer use increases, we need skilled workers who are reliable and competent, that is, we need "workers who are smarter, better trained, and more conscientious," writes Tyler Cowen (Average Is Over). "The premium is on conscientiousness." But, this is not the narrative we are fed.]
Arithmetic That Is Arithmetic: 1912
 |
| 8th Grade Exam 1912 (The Arithmetic Part), Bullitt County Schools, KY |
In 1912, kids did arithmetic using standard algorithms and paper pencil. No calculators, of course. In short, they were taught to calculate quickly, recognize key problem types, and apply straightforward arithmetic to solve questions.
[Aside. Here is a mental arithmetic question for 3rd/4th graders from Ray's Intellectual Arithmetic (1877), a 140 page textbook for 3rd/4th grade combined: If 12 peaches are worth 84 apples and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches? Indeed, a 140 page textbook for two grade levels is a novel idea compared to today's 4th grade 500-page enVisionMath.]
Understanding Is a Matter of Degree.
A child's understanding of something is not the same as an adult's. A 1st grader's understanding of place value is not the same as a 5th grader's, etc.
Understanding should be inferred via a student's ability to solve arithmetic questions, that is, by doing arithmetic, and not on selecting a nonstandard, "understanding-type" algorithm, or making a drawing, or writing an explanation, which are arguable points of reform math, such as in Common Core. And, to do arithmetic well presupposes that the student knows arithmetic well through study, memorization, and practice (drill for skill). Standard arithmetic knowledge in long-term memory, both factual, conceptual, and efficient procedural, is imperative, yet, kids, today, are not always required to master standard arithmetic. For example, standard algorithms are often delayed, marginalized, or not practiced enough. Typically, what is taught under the yoke of Common Core is the latest revision of NCTM reform math--a progressive ideology of sameness or equalizing downward and an ed theory of constructivism--via inefficient minimal guidance methods, such as discovery, or project, or problem-solving learning in group work; complicated, cumbersome multiple models or strategies to do simple arithmetic; so-called real-world questions that require calculators, etc. Below is an example of a 5th-grade parents guide to Common Core.
#3. Multiply 5.3 by 2.4 using the area model (5th Grade Quiz). Show our work.
[Aside. Unfortunately, educators are told to teach multiple models [many ways] to do simple arithmetic, not the standard algorithms that are efficient, easy to learn, and always work. Standard algorithms should be taught first, not put on the back burner. The area model shown below is total nonsense. When would a student use an area model to calculate products? It is pointless, useless, and ridiculous.]
 |
| Screenshot above from my Math Notes in 2014: http://thinkalgebra.blogspot.com/2014/07/multiplemodels.html |
The New SAT Continues the Common Core Reform Math Brand.
The new SAT (2016) is a product of Common Core. It has nearly twice as many calculator questions as non-calculator questions. The heavy use of graphing calculators reflects the overall Common Core scheme: Let's concentrate school math on solving real-world problems so that kids use calculators. Calculator use in elementary school, as early as kindergarten, dates back to the failed NCTM reform math standards of 1989. The new SAT of 2016, which locksteps to K-12 Common Core, also overemphasizes data analysis, probability, and statistics. Students must rely on TI-84 graphing calculators for these topics, especially the statistics functions. The inclusion of these topics (and others) is another tactic used by reformists to defend calculator use among young students, even if their arithmetic and algebra knowledge and skills are weak.
The NEW SAT question type (#16) is, in my opinion, a 7th-grade pre-algebra level question, not a high school level. Note. A few of my Title I fifth graders in my Teach Kids Algebra program could figure this out, too. A well-prepared 7th-grade pre-algebra student should find the answer simply by examining the graph. No calculator is needed, just knowledge. The answer has to be either C or D because both have a y-intercept at -4. A quick "rise to run" check (1 to 3) means the slope is 1/3, not 3. In short, no calculator is needed, so why is this a SAT calculator-allowed question? I can only guess, but, apparently, under Common Core reform math, the expectation is that most high school students will not gain sufficient knowledge of algebra fundamentals to figure this out without using a graphing calculator. It This is yet another example of dumbing down the math. The NEW SAT has nearly twice as many "calculator allowed" questions as "no calculator allowed" questions. It is cause for alarm! (Question Source: Kaplan 2016 SAT)
[Late Note. Perhaps, the question is considered a nonroutine problem because you need to know stuff, such as the y-intercept, slope, linear equation form, etc. and know how to figure these out from a visual as you apply the concepts to develop an equation in y = mx + b form. Gee that is knowing and using what you know. And it starts with knowledge.]
In my opinion, this calculator problem (#16) clearly illustrates the sharp difference between knowing basic math in long-term memory (and applying it) and Common Core's expectation of a calculator-dependent-dominated math curriculum. Unlike their peers in top-performing nations, American students are lost without calculators. They don't know math. They can't do simple calculations, such as 2.54 x 1000 or -7 + (13/17) + 7 without reaching for the calculator. (Answers: 2540; the fraction 13/17) Over the years, calculators have dumbed down math content via US reform math programs. [Note. For -7 + (13/17) + 7, the calculator spits out .7647058824. What does that mean? The student didn't know that -7 and 7 are opposites (inverses) and equal zero when added via the commutative and associative rules: -7 + 7 = 0 or 7 + -7 = 0.]
[Aside. Common Core, like NCTM reform math, believes that real-world problem solving should be the focal point for K-12 school mathematics. What a terrible idea! Of course, this reform ideology "justifies" the frequent use of calculators, especially graphing calculators. What is lost? Fundamental math ideas and skills (factual, conceptual, and procedural knowledge) that are not linked directly to the real world problems are marginalized. With a calculator, it is now possible to calculate problems without knowing much math in long-term memory, so why memorize, or practice, or drill for skill? It is nonsense, of course. It is not the way novices master mathematics. Reform math, in my opinion, is an anti-knowledge approach.
Indeed, calculators have dumbed down content in mathematics. In short, the graphing calculator has replaced knowledge and skills. Our kids are weak in math, and we made them that way. H. Wu adds that under reform math, there is "a serious lack of essential technical facility—the ability to undertake numerical and algebraic calculation with fluency and accuracy." Common Core follows the same reform trend and avoids or delays standard algorithms.
-------------------------------
Pattern Recognition
Pattern recognition comes from the experience of doing and studying many types of math questions. "Oh, this is a percent of change problem." Understanding the problem (pattern recognition) is much more important than understanding "why" an algorithm works, and I think, this is what reform math people and Common Core fail to grasp, but G. Polya (How to Solve it) did. He lectured, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." Polya points out: "[Give students] plenty of opportunity for imitation and practice.... You learn to do problems by doing them." Pattern recognition is the key to solving math questions. Understanding a standard algorithm is ancillary because understanding develops slowly. Understanding does not produce mastery; practice does!
Students need to learn patterns for different types of problems. Learning patterns requires substantial imitation, practice, and experience. Pattern recognition is the key to problem-solving, observes Ray Kurzweil, MIT, How to Create a Mind. Moreover, kids are novices and do not need to learn multiple model stuff or do discovery lessons in groups. As Richard Feynman once exclaimed: Why do elementary school children learn or practice stuff that has little value? Furthermore, K-6 teachers are generalists and don’t know enough math content to teach standard arithmetic well (or Common Core reform math) contends Dr. H. Wu in a recent article. Additionally, parents are confused and baffled because they cannot understand the convoluted reform math or what is going on. Likewise, kids are confused, frustrated, and don't get it.
Speed, Efficiency, & the Correct Answer
Furthermore, I have always thought that the purpose of learning standard arithmetic was to solve questions as efficiently and quickly as possible, but not according to reform math (via Common Core), which stresses learning many different ways to solve the same problem, most of which are time-consuming, confusing, and more complicated than standard arithmetic. When Common Core is interpreted as reform math, the standard algorithm is merely one of many ways to solve math questions, and it is often pushed to the back burner to focus on what I call the “understanding” ways to do math, which include drawings, writing explanations, nonstandard "understanding" procedures, invented algorithms, intermediate algorithms, etc. I think most of this "understanding stuff" is math education for teachers rather than standard arithmetic for kids. We should not be training kids to be little math teachers or young mathematicians.
Unfortunately, children seldom practice for mastery the best strategies, which typically are old-school arithmetic, such as the standard algorithms. The importance of standard algorithms has been marginalized by reformers, even though the intrinsic merits and fundamental importance of automating standard algorithms for novices have been substantiated by many mathematicians, including W. Stephen Wilson, H. H. Wu, and so on. While math questions can often be solved in different ways, teachers should emphasize speed and efficiency to get the right answer. In short, novice students should learn the most efficient ways to solve arithmetic questions from the get go. Don't clutter the minds of beginners with a bunch of non-standard algorithms (many ways) or ask them to make drawings or write explanations. Instead, teach students pattern recognition of problem types and calculating using standard algorithms first and later some tricks (shortcuts).
Unfortunately, students are often asked to make drawings, or write explanations, or use inefficient algorithms or models that purport to show their understanding, such as the area model for multiplication or the partial quotient method for a division, etc. But, I think, this should not be the reason to study arithmetic. Usually, straightforward standard arithmetic or algebra is the best strategy, but it has been marginalized in Common Core and early reform math programs. Indeed, the “many ways” for “understanding” do not stand up to the scrutiny of cognitive science or even common sense. Why make arithmetic harder than it is?
I agree to disagree.
Wagner & Dintersmith (Most Likely to Succeed) focus on preparing kids for the innovation era as if the 20th century were not an innovation era. It is what I think: We are not producing enough home-grown talent. Our exploding tech companies are short on STEM talent; hence, for decades, major high-tech companies have imported foreign talent because many could not find enough home-grown talent. Furthermore, many high-tech companies locate branches where the talent is: Asian nations. And, while our STEM graduate schools are simply the best in the world--at least for now--they still attract a host of international students. Some stay in the US to work while a growing number return to their native roots.
I agree with Wagner & Dintersmith that the business model and standardized test approach are counterproductive and should be discontinued. But, sadly, Dintersmith repackages progressive child-centered-ideology that failed in the past—child driven discussion and child-centered assessment, etc. Lectures (explaining with worked examples) are out and replaced by a project-based approach in which teachers are facilitators, not academic leaders who know the content. Dintersmith's approach is anti-knowledge. He thinks content [knowledge] people are old school.
I agree with Dintersmith that children should master core academic content, but I do not agree with his approach, which is child-centered-project-based learning. The progressive ideology of the four Cs (critical thinking, collaboration, communication, and creativity) has displaced memorization, repetition, and practice. In short, gaining knowledge in long-term memory is not that important. Frankly, kids don’t study arithmetic because it will make them more creative or collaborative, and so on. Ordinary thinking makes them creative. Kids study arithmetic to learn it (automate it) and to use it to solve problems.
Sameness Ideology
Progressives [aka liberals] postulate that fairness and equality should dominate education, not individual achievement or excellence. Thus, in my opinion, sameness or uniformity has become the mantra of the education business, which is test-driven. Indeed, Common Core is the centerpiece of the test-based reforms, which drives both curriculum and assessment and makes lofty promises of college and career readiness without evidence. Kids are told to follow their passion in college, but many kids end up without a job, crushing student debt and live with their parents.
"It's Your Brain That Count!"
I am not sure we know what to do to change the culture in low-income Title 1 schools. The liberal answer has been to put more money into these schools (e.g., Title 1 funds). But, we have neglected the most important reform, which is equalizing funding. But, even if this would happen, I am skeptical that there would be leapfrog-type improvements. I am sure of one thing, however. I have found that teaching algebra lessons in the early grades (1-5) in a low-income Title 1 school works well when I explicitly teach complex content through worked examples with practice sheets I make up. And, even though these kids had come from low-income families, I discovered a lot of math talent distributed among them. It’s there, but it just isn’t being developed through test-based accountability reforms such as Common Core’s EnableNY. I agree with Amanda Ripley, who asserts, "Poor kids could learn more than they were learning." But, I disagree with the narrative from Diane Ravitch and others that the main problem in our schools is poverty. In Finland, Heikki Vuorinen says the opposite. He is quoted by Amanda Ripley: "Wealth doesn't mean a thing. It's your brain that counts." Vuorinen's message conflicts with the popular poverty narrative found in the US. The Finns didn't wait until poverty was cured to change their educational system.
Smart low-income kids cannot get to real Algebra 1 in middle school because Common Core reform math pushes Algebra 1 to high school. Also, CC is not set up for STEM. When I taught my algebra enrichment program (Teach Kids Algebra--TKA), I found many bright minority students, but, under the grip of Common Core in which everyone gets the same, they won't get to real Algebra 1 in middle school. No students will.
Fluency
What does fluency mean in Common Core or to progressive reformists? It doesn’t mean practicing the standard algorithms for automation. To the Common Core people, fluency means to do something in many different ways, which allegedly implies a deeper understanding. In Common Core, as I understand it, the standard algorithm is merely one of the many ways, often not the preferred way. That said, standard algorithms are not practiced for mastery. The fact that standard multiplication algorithm is pushed from 3rd to 5th grade validates the motives of the writers that the standard algorithms are just not all that important, which is a shift away from standard arithmetic. All those cognitive models, or strategies or many ways in Common Core's EngageNY scripted curriculum apparently are much more important because they allegedly demonstrate fluency and deeper understanding, even though evidence supporting such claims is lacking when placed under scrutiny. In contrast, standard arithmetic, when explained well and taught for automation (fluency), has always worked well for most kids.
Innovation
Coming Soon
©2015 LT/ThinkAlgebra.org
Credits: Caitlin
