ThinkAlgebra strongly recommends the teaching of a "standard arithmetic" curriculum that focuses on the critical foundations of algebra as stated by the National Mathematics Advisory Panel (2008), not a version of NCTM "reform math"--via Common Core or state standards--that does not. Students need a standard arithmetic curriculum to advance, not a reform math curriculum.
The teaching of standard arithmetic isn't going back to something that did not work in the past; it is boosting and revitalizing something that did work. Indeed, content knowledge in arithmetic and algebra (aka skills, ideas, uses) is the foundation for doing the math. Knowledge in long-term memory enables clear thinking, problem-solving, and creating.
The "understanding" of math is in the "doing" of math. For example, in 1st grade, students begin to understand the idea of addition (and perimeter) by calculating the perimeters of squares, rectangles, and other polygons using memorized facts and the efficient procedures of standard arithmetic. The number line is helpful at first.
The teaching of standard arithmetic isn't going back to something that did not work in the past; it is boosting and revitalizing something that did work. Indeed, content knowledge in arithmetic and algebra (aka skills, ideas, uses) is the foundation for doing the math. Knowledge in long-term memory enables clear thinking, problem-solving, and creating.
The "understanding" of math is in the "doing" of math. For example, in 1st grade, students begin to understand the idea of addition (and perimeter) by calculating the perimeters of squares, rectangles, and other polygons using memorized facts and the efficient procedures of standard arithmetic. The number line is helpful at first.
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| Elementary students also need science, geography, music, art, history, etc. |
But students need more than knowledge of standard mathematics; they also need knowledge of history, geography, science, art, and music--subjects that are often shortchanged in elementary school. Moreover, students need more than mere exposure to these subjects; they need knowledge in long-term memory. Understanding in reading requires broad knowledge in long-term memory. Reading comprehension, vocabulary development, and reading-to-learn open up the child's future. Put simply, knowledge is the key to clear thinking.
Teaching kids "how to think" as a generalized skill has been a failed methodology in education for decades. We need to make sure students have gained sufficient domain knowledge upon which to think. Kevin Ashton (How To Fly A Horse) writes, "Creation is a result--a place thinking may lead. Before we can know how to create, we must know how to think. Having ideas is not the same thing as being creative. Creation is execution [work], not inspiration. We are not all equally creative, just as we are not all equally gifted orators or athletes. But we can all create." The quality of one's thinking is a function of one's knowledge, and the quality of one's creativity is a function of one's thinking. The base of thinking well and, therefore, creating, is knowledge! The crux of the matter is that many educators do not think knowledge in long-term memory is that important. They are dead wrong! Downplaying knowledge is contrary to the fundamentals of the cognitive science of learning. (Read the next paragraph on reform math via Common Core.)
Reform Math via Common-Core-Influenced State Standards
Slows the Learning of Standard Arithmetic.
SB, a journalist for TH, reports on Common Core’s use in a local school district (January 23, 2016). She writes, “Multiple strategies, versus a single algorithm, are taught. Common Core expects students to conceptually understand math. Students, for instance, are not taught rote memorization of multiplication tables. BH, the district’s PreK-12 mathematics coordinator, said students instead are taught reasoning and conceptual understanding to be fluent in multiplication facts.” Note. Reform math doesn't get kids to the level they need to be for success in algebra-1 in 8th grade or sooner. Not all kids will get there, but the students who learn math faster than others should, but most do not under a Common-Core-based curriculum. Also, in mathematics, the beginner is often overloaded cognitively with the extras of reform math (e.g., multiple strategies, making drawings, writing explanations, etc.) and with prep for mandatory testing (Every Child Succeeds Act); consequently, the student makes slow progress. Students should not waste time on the area model or strategy for division or multiplication, and most of the other strategies. In the modern classroom of group work, reform math, and test prep, there is little time for the teaching of standard arithmetic, which, for the reformers, is not a high priority. Also, other subjects are pushed aside or undervalued, such as science, geography, library time, music, history, art, etc.
Really? Frankly, many of the so-called math reforms reported by journalist SB above are fads or practices with sparse evidence. The Common Core way sounds so inviting and appealing: Kids can reason the facts into memory. All you need is understanding. No need for rote memorization and all that old school stuff. No more drill for skill. Practice is out. It sounds too good to be true, and it is! On average, the math reforms have not worked as expected. The reason is that most math reforms have been based on anecdotal evidence and faddish ideas. That is, most reforms were not anchored in valid scientific evidence. The reformers seem to ignore the science of learning. The math reforms are unsatisfactory for teaching standard arithmetic to young children, yet they persist. They often disregard the cognitive science of learning and the major role of long-term memory in learning new ideas and problem-solving. The "multiple strategies" approach often confuses novices, increases cognitive load, and alienates parents. In the real world, students learn new stuff from old stuff they already know. Hence, knowledge in long-term memory is the key to learning more math, faster; it enables clear thinking and problem-solving (i.e., critical thinking in math). Consequently, gaining essential factual and efficient procedural knowledge in long-term memory as quickly as possible should be a primary goal of math education, but, apparently, it isn't.
The consequence: math achievement has been stalled for decades. Recent data corroborate a continuing trend of poor math achievement. The 2015 NAEP math scores for 4th and 8th-grade students are lower than in 2013. According to the latest ACT scores, most students are still ill-prepared for college. It is disappointing, but not surprising. (NCTM math reforms since the early 90s combined with NCLB test-based accountability reforms since the beginning of 2000s have led to substandard math achievement. So have fads, progressive policies of sameness, and weak teacher training.) Nothing new. Diane Ravitch writes in her blog (1-31-16), "The bad part about ESSA [the new Every Child Succeeds Act] is that it preserves the mindset of NCLB, a mindset that says that standards, testing, and accountability are the keys to student success. They are not."
Slows the Learning of Standard Arithmetic.
SB, a journalist for TH, reports on Common Core’s use in a local school district (January 23, 2016). She writes, “Multiple strategies, versus a single algorithm, are taught. Common Core expects students to conceptually understand math. Students, for instance, are not taught rote memorization of multiplication tables. BH, the district’s PreK-12 mathematics coordinator, said students instead are taught reasoning and conceptual understanding to be fluent in multiplication facts.” Note. Reform math doesn't get kids to the level they need to be for success in algebra-1 in 8th grade or sooner. Not all kids will get there, but the students who learn math faster than others should, but most do not under a Common-Core-based curriculum. Also, in mathematics, the beginner is often overloaded cognitively with the extras of reform math (e.g., multiple strategies, making drawings, writing explanations, etc.) and with prep for mandatory testing (Every Child Succeeds Act); consequently, the student makes slow progress. Students should not waste time on the area model or strategy for division or multiplication, and most of the other strategies. In the modern classroom of group work, reform math, and test prep, there is little time for the teaching of standard arithmetic, which, for the reformers, is not a high priority. Also, other subjects are pushed aside or undervalued, such as science, geography, library time, music, history, art, etc.
Really? Frankly, many of the so-called math reforms reported by journalist SB above are fads or practices with sparse evidence. The Common Core way sounds so inviting and appealing: Kids can reason the facts into memory. All you need is understanding. No need for rote memorization and all that old school stuff. No more drill for skill. Practice is out. It sounds too good to be true, and it is! On average, the math reforms have not worked as expected. The reason is that most math reforms have been based on anecdotal evidence and faddish ideas. That is, most reforms were not anchored in valid scientific evidence. The reformers seem to ignore the science of learning. The math reforms are unsatisfactory for teaching standard arithmetic to young children, yet they persist. They often disregard the cognitive science of learning and the major role of long-term memory in learning new ideas and problem-solving. The "multiple strategies" approach often confuses novices, increases cognitive load, and alienates parents. In the real world, students learn new stuff from old stuff they already know. Hence, knowledge in long-term memory is the key to learning more math, faster; it enables clear thinking and problem-solving (i.e., critical thinking in math). Consequently, gaining essential factual and efficient procedural knowledge in long-term memory as quickly as possible should be a primary goal of math education, but, apparently, it isn't.
The consequence: math achievement has been stalled for decades. Recent data corroborate a continuing trend of poor math achievement. The 2015 NAEP math scores for 4th and 8th-grade students are lower than in 2013. According to the latest ACT scores, most students are still ill-prepared for college. It is disappointing, but not surprising. (NCTM math reforms since the early 90s combined with NCLB test-based accountability reforms since the beginning of 2000s have led to substandard math achievement. So have fads, progressive policies of sameness, and weak teacher training.) Nothing new. Diane Ravitch writes in her blog (1-31-16), "The bad part about ESSA [the new Every Child Succeeds Act] is that it preserves the mindset of NCLB, a mindset that says that standards, testing, and accountability are the keys to student success. They are not."
In its 2008 report, the National Mathematics Advisory Panel (NMAP) stated the Critical Foundations of Algebra. Dr. William Quirk, Ph.D. in mathematics, noted that some of the critical foundations are missing from Common Core, now state standards. They included the automatic recall of single-digit math facts, the automatic execution of the standard algorithms, the proficiency with fractions, including decimals, percentages, and negative fractions, and the geometry formulas to analyze the properties of 2- and 3-dimensional shapes (perimeter, area, volume, surface area). Also, the NMAP made clear that "practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees up working memory for more complex aspects of problem-solving." That is, the NMAP acknowledged a fundamental idea of the science of learning. As expected, the main stumbling blocks in algebra are the automatic recall of number facts, the automatic execution of the standard algorithms, and the fraction equivalents and operations, including decimals and percentages. In summary, a primary reason students stumble in algebra is that they are weak in standard arithmetic.
Other Comments
Reformers believe that kids can "reason" the multiplication facts into memory. No need for rote memorization, which is Old School. The reformers are wrong, of course. It’s illogical to think this way. Rote does not preclude some level of conceptual understanding. The concept of multiplication is simple and easily shown on a number line, and the meaning of single-digit facts, such as 2 x 5, which is two sets of 5 in each set, is also uncomplicated for 2nd graders to grasp. Indeed, students should learn essential facts and step-by-step procedures by repetition, which is drill for skill.
The people who wrote the standards, policymakers, so-called “math educators,” professors from schools of education, and many others who advocate the reforms in math have little knowledge of the science of learning. Readiness for an idea or topic is not a matter of age but a function of mastering prerequisites in long-term memory. Students learn new ideas that are linked to old ideas they already know. They must memorize the times tables and work with the standard algorithms to advance to the next level. They need to learn to reduce fractions to their lowest terms, add numbers with different signs, line up decimal points, and so on. In short, students must master standard arithmetic to advance to the next level, which is algebra in middle school. Reform math via Common Core doesn't get them there.
The Common Core Way slows the learning of standard arithmetic and leads to increased cognitive load. The "multiple strategies" approach in Common Core or state standards confuses students, alienates parents, increases cognitive load, and makes learning standard arithmetic more elusive and needlessly complicated. To move forward, students must master standard arithmetic first. However, under the yoke of Common Core, students' minds are cluttered with multiple convoluted strategies to do simple arithmetic, which stalls achievement. In reform math, the standard algorithm has been put on the back burner, which is contrary to the science of learning and the advice of mathematicians, e.g., W. Stephen Wilson, H.H. Wu, James Milgram, and many others, like Sandra Stotsky, who was on the National Mathematics Advisory Panel (2008) and validation committee.
The "understanding" of arithmetic is in the "doing" of arithmetic and includes skills, ideas, and uses. Indeed, to do arithmetic well, students must be competent in calculating the standard algorithms first. Moreover, students need to grasp when to add, subtract, multiply, and divide, which is pattern recognition. Also, learning standard arithmetic requires three primary ingredients: (1) competence in computational skills, (2) key ideas of K-6 mathematics (rules, fractions, variables, equations, functions, negative numbers, graphs), and (3) uses of mathematics (applications in science, business, finance, economics, tech, engineering; also area, volume, perimeter, etc.). [Notes. "Understanding grows gradually," writes the late mathematician Robert B. Davis (The Madison Project, 1957). Incidentally, my Teach Kids Algebra program (TKA) focuses mostly on the fundamental ideas of elementary school mathematics. The biggest drawback is that students do not learn enough standard arithmetic in their regular classrooms. It has been a decades-old problem. Moreover, the influence of the classroom teacher has dwindled over the decades due to reforms and fads imposed on schools. Teachers have little say in curriculum or instruction. They are often told to teach to the test. They did not create the mess in education and should not be blamed for it. Put simply, bad ideas and fads are often imposed on teachers who have little say.]
In Common Core, rote learning, which is learning by repetition, has been replaced by conceptual understanding and, apparently, reasoning, both of which are difficult to pin down or measure (quantify). Indeed, how do reasoning and understanding make students "fluent" in the times tables or the standard algorithm? It's a leap of faith. They don't! The assumption from reform math is contrary to the science of learning. Furthermore, “fluent” or “fluency” in reform math via Common Core or state standards is not explicitly defined. What does fluent mean in this context?
Rote learning means learning by repetition and is a fundamental technique for learning anything. The concept of multiplication and the meaning of the multiplication facts (3 x 5 means three sets of five in each set or a total of 15) are easy to grasp when practiced well. Also, the standard algorithm always works and is easy to learn when the single-digit multiplication facts are automated in long-term memory. Learning multiplication or standard arithmetic well requires ample practice, that is, drill for skill. The notion of math reforms is that reform pedagogy and group work are much more important than "mathematical substance." The teacher doesn't teach; the teacher facilitates. Long-term memory knowledge is not that important, say the reformists. The reformers are wrong!
Memorizing the single-digit multiplication facts (i.e., automating them in long-term memory) makes them instantly available for learning new content (factual and procedural knowledge), new concepts, and the standard algorithms. They enable problem-solving. Instead of memorizing the facts, Common Core wants kids to "reason the facts," that is, calculate them from known facts, which, presumedly, they had memorized. It is illogical. Indeed, a child cannot "reason something" without first gaining sufficient knowledge in long-term memory. Also, calculating the multiplication facts (as needed) wastes time and needlessly clutters and reduces memory space needed for problem-solving and learning new content. (Note. "Working Memory" space is very limited.) In short, the so-called Common Core way often ignores the intrinsic relationship between working memory and long-term memory and the critical importance of gaining factual and standard procedural knowledge in long-term memory to enable problem-solving, understanding, and illumination.
Indeed, it is fantasy, even bizarre to believe or postulate that those students through reasoning and understanding will acquire fluency, which is to "get" the times table into long-term memory. Consequently, most kids are not good at calculating (i.e., the skills, ideas, uses of arithmetic). Facts and efficient procedures must be automated to do arithmetic well. A visually-moderated sequence (VMS) of steps, if practiced enough, becomes automatic in long-term memory. Memory is better (and faster) than generating single-digit math facts by calculating (reasoning the facts) or using an inefficient area model to multiply or divide, and so on.
Lastly, Richard E. Nisbett (Mindware) points out, "Although the unconscious mind [long-term memory] can compose a symphony and solve a mathematical problem that's been around for centuries, it can't multiply 173 by 19." The conscious mind [working memory] can do arithmetic, but it follows the rules automated in the unconscious mind. Nisbett points out, "The unconscious mind operates according to rules. I know the rules of multiplication, I know the number 173 and 19 are in my head, I know I must multiply 3 by 9, save the 7 and carry the 2, and so forth. I can check that what's available in my consciousness is consistent with the rules that I know to be appropriate. But none of this can be taken to mean that I am aware of the process by which multiplication is carried out." Nisbett sums up, "Don't assume that you know why you think what you think or do what you do." 1-30-16 To Be Revised
Reformers believe that kids can "reason" the multiplication facts into memory. No need for rote memorization, which is Old School. The reformers are wrong, of course. It’s illogical to think this way. Rote does not preclude some level of conceptual understanding. The concept of multiplication is simple and easily shown on a number line, and the meaning of single-digit facts, such as 2 x 5, which is two sets of 5 in each set, is also uncomplicated for 2nd graders to grasp. Indeed, students should learn essential facts and step-by-step procedures by repetition, which is drill for skill.
The people who wrote the standards, policymakers, so-called “math educators,” professors from schools of education, and many others who advocate the reforms in math have little knowledge of the science of learning. Readiness for an idea or topic is not a matter of age but a function of mastering prerequisites in long-term memory. Students learn new ideas that are linked to old ideas they already know. They must memorize the times tables and work with the standard algorithms to advance to the next level. They need to learn to reduce fractions to their lowest terms, add numbers with different signs, line up decimal points, and so on. In short, students must master standard arithmetic to advance to the next level, which is algebra in middle school. Reform math via Common Core doesn't get them there.
The Common Core Way slows the learning of standard arithmetic and leads to increased cognitive load. The "multiple strategies" approach in Common Core or state standards confuses students, alienates parents, increases cognitive load, and makes learning standard arithmetic more elusive and needlessly complicated. To move forward, students must master standard arithmetic first. However, under the yoke of Common Core, students' minds are cluttered with multiple convoluted strategies to do simple arithmetic, which stalls achievement. In reform math, the standard algorithm has been put on the back burner, which is contrary to the science of learning and the advice of mathematicians, e.g., W. Stephen Wilson, H.H. Wu, James Milgram, and many others, like Sandra Stotsky, who was on the National Mathematics Advisory Panel (2008) and validation committee.
The "understanding" of arithmetic is in the "doing" of arithmetic and includes skills, ideas, and uses. Indeed, to do arithmetic well, students must be competent in calculating the standard algorithms first. Moreover, students need to grasp when to add, subtract, multiply, and divide, which is pattern recognition. Also, learning standard arithmetic requires three primary ingredients: (1) competence in computational skills, (2) key ideas of K-6 mathematics (rules, fractions, variables, equations, functions, negative numbers, graphs), and (3) uses of mathematics (applications in science, business, finance, economics, tech, engineering; also area, volume, perimeter, etc.). [Notes. "Understanding grows gradually," writes the late mathematician Robert B. Davis (The Madison Project, 1957). Incidentally, my Teach Kids Algebra program (TKA) focuses mostly on the fundamental ideas of elementary school mathematics. The biggest drawback is that students do not learn enough standard arithmetic in their regular classrooms. It has been a decades-old problem. Moreover, the influence of the classroom teacher has dwindled over the decades due to reforms and fads imposed on schools. Teachers have little say in curriculum or instruction. They are often told to teach to the test. They did not create the mess in education and should not be blamed for it. Put simply, bad ideas and fads are often imposed on teachers who have little say.]
In Common Core, rote learning, which is learning by repetition, has been replaced by conceptual understanding and, apparently, reasoning, both of which are difficult to pin down or measure (quantify). Indeed, how do reasoning and understanding make students "fluent" in the times tables or the standard algorithm? It's a leap of faith. They don't! The assumption from reform math is contrary to the science of learning. Furthermore, “fluent” or “fluency” in reform math via Common Core or state standards is not explicitly defined. What does fluent mean in this context?
Rote learning means learning by repetition and is a fundamental technique for learning anything. The concept of multiplication and the meaning of the multiplication facts (3 x 5 means three sets of five in each set or a total of 15) are easy to grasp when practiced well. Also, the standard algorithm always works and is easy to learn when the single-digit multiplication facts are automated in long-term memory. Learning multiplication or standard arithmetic well requires ample practice, that is, drill for skill. The notion of math reforms is that reform pedagogy and group work are much more important than "mathematical substance." The teacher doesn't teach; the teacher facilitates. Long-term memory knowledge is not that important, say the reformists. The reformers are wrong!
Memorizing the single-digit multiplication facts (i.e., automating them in long-term memory) makes them instantly available for learning new content (factual and procedural knowledge), new concepts, and the standard algorithms. They enable problem-solving. Instead of memorizing the facts, Common Core wants kids to "reason the facts," that is, calculate them from known facts, which, presumedly, they had memorized. It is illogical. Indeed, a child cannot "reason something" without first gaining sufficient knowledge in long-term memory. Also, calculating the multiplication facts (as needed) wastes time and needlessly clutters and reduces memory space needed for problem-solving and learning new content. (Note. "Working Memory" space is very limited.) In short, the so-called Common Core way often ignores the intrinsic relationship between working memory and long-term memory and the critical importance of gaining factual and standard procedural knowledge in long-term memory to enable problem-solving, understanding, and illumination.
Indeed, it is fantasy, even bizarre to believe or postulate that those students through reasoning and understanding will acquire fluency, which is to "get" the times table into long-term memory. Consequently, most kids are not good at calculating (i.e., the skills, ideas, uses of arithmetic). Facts and efficient procedures must be automated to do arithmetic well. A visually-moderated sequence (VMS) of steps, if practiced enough, becomes automatic in long-term memory. Memory is better (and faster) than generating single-digit math facts by calculating (reasoning the facts) or using an inefficient area model to multiply or divide, and so on.
Lastly, Richard E. Nisbett (Mindware) points out, "Although the unconscious mind [long-term memory] can compose a symphony and solve a mathematical problem that's been around for centuries, it can't multiply 173 by 19." The conscious mind [working memory] can do arithmetic, but it follows the rules automated in the unconscious mind. Nisbett points out, "The unconscious mind operates according to rules. I know the rules of multiplication, I know the number 173 and 19 are in my head, I know I must multiply 3 by 9, save the 7 and carry the 2, and so forth. I can check that what's available in my consciousness is consistent with the rules that I know to be appropriate. But none of this can be taken to mean that I am aware of the process by which multiplication is carried out." Nisbett sums up, "Don't assume that you know why you think what you think or do what you do." 1-30-16 To Be Revised
Special Interests, Philanthropists, & Government Rule Education
The latest technologies, innovations, fads, and reforms have not energized rapid gain in K-12 math and science achievement. Moreover, they will not revive rapid economic growth either, asserts economist Robert J. Samuelson, who says that our economic growth lags behind regardless of innovation. (Economic growth was about 2% in 2015 compared to China's growth of nearly 7%.) Our education system has stalled for decades and so has economic growth. Are the two strongly correlated or is it merely a coincidence? Also, it appears that special interests (school publishers, test makers, tech companies, etc.) have highjacked control of K-12 education with the assistance of government mandates and money ($$$$$) from both government and philanthropists who think they know how to fix education. The government at all levels (federal, state, and school district) and philanthropists (e.g., Gates, Zuckerberg, et al.) are often dead wrong. Just as in government, education has become a giant bureaucracy empowered by special interests.
The latest technologies, innovations, fads, and reforms have not energized rapid gain in K-12 math and science achievement. Moreover, they will not revive rapid economic growth either, asserts economist Robert J. Samuelson, who says that our economic growth lags behind regardless of innovation. (Economic growth was about 2% in 2015 compared to China's growth of nearly 7%.) Our education system has stalled for decades and so has economic growth. Are the two strongly correlated or is it merely a coincidence? Also, it appears that special interests (school publishers, test makers, tech companies, etc.) have highjacked control of K-12 education with the assistance of government mandates and money ($$$$$) from both government and philanthropists who think they know how to fix education. The government at all levels (federal, state, and school district) and philanthropists (e.g., Gates, Zuckerberg, et al.) are often dead wrong. Just as in government, education has become a giant bureaucracy empowered by special interests.
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