It's summertime, and time to think about critical thinking, re-evaluate our beliefs and cogitate on the fact that "thinking" and "knowing facts" are intrinsically intertwined.
2nd & 3rd Grade Times Table
Unfortunately, fact learning and memorizing are disparaged in many progressive schools, even though knowledge is the goal of learning and the basis of critical thinking.
2nd & 3rd Grade Times Table
Unfortunately, fact learning and memorizing are disparaged in many progressive schools, even though knowledge is the goal of learning and the basis of critical thinking.
Immanuel Kant reminds us that thought without content is empty. For decades, knowledge has been downgraded, even though "both factual knowledge and thinking skills are essential for students to be able to solve meaningful problems." Daniel T. Willingham reminds us that, "Factual knowledge must precede [thinking] skill." He explains, "The ability to analyze and to think critically [i.e., thinking skills] require extensive factual knowledge." Fact learning is important.
What's missing in U.S. math programs is the mastery of number facts and calculating skills that support the concepts. If you can't calculate it, then you don't know it. For novices, standard calculating skills on paper are essential for solving arithmetic and algebra problems--not calculators, which divert the attention of students from automating vital factual and procedural knowledge, starting in the 1st grade.
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| Calculators often cover up the weak teaching of basic arithmetic skills. |
On average, many U.S. children are not mastering basic arithmetic. The way we have been teaching math can block a child's future. The math reforms and progressive methods of instruction downplay memorization, standard algorithms, and traditional instructional methods such as drill-to-develop-skill. The progressive reforms and methods are flawed guidelines and a recipe for mediocrity.
The mistaken assumption has been that students would use calculators later, so why learn formulas (i.e., rules expressed as symbols) or paper arithmetic, such as whole number arithmetic, fraction arithmetic, or integer arithmetic? The reformers claim that acquiring knowledge, both factual and procedural, in long-term memory is not that important, but they are wrong!
We are not teaching children for the future; we are teaching for a test.
Reform math confuses students and frustrates parents. Calculating skills, which are needed to solve problems in math, are weak. The leading conflict in education has been the "teaching," as the late Zig Engelmann had said, repeatedly. But, many teachers, educationists, politicians, and administrators don't think that way. Instead, they give excuses to justify bad performance in math, such as poverty, insufficient money, poor self-esteem, or lack of equity. Diane Ravitch wrote that American students never did well on international tests. But, it is never the teaching!
Test Scores Are A Major Disappointment in Arizona
It's not okay. It's tragic!
Compared to the Asian level, U.S. math is taught poorly!
What is disturbing in Arizona, for example, is that half the students failed the 3rd-grade reading test and half failed the math test. At the 8th-grade level, nearly 70% failed the math test, and over 60% failed the reading test. Stand For Children Arizona "believes many solutions lie in solving Arizona's education funding and teacher crisis." These same-old excuses have been around for at least 50 years. The problem is the "teaching" in the schools. The reformed math curriculum and instructional approaches of the progressive ideologues don't work, but many reform-minded educationists, including the professors at schools of education, won't admit fault for a failed system they have helped to create. And, they won't take the steps necessary to fix it.
Observations
Not all "good students" want to excel or be accelerated. Some students who are good at reading don't like to read books. Likewise, some students who are good at math don't want to do math. They would much rather play games, text, or do social media on their smartphones. These observations are similar to those from instructors at the Johns Hopkins Center for Talented Youth (CTY). Also, most children are average, not exceptional, but there are a handful of children who are exceptional in math and music, even at age 4. So, I would withhold characterizations such as talented, gifted, prodigy, and so on. Stop praising kids for no good reason. There will always be kids who are better in math than others for an assortment of reasons, but better should not mean gifted or the next Feynman.
Here are a few important ideas from Daniel T. Willingham's book (When Can You Trust the Experts?) that may help teachers gain a different standpoint. Many statements below are direct quotes or paraphrases.
It's not okay. It's tragic!
Compared to the Asian level, U.S. math is taught poorly!
What is disturbing in Arizona, for example, is that half the students failed the 3rd-grade reading test and half failed the math test. At the 8th-grade level, nearly 70% failed the math test, and over 60% failed the reading test. Stand For Children Arizona "believes many solutions lie in solving Arizona's education funding and teacher crisis." These same-old excuses have been around for at least 50 years. The problem is the "teaching" in the schools. The reformed math curriculum and instructional approaches of the progressive ideologues don't work, but many reform-minded educationists, including the professors at schools of education, won't admit fault for a failed system they have helped to create. And, they won't take the steps necessary to fix it.
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| Students are confused with the way math is taught today. |
Not all "good students" want to excel or be accelerated. Some students who are good at reading don't like to read books. Likewise, some students who are good at math don't want to do math. They would much rather play games, text, or do social media on their smartphones. These observations are similar to those from instructors at the Johns Hopkins Center for Talented Youth (CTY). Also, most children are average, not exceptional, but there are a handful of children who are exceptional in math and music, even at age 4. So, I would withhold characterizations such as talented, gifted, prodigy, and so on. Stop praising kids for no good reason. There will always be kids who are better in math than others for an assortment of reasons, but better should not mean gifted or the next Feynman.
Here are a few important ideas from Daniel T. Willingham's book (When Can You Trust the Experts?) that may help teachers gain a different standpoint. Many statements below are direct quotes or paraphrases.
1. Critical thinking is so difficult to measure.
2. Practice is necessary to improve.
3. The spiral curriculum (J. Bruner) failed.
4. Practicing math facts will help with long division.
5. We have tests that measure content knowledge in major subject matter areas (e.g., math, science, etc.). But, we don't have good tests to measure student's analytic abilities, creativity, enthusiasm, wisdom, attitudes toward learning.
6. Children need feedback so that they can make corrections.
7. Our ability to measure these qualities (creativity, collaborativeness, critical thinking) is limited. Measurement is the key to science. Opinion is not based on science.
8. If it disagrees with experiment, it's wrong (Feynman)
9. Solving a problem is not a matter of critical thinking, which is difficult to measure. It is a matter of recognizing the problem type.
10. Both factual knowledge and thinking skills are essential for students to be able to solve meaningful problems.
Notes.
1. Learning objectives should be specific, measurable, and achievable (Gagne).
2. Engagement is not the same as learning.
Thinking Well Requires Knowing Facts in Long-Term Memory
Daniel T. Willingham, a cognitive scientist, points out that factual knowledge must precede skill. He explains, "Thinking well requires knowing facts, and that's true not simply because you need something to think about. The very processes that teachers care about most--critical thinking processes such as reasoning and problem-solving--are intimately intertwined with factual knowledge that is stored in long--term memory (not just found in the environment). In short, critical thinking processes are tied to background knowledge." Learning facts should start in preschool.
Notes.
1. Learning objectives should be specific, measurable, and achievable (Gagne).
2. Engagement is not the same as learning.
Thinking Well Requires Knowing Facts in Long-Term Memory
Daniel T. Willingham, a cognitive scientist, points out that factual knowledge must precede skill. He explains, "Thinking well requires knowing facts, and that's true not simply because you need something to think about. The very processes that teachers care about most--critical thinking processes such as reasoning and problem-solving--are intimately intertwined with factual knowledge that is stored in long--term memory (not just found in the environment). In short, critical thinking processes are tied to background knowledge." Learning facts should start in preschool.
Vague generalizations, faulty beliefs, and epistemic statements should not drive educational decisions but often do.
Many statements are based on epistemic assumptions that are statements about the nature of learning and are often confused with empirical generalizations, which are actual observations of what children do (Willingham).
E. D. Hirsch (Why Knowledge Matters: Rescuing Our Children from Failed Educational Theories) points out three prevailing ideas that have led to the decline of knowledge: (1) developmental appropriateness, (2) child-centered, and (3) skill-centrism (i.e., thinking without content).
Education is loaded with wrong assumptions and contentious ideas that are not evidence-based. Here are some:
1. learning is social (kids learn best in group work)
Many statements are based on epistemic assumptions that are statements about the nature of learning and are often confused with empirical generalizations, which are actual observations of what children do (Willingham).
E. D. Hirsch (Why Knowledge Matters: Rescuing Our Children from Failed Educational Theories) points out three prevailing ideas that have led to the decline of knowledge: (1) developmental appropriateness, (2) child-centered, and (3) skill-centrism (i.e., thinking without content).
Education is loaded with wrong assumptions and contentious ideas that are not evidence-based. Here are some:
1. learning is social (kids learn best in group work)
2. everyone learns differently (learning styles theory)
3. knowledge is constructed (constructivist theory)
4. learning is natural
5. learning must be fun to be effective
6. "Without struggle, there is no learning."
7. laptops for all, college for all, algebra for all, and other inane sayings, such as "teaching for understanding." Really? How do you measure or quantify understanding or creativity? Can you code understanding on a computer?
8. learning depends on self-esteem
10. imagination is the source of innovation (No, it is knowledge!)
11. thinking is independent of knowledge
12. memorization and drill are bad for kids
13. children develop in cognitive stages (Piaget)
14. children learn best with hands-on stuff and inquiry-discovery learning
15. the teacher should be a facilitator so as not to disrupt a student's natural learning.
16. children need calculators and graphing calculators to learn arithmetic and algebra
16. children need calculators and graphing calculators to learn arithmetic and algebra
Science does not prove things right.
It is a method that eliminates wrong ideas. (Richard Feynman)
There are empirical findings that work in the classroom. One that all scientists agree on is "practice helps memory," reports Daniel T. Willingham. Also, "If a child is not cognitively ready to take on particular work, it's not because she has not yet reached the right developmental stage. It's because she doesn't have the background knowledge to make sense of the work." In short, Piaget's theory of stages doesn't work. Willingham writes, "Empirical generalizations, in contrast [to theoretical statements], are not predictions but are summaries of things that scientists have observed. Theories come and go." Theories are never complete or absolute and are revised over time as more observations are made.
Charles J. Sykes writes, "There is little or no research to justify the most sweeping changes in classroom practice." And, when there are empirical findings, reformers seem to ignore them. Educationists stick to flawed theories such as Piaget's stages, but, intrinsically, no content is "developmentally inappropriate," which has been an excuse not to teach specific content.
Thinking Is Domain-Specific!
Teachers should know the cognitive science of learning and how it applies to education, but they do not. Schools of education don't teach it. Nor do they teach that thinking is domain-specific, which means that thinking in math is different from thinking in science, and so on. (Incidentally, thinking in math is called problem-solving.) You cannot solve an algebra problem without knowing the requisite algebra content.
Minimal Guidance = Minimal Learning
Often, students are asked to work in groups to do math-like activities that result in little learning. (Kirschner, Sweller, and Clark: Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching) Teachers should switch from minimal guidance to more effective methods of instruction (i.e., explicit teaching that involves clear explanations and background knowledge). It means that elementary and middle school teachers should know the content, not just grade-level content, but higher-level math, such as precalculus.
To learn something is to remember it--not just for a test but for the future.
Math Knowledge Is Cumulative.
"One idea builds on another" (Ian Stewart). It takes hard work to learn math. Moreover, you "can't think your way to a solution of an algebra problem without knowing the algebra" (C. J. Sykes). If you can't calculate it, then you don't know it. For example, you cannot figure out perimeter problems if you don't know how to do sums. Also, you can't solve a trig problem without knowing trig. You can't translate Latin without cumulative background knowledge in Latin translation, grammar, and vocabulary. In short, there are prerequisites (i.e., background knowledge) students must know in long-term memory.
"One idea builds on another" (Ian Stewart). It takes hard work to learn math. Moreover, you "can't think your way to a solution of an algebra problem without knowing the algebra" (C. J. Sykes). If you can't calculate it, then you don't know it. For example, you cannot figure out perimeter problems if you don't know how to do sums. Also, you can't solve a trig problem without knowing trig. You can't translate Latin without cumulative background knowledge in Latin translation, grammar, and vocabulary. In short, there are prerequisites (i.e., background knowledge) students must know in long-term memory.
DNA
Academic achievement is linked directly to academic ability and knowledge!
Academic achievement is linked directly to academic ability and knowledge!
Also, DNA accounts for about 60% of the variation in school achievement (R. Plomin). Genetic variation means that children, even from the same family, do not have the "same abilities to learn the things that schools teach," such as mathematics (Murray, Plomin, Sykes, Willingham).
Robert Plomin (Blueprint) points out that variation in school achievement is mostly DNA. Furthermore, the percentages are "what is" and do not predict "what could be," says Plomin. They are not deterministic. If 60% is genetic, then 40% of the variation in school achievement is non-genetic.
Indeed, most children can learn the basics of arithmetic and algebra at an acceptable level, even though some children learn math skills and knowledge faster than others. Positive attitudes, motivation, persistence, parents, and teachers play essential roles in learning. Also, review and "practice [are] crucial to long-term retention" of knowledge (Willingham).
The Flight from Knowledge and, Consequently, Critical Thinking.
For years, I have heard the mantra of critical thinking without content. But, critical thinking is domain-specific. It is a function or product of knowledge. You must master chemistry concepts, such as atomic structure and stoichiometry, and certain math skills, such as logarithms, variables (algebra equations), significant figures, scientific notation, quadratic equations, dimensional analysis, exponents, graphs, formulas. Thinking skills are domain-specific.
Reform Math
In my opinion, the reform math movement has been responsible for disrupting the teaching of mathematics, starting in the 1st grade. Reform math educationists advocated the early use of calculators and de-emphasized computation skills and memorization of math facts. More importantly, the role of the teacher had changed from an academic leader to a mere facilitator. No wonder teachers are undervalued. They majored in education instead of an academic subject.
Old School Is Out. Anything Digital Is In
Anything that was old school was kicked out as obsolete by radical reformists. It's the 21st century, so anything that involves digital stuff is good for learning, such as computers, calculators, laptops, tablets, learning software, the Internet, etc. Blackboards were replaced with whiteboards, which were replaced with costly smart boards.
Within the reform math framework, children are asked to invent their own solutions and debate the solutions in groups. Really? With calculators, educationists insist that children could move straight to doing math (e.g., real-life problems) without formally teaching basics, which students would learn along the way, as needed. Really? The unintended consequences include the steady decline of background knowledge and achievement.
It seems clear that the "reform math" educationists intended to replace the memorization of number facts and paper calculation skills with calculators. In short, calculators are substituted for basic skills. Is it any wonder why our kids stumble over simple arithmetic and have difficulty with number sense?
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| U.S. kids stumble over basic arithmetic. Calculators often cover up the weak teaching of basic arithmetic skills. |
The state tests don't measure competency in basic arithmetic content. Thus, parents don't know what arithmetic students know or don't know. Can the student do long division, add fractions with unlike denominators, solve basic equations with inverses, operate on integers, use percentages, solve a proportion, graph a function, or apply the Pythagorean theorem? Furthermore, some test questions allow calculators. The state test is mostly a reading test and reasoning test (so-called higher-level thinking). The fact is that students cannot do higher-order thinking in math without knowing math. And, they cannot learn new math knowledge without connecting it to old math in long-term memory, that is, knowing the prerequisites. Empirical findings show that students must know content knowledge, which is the basis of critical thinking (i.e., problem-solving in math).
Multiplication Scope
In 1st grade, students learn multiplication as repeated addition: 3 x 4 = 4 + 4 + 4 = 12.
In 2nd grade, students memorize half of the multiplication table for instant recall and use multiplication ideas to solve word problems.
In 3rd grade, students memorize the rest of the table for instant recall and learn the standard algorithms for both long multiplication and long division.
In 4th grade, the focus should be on fractions, decimals, percentages, and ratios/proportions.
In 2nd grade, students memorize half of the multiplication table for instant recall and use multiplication ideas to solve word problems.
In 3rd grade, students memorize the rest of the table for instant recall and learn the standard algorithms for both long multiplication and long division.
In 4th grade, the focus should be on fractions, decimals, percentages, and ratios/proportions.
To Be Revised
This posting is information (bits and pieces).
6-17-19, 6-18-19, 6-19-19, 6-20-19, 6-25-19, 6-27-19, 6-28-19, 6-29-19,
7-2-19
7-2-19
Comments: ThinkAlgebra@cox.net
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