The late Zig Engelmann said that if you want to know if an idea works, then go into a classroom, teach it, and see what happens, which is what I did in the winter of 2011 as a volunteer guest algebra teacher for two 1st grades, two-second grades, and one-third grade. It was clear to me that we underestimate the content children can learn.
Eric A. Nelson ("Cognitive Science and the Common Core Mathematics Standards" 2017) writes, "Between 1995 and 2010, most U.S. states adopted K‐12 math standards which discouraged memorization of math facts and procedures [NCTM]. Since 2010, most states have revised standards to align with the K-12 Common Core Mathematics Standards (CCMS). The CC does not ask students to memorize facts and procedures for some key topics and delay work with memorized fundamentals in others." Because of the limited space in the working memory (WM), Nelson observes, "When solving math problems of any complexity, due to WM limits, students must rely almost entirely on well‐memorized facts and algorithms." It is what the science of learning is all about. Kids are novices, not experts. They need to memorize essential stuff in long-term memory to solve math problems in the Working Memory. They need to learn content.
The late Zig Engelmann observed: "You learn only through mastery!"
Paying attention is important in learning. If you are distracted by something, then you diminish your working memory's thinking capacity. Daniel Kahneman (Thinking Fast and Slow, 2011) writes, "Anything that occupies your working memory reduces your ability to think," which is not good for learning and problem-solving. For example, sitting in groups is a distraction for some kids.
Attention in the classroom is often inconsistent, so is learning.
Attention in the classroom is often inconsistent, so is learning.
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| It takes effort to pay attention in class! |
Some basic ideas in the cognitive science of learning:
Note. Some of the content in this section has been taken directly or rephrased from The Science of Learning, 2015.
1. Students learn new ideas that are linked to old ideas they already know.
2. To learn, students must transfer information from working memory (where it is consciously processed) to long-term memory (where it can be stored and later retrieved). Learning is a change in long-term memory. Students have limited working memory space that can easily be overloaded. Worked examples reduce the cognitive burden. A worked example is a step-by-step demonstration of how to perform a task or solve a problem.
3. Grasping new ideas can be impeded if students are confronted with too much information at once.
4. Cognitive development does not progress through a fixed sequence of age-related stages [as Piaget had thought]. The mastery of new concepts happens in fits and starts. Readiness is not age-dependent; it is determined by the student's mastery of the prerequisites. (A college student who has not mastered algebra will have difficulty with calculus, while a 10-year old who has mastered algebra/trig (yes, they do exist) will have little trouble with calculus. Readiness is a function of mastering prerequisites.)
5. Each subject area has some set of facts that, if committed to long-term memory, aids problem-solving by freeing working memory resources and illuminating contexts in which existing knowledge and skills can be applied. Memory [of math facts, for example], is much more reliable than calculation.
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Note. Inferences are built on observations, which are measurements. An interpretation of the measurements is called an inference. The idea of observation/inference was a fundamental part of a K-6 program called Science--A Process Approach (SAPA, 1967). Today, there is nothing like SAPA. An inference is not a fact, yet textbook writers and journalists often treat inferences and correlations as facts.
All scientific theories are tentative; they are subject to change or correction with additional data. Almost all studies in education have not been replicated for one reason or another. Repeatability is a basic idea in scientific research. If an experiment is repeated and gives different results, then there is something wrong. If a study has not been repeated, which is often the case in education, it cannot be trusted because its claims have not been verified. Also, it is not surprising that 82% of the government grants for classroom innovations (experiments) issued by the U.S. Department of Education failed to improve reading and math achievement. Unfortunately, there is a lot of junk in education with no basis in science. If you cannot measure something, then it is not science. It's speculation, and we have a lot of that going around in education.
Some of the difficult to measure ideas include understanding, creativity, innovation, collaboration, critical thinking, self-esteem, analytic ability, enthusiasm, persistence, and so on.
In progressive education schools, many teachers have been taught idealistic, unrealistic learning theories, such as constructivism, rather than the cognitive science of learning, which is practical. Often, teachers are trained to be facilitators of learning, not academic leaders. They frequently implement test prep and minimal guidance instructional methods, which are not supported by evidence as being effective. These are among the least effective methods. Is it any wonder that our kids aren't learning arithmetic like students in other countries? Teachers are caught in the middle of a mess they did not create. Bad ideas and fads are often imposed on teachers who have little say.
I think teachers realize that a one-size-fits-all approach (Common Core or state standards) is fundamentally flawed and counterproductive. We need to stop blaming teachers for our education shortcomings and start blaming the schools of education, a progressive reform agenda, a plethora of untested fads and unproven reforms, and the numerous state and federal policies, mandates, and laws that interfere with and impede the efficient education of our children.
I reject the progressive utopian ideology and agenda of sameness (equalizing downward), displacing essential content for thinking skills, well-intentioned fads (evidence doesn't matter), and a "radical constructivist view of mathematics learning," which are embodied in the “reform math movement [then NCTM, now Common Core] that stresses an undefined conceptual understanding and student-created algorithms," along with early calculator use. The notion in reform math is that 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.
In contrast, I agree with Sandra Stotsky and the 2008 National Mathematics Advisory Panel that endorse an "academically stronger mathematics curriculum as well as for fluency in students’ computational skills with whole numbers and fractions" using standard algorithms. In other words, the Panel argues for explicit teaching of substantive "standard arithmetic" to prepare more kids for algebra. Indeed, one of the Panel's recommendations was that all schools should offer a valid Algebra-1 course no later than 8th grade, which implies that schools need to prepare more elementary students for success in the course by ensuring they know standard arithmetic well. (Quotes from Sandra Stotsky, National Mathematics Advisory Panel)
Teaching in the 21st Century has become unnecessarily difficult, complicated, and perplexing, especially with the extra burdens of Common Core, government mandates and policies, unproven reforms (fads), and pervasive progressive ideology of "sameness" and of "thinking skills displacing essential content."
One problem is that kids widely vary in academic ability, yet they are grouped together in the same classroom (inclusion policies). Another is that thinking skills, without substantial knowledge supporting them, are at best superficial. Improving inference-making skills implies substantially improving the student's knowledge and experience on a particular topic or subject. Having content knowledge in long-term memory is the key to profound and intelligent thinking. Googling and reading a few paragraphs and bits and pieces do not cut it. Kids are novices and need accurate knowledge in long-term memory to think well, not vice versa. For example, to solve math problems, students need basic knowledge, both factual and procedural, in long-term memory. The knowledge also has to be specific; you can't solve trig problems without knowing and applying trig. In summary, the quality of your thinking is a function of the quality of your knowledge.
One problem is that kids widely vary in academic ability, yet they are grouped together in the same classroom (inclusion policies). Another is that thinking skills, without substantial knowledge supporting them, are at best superficial. Improving inference-making skills implies substantially improving the student's knowledge and experience on a particular topic or subject. Having content knowledge in long-term memory is the key to profound and intelligent thinking. Googling and reading a few paragraphs and bits and pieces do not cut it. Kids are novices and need accurate knowledge in long-term memory to think well, not vice versa. For example, to solve math problems, students need basic knowledge, both factual and procedural, in long-term memory. The knowledge also has to be specific; you can't solve trig problems without knowing and applying trig. In summary, the quality of your thinking is a function of the quality of your knowledge.
Teaching children "math practices" or "science processes" is not the same as teaching essential standard arithmetic or science content. Children are not pint-sized mathematicians or junior scientists. They are novices who need to master content (facts and efficient procedures) from the get-go. When asked why Asian nations do a better job teaching math and science, Arthur Levine, former president of Columbia University's Teachers College, says they "start earlier, work longer, and work better." He explains, "Kids are capable of learning about mathematics much early than we thought." We have a weak curriculum, even under Common Core, which is not world-class.
Furthermore, most US teachers do not know or use the "cognitive science of learning" in their teaching. For example, starting in 1st grade, children are not required to memorize math facts and practice standard algorithms for automation in long-term memory under reform math. They seldom master the basic rules that govern the behavior and meaning of numbers and operations.
Essential facts and standard procedures (some of the fundamentals of arithmetic) need to be automated in long-term memory (a vast storehouse) for direct use in problem-solving that occurs in the limited mental space of working memory. Daniel Willingham also states, "The brain is not designed for thinking ... Humans don't think very often (i.e., solving problems, reasoning, reading something complex, or doing any mental work that requires some effort) because our brains are designed not for thought but the avoidance of thought."
Thinking requires focused attention and effort. In the classroom, "factual knowledge is important," says Willingham, and "practice is necessary." These ideas are key fundamentals in the cognitive science of learning.
Thinking requires focused attention and effort. In the classroom, "factual knowledge is important," says Willingham, and "practice is necessary." These ideas are key fundamentals in the cognitive science of learning.
Regrettably, reform math via Common Core pays little attention to the science of learning and often exposes novices to cognitive load problems and frustration. "Teaching strategies [such as nonstandard algorithms in Common Core reform math] instead of knowledge has only yielded an enormous waste of school time," writes E. D. Hirsch Jr. [Aside. In Common Core reform math, strategies refer to many different, complicated, inefficient, nonstandard, or invented algorithms to do simple arithmetic. Why not teach kids efficient, standard algorithms to begin with?]
Each subject (e.g., math, science, history, literature, the arts, etc.) has its unique background knowledge, language, and way of thinking, so generalized skills do not transfer easily, which is why Hirsch says that generalized skills are a waste of classroom time.
Furthermore, the thinking skills required for solving an equation in algebra, a physics problem, or the exegesis of a literature text are not always the same. (Deductive reasoning from well-established rules (facts) is used in math, but a counterexample invalidates the rule unless conditions are defined, such as a ÷ b, b ≠ 0, etc. Math is based on facts, logic, and true statements.
In contrast, inductive reasoning is used in science, but a few conflicting measurements (data) do not invalidate a relationship. For example, linear regression analysis can calculate a correlation coefficient and build a linear equation that models the relationship between two variables. Incidentally, regression analysis is a statistical tool used in many different disciplines.
Furthermore, the thinking skills required for solving an equation in algebra, a physics problem, or the exegesis of a literature text are not always the same. (Deductive reasoning from well-established rules (facts) is used in math, but a counterexample invalidates the rule unless conditions are defined, such as a ÷ b, b ≠ 0, etc. Math is based on facts, logic, and true statements.
In contrast, inductive reasoning is used in science, but a few conflicting measurements (data) do not invalidate a relationship. For example, linear regression analysis can calculate a correlation coefficient and build a linear equation that models the relationship between two variables. Incidentally, regression analysis is a statistical tool used in many different disciplines.
Moreover, a high correlation coefficient between variables should not imply causation. Also, you should not extrapolate beyond the actual data, which is why mutual fund companies caution investors that past performance is no guarantee of future performance. That said, the ability to use formal logic, that is, logical reasoning is critically important in math, science, and other academic disciplines.
Our personal decisions on national issues, buying a car, investing, or selecting a school are seldom based on logical reasoning because we don't bother to optimize our choices. Still, neither do the so-called policymakers and decision-makers at the local, state, and federal government levels. For example, the content of the Common Core standards, the costs of their implementation, or their impact on teachers and students were not thoroughly examined, analyzed, or critically reviewed by policymakers and the powers that be, that is, the decision-makers, who hurriedly and blindly adopted Common Core long before the final draft was written. In short, the people-in-charge did not bother to optimize their decisions when making critical choices that have affected every child and teacher in the public schools.
Common Core was plunged down our throats by higher-ups. The effort and time we spend on making a decision should be proportional to its importance, says Nobel Prize-winning economist Herbert Simon. Hurried or hasty decisions in education are often counterproductive and have unintended consequences. It seems that our policymakers and powers that be have not followed Simon's principle in K-12 education. Moreover, as it turns out, Common Core math standards are often implemented as a repackaged flavor of NCTM reform math. Reform math, based on discredited constructivist learning theory and inefficient minimal teacher guidance for instruction, disregards some of the science of learning's fundamental principles.
We often make assumptions (inferences) or favor innovations without scientific evidence (e.g., Common Core standards will improve test scores and make all kids ready for college/career, or the latest technology will rescue our lagging math and reading achievement, etc.). We often make decisions on insufficient information, anecdotal evidence, intuition, expediency, trendiness, feelings, opinions of friends, celebrities, politicians, so-called experts, social media, etc. We sometimes believe dubious or far-fetched claims made by alarmists. The effort and time we spend on making a decision should be proportional to its importance, says Herbert Simon, but that's not how we have in the real world.
Some of the most popular practices and programs in education are not evidence-based and don't work well because they go against learning science. Furthermore, having a steady diet of test prep is not education. Keith Devlin writes, "Decision-making [something computers do well] and thinking [something humans do well] aren't the same, and we shouldn't confuse them." We humans don't bother to optimize our decisions. We are not very good at making good decisions.
Teaching children "math practices" or "science processes" is not the same as teaching actual arithmetic or physical science. Children are not pint-sized mathematicians or junior scientists. They are novices who need content to think well! Furthermore, content should be taught explicitly or directly, that is, through strong teacher guidance. The content children learn from manipulatives or discovery activities in group work is trivial compared to the content children learn when taught explicitly. Students should not be required to make drawings (visuals) or use nonstandard algorithms to do simple arithmetic. Students should practice and use standard algorithms from the get-go.
In school, the primary objective should be "fill the mind with specific content" to enable smart thinking. Indeed, "training the mind to think" is essential, but smart thinking depends on gaining knowledge in long-term memory via a content-rich curriculum. Furthermore, the content should be a broad-based, liberal arts curriculum, including the arts, humanities, math, and science.
The quality or merit of a student's problem solving, critical thinking, or inference-making skills is a function of substantial knowledge of the subject at hand (Science of Learning, 2015). This fundamental idea of the cognitive science of learning is not new. For example, Immanuel Kant (1724-1804) once remarked, "Thoughts [critical thinking] without content [knowledge] are empty."
To base a curriculum on problem-solving or critical thinking without acquiring sufficient content knowledge in long-term memory, let's say in mathematics, has little merit.
"The reality is that you can think critically about a subject only to the extent that you are knowledgeable about the subject," writes Paul Bruno (Edutopia Blog). In short, problem-solving, critical thinking, or inference-making skills depend heavily on your knowledge of and experience with a particular topic, subject, or domain.
To base a curriculum on problem-solving or critical thinking without acquiring sufficient content knowledge in long-term memory, let's say in mathematics, has little merit.
"The reality is that you can think critically about a subject only to the extent that you are knowledgeable about the subject," writes Paul Bruno (Edutopia Blog). In short, problem-solving, critical thinking, or inference-making skills depend heavily on your knowledge of and experience with a particular topic, subject, or domain.
In contrast to Common Core, you cannot do quality critical thinking well by reading a few paragraphs. Bruno explains, "This is in stark contrast to the common desire among educators and policymakers to teach so-called [generalized] thinking skills that can be applied in any situation." Thinking skills are unique to a subject or domain. Improving inference-making skills implies improving students' knowledge base and experience in a particular area, subject, or field.
The so-called strategies in Common Core reform math, that is, nonstandard or invented algorithms, should not replace, delay, or hinder the memorization of basic number facts or standard algorithms' learning through practice in first grade or any grade. Memorization and practice to gain factual and procedural knowledge in long-term memory do not squelch creativity and learning, as some erroneously think. Knowledge gaining engages and enables problem-solving, creativity, and innovation.
Instruction
Many widespread, favored "classroom practices" are not supported by scientific evidence and are among the least effective. Kirschner, Sweller, & Clark (2006) point out that minimal guidance methods, in which the teacher is a facilitator or coach of learning, are typically the least effective classroom practices. These constructivist-based minimum guidance practices 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 or learning, not popularity, intuition, or ideology. Current cognitive research supports direct [explicit] teacher guidance. "Direct instruction involving considerable guidance, including examples, resulted in vastly more learning than discovery," writes Kirschner, Sweller, & Clark.
Kirschner, Sweller, & Clark explain (long quote), "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 straightforward teacher guidance and encouragement, especially with new, more complex content. Furthermore, Kids don't need to work in groups. Kevin Ashton (To Fly A Horse) points out, "Working individually is more productive than working in groups."
Kids are novices, not experts, so they need straightforward teacher guidance and encouragement, especially with new, more complex content. Furthermore, Kids don't need to work in groups. Kevin Ashton (To Fly A Horse) points out, "Working individually is more productive than working in groups."
More principles of the science of learning.
1. Memory is more reliable than calculating.
Calculating single-digit facts clutter a student's limited working memory and leave less space for thinking. For example, add 6 + 9 using make ten. The student adds 1 to 9 to make 10, then subtracts 1 from 6 to make 5 (to compensate, we are told), then adds the two results: 10 and 5 to get 15. That's a total of three calculations the student must do and hold in working memory. A few students can do this, but many students got confused and mixed up. To avoid the clutter and cognitive overload, simply memorize 6 + 9 = 15 (for instant retrieval from long-term memory). When the situation of 6 + 9 or 9 + 6 arises in a question or word problem, the student instantly retrieves 15 without thinking or calculating. It is automatic. Thus, one of the basic precepts of the science of learning is the relationship between working memory (limited) and long-term memory (vast) and why storing arithmetic essentials in long-term memory frees mental space for thinking, that is, "memory is more reliable than calculating."
2. Another principle of learning is explaining a worked example on the whiteboard step-by-step-by-step. Explaining something in small steps lessens the chance of cognitive overload in the student's working memory.
3. Another principle is that kids are novices, not experts or adults. Kids need straightforward, explicit instruction, not group work or collaboration. They need to learn [absorb] in-depth content in long-term memory to learn to think well. Furthermore, kids don't think like adults, and they are not pint-sized mathematicians or little scientists. Let me repeat, kids are novices and need to transfer content into long-term memory to think well. The transfer of material from working memory to long-term memory storage requires memorization, repetition, practice, and study.
Many poor decisions (with good intentions) have been made in education over the decades, and the progressive reformists continue making them. The policymakers (those in charge) wrongly decided to "push our existing system harder for incremental improvements and rely on policies calling for curriculum homogeneity, more standardized testing, and teacher accountability tied to student test score performance," write Wagner & Dintersmith (Most Likely to Succeed), to chase after higher test scores. I agree with Wagner & Dintersmith that the mastery of core academic content is essential. Still, I reject the solution of minimal teacher guidance methods (project-based learning) for accomplishing this. Knowledge is not all that relevant. It has always been true that children need to memorize and practice to master the fundamentals of math in long-term memory.
Frankly, I don't think kids who know bits and pieces of a reduced curriculum will be successful because skillful thinking in a subject without substantial knowledge of that subject is empty. Maybe, one day, the science of learning and an excellent math curriculum taught by teachers who know math will be implemented in our schools, but not under Common Core, its burdensome baggage, and goal to score higher on tests. In short, our schools under Common Core and state standards are test-driven. In ed schools, most teachers have been taught idealistic, unrealistic learning theories rather than the science of learning, which is practical. Like Sandra Stotsky, I reject the "radical constructivist view of mathematics learning," which is a modern “reform math movement [that stresses] an undefined conceptual understanding and student-created algorithms." (Quotes from Sandra Stotsky, National Mathematics Advisory Panel)
Note. This page is a work in progress. It still is. It is not an essay. It is far from complete. Please excuse typos and other errors. October 21, 2015, November 1, 2015, November 12, 2015, 11-19-19, 10-17-20
Draft 1.
Models: Hannah
Models: Hannah
©2015 - 2016 LT/ThinkAlgebra
