Sunday, November 24, 2019

Rabbit Hole Math

Falling Down The Rabbit Hole 
Alice exclaims, "I never heard of uglification." 
Like Alice, we have fallen down the rabbit hole where kids learn the different "branches of Arithmetic--Ambition, Distraction, Uglification, and Derision" as Mock Turtle points out. Mock Turtle is not a teacher; he is a facilitator, which is an important aspect of progressive pedagogy
In contrast, Alice explains, "I never heard of uglification." She's not alone! Down the rabbit hole, Alice encountered an alternative, unorthodox, cumbersome, constructivist form of arithmetic--not the straightforward, standard arithmetic that she had memorized, studied, and learned. 

Note. In math, American high school students are at the bottom (PISA, 2018).
Click: China is at the top.

Reform math is upside down and backward; that is, it grossly misinterprets Bloom's taxonomy by jumping to higher-level thinking skills before the fundamentals had been learned in long-term memory. Indeed, today, math education has become bizarre, cluttered, and counterproductive. Reform math makes little sense. Memorization is considered old school. "The prevailing theory is that students must engage in constructing their own knowledge rather than memorizing facts that will only bore them and that they don't really understand," writes Natalie Wexler (The Knowledge GAP). It's the wrong approach. She observes that "skipping the step of building knowledge doesn't work."

It is common sense that if kids don't learn the fundamentals of arithmetic, starting in the 1st grade, then they are blocked from higher-level math (Engelmann)

Aim for Competency!
If we want students to become competent in arithmetic and algebra, then they need to be more like ballet dancers, gymnasts, swimmers, violinists, chess players, etc. That is, students need to practice and review so that the fundamentals of math stick in long-term memory (i.e., for automaticity). We don't do that! In fact, what we often do in the classroom is the opposite of the cognitive science of learning. 

The Knowledge Gap
The knowledge gap has been known for decades and disregarded. In math, there is a considerable knowledge gap, too, because children don't memorize times tables or learn the mechanics of the standard algorithms like they used to 50 years ago. Hence, calculating skills are weak, and learning is not coherent, systematic, or hierarchical.  

Calculating skills are conveyed to concepts or ideas in math. Even though the ideas in math, themselves, are abstract, such as operations like addition or the concept of perimeter, they are learnable even in 1st grade. I know; I did it! 

Math Knowledge means math skills. Having math knowledge is the ability to convey the proper math skills to the appropriate problem type and solve the problem by paper calculating. Getting the correct answer has always been critical in applying math, but it has been marginalized in current math teaching, which is a mistake. 

To advance, kids must know math facts and standard mechanics in long-term memory.
The math fact 3 + 8 = 11 is the solution to a vast number of word problems. Because the math fact is abstract, it can be used again and again. That's the power of abstraction. If the child memorizes the 3 + 8 fact, suddenly she has the solution to hundreds of math problems. It's math power! Even though a student may have some idea of perimeters, she cannot calculate a perimeter unless she knows sums. If you can't calculate it, then you don't know it!  The memorization of facts is good for novices.

Calculating skills are intrinsic skills for solving problems in math. 

Education, today, does not work. Math needs a radical break from the status quo, which has been reform math. Teachers should embrace a knowledge approach that is supported by the cognitive science of learning. Also, in math, knowledge is math skills. There is a lack of content in elementary school math. Common Core is a continuation of a content-free approach. It is not world-class. Schools fail to build knowledge, not only in reading but also in math, science, and other subjects. 

In addition to calculating and logic, math skills include knowing vocabulary, applying concepts and mechanics, recognizing patterns (e.g., problem types), and grasping the meaning of structure (e.g., 2n means 2 times the number n; 3^4 means the product of 4 copies of 3, which is 3 x 3 x 3 x 3 = 81; ab = ba, and so on). In other words, students must know math stuff in long-term memory to perform math. Furthermore, knowledge is the basis of problem-solving. 

You can't solve trig problems without knowing some trig. You can't translate Latin without knowing some Latin. You can't grasp literature without knowing some textual criticism. All fields of study require the memorization of the basics and experience. It's a process of building knowledge. There is no generalized thinking skill. Thinking is domain-specific. 

Study hard enough to become Smart enough! 
(South Korean Motto)

©2019 - 2010 LT/ThinkAlgebra 

Friday, November 15, 2019

Cognitive Science

Cognitive Science

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.

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.
---------------------- End of Section

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.      

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.   
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.

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.

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.

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."  

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

©2015 - 2016 LT/ThinkAlgebra

Friday, November 1, 2019

NAEP Test Scores

Lost in Math. You are not alone.
66% of 8th-grade students are not proficient in math.
66% of 8th-grade students are not proficient in reading.
The latest NAEP scores show that students are not getting any better.  Many students are regressing in readiness. What can we do to reverse the trend of stagnation? Teachers and schools are not to blame, but the progressive policymakers who have never been in the classroom are. 

NCTM: National Council of Teachers of Mathematics
OECD: Organization for Economic Cooperation and Development
NAEP: National Assessment of Education Progress
WM: Working Memory
CC: Common Core

Only 35% of 4th graders are proficient or above in reading. Also, a child rated as proficient in reading does not mean the child is reading on grade level, warns the NAEP. It is the same story in math--only 41% of 4th graders reach the proficiency cut scores. Again, a student who is rated as proficient does not mean the student is doing the math on grade-level. Indeed, kids are poorly trained for college-level work. And many blame Common Core and state standards that are primarily based on CC. Betsy DeVos slammed the K-12 education establishment for allowing students to fall behind in math and reading (Newsweek). I blame the reform math curriculum, along with faulty or inefficient math instruction (the teaching) that often conflicts with cognitive science. Teachers are not teaching conventional arithmetic; they are teaching reform math leftover from the failed 1989 NCTM math standards, which, unfortunately, were resurrected via CC. 

The K-5 Teacher Conundrum 
Elementary school teachers tell me that they have to teach many of the strategies because they are on the state test and in materials they are told to use. Where are the standard algorithms? 

Strategies Crowd the Curriculum
Standard Algorithms Are Not a Priority
Memorization and Drill Are Downgraded 
"For each of the four multi-digit operations, the Common Core standards ask students to practice multiple strategies for one year or two before learning the standard algorithm," writes Eric A. Nelson ("Cognitive Science and the Common Core Mathematics Standards"). The strategies crowd the curriculum, along with a bunch of other stuff, such as the eight standards for mathematical practice, social-emotional learning, self-esteem and mindfulness activities, and lots of group work (e.g., discovery learning, project-based learning, and so on). The consequence has been to downgrade the importance of the standard algorithms. "Students are taught under math standards that discouraged initial memorization for math topics ... [thus, they] will have significant difficulty solving numeric problems in mathematics, science, and engineering." Nelson points out that U.S. 16-24-year-olds ranked dead last on a recent OECD assessment of numeracy skills among 22 developed-world nations. Nelson writes, "The CC standards do not ask students to memorize facts and procedures for some key topics and delay work with memorized fundamentals in others." 

The CC math standards do not ask students to memorize the subtraction and division facts. Furthermore, many students coming into the 4th grade and 5th grades have not mastered the multiplication facts. Teachers have been taught to decrease memorization and drill (aka practice). Doing this is contrary to the cognitive science of learning. 

The multiplication table and the standard algorithms for both multiplication and long division should have been learned no later than the 3rd grade so that students can engage in problem-solving. 

"When solving math problems of any complexity, due to Working Memory limits, students must rely almost entirely on well-memorized facts and algorithms," writes Nelson. CC does not require students to memorize half of the math facts. Subtraction and division facts are not remembered and are calculated as needed using strategies, which is a backward approach. 

Nelson writes, The 1989 NCTM standards called for "increased attention" to "reasoning" and decreased attention" to "memorizing rules and algorithms," "manipulating symbols," and "rote practice." He points out that there has been a sharp decline in student test scores in math computation.

In short, to improve student problem solving (avoiding the limits of the Working Memory), students should automate (memorize) the math facts and learn the standard procedures first, not a bunch of strategies that interfere with and downplay the standard algorithms and math facts. Why waste a couple of years of instructional time teaching strategies instead of what students must know for problem-solving: math facts and standard algorithms?   Who uses the array, lattice, or area strategies to multiply numbers? Is it any wonder that American students are behind international math benchmarks?

FYI: Incidentally, the student at the top of the page (Kailey), when she was in 7th grade, solved quadratic equations by completing the square. All 7th graders took Saxon Algebra One. Most of the children who model for my illustrations are excellent math students. 


©2019 - 2020 LT/ThinkAlgebra