Radical Ideas 2

Radical Ideas 2

Merry Christmas

Link to Radical Ideas 3, which will soon replace Radical Ideas 2.

After eight years of Common-Core-based reform math, a meager 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Common Core reform math doesn't work, so why are we still teaching standards and progressive ideas based on Common Core? When big decisions are made in education, the mistakes are not small, and the unintended consequences run deep, such as 76% not proficient in math. Common Core has not made our students better in math.

Remote makes some kids sad,
angry, and frustrated. 

Remote schooling is leaving some children sad and angry, writes Hannah Natanson and Laura Meckler at the Washington Post. For example, a 9-year-old student is stuck in Zoom school and hasn't seen a friend since March. She cries when she gets angry and frustrated with remote schooling and fears she isn't learning enough new material to pass the 4th grade.

Remote often hurts kids more than it helps. It is a bust. How many kids are self-motivated to pay attention and stare at a screen much of the day and do homework?

Musing

November 27, 2020

✓  "If you can't explain difficult mathematics to little kids, then you don't know it well enough."  (Or, you don't know how children learn math.) H. Wu, a mathematician at UC-Berkeley, has been teaching workshops and courses for K-8 teachers for decades. Wu wrote that most teachers don't know enough math to teach Common Core math. Many teachers are just average and have difficulty explaining complicated math to students. Many teachers take my course in the summer, says Wu, but when teachers return to the classroom, they teach the same old reform math with minimal guidance group work methods. Learning more content doesn't always alter the way teachers teach, noted Wu, but it is a step in the right direction. The premise has been that teachers know best how to teach math and reading. Really? National and international test scores don't support the premise. 

✓  Note: Common Core is not world-class because it does not meet international benchmarks in math. By 4th or 5th grade, students are about two years behind their peers from top-achieving nations. The advent of Common Core in 2011 revived reform math and its minimal guidance methods, which have been promoted by math educators from schools of education. Teachers seem to be teaching pedagogy, such as 5 or 6 different ways to multiply, not content. The standard algorithms are pushed to the side.  

✓  The reformers want to stamp out Old School, such as traditional arithmetic, memorization, and practice-practice-practice. They say they know best how to educate your child. Well, I think not. Reform math has been a bust, along with fads such as Common Core, remote and hybrid learning, minimal guidance methods of instruction (e.g., group work, discovery learning, project-based learning, etc.), equalizing downward by lowering those at the top (Thomas Sowell: a crazy idea taught in schools of education), mathematical practice standards (from the NCTM people), below world-class math standards, grade inflation, and low expectations, all of which make K-12 math education inferior and mediocre. 

😧  After eight years of Common-Core-based reform math, a meager 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Our K-12 math teaching is horrible. The problem originates in 1st grade with inadequate teaching of basic arithmetic. In my analysis of 2011, the Common Core 1st grade math standards were significantly behind the Singapore 1st-grade syllabus of arithmetic content. Singapore 1st-grade students learned a lot more arithmetic content than American students, such as memorizing arithmetic facts, using formal algorithms, learning multiplication, and writing equations to solve word problems. In short, Common Core math standards were not world-class, so why did states eagerly adopt them?    

Note: This page has grown, so I will likely delete major parts by Christmas or start "Radical Ideas 3" for the overflow. The redundancy is intentional. 

Algebra in 1st grade: Teach Kids Algebra

✍️ The late Richard Feynman wrote, "I would rather have questions that can't be answered than answers that can't be questioned." It is why I oppose those in education who think they know best how to teach children math, such as Jo Boaler, a so-called math educator. If they had known how best to teach math, then almost all our kids would learn algebra in the 1st grade. Frankly, when asked how I teach 1st and 2nd graders real algebra, I reply, "I don't know how I explain complex material to children ... But I do know that if I can't explain difficult math to students, then I don't know it well enough." Also, I think in terms of prerequisites (Gagne), not stages (Piaget), and write coherent "worked-out examples" that are performance-based (Mager's behavioral objectives). Is there a more straightforward way to get from A to B? 

✓  Why make learning arithmetic harder than it is? It is another reform math radical idea that makes no sense. Learning arithmetic fundamentals should not be confusing, hard, or complicated. 

✓ I agree with Daniel Willingham that "children are more alike than different in terms of how they think and learn." Learning styles are not supported, that Jane learns better in one way that would be bad for Bill. No, both Jane and Bill think and learn in the same way through practice that makes memory long-lasting.  

📌 Association Theory

New math content is learned faster and easier when associated with or linked to existing math knowledge in long term memory. Prerequisite knowledge is essential, not stages of development.  

✍️ Educators need to "make learning easier: more user-friendly and far more accessible," writes Sanjay Sarma at MIT (Grasp, 2020) and abandon the idea that "serious learning should be difficult." Reform math is an excellent example of making arithmetic harder to learn with at least five multiplying methods and little attention paid to standard algorithms or memorizing math facts. Toss into the mix the so-called standards for mathematical practice, social-emotional and self-esteem stuff, minimal guidance instructional strategies (i.e., group work, discovery learning, etc.), and a lot of grade inflation. What a mess! Is it any wonder that most kids never become good at arithmetic, which is the backbone of algebra? 

Chris Ferrie has the ability to explain complex science to little kids. 

✍️ "Science doesn't prove things right. It is a method that eliminates wrong ideas." We don't have anything like that in education or in our science textbooks. It is difficult to find control groups for experiments. Often, instructional time is not controlled but should be. Consequently, educators seldom weed out bad ideas. If an idea, belief, or theory doesn't work in the classroom with real kids, it is wrong. If it doesn't agree with experiment, it is wrong, says Feynman. How many teachers can give simple explanations of complex ideas?

✍️ Fairness Policies Downgrade Individual Excellence

Unfortunately, the reform math people have dominated mathematics education in our schools for decades and believed that their ideas (pedagogy, ideology, & philosophy) are superior regardless of evidence. Their "fairness" policies are flawed because they downgrade excellence and individual achievement. They think that "equalizing downward by lowering those at the top"(1) is good policy, but I believe it has been a destructive policy. The "fairness crusades" have marginalized individual achievement, hard work, and excellence in our schools. (1) The quote is from Thomas Sowell (Dismantling America). 

It's true!!!!!!!!
The more I know, the more I can learn, the faster I can learn it, 
the better I can think and solve problems. WOW, isn't cognitive science great?

Note: This website counters reform math, discovery learning, project-based learning, and other minimal guidance, constructivist methods (via group work). It also opposes the status quo, including teacher unions that tell teachers not to teach as they demand more money. Schools are loaded with beliefs, fads, and policies, such as remote and hybrid learning, which have scant evidence supporting them. This website voices my opinion. It includes Radical Ideas and who endorses them. Sameness is a basic tenet of socialism, which has gripped our schools for decades. The latest strategy is Common Core reform math that started in 2011. After eight years of Common-Core-based reform math, only 24% of 12th graders were proficient in math (NAEP 2019), an awful statistic. Common Core and state standards have failed to deliver career and college readiness for 3/4 of the students. 

The math curriculum is not world-class, according to my 1st-grade analysis of 2011. By 4th grade, most of our students have fallen two years behind in math achievement. Moreover, grade inflation dominates our public schools. Kids get good grades for no good reason to boost their self-esteem, we are told by education leaders. It is nonsense, and it was replaced by so-called social-emotional lessons. Really?    

💡 Thomas Sowell (Discrimination and Disparities, 2019) points out the toxic idea of "equalizing downward by lowering those at the top." He describes it as a crazy idea taught in education schools. The quest for sameness by any means, even if kids learn less, has been a "fallacy of fairness in education," explains Sowell, a noted black scholar. He writes that not all disparities are discrimination. Many are not.

✍️ Notes from Daniel Willingham, Richard Feynman, Amy Chua, Stanislas Dehaene, and others.

• To learn something is to remember it.
• Engagement is not the same as learning. 
• Critical thinking is difficult to measure.
• Practice is necessary to improve.
• The spiral curriculum of J. Bruner failed.
• Practicing math facts will help with long division.
• Children need feedback so that they can make corrections.
• We have good tests that measure content knowledge.
• Our ability to measure creativity, collaboration, or critical thinking is limited.
• "Thinking well requires knowing facts.  Factual knowledge must precede [higher level thinking] skill."  
• Opinion is not science.
• Your mind is lazy and doesn't want to think.
• You have to force yourself to recall a fact or a procedure in arithmetic or algebra.
• To learn effectively, students should quiz themselves at home and school on new content Flashcards give instant feedback. (Stanislas Dehaene, How We Learn, 2020).
• "Your ideas will never be more effective than your ability to make others grasp them." (Thomas Oppong, 50/50) In short, you have to explain complex stuff so those very young children can begin to grasp it. Many teachers cannot teach content well. (This is similar to a saying by the late Richard Feynman, who, in physics, was the "Great Explainer.")
• With a good education, you increase differences. (Feynman)
• "Nothing is fun until you are good at it. Rote repetition is underrated in America" (Amy Chua)
• A Chinese mother believes that "Schoolwork always comes first; an A-minus is a bad grade; Your children must be two years ahead of their classmates in math." And, "No matter what, you don't talk back to your parents, teachers, elders." (Amy Chua, Battle Hymn of the Tiger Mother, 2011). 

✍️ Radical Ideas 

Let's Equalize Outcomes, Inflate Grades, Teach Less Content, Dump the SAT/ACT for Diversity & Equity, Change OBE (Outcome-Based Education) from academic goals to "feelings" goals, Dump traditional arithmetic (Old School) for reform math that is to world-class, Dump explicit teaching for ineffective Minimal Teacher Guidance During Instruction methods, Dump paper-pencil calculating for calculators, Dump memorization and repetition for content-free critical thinking, etc. The influential leader of these and other radical ideas is Jo Boaler, a "math educator" at Stanford University, not a mathematician or scientist.

✍️ The use of worked examples can substitute for problem-solving in learning Algebra (Sweller). Over many decades, I found that explaining carefully selected worked examples on the board can boost K-5 arithmetic learning (i.e., skills,  ideas, and uses) and middle school algebra, especially when joined with flashcards for retrieval practice from long-term memory (Dehaene). Flashcards have many learning applications: math facts/definitions/procedures, spelling, definitions, vocabulary, procedures, language, and so on. Why do progressive teachers ignore flashcards? (Note: ZOOM is a fad that will fade, but flashcards are not and will come back because they help children learn.) 

✍️ Problem-solving is a function of knowing stuff. Try solving a trig problem without knowing trig or translating Ovid's Metamorphoses without knowing Latin vocabulary and structure.

✍️ There is no substitute for knowledge in long-term memory and the practice that gets it there. It's the cognitive science of learning.

✍️ Children can learn much more math content than taught. The problem begins in K-5 with standards that do not meet international benchmarks, a reform math curriculum that marginalizes basic arithmetic and standard algorithms, and minimal guidance teaching methods that are inefficient (group work). It's the teaching! Math achievement has stalled. For at least a decade, our students have not been getting any better in math. Students fall behind, and remote and hybrid teaching exacerbates the problem. The curriculum, which is based on Common Core or Common-Core-like standards, is not world-class.

✍️ Students are not prepared. Like math, reading scores have stagnated over the past decade, too (NAEP). Only 37 percent of 12th-graders were proficient in reading, 24 percent in math. The results of the 2019 NAEP tests are bad news for the reformers. It means most 12th graders are not prepared to pass entry-level college courses. But the deficiencies start in K-5. The fundamentals of arithmetic and reading have been taught poorly.

✍️ Alice Crary and W. Stephen Wilson in the New York Times (2013) point out that reform math programs have killed traditional math. They write, "The standard algorithms are either de-emphasize to students or withheld from them [students] entirely." Moreover, "The staunchest supporters of reform math are math teachers and faculty at schools of education," where teachers are trained. Now you know why reform ideas persist in our classrooms. Many of the reform ideas taught in schools of education are rooted in NCTM standards, dating back to December 1989. In reform math, the reasoning is much more important than learning content knowledge. But it isn't. 

There is no substitute for knowledge in long-term memory and the practice that gets it there. Problem-solving is a function of knowing stuff. To perform arithmetic and algebra well, students must know facts, procedural knowledge, and problem types in long-term memory. But, reformers think that facts and techniques don't matter much. Really? All the reasoning (i.e., critical thinking) in the world won't help you solve a trig problem unless you know trig. Incidentally, the latest surge of reform math correlates with Common Core or Common-Core-like state standards.  

√ Joye Walker, High School Math Teacher

The dire results of reform math show!

"I am in my 22nd year of teaching high school math and am astonished when kids can't multiply.  One time I saw a student doing 26+26+26+26+26+26+26 on an algebra 2 quiz (no, I didn't let them use calculators) and asked him why he didn't just multiply 26x7.  He replied, "I really never learned to multiply."  I have seen 1000+1000+1000+1000 on people's scratch paper. No wonder they can't [multiply fractions]! What repeated addition works there? Just shouldn't happen." 

I saw the same nonsense in 4th-grade when some kids did repeated addition, lattice, and other methods to multiply, even though I demonstrated the standard algorithm (3 x 75). Sadly, the standard algorithms for long multiplication and long division are not taught in many schools, much less mastered, by age 10, the international benchmark. Kids use calculators, so they stumble over simple paper-pencil arithmetic. Arithmetic is the basis of algebra, but most K-8 teachers don't get it.    

 

✍️ Pre-Algebra is for average 7th-grade students who are prepared. It is not advanced math. It's arithmetic with variables, expressions, and equations. Thus, students need to master arithmetic beginning in 1st grade. Unfortunately, almost all middle school math textbooks, including pre-algebra texts, and many elementary math programs, use calculators, even in 1st grade. One popular elementary school program states that if some students have difficulty learning math facts or algorithms, then let them use calculators, which is a bad idea.

Here is a good idea: dividing by a number (other than zero) is multiplying by its reciprocal. It is basic arithmetic that should be taught in 4th grade with fractions. Thus, 8 ÷ 4,  by definition, is 8 x (1/4). Therefore, a division is actually a multiplication by reciprocal. Zero doesn't have a reciprocal because the product of reciprocals must always equal 1 by definition (5 x (1/0), oops, one can't divide by zero.  (Model: GabbyB, a Middle School Student)

✍️ Prealgebra is for average 6th, or 7th-grade students provided they are prepared properly in K-5 math (e.g., standard algorithms, fractions-decimals-percentages, ratios, reciprocals, equations, integers, exponents, etc.). The prealgebra textbooks presume students are ready and have competent calculating skills across operations, especially whole numbers and fractions/decimals. Almost all prealgebra textbooks require calculator use, so calculating skills must be learned well in K-5, but they are not. The crux is that reform math based on Common Core or state standards derived from CC does not adequately prepare students for a 7th-grade prealgebra course. Reform math marginalizes calculating skills, including the standard algorithms. It does not prepare students for prealgebra or algebra courses in middle school. Calculators should not replace weak calculating skills. Students with weak arithmetic skills struggle with algebra and high school math. 

✍️ A popular elementary school program (EM) encourages the early use of calculators, a terrible idea. "In fact, 4-function calculators are sufficient in Kindergarten through Grade 3, and scientific calculators are sufficient in Grades 4 through 6." According to the Teacher's Reference Manual, some studies suggest that "calculator usage does not hinder the development of paper-and-pencil skills." There is a "preponderance of evidence." Really? These so-called studies, like so many that pop up in education, are mostly bunk. Reform math, I believe, is why U.S. students never seem to master arithmetic and algebra. Memorizing math facts to support standard algorithms is not stressed. Reformists claim that traditional arithmetic skills are obsolete; they are not. The skills, including long-division, are essential for algebra. (Note: EM is short for Everyday Mathematics, part of The University of Chicago School Mathematics Project, which is very popular.)

Contrarily, kids are novices and should not use calculators or calculating software for anything in K-5. Calculator usage marginalizes the importance of learning math facts and standard algorithms. After 8 years of Common Core reform math, the result is that 76% of 12th graders are not proficient in math (NAEP 2019). In short, for at least a decade, our students are not getting any better in math. Most kids who want to go to college end up in remedial math classes (mostly high school level algebra) at community colleges. ("But, I got As and Bs in high school algebra!" Regrettably, many high school courses were dumbed down and called "college prep" in name only, but not in content.) 

✍️ Capable students in the 6th or 7th grade should take a rigorous Pre-Algebra course to prepare for Algebra-1 in middle school--not reform math from Common-Core-like K-8 standards that push Algebra-1 into high school. Algebra-1 is for average middle school students who are prepared. Many traditional algebra topics are introduced in several Pre-Algebra courses, including right-triangle trig. K-5 students must be prepared much better in standard arithmetic for pre-algebra and algebra courses in middle school. My Teach Kids Algebra program (2011-2019) introduced important algebra ideas to average students as young as 1st grade via standard arithmetic. 

Note: Likewise, AP Calculus is for average high school students who are well prepared, explains Richard Rusczyk, the Art of Problem Solving. Since most textbooks were written for average students, he decided to write his own textbook series for the very best math students. Rusczyk's challenging textbooks for advanced, gifted math students start with 6th-grade Prealgebra. The texts are not for average or even above-average students. They are for the cream of the crop and require the best teachers, too.

✍️ No later than the 4th or 5th grade under Common-Core-like state standards, US students have fallen behind their Asian peers by about two years in math achievement. Can they catch up in high school? I think not. Only a meager 24% of 12th graders--who have had eight years of Common-Core-like reform math via state standards--are proficient at math (NAEP 2019). In short, our students were not getting any better. The hype and promise of the Common Core's primary objective of college/career readiness went bust. 

Common-Core-like state standards keep our kids below international benchmarks. Students start behind and stay behind in progressive, reform math classrooms filled with radical ideas. One radical idea is that everyone can create original mathematics. Really? Kids are novices, not little mathematicians, and should not approach math as working mathematicians. 

Note: Only 24% of 12th-grade students reached the Proficient Level or above in mathematics (NAEP, 2019): 3% rose to the Advanced Level, but a whopping 40% scored below the NAEP Basic Level. In other words, K-12 teachers have not been teaching the arithmetic and algebra that kids must master. They are pretending to teach math by using calculators and group work. But, the basics are for everyone. There is no reason that students coming into 4th grade don't know the single-digit multiplication facts and standard algorithms for both long multiplication and long division. Students stumble over simple arithmetic because of the way math is taught nowadays as reform math. The late Zig Engelmann explained that inadequate math and reading achievement boils down to "the teaching." For example, one reform math idea is that learning should always be fun, but the reality is that learning the basics of arithmetic and algebra is hard work for most students, not play. 

Furthermore, suppose you don't know enough arithmetic. In that case, you will struggle to learn algebra (e.g., quadratic and exponential equations), higher-level mathematics (e.g., calculus), and algebra-based chemistry and physics, much less earn a degree or certificate. STEM is out, too. And, your financial education will likely suffer immensely, living paycheck to paycheck, investing $zero. Why don't K-5 teachers grasp that traditional arithmetic is essential to the student's future? So, teach arithmetic, not reform math! 

If you enroll at a community college, the placement test will probably sort you into remedial courses (high school algebra). Still, very few kids make it through remedial math mud. You will likely drop out. Elementary school teachers must make sure that kids learn standard arithmetic well in grades 1 to 5. It may not be much fun, but the student's future depends on it. The crux is that most teachers teach reform math, not traditional arithmetic. They need to switch to standard arithmetic and accept Old School ideas such as memorization, repetition (drills), and flashcards, but it won't be easy.   

✍️ K-5 math should get kids ready for algebra, but K-5 reform math doesn't. Dr. W. Stephen Wilson, a mathematician, writes, "The place value system, fractions, and standard algorithms all contribute greatly to algebra readiness." Reform math, which replaced traditional arithmetic decades ago, does not emphasize standard algorithms. It also asks students to invent algorithms, use calculators, and solve math problems via critical thinking without knowing math. These are radical ideas promoted by the far-left. Immanuel Kant (1724 - 1804) wrote that thought (i.e., critical thinking) without content knowledge is empty. Why do most teachers skip the knowledge-learning step? After all, a student cannot do percent problems without knowing the math of percentages and equation-solving in long-term memory. 

Natalie Wexler (The Knowledge GAP, 2019) writes, "But skipping the step of building knowledge doesn't work." She points out that the real problem is the "failure of elementary schools to build knowledge." Building knowledge is not in state standards, we are told. But, teachers changing from reform math to standard arithmetic won't be easy. I think it is next to impossible because of liberal indoctrination common in the profession. Parents may need to teach basic arithmetic to their children, hire a tutor, or enroll in Kumon. Parents are ultimately responsible for their children's education, not the schools. 

Reform math, which permeates most of today's progressive classrooms, marginalizes arithmetic. I have stated it several times. I am also told that some kids can't memorize the multiplication table; thus, if some can't, then none should be required. In Everyday Mathematics (EM), kids use calculators for facts they have not memorized and "calculations beyond their skill level." In contrast, it has been my experience that kids can learn most of the x-table in the 2nd and 3rd grades and perform the standard multiplication algorithm as needed by age 10, including long-division. 

A 5th-grade teacher once said that she doesn't teach long division because students don't understand it. My response: "Do your students understand the nonstandard, partial quotient method? How about the lattice method for multiplying 87 x 3.85? Do your students understand that?" I no longer try to reason with teachers. In EM, "The addition algorithm is probably the best of the U.S. traditional computation algorithms. While EM does not focus on it, it is a viable alternative." [Alternative to what? Why do you not focus on it?] "If you decide to teach this algorithm [Apparently it is optional and not that important], be sure to treat it as one of several possibilities...." [Really? Is it any wonder that American kids are bad at basic arithmetic. It is not taught.]

✍️ Radical Ideas 

Let's Equalize Outcomes, Inflate Grades, Teach Less Content, Dump the SAT/ACT for Diversity & Equity, Change OBE (Outcome-Based Education) from academic goals to "feelings" goals, Dump traditional arithmetic for inferior reform math, Dump explicit teaching for Minimal Teacher Guidance During Instruction methods, Dump paper-pencil calculating for calculators, Dump memorization and repetition for content-free critical thinking, etc. The influential leader of these and other radical ideas is Jo Boaler, a "math educator" at Stanford University, not a mathematician or scientist.

✍️ We are in an era of dumbed-down math and extremist, far-left ideas. Reform math is not world-class math—the cumbersome procedures of reform math screw up basic arithmetic. The views expressed on "Radical Ideas" are my opinions and subject to change. Note: My Teach Kids Algebra (TKA) algebra program for grades 1 to 4 is not a far-left radical idea. It expands the curriculum beyond traditional arithmetic, which is long overdue. TKA has its roots in The Madison Project (MP 1957) and Science--A Process Approach (SAPA 1967). The MP (1957), SAPA (1967), and my TKA (2011) show that very young children can learn much more content than the current math curriculum, beginning in the 1st grade.  

First-Grade STEM Lesson - An Equation Is a Balance

Teach Kids Algebra (TKA) by LarryT

• Equation Structure: Expression = Expression (x - 3 = 19)
• Equal-Arm Balance: Left Side = Right Side
• Solving (Guess & Check): x - 3 = 19 is true only when x = 22 
• Balanced: 19 = 19

Note: Some kids in the 1st and 2nd grades figured out x (the unknown) by adding 3 to 19: x = 22. (Inverse Idea: to undo -3, just +3 to both sides of the equation to isolate x.)

My "Teach Kids Algebra" STEM Program

✍️ Educators grossly underestimate the content knowledge kids can learn given proper instruction. Starting in 1st grade, elementary school students learned algebra (TKA) via worked examples (explicit teaching)--not discovery lessons, group work, or calculators. (Click: First Grade)  

Students in grades 1 to 4 need to know math facts, ideas, rules, and uses in long-term memory to free space for solving problems in working memory. For example, memorizing half the multiplication facts should be a major part of the 2nd-grade curriculum, but it isn't. The rest of the times tables and long multiplication and long division should be learned in 3rd grade (by age 10). 

In TKA, I focused on algebra's big ideas, such as variables, expressions, integers, and the three models of functions: equation-table-graph. (TKA, created in 2011, was a reaction against reform math by reinforcing the importance of standard arithmetic.) "Learning algebra requires a lot of basic knowledge and [step-by-step] techniques." (Quote: Ian Stewart, Mathematician)

✍️ Classroom teachers should focus on the mastery of essential content, not state test-based proficiency. Daniel Willingham writes, "Don't expect novices to learn by doing what exerts do." Kids are not pint-sized scientists, mathematicians, or scholars. They are beginners. Even a Calculus 101 student in college is a novice in calculus. However, learning calculus is not the same as learning to think using calculus (Sanjay Sarma, Grasp, 2020), which takes a lot of experience. Learning addition in 1st grade is not the same as learning to think using addition. But the first steps are learning place value, memorizing single-digit addition facts, and practicing the standard algorithms. In short, background knowledge. 

✍️ Links

Link1: How I Shoot Photos

Commentary

Link2: Gifted Programs

Our best math students often go unchallenged and fend for themselves in mixed classrooms. Some students need advanced-level content and fast-paced math classes because they are a couple years ahead of their peers in math achievement, but their academic needs are seldom met in elementary and middle school.

Link3: Teach Kids Algebra (TKA)  

STEM Math for Grades 1-5 

TKA started in 2011 as a reaction against Common Core reform math. I fused basic algebra ideas with standard arithmetic, not reform math. The importance of traditional arithmetic was stressed, starting with the automation of single-digit math facts that support the standard algorithms. It starts in 1st grade. 

Link4: PISA 2018  

American students did poorly in math. Gee, I wonder why?

Link5: Cognitive Science

Students learn new ideas linked to old ideas they already know. Readiness is not age-dependent; it is determined by the student's mastery of the prerequisites. 

Link5: Science

Science doesn't prove things right. It is a method that eliminates wrong ideas. Correlation should never imply causation.

Link6 Early Math

Early math is just as important as early literacy but is often neglected. The crux is that kids need explicit teaching from K-5 teachers who know math, but many elementary school teachers are weak in math.

 

✍️ My early algebra program for young children (Teach Kids Algebra) in grades 1 to 4 was acknowledged in a blurb on The Hechinger Report, the mid-January 15th (2020) issue. At the time, I had two 4th-grade classes and a 2nd-grade class. Very young students can learn symbolic arithmetic (aka algebra). By April 2020, I wrote several 1st-2nd-grade lessons for The Old SchoolHouse. The problem has been that the so-called advanced content is never introduced because it is deemed developmentally inappropriate. Piaget's theories don't work in the classroom. We grossly underestimate the math content very young children can learn. Piaget's view has been an excuse for not teaching certain content or limiting content by grade level, branding it as developmentally inappropriate.

First Graders should be discouraged from using fingers and encouraged to memorize addition facts.

✍️ Algebra-1 is a middle school subject for students who are prepared. Preparing for Algebra-1 starts in the 1st grade. Students won't be ready for Algebra-1 by middle school unless we upgrade the curriculum to world-class and boot out popular constructivist instructional methods (e.g., discovery learning, project-based learning, and other minimal guidance or group work methods) and theories that don't work; e.g., Dewey and Piaget.

Weekly Algebra Lessons - STEM Math - 4th Grade 4/3/2012

Explicit Teaching Via Worked Examples

😇 Teach Kids Algebra (TKA)

✍️ It's Not That Hard (If Kids Are Prepared)

Educators often make learning math hard, but it isn't that hard when explained well and linked to basic arithmetic students already know. Students have a vast capacity to learn, but educators are not taking advantage of it. I decided to teach introductory algebra to 6 and 7-year-olds (1st and 2nd-grade students) at an urban, Title-1 school of mostly minority students. Skills, ideas, and uses were introduced with worked examples. I fused algebra ideas with standard arithmetic. For example, 5 + 7 = 12 is a true equation or math fact shown on the number line, while  x + 7 = 12 is an equation in one unknown. What value of x makes a true statement?  Solving an equation for an unknown (x) means finding the value of x to make a true statement. So mathematics such as arithmetic or algebra "requires a lot of basic knowledge and technique," points out Ian Stewart (Letters to a Young Mathematician, 2006). Students must learn proper "technique" as they gain "knowledge" beginning in 1st grade. How does a 6-year-old solve simple equations? They use Guess and Check, based on memorized math facts and rules from long-term memory. Note: I show inverse equation-solving techniques (Undo) in the 3rd grade, but I would like to try it in 2nd grade.

Second and third graders also need to memorize the multiplication table. Thus, according to Edward Thorndike, 3 x 7 is associated with 21 and is easily shown on a number line. For example, a linear equation in y = mx + b form can be related to an x-y table of values, making a picture on a graph by plotting (x,y) points—one thing associates with another, with another, and so on. All three models of a function are interrelated. (Note: Slope/similar triangles, the meaning of y = mx+b; quadratic and exponential equations: 3rd to 6th grade). 

 

😢 Teach Kids Algebra STEM math lessons were discontinued for grades 1 to 4. The TUSD schools are still closed, and the hybrid model has been delayed (again) until after Christmas, so kids will remain in remote learning, which is a lousy educational choice. Kids fall behind, but no one seems to care that these kids' education has been ruined.    

😇  But things may get better if and only if we forge ahead by bringing kids back to the classroom full-time. Instead of "fixing the blame," let's "fix the problem." (the late Dr. Robert Schuller) In the 80s, I had Schuller's sign in my primary TAG classroom, and I would direct students to it as needed. It worked well. We in education should be beacons of hope for students, so drop negative rhetoric aimed at America by some teachers and citizens.

✍️ Reform Math Permeates Our Schools: It Failed in the Past.  

Reform Math resurged with the advent of Common Core, but math achievement has stagnated ever since. Why should math be taught the way mathematicians work? (Children are novices, not experts.) In reform math, students invent their own algorithms, use calculators and manipulates, and engage in group work. For equity, every student gets the same grade-level instruction (e.g., Eureka Math: Common Core reform math), regardless of achievement. Is this the way to teach beginners arithmetic? I think not. Children are novices, not experts. They have not automatized necessary procedural knowledge and facts and need to follow exact steps for problem-solving. Therefore, teaching the fundamentals of standard arithmetic should be the teacher's main focus. (Note: Test prep may increase state test scores but is an inferior math curriculum.)  

Reform math (Common Core) is not world-class math, which puts our kids behind their Asian peers. It doesn't stress the early mastery of essential content, such as memorizing the multiplication facts or learning standard algorithms in long-term memory. Jo Boaler, an influential reform math leader, marginalizes standard or traditional arithmetic and advocates "teacher as facilitator" and minimal teacher guidance methods during instruction (i.e., reform pedagogy, group work, etc.). The methods are ineffective, according to Paul A. Kirschner, John Sweller & Richard E. Clark, 2006, "Why Minimal Guidance During Instruction Does Not Work." In my opinion, the left has messed up math education badly for at least two generations.  

Minimal Guidance Methods = Minimal Learning

The Kirschner-Sweller-Clark Equation by LT

If Singapore 1st-grade students can drill and memorize the addition facts, learn formal (standard) algorithms, and use multiplication as repeated addition to solve problems, why can't our 1st graders do the same? The reformists wrongly believe that mathematical problems can be solved by jumping up to higher-level thinking, skipping over prerequisite background knowledge. Kids need content knowledge and efficient calculating skills in long-term memory to solve math problems. Try solving a trig problem without knowing trig and equation-solving methods (calculating). 

✍️ Knowledge is the basis of critical thinking, creativity, problem-solving, and innovation. Furthermore, critical thinking or problem-solving is domain-specific.

✍️ Study hard enough to become Smart enough! (S. Korean Motto)
I believe that most students can learn arithmetic and algebra at acceptable levels with study, practice, and effort when they are taught traditional math explicitly, not reform math using minimal guidance methods, group work, alternative algorithms, or test prep. We keep lowering the bar instead of encouraging students to do better by working harder, longer. For example, the math curriculum, based on Common-Core-like state standards, is below international benchmarks. Our 4th or 5th graders, on average, are two years behind their peers from Asian nations. In short, the math education that most kids get is substandard, starting in 1st grade. The goal should be the early mastery of basics, not state test-based proficiency. It has always been true that children need to memorize and practice (drill) to master the fundamentals of math in long-term memory, which requires a significant curriculum upgrade for all students. 

✍️  Correlation Is Not Causation

Diane Ravitch (Reign of Error, 2013) asserts that poverty and racial isolation are the root causes of low academic achievement, but correlations are not causations. Sandra Stotsky disagrees with Ravitch and illustrates that closing the achievement gap is not an unworkable educational goal; however, improving all students' curriculum should be the primary goal. The late Richard Feynman pointed out, "In education, you increase differences." The late Zig Engelmann also observed that the achievement problem links directly to "the teaching" in the classroom. Moreover, remote and hybrid models are replacing teaching with technology. Ravitch writes, "Yet with all its great potential, technology can never substitute for inspired teaching." She is right. Kids need in-person teaching full time. "Virtual learning is a poor substitute for real teachers and real schools." Lastly, Thomas Sowell points out that "equalizing downward, by lowering those at the top" is a "fallacy of fairness" and a terrible idea taught in education schools. (Note: The K12 math curriculum must be upgraded to meet international benchmarks. Reform math doesn't cut it.) 

✍️ Learn the steps of procedures first.

Breznitz and Hemingway (Maximum Brainpower) write, “People master a skill initially by following a set of rules [such as in arithmetic and algebra] and later by using knowledge acquired situationally, through a number of particular instances—through experience.” Recognizing problem types is “case by case” and requires practice and experience.

✍️ Note1. Many U.S. students don't do well in math because they were taught reform math rather than traditional arithmetic. Jo Boaler, an influential radical reformer and "math educator" from Standford University, argued for reform math and against Old School traditional arithmetic. She also promoted the "teacher as facilitator" concept, the use of manipulatives and calculators, and minimal guidance methods during instruction (e.g., group work). Reform math instruction resurged with Common Core, but math achievement has stagnated ever since. The NCTM math reforms that took hold in the early 1990s have been a fiasco.

Common Core reform math confuses and frustrates students, holds them
back and befuddles angry parents.

✍️ Note2. Jo Boaler downplays the importance of memorizing multiplication facts that support standard algorithms (i.e., traditional arithmetic) and substitutes reform math to equalize outcomes and curb math anxiety. She boasts that she never memorized the multiplication table yet has a Ph.D. in math education (not mathematics). Boaler's radical ideas are found in almost all progressive classrooms across America. The Boalerization of math education is deeply rooted and widespread, in my opinion. It means that students are not advancing because reform math is based on standards below world-class benchmarks. Children can learn much more content than the current math curriculum. The popular, minimal instruction methods (i.e., liberal pedagogy) and "teacher as a facilitator" are ineffective. One of the consequences has been poor paper-pencil calculating skills that are essential for solving math problems. Students stumble over simple arithmetic. Additionally, achievement in basic arithmetic has stagnated for over a decade (NAEP), coinciding with the advent of Common Core reform math and its state variants. Kids have lost a decade of academic progress in math, observes Michael Petrilli of the Fordham Institute--even more ground with remote and hybrid learning.

Furthermore, We should sort students by actual mathematical achievement on tests, not by ability (IQ). The mixing of low achieving kids in math with high achieving kids in math in the same classroom has been a catastrophe because it has lowered standards and expectations. It is the same old liberal "crazy idea" of "equalizing downward, by lowering those at the top," divulged by the eminent Thomas Sowell. 

Common Core math clutters the curriculum and a child's mind with complicated nonstandard algorithms that frustrate students and baffle parents. 

Note. Some students need advanced-level content and fast-paced math classes. They are a couple years ahead of their peers, but their academic needs are seldom met in elementary and middle school. Click Gifted Programs.

✍️ We keep doing the same things repeatedly with different twists and packaging and expect different results. It's fantasy thinking. Moreover, we blame disparities or achievement gaps on discrimination rather than the "teaching" governed by progressive pedagogy taught in ed school. The progressives have revamped the teacher's role to "facilitator," which does not work well. Teachers no longer teach, which is a radical idea. Thomas Sowell also points out that there are a host of reasons for disparities. (Sowell: Discrimination and Disparities, 2019)

✍️ Thomas Sowell (ThinkAlgebra Model: Emma, 10)

Thomas Sowell reminds us that not all disparities are discrimination. 
Many are not!    

✍️ For the past 50 years of America's progressive education movement, the most radical idea has been the teacher's role, from an academic leader using explicit instruction via worked examples to a facilitator using ineffective minimal teacher guidance methods (group work) and test prep. Consequently, memorization and repetition (e.g., old-fashioned drills and flashcards) have fallen out of favor in many progressive classrooms. Teachers were not taught the science of learning. 

In the 1950s, teachers, such as in the Catholic and independent schools and many public schools, focused on content and competency. Class size was not an issue. Beginning in the first-grade, students needed excellent calculating skills to solve problems. Memorizing math facts using flashcards was commonplace. 

✔︎ Another radical idea: Reform math has replaced traditional arithmetic; it confuses kids and baffles parents. In Common Core reform math, every student gets the same instruction despite achievement in the name of equity. But reform math has stagnated achievement for over a decade and kept most U.S. students significantly below international benchmarks. In short, according to national and international tests, many of our students are lousy at math because of the way it has been taught. 

 

✍️ Radical Ideas Dominate American Education

For children to achieve equal outcomes, they are often treated unequally, a radical idea that Thomas Sowell labels as a "fallacy of fairness." He writes that "equalizing downward, by lowering those at the top," is another radical idea "taught in schools of education." The quest for equivalent results (i.e., equalization) has not worked because "children differ in their ability to learn academic material," explains Charles Murray (Real Education, 2008). Closing the achievement gap should not be a goal in education, observes Sandra Stotsky (The Root of Low Achievement, 2019). 

STEM Math for elementary school is not a radical idea. It had its origin in the Madison Project (1957) and Science--A Process Approach (SAPA 1967). The math needed to do the science was built into K-6 SAPA. For example, four of the six processes in 1st-Grade SAPA were math or math-related. In contrast, arithmetic and geometry are not used in today's K-6 science classes. Students stumble over basic arithmetic. 

✍️ The Madison Project

Robert Davis, Syracuse University, wrote, "The Madison Project seeks to broaden this curriculum by introducing, in addition to arithmetic, some of the fundamental concepts of algebra (such as variable, function, the arithmetic of signed numbers, open sentences, axiom, theorem, and derivations), some fundamental concepts of coordinate geometries (such as a graph of a function), some ideas of logic (such as implication), and some work on the relations of mathematics to physical science. Arithmetic becomes evident as one sees it in relation to algebra and coordinate geometry."  

The purpose of my Teach Kids Algebra (2011-2019) program, like The Madison Project half a century ago, seeks to expand the elementary school curriculum beyond arithmetic. It was also a reaction against reform math, which often downplays the importance of memorizing math facts and learning the standard algorithms for operations.  

✍️ STEM Math, Grades 1 to 4

I fused basic algebra ideas to standard arithmetic in my Teach Kids Algebra (TKA) program for grades 1 to 4 at an urban Title-1 school of mostly Hispanic and black students. I thought that algebraic topics would be accessible to very young children, even 1st-graders, through standard arithmetic. Kids didn't have trouble with algebra; they had difficulty with basic arithmetic. Kids are taught (Common Core) reform math, which downplays the memorization of math facts that support standard algorithms, not traditional arithmetic; however, TKA reinforced the importance of standard or traditional arithmetic, starting with rules [e.g., n + 0 = n (add zero), n + 1 = n' (add one), a + b = b + a (commutative), distributive rule, etc.], number facts (e.g., addition facts, multiplication facts), and standard algorithms. 

Some of the algebraic concepts were: expressions and variables, integers, the order of operations, equations (true/false), solving equations in one variable using guess and check and later inverses, building x-y tables, plotting (x,y) points in Q-I, etc. Individual attention was an essential feature of the program. The last sessions were held in 2019 (before the virus) with a 2nd-grade class and several 4th-grade classes. The teaching method was explicit with worked examples, not minimal teacher guidance methods (group work). 

✍️ Equations can be true or false. 

I taught this idea in the first lesson of TKA for 1st-graders.

2 + 18 = 25 - 5 (True Equation; 20 = 20; Think, like a Balance)

5 - 4 = 10 (False Equation)

5 - 4 ≠ 10 (True Equation)

Note: The symbol = means equal to; ≠ means "does not equal to".

✍️ We should focus on the mastery of fundamentals, not learning for a test. In the U.S., instruction is the opposite. It is geared toward learning for the state test via a reform math curriculum that had failed in the past. No wonder our children can't do arithmetic or algebra well.

The goal should be the mastery of basics, not state test-based proficiency. Furthermore, math skills are domain-specific. Starting in 1st grade, some of the skills include memorizing the single-digit number facts, applying the place value system, using the rules, and learning the standard algorithms. These beginner skills are needed for doing arithmetic in 1st grade. So, why are our 1st-grade children not learning them?

✍️ Many students test into remedial algebra courses at community colleges or elsewhere because they have been taught reform math rather than traditional arithmetic from 1st grade on up. But, only a small percentage, the most capable, in my opinion, pass the algebra placement courses, often called Elementary Algebra (= Algebra-1) and Intermediate Algebra (= Algebra 2). Starting in 1st grade, public school teachers often paint an unrealistic picture of low academic ability children. "The lie is that every child can be anything he or she wants to be. No one really believes it, but we approach education's problems as if we did," writes Charles Murray (Real Education, 2008). Not everyone who passes College Algebra can pass Calculus 101, etc. Algebra-1 for all students in 8th grade is a broken idea. Indeed, academic ability varies a lot, explains Murray. "Children differ in their ability to learn academic material." He points out, "When the facts get in the way, we ignore them." 

✍️ Many assert that the root of low achievement is poverty. It's not poverty. "It's the teaching," explains the late Zig Engelmann. Often, report-card grades do not indicate the content students know or don't know because of rampant grade inflation, part of the feel-good for no good reason (self-esteem) movement that started in the early 70s. In The Roots of Low Achievement, 2019, Sandra Stotsky observes, "Most changes during the twentieth century reduced the content of academic coursework and time spend on this coursework by teachers and students." Low achievement and low expectations start in the 1st grade and run up the grades. Students can learn much more content than the reform math curriculum. My Teach Kids Algebra program is an example. 

First-Grade Algebra Lesson: x-y Tables and Graphs

✍️1st-Grade TKA Task (Culminating Lesson)

Given y = x + x + 2, build an x-y table of values for x = 0, 1, 2, 3, 4, then plot the (x, y) points! Like my self-contained 1st-grade class in the early 80s, these Title-1 students of 2011 memorized many of the addition facts (e.g., 5 + 7 = 12); consequently, they could 

• apply the addition standard algorithm to calculate sums quickly, 
• find the perimeters of polygons (formulas), 
• solve equations in one variable (12 + x = 25), 
• build x-y tables from a linear equation (y = x + x + 2), and 
• plot (x,y) points on graph paper, which were some of the action learning objectives that directed my teaching and led to student learning. 

Learning was an individual function, not a group task. Each 1st-grade student had to perform the objectives. They could not rely on group members or calculators. (Note: 1st and 2nd-grade students had seven lessons while older elementary school students had weekly, hour-long lessons for the entire school year.) Click: Teach Kids Algebra (TKA)

The Trio of Skills-Ideas-Uses

Successful math programs have three key aspects: 1. the computation skills of mathematics, 2. the ideas [concepts] of mathematics, and 3. the uses of mathematics. (Dr. Robert B. Davis, The Madison Project, 1957)

The "How" Before the "Why"

Tobias Dantzig ​(​Number: The Language of Science), a book highly praised by Albert Einstein, writes, "In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The strength of arithmetic lies in its absolute generality. ​a + b = b + a. Its rules admit of no exceptions: they apply to all numbers. Every number has a successor [add one]. There is an infinity of numbers."

✍️ Critical Thinking Is Domain-Specific

Critical thinking in math is different from critical thinking in science, critical thinking in literature, etc. Critical thinking is not a generalized skill; it is domain-specific! Kids cannot solve math problems unless they know math. Knowing is remembering from long-term memory.

Primary Learning Goal

In math, knowing facts, rules, and compact procedures (e.g., standard algorithms) in long-term memory builds the foundation for understanding, applying, and reasoning in math. Students cannot use something they don't know well in long-term memory. Indeed, the memorization of essential facts and efficient, compact procedures (i.e., the standard algorithms) play a vital role in performing math at an acceptable level. ​Standard arithmetic is simple and compact, but it is not easy to learn without memorizing single-digit number facts and practice-practice-practice. ​

The primary learning goal should be the mastery of necessary content in long-term memory, not proficiency on state tests. Also, Learning is remembering from long-term memory! "You don't know anything until you have practiced." (Richard Feynman)

✍️ Average Kids Can Do Math Well When Prepared

A student does not need to be gifted, or a genius to learn Algebra-1 in middle school, Calculus in high school, or grasp some algebra fundamentals in 1st grade, but they have to be prepared. Average kids can do these things when they are taught well and work hard to achieve. Zig Engelmann points out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning." The problem is that kids are not appropriately prepared, starting in grade 1. Unfortunately, American educators have bought into Piaget's cognitive development stages, but I show the stages are wrong every time I give an algebra lesson to young elementary school students. Also, performance on school achievement tests is roughly 60% inheritable (nature), which means that "environmental differences account for the remaining 40% of the variance (nurture)," observes Robert Plomin (blueprint, how DNA makes us who we are, 2018). He points out, "Inherited DNA differences are by far the most important systematic force in making us who we are." Plomin makes clear that the percentages are probabilistic, not deterministic. Abilities come from your parent's DNA, including academic ability to learn content schools teach.

Major Concept 

Learning is remembering from long-term memory. 

Learning is a change in long-term memory. If you can't instantly recall 6 x 7 = 42 (i.e., automaticity), then you haven't learned it. Memory (Long-Term Memory) is better and faster than calculating (Working Memory). Thus, to learn something is to remember it. Our ability to think depends on memory. Knowledge and skills are gained through memory. 

1877: Arithmetic in 1st and 2nd grade 

(Rays New Primary Arithmetic for K-2)

1. How many are 6x6-7? (Lesson 71)

2. When 3 lemons were selling for 15 cents, John gave 1 lemon and 5 cents in money for a book: What was the value of the book. (Lesson 75)

Note. The compact K-2 book by Ray (94 pages) covers addition, subtraction, multiplication, and division and extends significantly beyond today's regular 2nd-grade textbooks.

✍️ Mastery

Learning is remembering from long-term memory. It is a process that requires continual practice and review. It is mastery. Remember Zig Engelmann's quote, "If the students haven't learned it, then the teacher hasn't taught." Perhaps poorly, the content might have been taught, which was his point, but it wasn't learned (i.e., mastered). Learning in long-term memory is a process of continual practice and review. The result of the process is mastery. The fundamentals of arithmetic and algebra need to be mastered. In short, learning is remembering. For example, math facts like 6 x 7 = 42 must be automatic, not figured out each time you need to use 6 x 7, which means you don't know 6 x 7. Retrieval practice is essential. Use self-quizzing flashcards and repetition to memorize math facts so they stick in long-term memory.    

✍️ Knowledge is the "critical driver of thinking skill." In arithmetic, the best sequence matters a lot because "knowledge and skills build on one another." Daniel Willingham writes, "Children are more capable than we thought." I found this out when I taught algebra ideas fused to standard arithmetic to 1st and 2nd-grade students. The content children can learn or not learn is a function of what they already know (i.e., background knowledge). In short, prerequisite knowledge is consequential for learning new material. Learning algebra depends on knowing basic arithmetic, starting with math facts, properties of numbers, definitions, and efficient algorithms for calculating. If you can't calculate sums efficiently, then you can't find perimeters. Also, a student cannot figure out a percentage problem without calculating and equation-solving skills. (Quotes: Daniel T. Willingham, "How Can Educators Teach Critical Thinking Skills," 2020)

Not knowing single-digit math facts impedes standard algorithms' learning, fractions-decimals, percentages, ratio/proportion, algebra, geometry, and measurement. For example, the addition algorithm must be automatic, correct, and efficient. Novices should first learn the standard algorithms, which means that very young children need to memorize single-digit math facts supporting the standard algorithms. The reform math's alternative algorithms should be avoided because they are cumbersome, inefficient, and dead ends. Also, they needlessly clutter the curriculum as extras and increase the students' cognitive load.

✍️ Cogitate

• "Practice does not cause talent; it improves performance." (Ian Stewart)
• "You learn only by mastery." (Zig Engelmann)
• "Mastery requires memorization and repetition." (Stanislas Dehaene)
• "You don't know anything until you have practiced." (Richard Feynman)
• "The building blocks of understanding are memorization and repetition. (Barbara Oakley).

The quest for sameness (i.e., Common Core, equity, etc.) has been a "fallacy of fairness in education." Sameness is a poor fit for most students. It dilutes content. In mathematics, there is an inverse relationship between equity and rigor of content. (Quote marks: Thomas Sowell) Reducing content, shrinking grade-level curriculum, lowering test cut scores, and inflating grades are pseudo schemes to fix education. It is nonsense!

Graduation Rates

Online credit recovery, grade inflation, and watered-down courses have contributed to bogus high-school graduation rates. What a scam! 75% of high school seniors are not proficient in mathematics (NAEP). Also, strict schooling does not dent a child's curiosity, creativity, or motivation. Think, Einstein. "In education, you increase differences." Think, Feynman. 

✍️ Reform Math Is a Bust

Sometimes the easiest things in math are the most difficult to understand. Thus, much of the talk about understanding and alternative "reform math" algorithms called "understanding" algorithms is nonsense and a waste of instructional time. I would not worry much if your child doesn't understand the standard algorithms, especially long division. Most adults don't either. The most straightforward facts, such as 5 + 7 = 12, can be shown on the number line to Kinder children. Kids develop a "number line" understanding of algorithms that makes sense, while the alternative algorithms are complicated and complex and have little practical value. Kids are novices, not little mathematicians. Who multiplies 7 x 2.67 using the area model or the array model? The standard multiplication algorithm takes 15 seconds if that for 7 x 2.67. The caveat is that students need to automate the multiplication facts to support the standard algorithm. First-graders in Singapore learn multiplication as repeated addition. Thus, 3 x 4 = 4 + 4 + 4 or 12. Three jumps of four on the number line lands at 12. 

Remote Learning Is a Bust

Children learn very little through fads like remote learning or hybrid learning, which is slightly better.  

Note: Class size accounts for only 1% of achievement tests' variance, explains  Robert Plomin (blueprint, 2018). Class size does not matter much. 

Additional Notes 

Liberal educators want to revolutionize education by creating a utopian as normal; however, I think we should aim for the old normal first and then figure out which changes benefit students by implementing learning science. Among the likely changes would be to increase class size as it accounts for only 1% variance on achievement testing (Plomin), sort kids by achievement in each major subject, upgrade the math curriculum to correlate with international benchmarks, focus on teaching traditional arithmetic well, not reform math, starting in the 1st grade, and jettison standardized testing. 

Equity and academics vary inversely. Today's stress on equity has resulted in lower academic standards and achievement in primary subjects such as reading, math, science, and history. For decades, "equalizing downward by lowering those at the top" has been a widespread policy taught in education schools, writes Thomas Sowell. It is a "fallacy of fairness." We do not live in a perfect world.

As mathematician Keith Devlin describes it, functional understanding is "understanding a concept sufficiently well to get by for the present." It is "understanding that is defined in terms of what the learner can do" (i.e., apply it). Devlin also states, "I think many mathematical concepts can be understood only after the learner has acquired procedural skill in using the concept." Professor Devlin is saying that kids should have procedural fluency to foster functional understanding. But what does that mean? Performing math implies some level of understanding, but it can't be quantified. In school math (e.g., arithmetic, algebra, precalculus), the focus should be on the "how" first" Good math books have a bunch of worked examples in each lesson--the "how!" Most teachers spend too much time on understanding (the why). It is the wrong approach. The "why" should not come first. 

If you cannot calculate something, then you don't know it well enough. 

The more I know, the more I can learn, the faster I can learn it, the better  I can think and solve problems. 

Common Core reform math confuses and frustrates students, holds them back and befuddles angry parents.  

  

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