Tuesday, August 17, 2021


Back To School-1


  I reflect on education and often quote or paraphrase others who are more intelligent, knowledgeable, or wiser than I. (LT, Math Notes by ThinkAlgebra.org) 

Note: Some content has moved to Back To School - 2.

Much is taught, but little is learned.
(Maybe, not that much is taught.)


  • IQ is not fixed for life, explains Sanjay Sarma. Schooling increases IQ. 
  • To think math, kids must know math facts! 
  • Thomas Sowell stated in an interview, "We live in an age where rhetoric prevails, and no one cares about the facts." 
  • There is little evidence that students have mastered enough arithmetic and algebra (mathematics) or reading content, as test scores had remained flat for the last decade (2007 - 2017).  
  •   Schools have spent millions on technology; however, the new tech hasn't substantially improved achievement in math or reading. Tech is not the silver bullet or cost-effective. Many kids hate reading screens. Bring back (physical) textbooks. Remote cannot replace good, in-person teaching. 
  • Intensive tutoring is one way to help kids catch up--not reviewing or remediating. Why not? How can you review or remediate content the student has not learned?

Comment: The Chinese value excellence while the U.S. values equity, that is, equal outcomes. Equity math is a dumbed-down version. How can U.S. kids compete with kids in Asian nations when the arithmetic and algebra curricula are cut back. The prime concerns in U.S. schools are diversity, equity, and inclusion, not merit or quality. Excellence and competition aren't in the narrative. In contrast, the Chinese do not question the study of algebra or memorizing the x-table in the 2nd and 3rd grades. And they do not think that math is racist. 9-1-21

In the U.S., the equity agenda "equalizes down, by lowering those at the top," which discriminates against students who excel, especially Asian Americans. Some teachers are told to cut back on math topics to qualify for merit pay. Again, it is teachers first and kids last.

Note: Our best students often do not get a rigorous pre-algebra course in the 6th or 7th grade to prepare for a thorough Algebra-1 study in the 7th or 8th grade. We are not identifying or guiding mathematical talent early enough. Starting in the 1st grade, everyone gets the same curriculum and instruction--that's called equity. No student gets ahead. Children can learn much more than they are taught in most of our schools. 
Credit: "equalize down by lowering..." is a quote from Thomas Sowell. It is not new. Sowell wrote about equalizing over 30 years ago, probably earlier.  

Note: Most teachers and parents have never heard of domain specificity of skills, explains E. D. Hirsch, Jr. (Why Knowledge Matters, 2016). He points out that no generalized thinking skill applies to different domains. He observes, "Thinking skills cannot readily be separated from one subject matter and applied to other subject matters.” Many teachers, including gifted program teachers, do not recognize this important cognitive science finding. As a result, all-purpose skills do not exist.

Focus on Knowledge

Critical thinking in math (absolutes, deductive thinking) is different from critical thinking in science (observation/inference) is different from critical thinking in literature (textual criticism, vocabulary), etc. 

Why study algebra? The main reason is that it trains you for higher math, such as trig and calculus, which are needed for many STEM and financial careers, etc. You cannot be a successful math student without knowing algebra, which requires the mastery of arithmetic. Additionally, in algebra, symbols are used to represent numbers to solve problems. Therefore, knowing facts and techniques is needed to grasp algebra well. Also, learning to write and solve equations is vital in problem-solving.  8-31-21

The push for data science over algebra (or deemphasizing algebra), such as the new California Math Framework, is illogical and flawed because real algebra prepares students for higher math needed to enter the STEM fields. Data science, I believe, is an overt attempt to dumb down the math curriculum. AP Statistics, for example, is a TI-84 calculator course. Without algebra and trig, students are shut out of the hard sciences (chemistry, physics), finance, economics, mathematics, engineering, and many other great career opportunities. 

Note: "Data scientists use both multivariate calculus and linear algebra to perform their work." In the real world, algebra, trig, linear algebra, and calculus are needed. A data scientist combines mathematics with computer science and requires top-notch algebra and calculus courses in high school, which is not the purpose of the data science substitute for the algebra-precalculus-calculus track in the California Math Framework. Many kids have difficulty with algebra because of inadequate learning of basic arithmetic and the way algebra and math are taught in today's classrooms. Kids who had trouble with arithmetic are going to struggle in algebra. Teachers, stop teaching algebra like you teach social studies. Math students should be more like gymnasts and musicians: practice-practice-practice. (Practice, to get it right!) To solve math problems, they need to have factual and procedural knowledge in long-term memory, but they don't have enough knowledge or have unrelated bits and pieces.   

Failure in today's Classrooms
  • Remote has failed.
  • The Common Core reform math curriculum has failed. (It's not world-class, so why was it adopted by almost every state?) 
  • The Methods of Teaching basic arithmetic have failed. (Minimal guidance methods and group work do not work in math.)

"You need to put your kids in groups, you need to be using manipulatives, you need to deemphasize procedures and rote learning, you need to emphasize conceptual understanding." Really? These reform methods of teaching have failed to significantly boost student achievement. In fact, math performance has stagnated for at least a decade. Children lost a decade of learning (NAEP 2007-2017), observes Michael J Petrilli.  "There’s no way to sugarcoat these scores [NAEP 2017]; they are extremely disappointing." On top of that, most children have lost much of another year in remote learning. (Quote by Tom Loveless: "You need to put...) 8-31-21

The overuse of manipulatives and group work (discovery or project learning) in math instruction has failed to move our students forward. Students must know factual and procedural knowledge in long-term memory, but many don't. The head of the United Los Angeles teacher union says that it's okay that children didn't learn all the x-table (i.e., basic arithmetic). Instead, they learned resilience and survival. No--it is not okay! As of 9-1-21, Los Angeles kids are still not in school because the union puts teachers first. Who suffers? The students!

It's the teaching! In one school district (Tucson area), the 2019 state test pass rate for math was 30%, which dropped to 14% in 2021, a 53.3% decrease. Wait! Since when is a 30% pass rate acceptable in K-8? It's not a job well done! It's the teaching, as the late Zig Engelmann would often retort. Common Core reform math doesn't work. Yet schools, administrators, and teachers keep doing the same old stuff taught in schools of education. They listen to rhetoric, not facts. Unfortunately, reform math and its methods are often bolstered by influential education professors like Jo Boaler

Comment: Einstein participated in oral drills to acquire math skills when he attended a Catholic elementary school, just like the other students. A few years ago, I met two teachers who told students that Einstein was bad at math. Really? Quite to the contrary, Einstein, a physicist, was stellar in math. So why are these people allowed to teach? They definitely should read Lies, Damned Lies, and Science by Sherry Seethaler. (Scroll to the bottom of this page to see Einstein explain special relativity equations.)

Another major problem is that teachers promote that students can do critical thinking skills without sufficient content knowledge. No, they can't. Necessary thinking skills are domain-specific. There is no universal thinking skill. For example, thinking in math (deductive reasoning) is different from thinking in science (inferential reasoning) is different from thinking in literature (textual criticism), etc. 



Let's do some math arithmetic!

It's Arithmetic! What's that?

Some students have heard the word algebra but not arithmetic. The kids didn't realize that the math they were learning from their classroom teachers was largely arithmetic. Real arithmetic, however, requires memorizing single-digit facts to support the standard algorithms for efficient calculating of larger numbers. In contrast, many kids are instructed in reform math, which is "pretend math," a botched variant of arithmetic, often supported by influential education professors like Jo Boaler, who, at one time, boasted that she never memorized the multiplication table. Really?

In the name of what?
Cutting the math curriculum in the name of equity is no way to improve achievement. In my opinion, equity should not dismiss merit. Without merit or high-quality standards, there is little incentive for students to excel. Better teaching by applying a world-class curriculum and efficient instructional methods will boost achievement. The problem is that many K-8 teachers are weak in math, according to H. H. Wu, a mathematician at UC Berkeley, who conducts PD courses and summer classes for current and future teachers.

In education, equity now means equal outcomes for all students by lowering the content and inflating the grades so almost all students can pass a dumbed-down curriculum. But kids are not the same. Academic ability widely varies, points out Charles Murray (Real Education, 2008). Murray wrote that disruptive students should not be permitted to remain in class, and the Core Knowledge Curriculum should be taught to almost every student in grades 1 to 8, which leads to Algebra-1 in 8th grade. 

(Note: Core Knowledge, not to be confused with Common Core, just released a revised PreK-8 curriculum. I plan to examine it closely, starting with 1st-grade arithmetic.)

Automaticity! Automaticity! Automaticity!

"Learn skills all the way to automaticity!"

Doug Lemov (Practice Perfect, 2012) points out, "The power of learning things by rote is that it allows you to do them with unconscious efficiency. ... It's all but impossible to have higher-order thinking without strongly established skills and lots of knowledge of facts." Rote learning does not get in the way of higher-order thinking, as some claim. In arithmetic, students should practice getting it right. "While failure may build character and tenacity, it's not good at building skills. ... Many types of higher-order thinking are in fact founded on and require rote learning," explains Lemov. Thus, practice getting things right!

[Comment: Einstein participated in oral drills to acquire basic math skills when he attended a Catholic elementary school just like the other students.]

Also, rote learning should not imply a lack of understanding. At first, a child's understanding is what I call a "number line" understanding, in which all consecutive numbers are equidistant and form a linear scale. Add 1 to get the next number (3 + 1 = 4 or n + 1 = n'), from left to right. This foundational understanding is not taught well. Also, students should use the number line, which starts at zero, to calculate. The number line shows that the 3 + 4 is actually 7. In fact, the number line is important mathematics, but I don't see it used much in elementary school. Once simple combinations have been calculated on the number line, all the combinations that make 5, the sums should be committed to flashcards for practice and retrieval. At first, stick to whole numbers.    8-24,25-21 

Learning Through Worked Examples

Sanjay Sarma (Grasp, 2020) explains, "One upshot is that whenever something clogs up working memory [any distraction], your all-around problem-solving abilities take a hit. ... Learning via worked examples instead of solving a problem for yourself is one potential way past such working-memory roadblocks. ... Overlearn certain facts, like the multiplication table, so that summoning up those facts during problem-solving becomes undemanding." Sarma points out the critical importance of learning through worked examples to solve problems and memorizing the multiplication table so as not to clog your working memory.  

  Elementary School Mathematics Priorities  

by Dr. W. Stephen Wilson, a Mathematician

The five building blocks for higher mathematics: 

1. Numbers
2. Place value system
3. Whole number operations (i.e., Standard Algorithms)
4. Fractions and decimals
5. Problem solving

The problem as I see it is that the focus of reform math has been on conceptual understanding. Students need to learn certain concepts straightforwardly, such as the commutative rule, distributive rule, equations, the equal sign, division, convergence, linking addition and subtraction, graphs, perimeter, etc. But, they also must do and apply arithmetic (practical performance of knowledge) to a range of routine problems depending on the grade level. For example, well-taught 1st graders can calculate the perimeters of polygons. Problem-solving requires more than knowing concepts. It requires excellent math skills, that is, being able to calculate answers efficiently

Automating math facts, performing procedures correctly (primarily the standard algorithms, not reform math), using symbols for numbers and variables (unknowns and formulas), and recognizing problem types should be the foremost content for novices, starting in the 1st grade. Arithmetic requires practice-practice-practice to master, which is not stressed enough in reform math. Children should "drill for skill" at school and home. Flashcards work! Furthermore, students should practice the multiplication and long division standard algorithms no later than the 3rd grade (1st Semester is Multiplication, 2nd semester is long-division). The focus in the second semester of 4th-grade should be on common fractions and their operations. Students must acquire math skills quickly. They are not!  The skills should not be delayed as in Common Core. 

Delay, Delay, Delay 

Common Core math standards (or state standards built primarily on CC) delay the standard arithmetic of whole numbers. To do standard arithmetic well, students must automate single-digit math facts early in long-term memory. Use flashcard drills at school and home! The standard algorithms for multiplication and long-division should be taught and practiced no later than the 3rd grade. 

  Critics of standard arithmetic say that teachers should jump to the top of Bloom's Taxonomy of Learning (Critical Thinking) rather than focus at the bottom (Knowledge). But you can't think about stuff you don't know well. Also, 1st graders can intuitively grasp the idea division, but not the long division algorithm. Is it necessary to understand the long-division algorithm to perform it well? What students need to comprehend, I think, is the idea of convergence when practicing the steps of long division in 3rd grade. Students need to drill for developing skills. If math facts are not in long-term memory, then long division will be drudgery. 

Vivek Ramaswamy (Woke, Inc 2021) writes, "Being woke means obsessing about race, gender, and sexual orientation. Maybe climate change too. That's the best definition I can give." To show their wokeness, many corporations have yielded to so-called social justice ideas and radical diversity standards by supporting BLM and "infusing woke values into big business" or face retribution, according to Vivek Ramaswamy. "Corporations win. Woke activists win. Celebrities win. The losers are the American people." Woke diversity has nothing to do with the diversity of thought. 

 In education, there is little diversity of thought. Radicalized diversity has narrowed to race and gender metrics, not thought. As a contrarian, I challenge conventional wisdom. The liberal elites in schools of education, teacher unions, banks, corporate America, and tech companies have the power to dictate policy for the majority. Critical race theory is the opposite of civil rights leaders like Martin Luther King. Whatever happened to being judged by the content of your character? Not any more. You are identified by your race. 8-18-21

  Even math has become a weapon. Equity math cuts content to level down by lowering those at the top. Moreover, according to critical race theory activists, stressing the correct answer or showing your work is racist.  

  The significant change in our culture is that the people who have the money, such as the giant tech companies and corporate America, have the power to establish (dictate) the norm, not the citizens through the democratic process. Vivek Ramaswamy (Woke, Inc 2021) explains the new Golden Rule: "He who has the gold makes the rules." The idea that "every person's vote counts equally in our democracy" is no longer true, not "when [corporate] dollars mix with votes." It is hard to believe these things are true, but they are. An exception is mixing "so-called" social justice politics with sports. Some woke sporting events have lost fans, even the Olympics. For example, I stopped watching women's gymnastics.  

Reflections from the past...


"Children are not hurt by learning. However, standing still and lost motion kill." 

(Committee of 15 on Elementary Education 1895) The wisdom of the past smacks us right in the face. Kids are not merely standing still; many took steps backward. 

In 2017, I wrote:

  • Asian children are taught mechanics first with an explanation later, and it works. So why don't we do the same?   
  • The teacher's role is to help bright kids excel, not to let them fend for themselves.  
  • In education, you increase differences, explains Richard P. Feynman. 
  • Even though kids are not all equally intelligent, athletic, musical, motivated, or creative, most kids, starting in the 1st grade, can learn arithmetic and algebra at an acceptable level--if they are taught and practiced well. 



2017: The Standard Algorithm Is Primary!

Note: The 1st graders I had in 2011 for Teach Kids Algebra are now high school Seniors. The 2nd graders recently graduated from high school, and many will start college this Fall.

"Especially pernicious is the American Marxist's control over our public school and college classrooms, with the full support and active role of the two national teachers' unions ... where your children and grandchildren are being taught to hate our country and are brainwashed with racist propaganda." Mark R. Levin (American Marxism) comments on Critical Race Theory (CRT). 8-17-21

Furthermore, some students are taught equity math, a reduced content version of arithmetic, where "getting the correct answer" or "showing your work" is branded as racist by Critical Race Theory (CRT) radicals. Also, some teachers put their politics and ideologies into the classroom, which, in my opinion, are a form of indoctrination, not education. 

Even math has been a victim of such nonsense. Cutting the math curriculum in the name of equity is no way to improve achievement. Better teaching by applying a world-class curriculum and efficient instructional methods will boost achievement. The problem is that many K-8 teachers are weak in math. In education, equity now means getting equal outcomes for all students by lowering the content and inflating grades so almost all students can pass a dumbed-down curriculum. But kids are not the same. Academic ability widely varies, points out Charles Murray (Real Education, 2008). Murray suggested that disruptive students should not be permitted to remain in class, and the Core Knowledge Curriculum should be taught to almost every student in grades 1 to 8, which means Algebra-1 in 8th grade. (Note: Core Knowledge, not to be confused with Common Core, just released a revised PreK-8 curriculum. I plan to examine it closely, starting with 1st-grade arithmetic.)

Mathematics is not opinion; it is about facts.

Masks, etc. 

"Masking children, social distancing, hybrid schedules, plexiglass shields, and HEPA filters have little or no effect on the spread of coronavirus," writes David Zweig in the New York magazine. Open the windows for fresh air circulation works much better. 8-22-21

The U.S. didn't learn from European countries about the spread of coronavirus. Sweden, for example, never closed its K-8 schools and did not require students to wear masks. The lesson is, ventilate classrooms but do not mask kids. 

Summer sessions to catch kids up academically were a bust, too.


"Inflation is a hidden tax that takes away the value of money held by everyone at every income level," writes Thomas SowellInflation has been up substantially since the policies of the new president. Not good! But, there is something far more sinister in our schools. It's called critical race theory (CRT), which is divisive. Parents are irate about mask mandates and CRT shoved into schools. At a Colorado school board meeting, one articulate black parent, a descendant of slaves, denounces CRT in schools, "I am not oppressed, and I am not a victim." Many teachers, the media, teacher unions, the federal government, school administrators, politicians, state and local school district boards support CRT. The people who support CRT are wrongheaded 8-21-21

Remote has failed. 

Common Core has failed. 

The methods of teaching have failed.

"You need to put your kids in groups, you need to be using manipulatives, you need to deemphasize procedures and rote learning, you need to emphasize conceptual understanding." Really? Some of these explain why kids learn little arithmetic and algebra. 

  Some children have an extraordinary ability, whether math or figure skating, but developing it requires excellent teaching (coaching), practice-practice-practice, and uncompromising drive and persistence for years. Start earlyStill, ordinary kids with average ability can learn arithmetic, algebra, and even AP calculus, which is for average kids who are prepared, at acceptable levels, if they are taught better.   

What we need in our schools are better teaching and methods that work, not CRT.  We are not identifying our best students, not in math, etc. Enrichment has been the norm in most gifted programs, but the instruction should focus on content acceleration.  



Crazy math (?)

State test scores in the school district where I volunteer are:


2019 30% passed 

2021 14% passed, which is a dramatic decrease. 

(Note: 30% passing is shockingly low, to begin with, even with test prep.)

Percent Change 16% decrease. (It's wrong. The change is about 50%)

The information was copyrighted by a local TV station. (Reading is just as bad, down by about 28%.)  

Wait: The percent change of 30% down to 14% is -53.3% (not 16%, which is a difference: 30-14). You can estimate 50% by rounding: 14 is close to 15, which is half 30. How did the local TV news station calculate the 16%? They subtracted 30 - 14 = 16, but that is not the meaning of percent change in mathematics. The division step was missing. See below. 

FYI: Percent of Change is a calculation taught in a 7th-grade pre-algebra course, perhaps earlier. It is the amount of change divided by the original amount. Thus, going from 30 to 14 is a percent of decrease. The original is 30.   

(30 - 14)/30


.53 (change decimal to percent): 53/100 = 53% by definition.

The percent of decrease is about a whopping 53%.

Example: I bought a stock at $42/share. The market went down. Now the stock is worth only $25/share. The percent decrease is (42-25)/42 or about 40%. If I purchased 100 shares at $42/share ($4,200), then I lost about 40% of $4,200 or $1,680. I bought the wrong stock. 

But, let's assume that the current value of the stock suddenly increased by 40%. Did I get back all my investment? Many people believe is that if you lose 40%, you will gain it all back when the stock rises 40%. What is wrong with this logic?  Let's do some more arithmetic. (I lost $1680: 4200 - 1680 = $2520 amount left) Thus, 2520 + .40 x 2520 = $3528, not close to my initial investment of $4,200.  So, how much does the stock have to rise to get back the original investment? Would you believe 66.67%? (Summary: 40% gives me $3528, far short of $4200. But 66.67% gets me back to $4200!)

2520 + 2520x = $4,200, 

2520x =  4200 - 2520

2520x = 1680

x = 1680/2520

x = .6666666666 or 66.67%

Check: 2520 + .6666666666 (2520) = $4200

Many people, including teachers, don't understand how percentages work. 

What is 4 - 5? One 3rd-grade student said negative 1. She was correct. You can subtract a larger number from a smaller number. Also, it can be shown on an integer number line. You can also use debt as a negative number. Suppose you borrowed $5 from a friend. Later, you give your friend the $4. You are -$1 in debt. You still owe your friend $1.   

Low Early STEM Grades

Students who get Cs in early STEM coursework at the university, let's say in a rigorous chemistry course, will likely drop out of STEM. AP courses might not fly either. STEM majors who had passed AP Calculus in high school are often required to take the university's calculus courses. AP courses are for average students and not always equivalent to university courses in STEM math and science. We don't need mediocre engineer students who don't grasp calculus or statistics. AP Statistics is nothing more than a TI-84 calculator course. Even students in a college statistics course often fail to grasp some of the core concepts, according to Dr. Sam L. SavageThe Flaw of Averages, 2012.

Using Einstein's Equation

Mass: 1 kilogram

Speed of Light: 3 x 10^8 meters per second

Indeed, 1kg of matter can produce 9 x 10^16 J (joules) of energy. That's a lot of energy. The equation demonstrates that a massive amount of energy can come from a tiny amount of mass (1 kg). The equivalence of energy and matter or E = mc^2 is one of Einstein's most accessible equations, but deriving it is another story (Not shown).

Some of the complex math Einstein wrote to derive special relativity.

Much is taught, but little is learned!
Maybe, not enough math is taught in K-12.


Some education schools are teaching wannabe teachers that it is a matter of social justice when students do not grasp basic fraction ideas. The social justice baloney comes from education schools (e.g., Deborah Loewenberg Ball, former dean of the University of Michigan's ed school, cites "patterns of racism and marginalization.") Of course, students struggle with fractions, but this is not new. I believe the problem is the teaching, not social justice. Put simply, fractions are poorly taught.

Naming 1/3 on a number line (5th Grade ?) 

If 5th graders do not know that the mark is 1/3, how did they get to 5th grade? One student said it was 1/7. I taught this idea to 1st graders in a self-contained, desegregated, Title-1 urban classroom in the early 1980s. 

Ball writes, "Even classrooms that are rich in rigorous content and discourse are high-risk for reproducing patterns of racism and marginalization." Really? First, naming 1/3 on the number line is not rigorous content. Second, it has nothing to do with social justice. It is a matter of poor teaching. 

Ball argues, "Many taken-for-granted practices in classrooms reflect and reproduce patterns of marginalization and oppression." Ball cites a drill sheet of basic facts. Really? Memorizing math facts is essential arithmetic. Arithmetic is not racist. Ball seems to argue against kids learning arithmetic. Is it any wonder that kids don't learn enough math?


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8-28-21, 8-29-21, 8-30-21, 9-1-21, 9-5-21, 9-6-21, 9-8-21, 9-9-21   

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