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?


© ThinkAlgebra/LT


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

©2021 ThinkAlgebra.org, LT

Sunday, August 1, 2021

Summer2021 Part 2

Summer 2021 Part 2 [non sibi]

Welcome to the mad, mad world of math education. 😎

What has happened to math education in the U.S?
We need to teach the right math, the right way, to advance achievement, but we don't! Progress has stalled in both math and reading for over a decade. Instead of basic arithmetic, students get reform math--not enough memorizing, practicing, or teaching fundamentals. Understanding is over-emphasized and does not produce mastery; practice does! Also, being able to solve an equation is practical understanding! Dr. W. Stephen Wilson, a mathematician, establishes the priorities. 

Note: BackToSchool-1 is now available.
IQ is not fixed for life, says Sanjay SarmaSchooling increases IQ.  

  Elementary School Mathematics Priorities  ( Dr. W. Stephen Wilson)

The five building blocks for higher mathematics: 

1. Numbers

2. Place value system

3. Whole number operations (i.e., The Standard Algorithms)

4. Fractions and decimals

5. Problem solving 

In addition to dumbing down the math curriculum, using ineffective instructional methods (i.e., group work), and adhering to the "false equivalence between literacy and math skills," our classrooms have become battlegrounds for challenging issues, especially critical race theory, mask mandates, gender, and equity math. These issues and others have derailed education off course and stalled learning in math and reading. Parents are angry! 8-12-21

Children in elementary school should learn "as much knowledge as possible as quickly and efficiently as possible," writes Natalie Wexler (The Knowledge Gap, 2019). That's not happening! For decades and decades, educators "have vastly underestimated [and undersold] what their students were capable of." Consequently, by the 4th or 5th grade, U.S. students in math are at least two years behind their peers in top-performing nations. 

In the name of equity, Oregon has shelved its requirement that students demonstrate competency in math, reading, and writing to graduate. It is dumbing down the curriculum and part of the woke movement to end testing for graduation, rendering a high school diploma meaninglessIt happens when liberal buffoons run the state and department of education. Equity means no student gets ahead. All students get the same instruction regardless of achievement. Equity is based on equal outcomes, a radical idea. Consequently, some parents are fleeing public schools for private schools or homeschool. 8-10-21

Math reforms like Common Core have not worked. Dr. W. Stephen Wilson has established priorities for basic arithmetic. He argues against popular reform math programs like TERC, Everyday Mathematics (EM), and others based on reform math and minimal guidance methods. In math, teachers should not be facilitators. They should explain examples explicitly and make sure kids practice enough. 8-11-21 

  Understanding does not produce mastery; practice does
"To learn something is to remember it." If you can't retrieve the fundamentals from long-term memory, what have your learned? Your mind is an empty set. 8-11-21

Face it: Common Core has failed.
After a decade, the Common Core math standards have had minimal effect on student achievement, including State Standards built on CC. They were another flawed, top-down theory. So why are many states clinging to them in one form or another? A new assemblage of standards is not going to change things. Get out those old, dusty math textbooks and workbooks and put "group work" to rest. Teach arithmetic by explaining examples and practicing for automation. 

There are workbooks available at stores like Barnes & Noble: e.g., Spectrum, Harcourt, Kumon, etc. 

FlashCards work! (Stanislas Dehaene, How We Learn, 2020) Memorization of certain fundamentals such as math facts and efficient procedures is good for kids. Furthermore, the standard algorithms must be automated. Students should practice-practice-practice! 8-11-21

Kids need to memorize math facts and practice-practice-practice to make fundamentals, such as the standard algorithms, stick in long-term memory and be remembered for problem-solving. Students also need to recognize routine problem types and be able to solve them. They are beginners, not experts, so they need to practice a lot!. You can't solve problems in math without knowing how to calculate well. Kids should first learn standard algorithms for calculating, even in the 1st grade. Standard algorithms require the memorization of single-digit math facts. 8-11-21 

Students are novices and should not use calculators or be required to invent their own algorithms, as claimed by Everyday Mathematics (EM). Also, in EM students learn "focus algorithms" for each operation, which are substantially more complex and different from the standard algorithms that must be automated for problem-solving. The so-called focus algorithms, which are claimed to promote conceptual understanding, are rubbish. They confuse kids and hold them back. Instead of teaching basic arithmetic operations (i,e, standard algorithms) that kids must know for problem-solving, EM teaches pretend math. 8-11-21

Teachers tell me that they teach 5 or 6 different ways for multiplication and don't have time for the standard algorithm, which they never get to. Teachers do not realize that the standard multiplication algorithm is essential for problem-solving and should be given top priority. All these alternative, unconventional, or invented algorithms are not that important.    

I teach for understanding.
I believe in group work. 
It's not on the state test. 

One teacher told me that children don't understand long-division, so she doesn't teach it. I replied, "And your kids understand the lattice method for multiplication that I see on your bulletin board?" 8-11-21

Comment: Even 1st graders have an intuitive understanding of division. Also, Singapore starts multiplication in 1st grade as repeated addition: 3 x 4 = 4 + 4 + 4  or 12.  What do all the standard algorithms require? Automation of single-digit math facts. 8-11-21
All this reform math stuff has ruined U.S. math education, in my opinion. The reform leader has been Jo Boaler, who, at one time, stated that kids don't need to memorize the multiplication table that supports the standard algorithms and that teachers should be facilitators. The reform math idea is that kids should invent their own mathematics. Really? How many kid experts like Newton are out there? 8-11-21

***** Hung-Hsi Wu explains ("What's Sophisticated about Elementary Mathematics?"), "In other words, students' ultimate understanding of these [standard] algorithms must transcend place value to arrive at the recognition that all whole number computations are nothing but a sequence of single-digit computations artfully put together." 

Wu states that carrying is not the main idea of the addition algorithm. "The main idea is to break up any addition into the additions of single-digit numbers...." And with knowledge of place value, "put these simple computations together for the final answer." 8-11-21

"If I practice a lot, then arithmetic becomes easier for me, and I learn (remember) more, increasing my storage strength and retrieval strength. I also know that the mastery of fundamentals in long-term memory is essential to advance my STEM goals."
At School & At Home
FlashCards Work!

From The 74, "But a decade after they (Common Core or state standards primarily grounded in Common Core) were first adopted by states, little evidence exists to show that teaching or learning was significantly improved by the vast resources poured into implementing the standards," concludes researcher Tom Loveless (Between the State and the Schoolhouse: Understanding the Failure of Common Core, 2021).

Loveless writes in the book, "This is not a problem that another set of standards can solve. If standards came out tomorrow, and I agreed with every single word in them, I would still give them only a slim chance of being faithfully implemented — and less than that of moving the needle on student achievement. ... It's hard for the top of the system to have a large impact on what happens at the bottom of the system." 8-5-21


Teachers should focus on reading, writing, and doing arithmetic, not critical race theory (CRT), or diversity that excludes Asians, or equity that means equal outcomes by leveling content or "equalizing downward by lowering those at the top, or by lowering or dropping state requirements in math and reading/writing, etc. for graduation." Teachers should not depend on the Common Core reform math curriculum or minimal guidance methods (group work) to improve math achievement. There are workbooks available at stores like Barnes & Noble: e.g., Spectrum, Harcourt, Kumon, etc.

CRT and its many forms, including oppressed vs. oppressors, indoctrinate young children that race drives their future rather than hard work, perseverance, education, and character. If you are white, you are racist or an oppressor. Really? How dumb! Even math is called racist by CRT radicals. It's all wrong! In Arizona and other states, teaching CRT in public schools is illegal. Whiteness should not be described as a form of oppression. According to CRT radicals, in math, getting the right answer or showing your work is racist. No, it's not; it's good pedagogy.

Furthermore, according to the National Math Advisory Panel (2008), teachers should prepare more students for Algebra-1 no later than the 8th grade by stressing factual and procedural knowledge in long-term memory via memorization and practice-practice-practice. Algebra-1 is a middle school subject for average students who are prepared, and preparation begins in 1st grade. (You don't get good in math by not practicing math.) "You know nothing until you have practiced," explains Richard Feynman, Nobel Prize-winner in Physics.

Also, the U.S. mathematics curriculum needs upgrading to match international benchmarks, at the least. (Note: Common Core and almost all state math standards grounded in CC are not world-class. Our kids start behind and stay behind their peers from top-performing nations.) Instruction should be explicit and efficient--not a steady diet of minimal guidance group work. The standard algorithms for multiplication and long-division should be learned by students no later than the 3rd grade. Students must automate math facts in grades 1 to 3. (Credit: equalizing down..., a quote from Thomas Sowell8-2-21

Student in 7th-grade Algebra-1

  • Algebra-1 is a middle school subject, but not in most U.S. public schools.
  • Almost all students get the same dumbed-down reform math, not world-class content. 
  • Individual achievement does not count. Merit is pushed aside! It's the wrong approach!
  • Advanced kids get grade-level math like other kids. Why?
  Children need to do things that don't come easy for them. They should complete tasks and learn persistence in doing so. Also, practice does not cause talent; it improves performance up to a point. "You don't know anything until you have practiced," explains Richard Feynman

  Merit and learning are not valued enough. Education is no longer merit-based because grade inflation and group work (minimal guidance methods of instruction) squash it. Teachers no longer teach a coherent math curriculum. They facilitate group work. What's that? There has been a slow creep of American Marxism in education, a war against individualism and merit. For decades children have been praised and rewarded for no good reason. It is part of the racist Critical Race Theory, which is not new. 

✓  If students are at risk of algebra failure, then their arithmetic background has been weak. Another cause is the inadequate teaching of algebra. If I can teach 1st, 2nd, and 3rd graders the three representations of functions (equation-table-graph) and have them perform these skills on paper, then why can't teachers teach basic prealgebra math skills for mastery to prepare more students for a solid Algebra-1 course in middle school? I know why, but you won't like the answer. 

Note: Dump the new California math Framework in the waste can and return to the 1997 California math content standards before Common Core reform math replaced them for no good reason. The 1997 document, however, should be upgraded to match the 1st-grade Singapore standards, such as memorizing math facts and learning the standard algorithms. Also, no need for statistics, data analysis, and probability in 1st grade. The old CA standards are a starting point. So start!

Comment: We know that the best way to teach arithmetic or math is via explicit instruction when the teacher explains carefully selected worked examples. After that, the students need to recognize problem types and practice-practice-practice.

The math curriculum, which was built on Common-Core-like state standards, is not world-class math. Common Core ignored many of the well-established, international benchmarks in arithmetic and pushed algebra to high school. I wonder why states adopted Common Core so quickly, knowing this? (It was all about the money, money!) The gap between the content that Singapore 1st-grade students routinely learn and the content that U.S. 1st-grade students learn widens up the grades. Thus, by the 4th or 5th grade, our students, on average, are about two years behind their peers from top-performing nations, but no one seems to care that our kids are being out-educated. We are told that it is not that bad, but it is. 


My Teach Kids Algebra (TKA) project is STEM math for young elementary school students from the 1st grade on up. It started in 2011. I would often give 1st and 2nd graders equations such as x + x + 2 = 18 to solve. Find x. Students solved the equations by guess and check (i.e., trial and error), applied the equality idea (LeftSide=RightSide), and the algebraic rule for substitution: x must be the same number (e.g., x + x = 8, x can only be equal to 4). In the equation x + x + 2 = 18, x = 8. Thus, 8 + 8 + 2 = 18 and 18 = 18 (True Statement by Definition of Equal Sign). Solving an equation is critical thinking, starting with the idea that if the right side is 18, then the left side must make 18. Students must follow the substitution rule.  

The Baltimore Problem

The teacher writes in a blog, "I can teach math, but I can't pass the math test for full certification." The 3rd-grade "teacher" wrote that she had two master's degrees and gets excellent reports from her principal, parents, and students. Really? I wonder how she got into graduate schoolHow can you get a master's degree without knowing some college-level math? 

She works at a PreK-5 Baltimore elementary school where almost all students in grades 3 to 5 scored far below the state average--less than 50% proficiency in math (26%) and even less in reading (19%). It is terrible education with ineffective teaching. In my opinion, some teachers don't know how to teach kids basic reading, writing, and arithmetic. 

The "I can teach math" 3rd-grade teacher doesn't realize that her lack of content knowledge contributes to the innumeracy and illiteracy of students at the school. Like some other teachers, she has no business being in the classroom. 

At this school, almost all students are significantly below grade level. I can only think that most of the school's teachers have low expectations for these black students. Content knowledge is essential in math. Teachers in the lower grades need to know how to prepare students for Algebra-1 in middle school and above, but they don't.  

If teachers knew how to teach arithmetic, reading, and writing well, the scores would be significantly better, at least at the state averages, which, to begin with, aren't high quality. The K-8 teachers lack factual and procedural knowledge in math because education schools that train wannabe teachers don't require it. Instead, pedagogy is often substituted for math content. Also, getting two master's degrees in education does not make you a better teacher. It shows how weak the degrees in education and some other subjects have become. Many online degrees are not valid in my mind. The content and rigor are not there for many so-called degrees. Smart 12-year-olds (6th graders) could pass some of the "degree" courses. 

Advice: When tutoring high school students, I would often hear, "I understand the concept, but I can't work the problems," which is double talk. As a precalc tutor, I would help the student grasp how to do a procedure, step by step. Practicing the steps is important. "You don't know anything until you have practiced," explains Richard Feynman. Also, if the student can teach other students how to do the procedure, their knowledge is more profound.

Tutors must prepare, too. 

For example, I would work out all the precalc homework problems ahead of the tutoring session to guide the student to avoid mistakes. The tutoring was efficient and worked. If an answer didn't correspond to an answer in the back of the pre-calculus textbook, my tutoring students knew where they were mistake-prone, but more importantly, they also knew how to correct the errors without my help.

I kept getting the wrong answer to a complicated trig problem. The angle measurement could not be negative. The solution involved several calculations. My steps were right, but I kept getting the wrong answer. What's wrong? I rounded answers. When I didn't round and stored lengthy intermediate answers for use in other calculations, and so on, I ended up with the correct answer. Now you know why I always work the homework problems ahead of time. As it turned out, the student's math teacher ran into the same problem.

In a tutoring session, a 7th-grade pre-algebra student told me that his calculator kept giving the wrong answers in the trig problems and that his teacher said to get a new calculator. I took one look at the calculator and noticed it was set to radians, not degrees. I reset the calculator to degrees and tested tan (45) = 1. "You don't need a new calculator; you need a new teacher."

Idea: "If you really want to learn something, learn it with the view to explaining it to somebody else." (Sarah Flannery (In Code)

In short, you not only need to perform a skill correctly but be able to teach it, too. Teaching something complex is not easy. "You don't know anything until you have practiced," explains Richard Feynman, Nobel Prize-winner in Physics. If you can't calculate it, then you don't know it. In short, to learn something is to know it well enough to teach it to someone else.

Inflation is nothing more than a new tax.



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