Sunday, June 20, 2021


Welcome to Cogitatus4

Note: Some content was relocated to the bottom of this page: 
1. NovaLee, 9, confronts the school board! 
2. Critical race theory
3. Free Fall.
Special: How I shoot photos

Cogitatus4 continues on Summer2021
Some of the content here was repeated at the beginning of Summer2021.

✍️ ✍️✍️ Educational equity has changed to mean equal outcomes, which is not possible, and also discriminates against high-achieving students (e.g., Asian Students). To achieve comparable results is to dumb down the curriculum, teach less content, and inflate grades to pass all students, especially low-achieving or disadvantaged students. An example of feel-good equity math is when high-achieving math students have the same grade-level math program as the other kids. It is "equalizing down by lowering those at the top," points out black scholar Thomas Sowell. Equity is a "fallacy of fairness." 7-1-21

Teaching less content is a recipe 
for mediocrity.

The primary reason our students are behind is the teaching of a weak, reformed arithmetic curriculum using ineffective minimal guidance methods (group work). Very young students can learn much more content than being taught and much faster by employing the explicit teaching of essential examples and practicing for mastery. Explicit teaching and practice have been used for centuries to teach kids arithmeticSadly, memorization and drills, which are the basis of learning arithmetic fundamentals well, have been considered harmful pedagogy by reformers in today's progressive schools when they are not.

 The NEA, the most influential teachers union, which at one time voted to boycott Walmart, resolves that critical race theory is "reasonable and appropriate" and must be taught in all schools. What bunk! Sadly, education has become an unsound mixture of equity, diversity, inclusion, identity, and culturally relevant lessons, even in arithmetic. Almost everything in school is race-based. In my opinion, schooling is no longer about teaching kids to read, write, and do arithmetic. 7-4-21

  Critical race theory is "garbage," says Dr. Ben Carson. “It’s an attempt to use race as a mechanism for redefining our society. ...It wants our people to believe that your race is the most critical determinant of who you are and what happens to you in our society. In other words, it’s a bunch of garbage." In my opinion, schooling is more about race than academics, more about indoctrination than education. Today, equity means "equal outcomes," an impossibility unless the content is downgraded (dumbed down) to the lowest level and grade inflation continues so all students pass. But, not all teachers are Marxist or subscribe to critical race theory or black lives matter movements. America has been and still is the land of opportunity. 7-4-21

  Teacher Quality: Gains in Learning (Academic Achievement), Not Degrees or Experience

"The quality of a teacher is best judged by performance in the classroom as reflected in the gains in learning by students," not qualifications, degrees, experience, class size, and so on, according to Eric Hanushek (Waiting for Superman). We don't need more mediocre teachers or teachers who acquire master's degrees just to jump up on the pay scale. (Note: A master's degree in education does not improve teaching in the classroom.) We need knowledgeable teachers who are content-driven and can facilitate gains in reading, writing, and arithmetic. Good teachers who know content are hard to come by, as schools of education keep churning out mediocre teachers. 

There are few academic requirements for most elementary school and middle school teachers. They are not required to take college-level courses in precalculus/calculus, chemistry, physics, history, literature, philosophy, etc. Instead, future teachers major in education, which is not an academic discipline like history or chemistry. In my opinion, future teachers should major in regular academic discipline first. Unfortunately, education schools rarely attract the best college students according to SAT scores. Students can take education courses after they receive a bachelor's degree. 

According to equity advocates, if students are required to get the correct answer and show their work, that's whiteness (i.e., racism). It's bunk! What an offensive, radical idea, but that's the way advocates of equity projects think, such as the one at Education Trust where the goal is to "dismantle [alleged] racism in math instruction." Really? What nonsense! The radical advocates of equity math do not know math or how kids learn arithmetic. Their assumptions of whiteness are false, misleading, and ridiculous. Note: An Equity Trust project--a workbook for middle school that eliminates racism in math--was financed by Gates. Are you kidding? It is controversial and ludicrous. (Gee, I always thought 2 + 3 = 5. How can math facts and standard procedures be racist when they are not?) 6-24-21

Perpetuating Mediocrity 
Standards, testing, technology, teacher education, and professional development (PD) have not significantly boosted math achievement over the years, observes Beverly Jobrack (Tyranny of the Textbook), but perpetuate mediocrity. She is mostly correct and also points out that "learning is work." Many schools are removing acceleration or honors classes for equity, which is a terrible idea. 

Fallacy of Fairness
As the late Zig Engelmann would retort, "It's the teaching!" Unfortunately, teachers are urged to teach less content and inflate grades, so everyone passes. Teaching less content or inflating grades is not equity. It's irresponsible and a "Fallacy of Fairness," observes Thomas Sowell.

  It's The Teaching!

According to NAEP 2019, most 4th and 8th grade students lack adequate math skills and reading comprehension. The "Basic Level" isn't good enough. Most students should be at the NAEP Proficient Level but are not. It's the teaching. The problem begins with 1st-grade arithmetic and low expectations for all students. If teachers aim for Basic, then that's where students end up. Also, keep politics out of the classroom and school.   

Suppose educators want to break the correlation between poverty or race and achievement. In that case, teachers need to break from reform math (the status quo curriculum leftover from Common Core days) and teach traditional arithmetic, both factual and procedural knowledge, for mastery in 1st grade on up to prepare most students for algebra in middle school. Using old school methods (explicit teaching and lots of practice) rather than minimal guidance methods (group work) is a key change. But that's not what education leaders want to do. They want equity math, a leveling downward in content so that almost every student can pass, which is a fallacy of fairness, observes Thomas Sowell. Equity math does not challenge kids; it dumbs them down. Poverty does not cause low achievement. Correlation is not causation. 6-22-21

The Wonders of Calculus

"Without calculus, we wouldn't have cell phones, TV, GPS, or ultrasound. We wouldn't have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket," writes mathematician Steven Strogatz (Infinite Powers). If we want to continue innovating new technologies, more U.S. students, not fewer, should take calculus in high school to prepare for STEM calculus courses in college. Importing talent from Asian nations won't last forever.  We need to develop more of our own talent. 6-16-17-21 

AP Calculus is for average high school students who are prepared, writes Richard Rusczyk (the Art of Problem Solving). It's not equivalent to college-level calculus, which is why many universities and colleges don't accept AP for credit toward a STEM career. In addition, some of the high school AP teachers are not that good. (When you take courses at a Community College, be sure the credits are transferrable to a 4-year university if your goal is to acquire a bachelor's degree.)  

Przemek Chojecki ( writes, "Mathematics influences every aspect of our life and is behind all the recent breakthroughs, even though we may not be aware of it."

Mathematics starts with counting in pre-K and standard [traditional] arithmetic no later than 1st grade. Learning arithmetic as factual and procedural knowledge requires memorization, practice-practice-practice, and continual review. Memorization is good for kids! Math is abstract and more complicated than other subjects, and learning it well is not always fun. Students need to work hard to learn math. Unfortunately, the progress of many students has been slowed by minimal guidance methods (group work), remote instruction, a math curriculum that is substantially below world-class, and inadequate teacher academic requirements, especially in math and science. Our kids lag behind at least two years by the 4th or 5th grade mainly because of the teaching in the classroomEducators should stop giving excuses. Also, some students lack the task commitment or self-discipline to learn arithmetic and algebra. 

"It's the teaching," explains the late Zig Engelmann, coupled with an attitude of low expectations. At earlier ages, kids can learn much more arithmetic than is taught. We found that out in the early 1950s with The Madison Project. Today, kids are left in the dust compared to peers in some other nations, regardless of money spent.  6-18-21

Task Commitment & Self-Discipline

Michael E. Martinez (Future Bright) suggests that a pivotal element to predictive success in school or on the job is conscientiousness, which adds to IQ. "Conscientious people are achievement-oriented, pay attention to details, persist in solving complex problems, follow through when working independently," and other traits of task commitmentIQ is the key predictor of success, and conscientiousness adds to it. Unfortunately, American students tend to give up if they can't work a [math] problem immediately. Persistence and delayed gratification need to be modeled at home and taught in the classroom. 

"Conscientiousness does not link to IQ. [The correlation is practically zero.] Instead, it adds its predictive power to that offered by IQ," writes Martinez. Thus, "beyond the IQ score, conscientiousness also predicts students' academic achievement." A related personal trait to task commitment is self-discipline, which predicts academic achievement better than IQ scores. Signs of task commitment include the "capacity for perseverance, determination, hard work, and dedicated practice." These are traits that should be instilled in children from the 1st grade on up, even earlier. 

If you want to get better at arithmetic (or any task), then you need the "capacity of perseverance, determination, hard work, and practice." 


 Understanding does not produce mastery; practice does!

Calculating Skills Must Be Sharp & Automatic

Like physics, arithmetic is skill-based. To learn arithmetic well, specific factual and procedural knowledge must be memorized and practiced to automation. You cannot solve math problems without good calculating skills. A shortcoming in arithmetic leads to a weakness in algebra. Therefore, facility in the standard algorithms and supporting single-digit number facts, beginning with 1st-grade addition and subtraction, is vital for solving math problems and higher mathematics. 6-12-21

"The standard algorithms are among the few deep mathematical theories that can be taught to elementary school students," writes mathematician W. Stephen Wilson. Also, Wilson points out that standard algorithms must be learned with fluency. 6-14-21

Elementary School Mathematics Priorities (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 

Some elementary school teachers seem unaware of the consequences of not teaching children standard arithmetic for mastery and ignoring the importance of memorization and practice of fundamentals, starting in grade 1. Reform math with its minimal guidance methods (group work) doesn't cut it. U.S. students fall behind international leaders starting in the 1st grade. We don't expect 1st graders to learn to read, write, and do basic arithmetic, especially the standard algorithms, or 8th graders to learn Algebra-1. U.S. kids are shortchanged. 

For example, Singapore 1st-grade students learn multiplication as repeated addition and solve multiplication word problems. They write and solve equations in one unknown from word problems. "The translation of words into mathematics and the skill of solving multi-step problems are crucial, elementary school forms of critical thinking. Developing critical thinking is an essential goal of mathematics education," writes W. Stephen Wilson. Solving an equation is critical thinking. 6-14-21

Michael E. Martinez (Future Bright, 2013) states that education cultivates intelligence. He writes, "Intelligence is not simply a raw material for education; it is also a product of education. We can even quantify the impact of education in IQ: For every year of education, the counterpart gain in IQ is about 1/2 point." It is another reason that kids should be in school with intelligent in-person teachers. Students have lost about a year of education, some more, others less, so the student's IQ didn't gain; it stagnated, maybe went down. 6-15-21

Martinez also writes, "Early mathematics achievement now appears to have surprising power to predict student academic achievement in high school--both in mathematics and in reading." It supports early teaching of algebra linked to standard arithmetic. First graders should memorize math facts and efficiently perform standard algorithms to solve problems. They should also write equations in one variable to solve world problems. It is not that difficult. (Jill has 16 pieces of candy, then gives some to Bill. Now she has 11 pieces left. How many pieces of candy did she give to Bill? 16 - x = 11) Use guess and check, rules, and math facts to solve the equation. Writing and solving equations is critical thinking. Why are our 1st-grade students not doing this? They also need to know standard algorithms to accommodate larger numbers. Early achievement in arithmetic works! 

When the numbers are larger, elementary school students need standard algorithms, not calculators.


My Teach Kids Algebra (TKA) project is STEM math for young elementary school students beginning in the 1st grade. 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 = 18x = 8. Thus, 8 + 8 + 2 = 18 and 18 = 18 (True Statement: 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. 

According to the "fs blog" (Shane Parrish), the late Richard Feynman "proposed that kids be given simple algebra problems (2 times what plus 3 is 7) and be encouraged to solve them through the scientific method which is tantamount to trial and error. This, he argued, is what real scientists do." Exactly! Feynman fused algebra ideas to standard arithmetic, which is what I have done in TKA. In contrast to reform math, Richard Feynman points out that getting the right answer is essential. What is the purpose of math if not to get the correct answer and to solve problems? Some so-called math educators claim that getting the right answer should not be stressed. It's not essential, they say. Really? Conversely, to reform math, students must learn to do the math correctly to get the right answer.  

At first, 1st-grade students grapple with true/false numerical equaions such as 3 + 4 + 1 = 6 + 2The equation is true because both sides are 8: 8 = 8; However, this equation (10 - 4 = 3 + 3 + 1) is false because 6  7. (Note: The inequality symbol () means "not equal to")   6-9-2, 6-12-21

Comments about my TKA algebra project:  Soon, from equations in two variables (y = mx + b), 1st and 2nd-grade students construct an (x-y) table of values and plot (x,y) points in Q-I. This is a step beyond Feynman. Also, by the 3rd or 4th grade, students learn regular equation-solving techniques. Again, the classrooms of students I worked with were from Title-1 schools.

Education leaders say that slowing down math by cutting back on content and eliminating acceleration will help all students gain a deeper understanding. Really? Reducing class sizes to 15, which requires more teachers, the goal of the teacher unions, does not help students learn more. The problem is that many teachers are average, even mediocre. Schools need good tutors, one-to-one, not more teachers for smaller classes.

Praising a student for no good reason is part of the popular feel-good, self-esteem movement and a subtle form of indoctrination. American kids feel good about themselves, but many can't read, write, or do arithmetic well enough, so how can that be? Often, students are discouraged from excelling by reducing content, lowering expectations, inflating grades, and delaying essential math. Test prep also limits the content taught in the classroom. Often, useful content is restricted or not taught because it is not on the State test. 

✓ A major problem has been low expectations for all students.

The Teaching

✍️ Cutting content to close achievement gaps is lousy education, so is grade inflationGap closing should not be an educational goal, observes Sandra Stotsky (The Roots of Low Achievement, 2019). Educators and policymakers link poverty to poor achievement, which is a correlation, not a cause. Unfortunately, poor math achievement has not been related to the teaching in the classroom where the curriculum is below world-class and instructional methods are ineffective (i.e., minimal guidance = minimal learning). Moreover, reducing class size and pumping more money into schools have failed, too. 

✓ The liberal agenda has been to dumb down our kids, e.g., cutting content, using substandard instructional methods, inflating grades, and more. Not all teachers have bought into this, but many have. 

✍️ Boosting the math curriculum in the lower elementary school grades as I have done with the Teach Kids Algebra Project (TKA), which stresses traditional arithmetic, is a start, not the last word. TKA works when I go into classrooms and teach it myself. 6-1-21

Note: Algebra-1 is not advanced math. It is middle school math for average kids who are prepared. Also, Richard Rusczyk (the Art of Problem Solving) points out that AP Calculus is for average high school students who are prepared. Unfortunately, many of our teachers are ill-equipped to get kids ready for middle school Algebra-1. Rusczyk wrote books for students who were way above average, starting with Pre-Algebra. He also developed rigorous online courses based on his textbooks. 05-26-28-29-2021, 6-6-21

Critical Race Theory is designed to divide.

Marxists have taken over education at all levels (Federal, State, and Local).  Dumping critical race theory (CRT) and radical far-left ideology into our classrooms is indoctrination, not education. Today, education is all about equity and race--not teaching kids to read, write, or do arithmetic. What counts are equity, diversity, inclusion, identity, and culturally relevant lessons, even in arithmetic, not education. CRT is divisive and racist. Parents need to stamp it out. A few states have!  Some critics of parents who want CRT abolished in schools say that they, the parents, don't understand CRT. It is not the case when CRT advocates say that getting the right answer and showing your work are signs of white supremacy. There is nothing white or racist about arithmetic and algebra and their effective techniques of teaching math. 6-24-21

  The primary purpose of learning and doing arithmetic is to solve problems to get the right answer

  The quality of a teacher is best judged by performance in the classroom as reflected in the gains in learning by students, not by qualifications, degrees, experience, class size, and so on, according to Eric Hanushek, Waiting for Superman.

Free Fall
Blending prealgebra skills with scientific ideas should be routine no later than middle school, but it is rarely done. Our students are shortchanged! Learning to use formulas such as F=ma is important mathematics and fundamental science. Students must know the order of operations, substitute correctly, and solve for an unknown variable. Indeed, students must know some algebra. 

For example, in the free fall equation (d = .5gt^2), the distance d an object falls from rest, such as a rock, is proportional to the time (squared) or t^2. After 1 second into the fall, the distance covered is 1/2 x 10 x 1 or about 5 meters. 10m/s^2 is the acceleration due to gravity. (g = 10m/s^2 was rounded for easier calculations.) How long does it take a diver to hit the water falling from the 10-meter platform? (Notes: The formula does not account for air resistance. Also, g = 9.8 meters per second squared. Sometimes g is written with a negative sign meaning that the motion is downward toward the earth's center. g = -9.8m/s^2.)  

Often, our K-8 students are shortchanged by not being exposed
to more complex applications.

Free Fall Without Air Resistance

A student was timed as she stepped off the 10-meter platform at the University of Arizona diving well. The stopwatch time and the formula calculations were close. The free fall formula was algebraically manipulated to solve for t (time in seconds). Middle school students also figured out the velocity (v) of the diver at impact with the water using another formula, which also involved a square root calculation. The science formulas were magic, to calculate things without doing the measurements--the power of mathematics. (Later, students learned trig ratios to solve for unknown heights, etc.) The free fall formula was also used for 5 meters, 7.5 meters, and 1-meter distances. (Note: I did not discuss "sig figs" at this time. Canceling common factors in 4th-grade fraction problems prepared students for canceling units in chem and physics problems.) 

Note: Some physics majors have difficulty comprehending the content-rich textbooks, regrets a professor. Chemistry, the same.  


Overt Duplicity

NovaLee, 9, confronts the school board!
"I am so mad." A 9-year-old berates the school board.
NovaLee, an articulate, passionate 9-year-old girl from Minnesota, blasts the school board for overt duplicity. Two weeks ago, you [the School Board] said "No Politics in Schools," then you made BLM posters and put them in schools. "You expect me to believe that you did not know what you were doing by making these posters?" NovaLee demanded that the BLM posters be taken down. "I am so mad. ...We all understand the meaning. ...It is a political message about getting rid of police officers, rioting, burning buildings down while king Governor Walz just sits on his throne and watches."

“I do not care or look at the color of skin, but [upon seeing the posters in my school] you make me think of it. ...I do not judge people by the color of their skin. I don’t really care what color their hair, skin, or eyes [are]. I judge by the way they treat me.” She quotes MLK's speech, "by the content of their character." His dream has come true. NovaLee explains, "I have Asian, Mexican, white, Chinese, black friends, and I don’t care [the color of their skin]. I like them because some of them make me laugh, some are sweet and kind, sporty, or share the love of God. They are just my friends. You [the school board] have lied to me, and I am very disappointed in all of you." She said that if the school board can't follow its own rules, she won't follow them either: "I refuse to wear a mask!" 

Comment: We should admire NovaLee for exposing the school board's charade! I have known bright kids like this in my talented and gifted programs. While her Mom may have given some guidance, most of this is NovaLee. Indeed, she is articulate and spirited!

What happened before the Big Bang? 

Color doesn't really exist outside of our brains, does it?