Cogitatus4

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 arithmetic. Sadly, 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. 

------------

✓  Whiteness

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 (medium.com/data-science-rush) 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 classroom. Educators 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 commitment. IQ 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.

TKA

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 = 18, x = 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 + 2. The 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 inflation. Gap 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.  

6-20-21

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?

©2021 ThinkAlgebra.org/LT 

 

Cogitatus3

 Welcome to Cogitatus3 

"Study hard enough to become Smart enough."

Note:  Some content has moved to Cogitatus4. 

Click to go to Cogitatus4.

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 commitment. IQ 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 needs to be modeled at home and taught in the classroom. 

"Conscientiousness does not link to IQ. [The correlation is practically zero.] 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, are vital for solving math problems and higher mathematics. 6-12-21

Scientific Integrity Lost

✍️ Today, I often hear "follow the science," whether it be Covid shots, climate, education, or other juicy topics that have been politicized; however, we should heed the stern warning of Richard Feynman, which he preached in lectures to students, that real scientists "bend over backward" to show that maybe their conclusions were wrong or inconclusive. 

Feynman writes, "I'm talking about a specific, extra type of integrity that is not lying, but bending over backward to show how you're maybe wrong, that you ought to have when acting as a scientist."

Furthermore, just because something is statistically significant doesn't mean it is consequential, useful, or correct. Science can change with the advent of new data or measurements. It's not absolute like mathematics. So, I am skeptical whenever someone evokes "follow the science." What does that mean?  Real science requires the replication of results, which is no easy task. Also, the media fails to tell us what we should hear, but tells us what they think we want to hear, suggests Sherry Seethaler (Lies, Damed Lies, and Science). Seethaler points out that data analysis is rarely simple and straightforward. Thus, it may be possible to draw more than one conclusion. So, what does "follow the science" mean? Experts may give their opinions, but an opinion is not fact. 

CRT

Critical Race Theory has no place in our schools. It teaches racism and indoctrinates children. "Critical race theory is a Marxist experiment to remake society based on class struggle. It is not an educational tool and certainly should not be funded with taxpayer dollars," write Rick Esenberg & Daniel Lennington. It's not about equity, not when equity means the fallacy of fairness. 6-16-21

Critical Race Theory: Equalizing Outcomes

✍️ There is a lot of talk about critical race theory (CRT), which, I think, is an impractical idea, and who better to blast it out of the water than Thomas Sowell, a black scholar, who, in 2010, probably earlier, wrote in an essay that equalizing down by lowering those at the top is a fallacy of fairness and a crazy idea taught in schools of education controlled by far-left liberals. Cutting content and lowering expectations hurt all students. Attempts to close achievement gaps by teaching less content is a red flag that CRT (equalizing outcomes) has crept in. CRT didn't just pop up. It has been around for decades in divers forms and is gleefully supported by teacher unions, the media, and a political party--in the name of equity, of course. Gee, I wonder which political party? (Comment: "Divers" was my favorite word in high school.)

Iowa Gov. Kim Reynolds states, "Critical Race Theory is about labels and stereotypes, not education. It teaches kids that we should judge others based on race, gender, or sexual identity, rather than the content of someone's character." 6-9-21

Equalizing outcomes is a fallacy of fairness, points out Thomas Sowell. 

Unchallenged students learn little!

The Teaching

✍️ Cutting content to close achievement gaps is lousy education, so is grade inflation. Gap 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, etc. Not all teachers have bought into this, but many have.  

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.

✍️ I don't believe in delaying content in the math curriculum (e.g., Common Core). I think we should accelerate it for almost all students and rise to world-class benchmarks. U.S. educators grossly underestimate the content young children can learn with good instruction. 

Common Core delays content (see chart below), especially standard algorithms, one of the five "building blocks of elementary school mathematics," explains mathematician W. Stephen Wilson. He writes, "Mathematics is not a collection of unrelated topics. Mathematics is hierarchical. ...Certain topics must be taught [and mastered] before others." 

Common Core: Delay of Content, 2011 by LT

I found this out when I taught my algebra project (TKA) to typical urban 1st and 2nd-grade classes in 2011. I was told that it would not be possible to teach algebra to little kids, but I proved the critics wrong. In just 7 lessons, students learned the algebra content linked to standard arithmetic because the expectations were high and the teaching was there (i.e., explaining worked examples, practicing, and reviewing). Students learned the algebra content (equation-table-graph) because they were expected to. Classroom teachers were astounded. Educators need to implement high expectations and teach accordingly. 

✓  In my algebra lessons (TKA), new content was explained, linked to old content, and practiced. 

✍️ Many teachers barely know standard arithmetic, much less integrating it with algebra ideas. Mathematics has a dual role in conveying concepts and thinking, and I believe this duality is often lost in classroom teaching, especially the "thinking" aspect. Also, standard arithmetic is more demanding than other subjects because it is abstract and requires memorization and extensive practice to master. Moreover, one idea is based on or supported by prerequisite ideas. Thus, you can't skip around; the sequence is important. U.S. educators are not teaching the math content that children need to know and learn, starting with 1st-grade arithmetic. Reform math and minimal guidance group work do not cut it. 

✓  Minimal Guidance = Minimal Learning

✍️ Math Courses Matter! 

Students who take a rigorous course in precalculus in high school are much more likely to complete a 4-year bachelor's college degree than students who take only Algebra-2 (74% vs. 40%, respectively). 

✍️ In teaching mathematics, leaders never look at brain research. The starting point in arithmetic is numbers used as symbols, so the number 6 is the concept of 6, which is easily visualized on a number line. Add 1 to 5 to get 6, the next whole number (i.e., integer), and so on. To learn arithmetic, students must memorize basic facts. Repetition and drill are required so that fundamentals stick in long-term memory and are usable immediately for operational procedures, such as the standard algorithms, and for problem-solving. 

Start With the Number Line to Calculate Basic Facts.

The Number Line is Mathematics!  

Drill-To-Develop-Skill.

Children are not asked to memorize without understanding. For example, asking students to memorize (automate) 7 + 5 = 12 is not without some level of understanding or knowledge of numbers, addition, magnitude, and place value, that 12 is 1ten+2ones, etc. The number line shows that 7 + 5 is 12. No other explanation is required. Children acquire a number-line understanding. The single-digit facts need to be automated in long-term memory, which involves drill-to-develop-skill. Memorizing factual knowledge and practicing standard algorithms are not obsolete. They are essential. 7 + x = 12. Find x.

Factual knowledge and procedural knowledge in long-term memory are needed to do arithmetic well. Problem-solving requires efficient calculating skills such as the standard algorithms. You don't learn arithmetic in group work (i.e., minimal guidance methods of instruction). You learn arithmetic through active, explicit instruction with flashcards, practice-practice-practice, and review.

Students, even in the 1st grade, should learn factual
and procedural knowledge to master arithmetic,
which requires memorization and practice-practice-practice.

Comment: When I teach algebra to small children beginning in the 1st grade, I don't think of cultural relevancy or CRT, even though 90% of the students are minorities. They are expected to learn the content, which is often abstract, such as the number 6, the expression x + 2, or the equation y = x + x - 3. Algebra ideas are firmly rooted in standard arithmetic. "Math knows no races," writes David Hilbert. Yet, so-called experts say that a white teacher, such as myself, can't teach children of color. They are wrong!

✍️ Bright students covet advanced math!

Unchallenged students learn little. 

Credit: Kailey

7th-Grade Algebra-1

If I can teach typical 1st graders the equation-table-graph sequence in 7 lessons (TKA Project), certainly by 6th or 7th grade, even earlier, Algebra-1 can be taught to students who are prepared, but that's not how the K-12 education system works. As a result, U.S. math education has been a "catastrophic failure," according to Deift, Jitomirskaya, and Klainerman. 

Many K-8 teachers are not equipped to teach the prerequisites for algebra; consequently, their students are not ready for Algebra-1 in middle school. In addition, many teachers don't know enough arithmetic to teach it well.  

Below: 4th-Grade students at an urban Tucson K-8 public school with a diverse student population learn algebra fundamentals in my Teach Kids Algebra (TKA) Project, which I started in 2011. The students I teach have different backgrounds; however, they have similar opportunities to learn algebra. (Similar should not mean identical.) Algebra ideas are fused with traditional arithmetic. TKA, which is STEM math for elementary school students, can begin as early as 1st grade. Instead of dumbing down the math curriculum, TKA advances the U.S. curriculum to bring it up to and surpass international benchmarks.  

4th-Grade TKA Practice Sheet on Linear Functions

Educators grossly underestimate the content very young children can learn when appropriately taught and practiced for retention in long-term memory. Math fundamentals must be memorized and practiced for mastery, starting with standard arithmetic in 1st grade. Unfortunately, U.S. students are taught reform math, not traditional arithmetic. Consequently, many middle school students stumble over fractions, a 4th grade topic in international grade placement. 

Math is abstract and uses symbols to convey concepts. In addition to knowing basic math symbols and concepts, students also need proficient math skills to solve problems, starting with memorizing the single-digit math facts and practicing standard algorithms.

✓ Understanding does not produce mastery; practice does!

It's hard to do math without knowing math. 

Note: Equity has come to mean equal outcomes, which is an impossible task. To narrow achievement gaps in math, teachers inflate grades and "dumb down the curriculum so that everyone can pass, but no one can excel." It's a false narrative. (Quote: Charles J. Sykes, Dumbing Down Our Kids) 

✍️ Failure: The K-12 system is not producing enough students who are prepared for "rigorous graduate-level work" needed for STEM careers. Deift, Jitomirskaya, and Klainerman ("America Is Flunking Math") point out that teacher education schools and courses are a significant part of the problem. They explain, "The vast majority of K-12 math teachers are graduates of teacher-preparation programs that teach very little substantive mathematics," but "are filled with courses such as social justice, identity politics, or, at best, methods courses with minimal math content." Moreover, teachers can't teach the content they don't know well. 

Why are some educators teaching children that math is racist? 

Crazy ideas like equalizing down or anti-racist mathematics, one of the latest fads that blends math with oppression, "threaten to further degrade the teaching of mathematics." Also, a student who majors in mathematics (not math education) or physics (not science education) should not have to go through a tedious, teacher-certification process to teach in public schools or any school. Even the late Richard Feynman, Nobel Prize winner in Physics, would not qualify to teach physics or math in our public schools. Who better than Feynman to explain complex ideas to children?  

Furthermore, the radical reformers want every student to take the same math no matter the student's level of achievement (one-track math). And, make sure to dumb it down enough (i.e., "equalizing down") so that all students or identities can pass. Kick Algebra-1 out of middle school and Calculus out of high school. For equity, no acceleration! It's sameness all the way, which is a drastic disservice to all students. Ill-advised, dumbed-down reforms often drive bright college students out of education and the classroom. (Note: "equalizing down, by lowering those at the top" is a "fallacy of fairness" is from an essay by Thomas Sowell, 2010.)

✍️ Some educators imagine that students have the same academic ability, which isn't true. Likewise, the same is true for athletic ability, artistic ability, or musical ability, etc.  Pretending that students are the same is a myth. Students do not have similar intellectual capacity, so "gap closing" should not be an educational goal, suggests Sandra Stotsky (The Roots of Low Achievement, 2019). Some kids are better at math than others. It's not going to change. The K-8 kids who are better at math need advanced content, but instead, they are given enrichment in so-called gifted programs, project-based stuff, or math for average students. Also, high-achieving math students are mixed with low-achieving students in the name of equity, a terrible idea. "Academic outcomes are not simply a function of money," writes Freddie deBoer. "Throwing money at our educational problems, while noble in intent, hasn't worked." Yet, we continue doing it. 5-22-21 

✓ Students are not equally creative or academically equal. 

✍️ Regression to Mediocrity

Our education system of mixed ability classrooms and minimal guidance methods in K-8 and even into high school is regression to mediocrity and a recipe for failure. Moreover, looking at a screen--remote learning--is not proper schooling; it is "pretend" learning. Kids need active in-person teaching all day without screens. Teachers should stand in front of the class and explain how things work, especially in arithmetic and algebra. Also, popular project-based learning (i.e., group work) is not effective in mathematics. U.S. education has been littered with low expectations and failed fads.

We don't expect most students to take algebra in middle school or calculus in high school. Why not? Dumbing down starts early! We don't expect 1st-grade students to memorize the addition and related subtraction facts, learn the standard algorithms, grasp multiplication such as 3 x 4 as repeated addition: 4 + 4 + 4 = 12, find perimeters, and introduce algebra such as linear functions: equation-table-graph. Why not? In today's progressive classrooms, memorization and drill (practice-practice-practice) are branded as poor pedagogy. They are not. They are essential for learning. Lastly, Common Core and most state math standards are not world-class. 5-21-21

1st-Grade Algebra Lesson #7 (Point Plotting in Q-I and Finding Perimeters)

Urban K-8 School (Tucson, AZ)  Note: My Teach Kids Algebra (TKA) Project is STEM Math for elementary school children. It's a start.

1st Grade TKA
We grossly underestimate the content students 
can learn given proper instruction.

✍️ Input ≠ Output 

The Educational Opportunity Project at Stanford University inferred that "Racial, socioeconomic, and gender disparities in academic performance and educational attainment are stubborn features of the U.S. educational system." Are these disparities the cause of low achievement? I think they are correlations or assumptions, if that. 

✓ Correlation is not causation. 

"Statistics are not facts," explains Daniel J. Levitin (A Field Guide to Lies, 2016). "They are interpretations. And your interpretation may be just as good as, or better than, that of the person reporting them to you." The data may suggest correlations, but correlations are not facts. (I can link big feet to spelling, but big feet do not cause good spelling. Similarly, good spelling does not cause big feet.) 

The problem is that the blame for the gaps should mostly fall on schools of education, the NCTM standards, Common Core, and teachers who, for decades, have been using ineffective curricula and flawed instructional methods. It's called reform math couched in minimal guidance methods (e.g., discovery, project, etc.). In short, group work. It has been a train wreck. 

Thomas Sowell (Discrimination and Disparities, 2019) states that equal input does not result in equal output. I'm not even sure that similar inputs (e.g., equal opportunities) can exist. Sowell warns not to confuse correlation and causation. Also, projecting beyond the actual data (extrapolation) is a flawed methodology, an idea I used to stress elementary school students in Science--A Proces Approach (1967). Statistics is not science, and inferences are not facts. 

Our kids are at least two years behind their Asian peers by the 4th or 5th grade in math. Why? Our students have not been taught the math that students in high-performing nations learn, a conclusion from The National Math Advisory Panel (2008).

✓ It is that simple! Students aren't taught!

✓  It behooves us to be aware that critical race theory (CRT) and Marxism have crept into our schools. Almost everything in education seems to be race-based or equity-based, which is inexcusable. Education is no longer about teaching kids to read, write, and do arithmetic. It's about equity, diversity, inclusion, identity, and culturally relevant lessons, even in arithmetic. The more stress put on equity by lowering standards, the less focus on academics, an inverse relationship. Mixing high-achieving students with low achieving students, inflating grades, and reducing the curriculum criterion to level the playing field result in a low-quality, low-expectation, mediocre education. Teaching less math content is a reckless, ill-advised strategy that hurts most kids. "It's crazy."

✓ Lowering standards dumbs down the curriculum and our kids. All students get the same math without regard to achievement or prerequisites, which had been the equity strategy of Common Core (CC) and state standards primarily based on CC. It's another stupid idea.

✍️ Teaching Less Content Is Not Equity. It's irresponsible and a

"Fallacy of Fairness," says Thomas Sowell. 

Experts tell us that kids are far behind in math due to at-home learning via remote or online. But they were far behind their Asian peers before the schools were closed. Some expert educators say that kids learn math, not at home but in the classroom where deep learning takes place--"it's how children learn best, which is often project base or done in groups," reports Valerie Cavazos. Really? The experts are wrong about learning, which has been insufficient for several decades before the virus because the dumbed-down curriculum was taught using minimal guidance methods. Now, with remote lessons, it is worse. 

The goal of "so-called" deep learning is often used "to justify teaching less content," writes Sandra Stotsky (The Roots of Low Achievement, 2019). The strategy of gap-closing is to teach less math content--by lowering those at the top. Stotsky explains that gap closing should not be an education goal. Kids were already two-grade levels behind Asian kids by the 4th or 5th grade with regular classroom teaching before the virus. Group work and project-based learning are reform math and inferior to explicit teaching via worked examples and practice. (Kirschner, Sweller, & Clark: "Why Minimal Guidance During Instruction Does Not Work," 2006)

✓ Minimal Guidance = Minimal Learning

✍️ Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich, says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe." There will always be achievement gaps because good education creates inequalities, explains the late Richard Feynman.

Sandra Stotsky points out that closing gaps should not be an educational goal. Students who study more, practice more, pay better attention in class, are more industrious, delay gratification, and have an optimistic attitude create inequalities. 

✓ Understanding does not produce mastery; practice does!

✍️ Today, education seems more about equity, diversity, inclusion, identity, and culturally relevant lessons than excellence, achievement, and teaching kids to read, write, and do arithmetic. Critical Race Theory (CRT), for example, teaches hate for America and its foundation and promotes bias and discord. Thankfully, some teachers don't buy into the "critical race theory" and its ramifications. Children are not and should not be indoctrinated as little activists; they are students. Parents, teachers, state legislatures, school boards, and leaders must block radical Critical Race Theory (CRT) from schools. 

 

✍️ Fairness Policies Downgrade Individual Excellence

There is a war on achievement and excellence!
Equity no longer means fairness!
Arithmetic is no longer just arithmetic. 

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

✍️ The proposed California math framework* cancels gifted programs, homogenous ability grouping, Algebra-1 in 8th grade, and high school Calculus, all in the name of equity. In short, no student should get ahead. That's equity? No, it isn't. It's a "fallacy of fairness," says Thomas Sowell, and an overt bias against bright students who study more and work harder to achieve. In my opinion, the new math framework is directed against high-achieving students, usually Asian students. I hope the CA Department of Education does not approve it. (*Some information in this post is from Steve Miller's Twitter.)

I need calculus for STEM. It's gone!

Angry parents (Virginia) are raging against equity math in which everyone gets the same math--no remedial classes to help weak students, no accelerative courses for advanced students. But, it appears that Virginia is backing away from equity math and the anti-tracking movement, according to Jay Mathews. I hope parents in California, Arizona, Oklahoma, and elsewhere do the same.

Equity math is a fallacy of fairness and originates from Critical Race Theory (CRT), which has been around for most of this century and before. CRT is far-left radicalism and should not guide the curriculum, but it has crept in. CRT teaches racism. It is anti-white, anti-grouping, anti-achievement, and anti-bright students who work hard to achieve. If you disagree with CRT, you are attacked and called a racist.

Robbie Soave at Reason, reporting on the CA math framework, writes, 

• "In fact, the framework concludes that calculus is overvalued, even for gifted students." [Really?]
• "The framework's overriding perspective is that teaching the tough stuff is college's problem: The K-12 system should concern itself with making every kid fall in love with math."  [Really?]
• "Broadly speaking, this entails making math as easy and un-math-like as possible. Math is really about language and culture and social justice, and no one is naturally better at it than anyone else, according to the framework." [Really?]
• "The entire second chapter of the framework is about connecting math to social justice concepts like bias and racism." [It's true! It's Critical Race Theory.]

✍️ The CA math framework wants to dumb down arithmetic and algebra so that all children can learn the same math. After all, according to the framework writers, everyone is just as good at math as everyone else. But, unfortunately, the assertions and assumptions made by radicals in the math framework demonstrate how deep Critical Race Theory has penetrated education and how bent the thinking of many education leaders, and teachers has become. And, it is coming to a public school near you.

✍️ The amalgamation of strategies, mixed classrooms, and minimal guidance methods to learn basic arithmetic has been a recipe for poor achievement. Yet, in my opinion, the new California math framework advocates more of the same with a twist of culturally relevant lessons. Really? Critical Race Theory (CRT) has crept into the curriculum, even in math.

✍️ "Almost all of us improve with good teaching and plenty of practice, whatever the activity. That's why education is worth the effort." (Ian Stewart, Significant Figures 2017)

✍️ To find talented kids, in contrast to enrichment-centered programs at many schools, the Center for Talented Youth (CTY) at Johns Hopkins requires elementary school students to take the School and College Ability Test or SCAT (verbal + quantitative) at least two grade levels ahead and score at the CTY level. Thus, a 3rd grader would take a test designed for 5th graders, and so on. In the 7th grade, middle school kids need very high scores on the College Board SAT to qualify. In addition, all summer and online courses for admitted students are academically accelerated. As a result, eligibility for CTY classes is not easy. "Johns Hopkins Center for Talented Youth (CTY) is the world leader in gifted and talented education, serving the world's brightest students." Also, visit Bright Now.

Gifted and advanced students crave advanced content (Click)

✍️ Tangents in 4th Grade

Trig Ratios in 4th Grade Math, An Extension of Ratios and Proportions
The equal sign should be an "approximately equal" sign. 

✍️ If I want to find out if something works in the classroom, I test it and learn from my mistakes, which is the approach that the late Siegfried Engelmann had used. Try it first; see what happens. For example, I decided to teach kids trig ratios in the 4th grade (the late 1990s) as an extension to ratio and proportion lessons. The trig lessons worked better than expected via worked examples, explicit teaching, and practice. No calculators! No group work. No minimal guidance methods. Even though the "trig ratio" lessons functioned well in my 4th-grade classroom, it is not proof they will succeed in other elementary school classrooms. It depends on the students' prerequisite knowledge and the teachers' math background.

✍️ On the other hand, what schools usually do is implement without knowing if something works, which is a stupid approach to education. School decisions often rely on testimonials, so-called studies or research, flashy, colorful materials or textbooks, and impressive "too good to be true" sales pitches. Some teachers might actually page through a textbook, but they don't read it. Colorful Graphics in textbooks and on classroom walls often distract children from the task at hand.

Schools buy expensive programs without testing them. "It's all in the implementation," educators are told. "If you do it right, kids will learn." Also, "We will train your teachers ($$$$$)." But kids don't learn much--little bang for the buck. The software or videos often lack credible instructional value. Why? The classroom teacher isn't giving the lesson. There is little follow-up to the video. Teachers can't ask probing questions, give a different explanation, or answer student questions. 

Reformers: "If we only had more funding and more time." 

Good intentions are not good enough! 

District: "We bought all these computers and software, so use them!" 

Your evaluation depends on it.

✍️ Marxist radicals have taken over education at the local, state, and federal levels. The only way to close gaps, they say, is to lower content standards and eliminate excellence. No child gets ahead. The radicals have redefined achievement as "privilege," says Thomas Sowell. These are terrible ideas! "Equalizing down, by lowering those at the top," is a crazy idea, a "fallacy of fairness," not equity, explains Sowell in his book of essays and his latest book below. 

Credit: EmB

✍️ When education cowers to activism, racism, and radicalism, you are no longer talking about education. You're talking about indoctrination! Today, much of education is not about teaching kids to read, write, do arithmetic or about achievement, merit, and excellence. Government schools advocate equity, diversity, identity, and inclusion. They are strongly supported by teacher unions, schools of education, federal and state departments of education, school boards, and others, including the media. I believe education has gone down the rabbit hole, an abyss. Commonly, there seems to be an inverse relationship between excellent academics and equity. The "equity" reformers have been dumbing down the curriculum for decades. 

✍️ Einstein based his theories of relativity on two impossibilities: 

1. "It’s impossible to exceed the speed of light."

2. "It’s impossible to tell the difference between gravity and acceleration." 

If either of these assertions were not true, the laws of physics would change radically, even fall apart. Physics is the most fundamental science, yet it is downplayed in the NextGen science standards in favor of issues-based science. Also, K-12 teachers are weak in chemistry, physics, and math. Science textbooks cannot be trusted and often imply a correlation, even a theory, as a fact. 

I am thinking about impossibilities in learning theory. Perhaps one is that it is impossible to have an empty slate. You were born as a learning machine with innate intelligence, abilities, and genetic differences that can widely vary from person to person, group to group, or even within children from the same family, etc. DNA determines who we are. (Quote: QuantaMagazine, 2021; DNA from Robert Plomin blueprint, 2018 MIT)

Think On These Things

"Finally, brethren, whatsoever things are true, whatsoever things are honest, whatsoever things are just, whatsoever things are honest, whatsoever things are just, whatsoever things are pure, whatsoever things are lovely, whatsoever things are of good report; if there be any virtue, and if there be any praise, think on these things." (St. Paul's Epistle to the Phillippians, KJV)

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Cogitatus2 

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Photography: Click How I Shoot Photos  

Algebra: How I teach elementary school students algebra--Teach Kids Algebra.

Reform Math: The Calculator Problem

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Email ThinkAlgebra@cox.net to make comments. 

How I teach elementary school students algebra: Teach Kids Algebra.

Note: I was not able to sign in to make changes on this page on the afternoon of 

6-8-21 for unknown reasons. Now, I am able to sign in to make changes or additions to this page. 

This page is too long, so I am thinking of starting a new page, perhaps Cogitatus4 or some other name. "Do you want to build a snowman?"