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.
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!
✍️ 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!
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.
"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!
✍️ 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!
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.)
- "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.]
|Trig Ratios in 4th Grade Math, An Extension of Ratios and Proportions|
The equal sign should be an "approximately equal" sign.
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)