Welcome to Cogitatus4
✓ 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:
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 = 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.
✍️ 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.
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.
“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!"
What happened before the Big Bang?
Color doesn't really exist outside of our brains, does it?